# On the action of the Weyl group on canonical bases

Fern Gossow  F. Gossow: Department of Mathematics, University of Oregon, USA
<EMAIL_ADDRESS>and  Oded Yacobi  O. Yacobi: School of Mathematics and
Statistics, University of Sydney, Australia<EMAIL_ADDRESS>
###### Abstract.

We study representations of simply-laced Weyl groups which are equipped with
canonical bases. Our main result is that for a large class of representations,
the separable elements of the Weyl group $W$ act on these canonical bases by
bijections up to lower-order terms. Examples of this phenomenon include the
action of separable permutations on the Kazhdan–Lusztig basis of irreducible
representations for the symmetric group, and the action of separable elements
of $W$ on dual canonical bases of weight zero in tensor product
representations of a Lie algebra. Our methods arise from categorical
representation theory, and in particular the study of the perversity of
Rickard complexes acting on triangulated categories.

##  1\. Introduction

###  1.1. Set-up

Let $I$ be a simply-laced Dynkin diagram of finite type, and $W=W_{I}$ its
associated Weyl group. In this paper we’ll be interested in the following
situation: we have a (complex) representation $\pi:W\to\mathrm{GL}(U)$
equipped with a canonical basis $\mathbb{B}\subset U$, usually of geometric or
categorical origin.

We ask the following:

######  Question 1.1.

Can we extract interesting symmetries of $\mathbb{B}$ from the action of $w\in
W$?

The origins of this question goes back to the 1990s, when the longest element
$w_{0}\in W$ was studied in this context: Lusztig investigated its action on
the canonical basis elements of weight zero in irreducible representations of
the Lie algebra [30], Mathas studied its action on the Kazhdan–Lusztig basis
of the Hecke algebra [33], and Berenstein–Zelevinsky and Stembridge studied
the case of the longest permutation acting on the Kazhdan–Lusztig basis of
irreducible Specht modules [2, 40]. More recently, Rhoades studied the case of
a particular Coxeter element in the symmetric group (the “long cycle”) acting
on Specht modules corresponding to rectangular partitions [34].

The main aims of our work are two-fold. Firstly, we show that these results
are very special cases of a much wider phenomenon: for a large class of
elements (the “separable” elements of $W$) and for a large class of
representations, the action of $w$ encodes interesting symmetries of
$\mathbb{B}$. These symmetries arise from an underlying crystal structure in
the sense of Kashiwara [23]. Secondly, our methods are rooted in categorical
representation theory, and in particular we show how perverse autoequivalences
of triangulated categories can be well-suited to study and explain interesting
combinatorial phenomenon arising in Lie theory.

###  1.2. The symmetric group

We consider first the prototypical examples of the kinds of results we have in
mind. Set $I=\mathrm{A}_{n-1}$, i.e. $W=S_{n}$ is the symmetric group, and for
a partition $\lambda\vdash n$ let $S^{\lambda}$ be the corresponding Specht
module. Recall that as we range over all partitions, these give a complete
list of the irreducible representations of $S_{n}$.

The Kazhdan–Lusztig (KL) basis of the Hecke algebra (cf. Appendix B) descends
to a canonical basis of the representations $S^{\lambda}$ (cf. Section 5). We
denote this basis by
$\mathbb{K}\mathbb{L}_{\lambda}=\\{C_{T}\;|\;T\in\mathrm{SYT}(\lambda)\\}$,
where $\mathrm{SYT}(\lambda)$ is the set of standard Young tableaux of shape
$\lambda$. Then, by Berenstein–Zelevinsky and Stembridge [2, 40], for any
$\lambda\vdash n$ and any $T\in\mathrm{SYT}(\lambda)$, we have $w_{0}\cdot
C_{T}=\pm C_{\mathsf{ev}(T)}$, where
$\mathsf{ev}:\mathrm{SYT}(\lambda)\to\mathrm{SYT}(\lambda)$ is the evacuation
operator (cf. Example 5.7). Thus Question 1.1 has an affirmative answer in the
case when $w$ is the longest element acting on the KL basis of a Specht
module.

Rhoades proved a similar result for the “long cycle” $c_{n}=(1,2,\ldots,n)\in
S_{n}$: in the case when $\lambda$ is rectangular, $c_{n}$ acts on the KL
basis by the promotion operator $\mathsf{pr}$, another well-studied symmetry
of $\mathrm{SYT}(\lambda)$ [34]. Using this, Rhoades established a cyclic
sieving phenomenon for the action of promotion on rectangular tableaux.

It’s easy to see that Rhoades’ Theorem is false for non-rectangular
partitions. Nevertheless, we were recently able to show that the theorem holds
for all partitions if one allows for a more general framework [17]. Indeed,
let $\lambda\vdash n$ be any partition, and define a preorder on
$\mathrm{SYT}(\lambda)$ by $R\leq T$ if the box of $R$ containing $n$ is in a
higher row than the box of $T$ containing $n$. We proved that $c_{n}\cdot
C_{T}=\pm C_{\mathsf{pr}(T)}$ up to lower-order terms.

Upon reviewing these results, it is natural to try to refine Question 1.1. For
this we define:

######  Definition 1.2.

Let $U$ be a vector space with basis $\mathbb{B}$, $w:U\to U$ an operator,
$\xi:\mathbb{B}\to\mathbb{B}$ a bijection, and $\leq$ a preorder on
$\mathbb{B}$. Then $w$ acts on $(U,\mathbb{B},\leq)$ by $\xi$ _up to lower-
order terms (l.o.t.)_ if for every $x\in\mathbb{B}$ there are
$a_{y}\in\mathbb{Z}$ such that:

$w(x)=\pm\xi(x)+\sum_{y<x}a_{y}\xi(y),$

and the sign of $\xi(x)$ only depends on the equivalence class of $x$ under
$\leq$.

In some cases, we will leave the bijection and preorder unspecified and simply
say that $w$ acts on $(U,\mathbb{B})$ by some bijection up to lower-order
terms. This is reasonable since the bijection is uniquely specified by the
action of $w$ on $\mathbb{B}$ (cf. Lemma A.1). There is also a compact
description of the above definition in terms of QR decompositions (cf. Lemma
A.2).

In light of this definition, we now ask:

######  Question 1.3.

Which permutations $w\in S_{n}$ act on
$(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$ by some bijection up to l.o.t.?
Under the association of $\mathbb{K}\mathbb{L}_{\lambda}$ with
$\mathrm{SYT}(\lambda)$, how can we describe the bijections which appear in
terms of tableaux?

Our first main result is that a large and well-studied class of permutations,
namely the separable permutations, act on the KL basis by bijections up to
l.o.t.

Separable permutations are constructed as follows. Given $x\in S_{m}$, $y\in
S_{n}$ with permutation matrices $[x],[y]$, define $x\oplus y,x\ominus y\in
S_{m+n}$ by:

$\displaystyle[x\oplus y]=\begin{pmatrix}[x]&0\\\
0&[y]\end{pmatrix},\quad\quad[x\ominus y]=\begin{pmatrix}0&[x]\\\
[y]&0\end{pmatrix}.$

The set $S_{n}^{\mathrm{sep}}$ of separable permutations is given by those
permutations obtainable from $(1)\in S_{1}$ by repeated applications of
$\oplus$ and $\ominus$ [26]. In particular $S_{n}^{\text{sep}}$ contains both
$w_{0}$ and $c_{n}$. Note that $S_{n}^{\text{sep}}$ can be alternatively
defined as the set of permutations avoiding the patterns $2413$ and $3142$,
and they are counted by the large Schröder numbers111Large Schröder numbers
appear as sequence A006318 in the OEIS: $1,2,6,22,90,394,1806,8558,\ldots$.

We prove:

######  Theorem 1.4.

Fix $w\in S_{n}^{\mathrm{sep}}$ and $\lambda\vdash n$ arbitrary. Then $w$ acts
on $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$ by a bijection up to lower-
order terms.

We discuss this theorem in detail in Section 5, where in particular we
describe the bijection and preorder associated to $w$ through the
combinatorics of tableaux. For an example of the new type of results that
arise from this, in Section 5.4 we specify an element $g_{n}\in S_{n}$ which
is the product of long elements for the tower of symmetric groups
$S_{n}\supset S_{n-1}\supset\cdots\supset S_{1}$. We show that $g_{n}$ acts on
$(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$ up to l.o.t. by an operation we
term “nested evacuation”.

We note that non-separable permutations don’t appear to encode any
permutations of standard Young tableaux (cf. Example 5.19). We conjecture that
in a precise sense Theorem 1.4 is the best possible solution to Question 1.3
(cf. Conjecture 5.20). If true, this gives a new and very different type of
characterisation of separable permutations.

###  1.3. Other Weyl groups

When $W$ is a simply-laced Weyl group, we use categorical methods to study a
much broader class of $W$-modules arising from representations of Lie
algebras. To describe this, let $I$ be the Dynkin diagram of $W$, and for a
subdiagram $J\subseteq I$, let $W_{J}\subseteq W$ denote the parabolic
subgroup. Let $w_{J}\in W_{J}$ denote the longest element. The separable
elements $W^{\mathrm{sep}}\subseteq W$ for general Weyl groups were first
defined by Gaetz and Gao [16]. Here we use an alternate characterisation [15,
Theorem 7]:

$W^{\mathrm{sep}}:=\\{w_{I_{1}}\cdots w_{I_{r}}\in W\mid I\supseteq
I_{1}\supseteq\cdots\supseteq I_{r}\\}.$

We’ll now exhibit the above phenomenon in $W$-modules arising from
representations of Lie algebras, such as tensor products of irreducible
highest weight modules. We prove that separable elements of $W$ act by
bijection up to lower-order terms on the canonical bases of weight zero.
Moreover, the canonical bases of such representations have a natural crystal
structure, and the bijection and preorder have an explicit crystal-theoretic
description.

Let $\mathfrak{g}$ be the complex semisimple Lie algebra with diagram $I$, and
let $X^{+}$ be the set of dominant integral weights. For $\lambda\in X^{+}$ we
let $L(\lambda)$ be the irreducible representation of $\mathfrak{g}$ with
highest weight $\lambda$. Given a list of dominant weights
$\underline{\lambda}=(\lambda_{1},\ldots,\lambda_{n})$, let
$L(\underline{\lambda})=L(\lambda_{1})\otimes\cdots\otimes L(\lambda_{n})$ be
the tensor product representation.

Tensor product representations $L(\underline{\lambda})$ carry various
canonical bases, constructed originally by Lusztig [31], which possess amazing
positivity properties. In this introduction we’ll focus on the dual canonical
basis, but similar results hold also for the canonical basis (cf. Section
4.3).

Recall that there is a natural action of $W$ on the zero weight space
$L(\underline{\lambda})_{0}$. Let $\mathbb{B}(\underline{\lambda})\subset
L(\underline{\lambda})_{0}$ denote the set of dual canonical basis elements of
weight zero.

######  Theorem 1.5.

Let $w\in W^{\mathrm{sep}}$ and let $\underline{\lambda}$ be a sequence of
dominant weights. Then $w$ acts on
$(L(\underline{\lambda})_{0},\mathbb{B}(\underline{\lambda}))$ by a bijection
up to lower-order terms.

We discuss this in detail in Section 6. The bijection and preorder are defined
using the crystal structure on the set of dual canonical basis elements. We
emphasise that as representations of $W$, the spaces
$L(\underline{\lambda})_{0}$ are very mysterious in general, even in type A
and for $n=1$.

###  1.4. Methodology

Our methods come from categorical representation theory. The starting point is
work of Khovanov-Lauda and Rouquier [25, 34] who developed the foundations of
this theory for $U(\mathfrak{g})$ and the quantum group $U_{q}(\mathfrak{g})$.

The definition of a categorical representation is recalled in Section 2.3. For
the present discussion, the salient points are as follows. We will consider
representations of $U(\mathfrak{g})$ on an abelian category $\mathcal{C}$. By
definition, the category is a direct sum of full subcategories
$\mathcal{C}_{\mu}$ indexed by weights, just as a finite dimensional
representation is a direct sum of weight spaces. From the categorical
structure, the complexified Grothendieck group $V=[\mathcal{C}]_{\mathbb{C}}$
is endowed with an action of $U(\mathfrak{g})$, and the classes of simple
objects in $\mathcal{C}$ descend to a distinguished basis of $V$. In the cases
of interest to us, this is the dual canonical basis.

Chuang and Rouquier constructed the Rickard complex, which is a functor of
bounded derived categories:
$\Theta_{i}\mathsf{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to
D^{b}(\mathcal{C}_{s_{i}(\mu)})$. Here $s_{i}\in W_{I}$ is the simple
reflection associated to a node $i\in I$. They proved that
$\Theta_{i}\mathsf{1}_{\mu}$ is an equivalence of triangulated categories, and
used this to prove Broué’s conjecture for symmetric groups [11]. Many
important equivalences in algebraic geometry and related areas turn out to be
Rickard complexes, for example the spherical twists that appear in Mirror
Symmetry.

The autoequivalences $\Theta_{i}\mathsf{1}_{\mu}$ on
$D^{b}(\mathcal{C}_{\mu})$ satisfy the braid relations (up to isomorphism)
[8]. Hence, given $w\in W_{I}$ we have an autoequivalence
$\Theta_{w}\mathsf{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to
D^{b}(\mathcal{C}_{w(\mu)})$, which is defined up to isomorphism.

Focusing now on the case $\mu=0$, the connection to our work arises since the
autoequivalence $\Theta_{w}\mathsf{1}_{0}$ categorifies the operator
$w:V_{0}\to V_{0}$. Therefore, to study the action of $w$ on canonical bases
of $V_{0}$, we’d like to understand the image of simple objects in
$\mathcal{C}_{0}$ under $\Theta_{w}\mathsf{1}_{0}$.

It turns out that the theory of perverse equivalences, also due to Chuang and
Rouquier [12], provides the correct setting to extract this information.
Indeed, a perverse equivalence is essentially a categorification of Definition
1.2 (cf. Lemma 3.4). More precisely, an equivalence
$\Theta:D^{b}(\mathcal{C})\to D^{b}(\mathcal{C}^{\prime})$ is perverse if it
preserves some filtrations on the domain and codomain categories such that on
each subquotient the induced equivalence is t-exact up to shift. Crucially,
from a perverse equivalence one can extract a bijection of the simple objects
by keeping track of the t-exact equivalences on each subquotient. The details
of this theory are explained in Section 3.1.

The main result that underlies our work is the following.

######  Theorem 1.6.

Let $\mathcal{C}$ be a categorical representation of $U(\mathfrak{g})$ and let
$w\in W^{\mathrm{sep}}$ be a separable element. Then
$\Theta_{w}\mathsf{1}_{0}$ is a perverse equivalence.

The case $w=w_{I}$ was studied in [20]. There is a subtlety in extending this
case to all separable elements, since compositions of perverse equivalences
are not in general perverse. In Section 3.3, we develop the necessary
techniques to prove Theorem 1.6, and in doing so we precisely describe the
necessary filtrations and resulting bijections. By investigating the
consequences of this theorem on the level of Grothendieck groups, we obtain
(generalisations of) Theorems 1.4 and 1.5.

###  1.5. Organisation of paper

We give a brief overview of the paper:

In Section 2 we recall some of the basic representation-theoretic objects
which play a key role throughout the paper. Particularly important is the
notion of an ordered $I$-filtration (Definition 2.1), which is a categorical
analogue of an isotypic filtration. This will be crucially used in the
construction of perverse equivalences that follow.

Section 3 is devoted to the theory of perverse equivalences. The main result
is Theorem 3.12, which is the precise formulation of Theorem 1.6 above.

In Section 4 we explain how to deduce combinatorial consequences from
perversity. Theorem 4.8 states a general result: in any $W$-module which
arises from categorification (cf. Equation (4.1)), separable elements act by
bijections up to l.o.t. We also explain how to deduce analogous results about
dual bases (cf. Corollary 4.10).

In Section 5, we prove Theorem 1.4 as a special case of Theorem 1.5. We then
develop the combinatorics of type A crystals in order to find explicit
descriptions for $\xi$ and $\leq$ on $\mathbb{K}\mathbb{L}_{\lambda}$ in terms
of tableaux. We consider a number of special cases, including the
generalisation of Rhoades’ Theorem concerning the action of the long cycle
(cf. Proposition 5.14).

In Section 6, we consider canonical bases of tensor product representations.
We give several examples of Theorem 1.5, including the action of the long
element of $S_{n}$ on the dual canonical basis in the $d$-fold tensor power of
the standard representation, and the actions of separable permutations
$(1,3),(1,2,3)\in S_{3}$ on the canonical and dual canonical basis of the
tensor square of the adjoint representation of $\mathfrak{sl}_{3}$.

Appendix A gives a number of general results about operators which act by
bijections up to l.o.t., including the connection to QR decompositions. In
Appendix B, we generalise the definition of separable elements to arbitrary
Coxeter groups, and extend known results about the Kazhdan–Lusztig cell
modules to prove an analogue of Theorem 1.4 for the left regular
representation of a Coxeter group (cf. Theorem B.1). This argument does not
require categorical methods.

### Acknowledgements

We thank Ben Elias, Christian Gaetz, Anthony Licata and Daniel Tubbenhauer for
helpful conversations which improved this work. O.Y. is supported by the
Australian Research Council under DP230100654.

##  2\. Representations, crystals and categorifications

In this section we recall the necessary background about representation theory
(classical and categorical), and associated combinatorial objects which we
will need in the rest of the paper. This section also serves to fix notation.

###  2.1. Basic objects

Let $I$ be a simply-laced Dynkin diagram of finite type. For nodes $i,j\in I$
we write $i\sim j$ if they are connected by an edge, and $i\not\sim j$
otherwise.

Let $\mathfrak{g}=\mathfrak{g}_{I}$ be the complex semisimple Lie algebra
associated to $I$, with Chevalley generators $e_{i},f_{i},h_{i}$ for $i\in I$,
and Cartan subalgebra
$\mathfrak{h}_{I}=\operatorname{span}_{\mathbb{C}}(\\{h_{i}\mid i\in I\\})$.
We denote by $X=X_{I}$ its integral weight lattice, and let $\langle
h,\lambda\rangle\in\mathbb{Z}$ denote the natural pairing between elements of
the integral Cartan subalgebra (i.e. integral linear combinations of the
$h_{i}$) and integral weights.

Let $X^{+}=X^{+}_{I}$ be the cone of integral dominant weights, and
$R=R_{I}\subset X$ the root lattice. The latter is a free $\mathbb{Z}$-module
generated by simple roots $\alpha_{i}$ for $i\in I$. For
$\alpha=\sum_{i}n_{i}\alpha_{i}\in R$ we let
$\mathrm{ht}(\alpha):=\sum_{i}n_{i}$. The simple roots define a partial order
$\preceq$ (denoted $\preceq_{I}$) on $X$ given by
$\mu\preceq_{I}\mu^{\prime}\Longleftrightarrow\mu^{\prime}-\mu\in\sum_{i\in
I}\mathbb{Z}_{\geq 0}\alpha_{i}$.

We will be working exclusively with complex finite-dimensional representations
of $\mathfrak{g}$. For such a representation $V$ and for $\mu\in X$, we let
$V_{\mu}$ denote its $\mu$ weight space, so that $V=\bigoplus_{\mu\in
X}V_{\mu}$ is the weight decomposition of $V$. For $\lambda\in X^{+}$ we let
$L(\lambda)$ be the irreducible representation with highest weight $\lambda$.
For any representation $V$, there is a canonical isomorphism

$\bigoplus_{\lambda\in
X^{+}}\big{(}L(\lambda)\otimes\operatorname{Hom}_{\mathfrak{g}}(L(\lambda),V)\big{)}\longrightarrow
V,$

and we let $\mathsf{Iso}_{\lambda}(V)$ denote the image of the
$\lambda$-summand under this map. This is the $\lambda$-isotypic component of
$V$, and we say $V$ is an isotypic representation of highest weight $\lambda$
if $V=\mathsf{Iso}_{\lambda}(V)$.

Given a subdiagram $J\subseteq I$, we consider the Lie subalgebra
$\mathfrak{g}_{J}\subseteq\mathfrak{g}$. For a representation $V$ of
$\mathfrak{g}$, we let $\mathsf{Res}_{J}(V)$ denote the restriction of $V$ to
$\mathfrak{g}_{J}$.

The Weyl group $W=W_{I}$ is generated by $\\{s_{i}\mid i\in I\\}$ subject to
the relations:

  1. (1)

$s_{i}^{2}=1$,

  2. (2)

$s_{i}s_{j}=s_{j}s_{i}$ if $i\not\sim j$,

  3. (3)

$s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j}$ if $i\sim j$.

Relations (2) and (3) are known as the braid relations, and they are the
defining relations of the braid group $B=B_{I}$ with generators
$\\{\sigma_{i}\mid i\in I\\}$.

Recall that the Weyl group naturally acts on the weight lattice. Let $w_{I}\in
W_{I}$ be the longest element. This element induces a diagram automorphism
$\theta_{I}:I\to I$, given by $w_{I}\cdot\alpha_{i}=-\alpha_{\theta(i)}$. For
a subdiagram $J\subseteq I$, we regard $W_{J}\subseteq W_{I}$ as a parabolic
subgroup. For a representation $U$ of $W_{I}$, we let $\mathsf{Res}_{J}(U)$
denote its restriction to $W_{J}$.

###  2.2. Crystals

We can canonically associate a combinatorial object called an $I$-crystal to
any representation of $\mathfrak{g}$. For us, an $I$-crystal consists of a
finite set $\SS$ together with “Kashiwara operators”
$\widetilde{e_{i}},\widetilde{f_{i}}:\SS\dashrightarrow\SS$ for all $i$ (the
$\dashrightarrow$ signifies a partially defined map), and a weight function
$\mathsf{wt}:\SS\to X$. These satisfy the following properties: for any
$b,b^{\prime}\in\SS$

  1. (1)

$\widetilde{e_{i}}(b)=b^{\prime}$ if and only if
$b=\widetilde{f_{i}}(b^{\prime})$,

  2. (2)

if $\widetilde{e_{i}}(b)$ is defined then
$\mathsf{wt}(\widetilde{e_{i}}(b))=\mathsf{wt}(b)+\alpha_{i}$, and if
$\widetilde{f_{i}}(b)$ is defined then
$\mathsf{wt}(\widetilde{f_{i}}(b))=\mathsf{wt}(b)-\alpha_{i}$.

Since we will only be interested in “normal” crystals, i.e. crystals arising
from finite dimensional representations of $\mathfrak{g}$, no additional data
is necessary to define the crystal (the functions usually denoted
$\varepsilon_{i},\varphi_{i}$ are already determined).

Kashiwara originally discovered the theory of crystals [23], and showed how to
construct a crystal associated to any integrable representation of the quantum
group. There are other means of constructing crystals, for example via
algebraic geometry [22] or category theory (which we recall below) [27].

An important point shared by all these approaches is that they are closely
related to the construction and existence of canonical bases of various types,
and the underlying set of the crystal is in natural bijection with such bases.
In particular, the crystal $\SS$ associated to the representation $V$
satisfies the property that
$\dim(V_{\mu})=\\#\\{b\in\SS\mid\mathsf{wt}(b)=\mu\\}$.

There are some basic operations on crystals which mirror analogous operations
on representations. For example, the direct sum $\SS\oplus\SS^{\prime}$ is a
crystal whose underlying set is obtained by disjoint union, and the additional
data is defined in the obvious way. Additionally, given a subdiagram
$J\subseteq I$, the restriction of an $I$-crystal $\SS$ is a $J$-crystal
denoted $\mathsf{Res}_{J}(\SS)$. The underlying set of $\mathsf{Res}_{J}(\SS)$
is the same as $\SS$, but we forget the Kashiwara operators indexed by $i\in
I\setminus J$, and the weight of $b\in\mathsf{Res}_{J}(\SS)$ is the
restriction of $\mathsf{wt}(b)$ to $\mathfrak{h}_{J}$. Crystal also carry the
structures of tensor products $\SS\otimes\SS^{\prime}$ and external tensor
products $\SS\boxtimes\SS^{\prime}$, where the underlying set is
$\SS\times\SS^{\prime}$ in both cases (we refer the reader to [7] for the
precise definitions).

We let $\mathbb{L}(\lambda)$ be the crystal of $L(\lambda)$. Note that there
are unique highest and lowest weight elements in $\mathbb{L}(\lambda)$, which
we denote $b_{\lambda}^{\text{high}}$ and $b_{\lambda}^{\text{low}}$. The
crystal associated to the representation $V$ is isomorphic to
$\bigoplus_{\lambda\in X^{+}}\mathbb{L}(\lambda)^{\oplus m(\lambda)}$, where
$m(\lambda)=\dim(\operatorname{Hom}_{\mathfrak{g}}(L(\lambda),V))$. We let
$\mathsf{Iso}_{\lambda}(\SS):=\mathbb{L}(\lambda)^{\oplus m(\lambda)}$. This
is the crystal analogue of an isotypic component.

We can picture an $I$-crystal as an $I$-labelled directed graph, whose
vertices are given by the underlying set of $\SS$, with an $i$-labelled edge
from $b$ to $b^{\prime}$ if $\widetilde{f_{i}}(b)=b^{\prime}$. See Figure 1 an
example in type A. Note that the crystal graph of $\mathsf{Res}_{J}(\SS)$ is
obtained from the crystal graph of $\SS$ by deleting all edges labelled by
$i\in I\setminus J$.

We recall the generalised Schützenberger involution $\xi_{I}:\SS\to\SS$, which
will play an important role in our work. To define this set map, it suffices
to consider a simple crystal $\mathbb{L}(\lambda)$, since as we described
above, $\SS$ is isomorphic to a direct sum of simple crystals. The map
$\xi_{I}:\mathbb{L}(\lambda)\to\mathbb{L}(\lambda)$ is determined by the
following properties:

  1. (1)

$\mathsf{wt}(\xi_{I}(b))=w_{I}\cdot\mathsf{wt}(b)$,

  2. (2)

$\xi_{I}(\widetilde{e_{i}}(b))=\widetilde{f}_{\theta_{I}(i)}(\xi_{I}(b))$,

  3. (3)

$\xi_{I}(\widetilde{f_{i}}(b))=\widetilde{e}_{\theta_{I}(i)}(\xi_{I}(b))$.

Note that by (1), $\xi_{I}$ swaps $b_{\lambda}^{\text{high}}$ and
$b_{\lambda}^{\text{low}}$, and $\xi_{I}$ is then determined on the rest of
the crystal by properties (2) and (3).

We now have the map $\xi_{I}:\SS\to\SS$, and so for any $J\subseteq I$ we also
get maps $\xi_{J}:\mathsf{Res}_{J}(\SS)\to\mathsf{Res}_{J}(\SS)$. Since these
are just set maps, we can write $\xi_{J}:\SS\to\SS$ without confusion.

###  2.3. Categorical representation theory

Categorical representations will play an important role for us. The
foundations of this theory are due to Rouquier [36] and Khovanov-Lauda [24].
Although there are some differences in their approach, later work of Cautis-
Lauda [9] and Brundan [5] has clarified their relationships, and the latter,
in particular, proves that their definitions agree.

In this work we will use categorical representations of $U(\mathfrak{g})$
following Rouquier’s approach. We note that everything we do has analogues in
the graded setting, i.e. for the quantum group and the associated Hecke
algebra, but that will not be our focus here.

Let $[\mathcal{A}]_{\mathbb{C}}$ denote the complexified Grothendieck group of
an abelian category $\mathcal{A}$, and recall that an exact functor
$F:\mathcal{A}\to\mathcal{A}^{\prime}$ induces a linear map
$[F]:[\mathcal{A}]_{\mathbb{C}}\to[\mathcal{A}^{\prime}]_{\mathbb{C}}$. Recall
also that a Serre subcategory $\mathcal{A}^{\prime}\subset\mathcal{A}$ is a
full subcategory such that for any exact sequence $A\to B\to C$ in
$\mathcal{A}$, if $A,C\in\mathcal{A}^{\prime}$ then
$B\in\mathcal{A}^{\prime}$.

Let $\mathbf{k}$ be the ground ring. A categorical representation of
$U(\mathfrak{g})$ consists of a family of $\mathbf{k}$-linear abelian
categories $\mathcal{C}_{\mu}$ where $\mu\in X$, exact linear functors
$\mathsf{E}_{i}\mathsf{1}_{\mu}:\mathcal{C}_{\mu}\to\mathcal{C}_{\mu+\alpha_{i}}$
and
$\mathsf{F}_{i}\mathsf{1}_{\mu}:\mathcal{C}_{\mu}\to\mathcal{C}_{\mu-\alpha_{i}}$,
and natural transformations:

  1. (1)

$X:\mathsf{E}_{i}\mathsf{1}_{\mu}\longrightarrow\mathsf{E}_{i}\mathsf{1}_{\mu}$,

  2. (2)

$T:\mathsf{E}_{i}\mathsf{E}_{j}\mathsf{1}_{\mu}\longrightarrow\mathsf{E}_{j}\mathsf{E}_{i}\mathsf{1}_{\mu}$,

  3. (3)

$\eta:\mathsf{1}_{\mu}\longrightarrow\mathsf{F}_{i}\mathsf{E}_{i}\mathsf{1}_{\mu}$,

  4. (4)

$\varepsilon:\mathsf{E}_{i}\mathsf{F}_{i}\mathsf{1}_{\mu}\longrightarrow\mathsf{1}_{\mu}$.

These satisfy relations, which depend on some further choices of scalars,
which we won’t directly use and so don’t specify. Details can be found in [36,
5]. In practice, we write $\mathcal{C}$ for a categorical representation,
leaving the remaining data implicit.

There are three important consequences of these relations which we mention:

  1. (1)

There are functorial isomorphisms

$\displaystyle\mathsf{E}_{j}\mathsf{F}_{i}\mathsf{1}_{\mu}\cong\mathsf{F}_{i}\mathsf{E}_{j}\mathsf{1}_{\mu}\text{
if }i\neq j,$
$\displaystyle\mathsf{E}_{i}\mathsf{F}_{i}\mathsf{1}_{\mu}\cong\mathsf{F}_{i}\mathsf{E}_{i}\mathsf{1}_{\mu}\oplus\mathsf{1}_{\mu}^{\oplus\langle
h_{i},\mu\rangle}\text{ if }\langle h_{i},\mu\rangle\geq 0,$
$\displaystyle\mathsf{E}_{i}\mathsf{F}_{i}\mathsf{1}_{\mu}\oplus\mathsf{1}_{\mu}^{\oplus-\langle
h_{i},\mu\rangle}\cong\mathsf{F}_{i}\mathsf{E}_{i}\mathsf{1}_{\mu}\text{ if
}\langle h_{i},\mu\rangle\leq 0$

  2. (2)

The functors
$(\mathsf{E}_{i}\mathsf{1}_{\mu},\mathsf{F}_{i}\mathsf{1}_{\mu+\alpha_{i}})$
are bi-adjoint. One adjunction is part of the definition (the unit and counits
are given by $\eta$ and $\varepsilon$), and the other is a consequence [5,
Theorem 4.3].

  3. (3)

The natural transformations $X$ and $T$ define an action of the quiver Hecke
(alias KLR) algebra on compositions of the $\mathsf{E}_{i}\mathsf{1}_{\mu}$
functors.

The first point above implies that from a categorical representation
$\mathcal{C}$ one can produce a classical representation. This is a
representation on the complex $X$-graded vector space
$V=[\mathcal{C}]_{\mathbb{C}}:=\bigoplus_{\mu}[\mathcal{C}_{\mu}]_{\mathbb{C}}$.
Define linear maps $\pi(e_{i}):V\to V$ by

$\pi(e_{i})([X])=[\mathsf{E}_{i}\mathsf{1}_{\mu}(X)],$

for $X\in\mathcal{C}_{\mu}$, and similarly $\pi(f_{i}):V\to V$. Letting
$\pi(h_{i})$ be the diagonal operator which acts by the scalar $\langle
h_{i},\mu\rangle$ on $[\mathcal{C}_{\mu}]_{\mathbb{C}}$, the resulting
representation is given by $\pi:U(\mathfrak{g})\to\operatorname{End}(V)$. We
say that $\mathcal{C}$ categorifies the representation $V$.

The third point can be used to show that there is an isomorphism of functors
$\mathsf{E}_{i}^{n}\mathsf{1}_{\mu}\cong(\mathsf{E}_{i}^{(n)}\mathsf{1}_{\mu})^{\oplus
n!}$, where $\mathsf{E}_{i}^{(n)}\mathsf{1}_{\mu}$ is a subfunctor of
$\mathsf{E}_{i}^{n}\mathsf{1}_{\mu}$. The functor
$\mathsf{E}_{i}^{(n)}\mathsf{1}_{\mu}$ is therefore called the divided-power
of $\mathsf{E}_{i}$.

Now, suppose $\mathcal{C}$ is a categorical representation which categorifies
a representation $V$. We recall that the crystal of $V$ can be extracted from
$\mathcal{C}$ ([11, Proposition 5.20] and [6, Theorem 4.31]). The construction
proceeds as follows: the underlying set $\SS$ of the crystal is the set of
isomorphism classes of simple objects in $\mathcal{C}$. Then for $[L]\in\SS$,
set $\mathsf{wt}([L])=\mu$ for $L\in\mathcal{C}_{\mu}$, and define the
Kashiwara operators by
$\widetilde{e}_{i}([L])=\text{socle}(\mathsf{E}_{i}(L))$ and
$\widetilde{f}_{i}([L])=\text{socle}(\mathsf{F}_{i}(L))$ (recall that the
socle of an object in an abelian category is its maximal semisimple
subobject).

If $\mathcal{C}$ categorifies an isotypic representation, we say that
$\mathcal{C}$ is an isotypic categorification. Analogous to the isotypic
decomposition of a representation, any categorical representation
$\mathcal{C}$ has an isotypic filtration [20, Section 3.3]:

######  Definition 2.1.

An $I$-filtration of $\mathcal{C}$ is a filtration by Serre subcategories

$0=\mathcal{C}(0)\subset\mathcal{C}(1)\subset\cdots\subset\mathcal{C}(n)=\mathcal{C}$

satisfying the following properties:

  1. (1)

For every $i$, $\mathcal{C}(i)\subset\mathcal{C}$ is a categorical
subrepresentation of $\mathfrak{g}_{I}$, i.e. it is invariant under all the
data defining the categorical representation on $\mathcal{C}$.

  2. (2)

The representation on the Serre quotient $\mathcal{C}(i)/\mathcal{C}({i-1})$
is an isotypic categorification.

An $I$-filtration is ordered if, letting $\lambda_{i}\in X_{+}$ denote the
highest weight of $\mathcal{C}(i)/\mathcal{C}({i-1})$, the list
$\lambda_{1},\ldots,\lambda_{n}$ consists of distinct highest weights such
that if $\lambda_{i}\prec_{I}\lambda_{j}$ then $i<j$.

Note that in an $I$-filtration of $\mathcal{C}$, different subquotients can be
isotypic categorifications of the same highest weight, but this cannot happen
in an ordered $I$-filtration.

Any categorical representation has an ordered $I$-filtration. Indeed, by [20,
Remark 3.9] an ordered $I$-filtration can be constructed from $\SS$. Suppose
that $\SS$ decomposes into components

$\SS=\mathsf{Iso}_{\lambda_{1}}(\SS)\sqcup\cdots\sqcup\mathsf{Iso}_{\lambda_{n}}(\SS),$

where $\lambda_{1},\ldots,\lambda_{n}\in X_{+}$ are distinct dominant integral
weights. Further, we arrange the weights so that
$\lambda_{i}\prec_{I}\lambda_{j}$ implies $i<j$. Let $\mathcal{C}(i)$ be the
Serre subcategory of $\mathcal{C}$ generated by simple objects $L$ such that
$[L]\in\mathsf{Iso}_{\lambda_{j}}(\SS)$ for $j\leq i$. Then
$0\subset\mathcal{C}(1)\subset\mathcal{C}(2)\subset\cdots\subset\mathcal{C}(n)=\mathcal{C}$
is an ordered $I$-filtration of $\mathcal{C}$.

Now we move on to consider triangulated equivalences that arise from
categorical representations. Let $D^{b}(\mathcal{C})$ denote the bounded
derived category. For $i\in I$ the Rickard complex
$\Theta_{i}\mathsf{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to
D^{b}(\mathcal{C}_{s_{i}(\mu)})$ was constructed by Chuang and Rouquier, who
initially used it to prove Broué’s abelian defect conjecture for symmetric
groups [11]. By definition, $\Theta_{i}\mathsf{1}_{\mu}$ is a complex of
functors concentrated in non-positive degrees:

$\Theta_{i}\mathsf{1}_{\mu}=\big{(}\cdots\longrightarrow(\Theta_{i}\mathsf{1}_{\mu})^{-1}\longrightarrow(\Theta_{i}\mathsf{1}_{\mu})^{0}\big{)},$

where for $r\geq 0$ and $\mu_{i}:=\langle h_{i},\mu\rangle$,

$(\Theta_{i}\mathsf{1}_{\mu})^{-r}=\begin{cases}\mathsf{E}_{i}^{(-\mu_{i}+r)}\mathsf{F}_{i}^{(r)}\mathsf{1}_{\mu}&\text{
if }\mu_{i}\leq 0\\\
\mathsf{E}_{i}^{(\mu_{i}+r)}\mathsf{F}_{i}^{(r)}\mathsf{1}_{\mu}&\text{ if
}\mu_{i}\geq 0\end{cases}$

The differential
$(\Theta_{i}\mathsf{1}_{\mu})^{-r}\longrightarrow(\Theta_{i}\mathsf{1}_{\mu})^{-r+1}$
is defined using the counit
$\mathsf{E}_{i}\mathsf{F}_{i}\mathsf{1}_{\mu}\longrightarrow\mathsf{1}_{\mu}$
arising from the adjunction. The fundamental results about Rickard complexes
are:

  1. (1)

For every $\mathcal{C},i\in I$, and $\mu\in X$, $\Theta_{i}\mathsf{1}_{\mu}$
is an equivalence of triangulated categories [11].

  2. (2)

The Rickard complexes satisfy the braid relations, i.e.
$\Theta_{i}\Theta_{j}\mathsf{1}_{\mu}\cong\Theta_{j}\Theta_{i}\mathsf{1}_{\mu}$
if $i\not\sim j$, and
$\Theta_{i}\Theta_{j}\Theta_{i}\mathsf{1}_{\mu}\cong\Theta_{j}\Theta_{i}\Theta_{j}\mathsf{1}_{\mu}$
if $i\sim j$ [8, Section 6].

Therefore, for any $w\in W$, we can define
$\Theta_{w}\mathsf{1}_{\mu}:=\Theta_{i_{1}}\circ\cdots\circ\Theta_{i_{\ell}}\mathsf{1}_{\mu}$,
where $w=s_{i_{1}}\cdots s_{i_{\ell}}$ is any reduced expression. This is
well-defined up to isomorphism.

##  3\. Perverse Equivalences

###  3.1. Definition and properties

We begin with the set-up for the definition of a perverse equivalence. Let
$\mathcal{A}$ be an abelian category, and let $\mathbb{B}$ be the set of
simple objects in $\mathcal{A}$ up to equivalence. For any collection of
objects $\mathbb{B}^{\prime}\subseteq\mathcal{A}$, we let
$\langle\mathbb{B}^{\prime}\rangle\subseteq\mathcal{A}$ be the Serre
subcategory generated by $\mathbb{B}^{\prime}$. Recall that
$\langle\mathbb{B}^{\prime}\rangle$ is the smallest full subcategory of
$\mathcal{A}$ containing $\mathbb{B}^{\prime}$ which is closed under quotients
and extensions.

Given a subcategory $\mathcal{B}\subseteq\mathcal{A}$, let
$D^{b}_{\mathcal{B}}(\mathcal{A})\subseteq D^{b}(\mathcal{A})$ denote the
subcategory consisting of complexes with cohomology supported in
$\mathcal{B}$. Equivalently, $D^{b}_{\mathcal{B}}(\mathcal{A})$ is the thick
subcategory of $D^{b}(\mathcal{A})$ generated by $\mathcal{B}$. We set
$D^{b}_{\mathbb{B}^{\prime}}(\mathcal{A}):=D^{b}_{\langle\mathbb{B}^{\prime}\rangle}(\mathcal{A})$.

Now suppose we have a chain of abelian categories
$\mathcal{C}\subseteq\mathcal{B}\subseteq\mathcal{A}$, and the induced chain
of triangulated categories:

$\mathcal{T}=D^{b}_{\mathcal{C}}(\mathcal{A})\;\subseteq\;\mathcal{S}=D^{b}_{\mathcal{B}}(\mathcal{A})\;\subseteq\;\mathcal{R}=D^{b}(\mathcal{A}).$

Let $Q:\mathcal{S}\to\mathcal{S}/\mathcal{T}$ denote the quotient functor, and
let $(\mathcal{R}^{\leq 0},\mathcal{R}^{\geq 0})$ denote the natural
t-structure on $\mathcal{R}$. We have induced t-structures on $\mathcal{S}$
and $\mathcal{S}/\mathcal{T}$ given by $(\mathcal{R}^{\leq
0}\cap\mathcal{S},\mathcal{R}^{\geq 0}\cap\mathcal{S})$ and
$(Q(\mathcal{R}^{\leq 0}\cap\mathcal{S}),Q(\mathcal{R}^{\geq
0}\cap\mathcal{S}))$. Note that the heart of $\mathcal{S}$ is $\mathcal{B}$
and the heart of $\mathcal{S}/\mathcal{T}$ is $\mathcal{B}/\mathcal{C}$.

Now suppose we have two chains of subsets of $\mathbb{B}$:

$\displaystyle\mathbb{B}(1)\subseteq\mathbb{B}(2)\subseteq\cdots\subseteq\mathbb{B}(s)$
$\displaystyle=\mathbb{B}$ (3.1)
$\displaystyle\mathbb{B}^{\prime}(1)\subseteq\mathbb{B}^{\prime}(2)\subseteq\cdots\subseteq\mathbb{B}^{\prime}(s)$
$\displaystyle=\mathbb{B}$ (3.2)

Setting $\mathcal{A}(i):=\langle\mathbb{B}(i)\rangle$ and
$\mathcal{T}(i):=D^{b}_{\mathcal{A}(i)}(\mathcal{A})$ and similarly for
$\mathcal{A}^{\prime}(i)$ and $\mathcal{T}^{\prime}(i)$, we have induced
chains of categories:

$\displaystyle\mathcal{A}(1)\subseteq\mathcal{A}(2)\subseteq\cdots\subseteq\mathcal{A}(s)$
$\displaystyle\mathcal{T}(1)\subseteq\mathcal{T}(2)\subseteq\cdots\subseteq\mathcal{T}(s)$

and similarly for the primed versions. Set $\mathcal{T}:=D^{b}(\mathcal{A})$.
Note that $\mathcal{T}=\mathcal{T}(s)=\mathcal{T}^{\prime}(s)$.

######  Definition 3.3.

Suppose $F:\mathcal{T}\to\mathcal{T}$ is an autoequivalence. Let
$p:\\{1,\ldots,s\\}\to\mathbb{Z}$. Then $F$ is a perverse equivalence with
respect to $(\mathcal{A}(\bullet),\mathcal{A}^{\prime}(\bullet),p)$ if for
every $i$:

  1. (1)

$F(\mathcal{T}(i))=\mathcal{T}^{\prime}(i)$, and

  2. (2)

the induced equivalence
$F[-p(i)]:\mathcal{T}(i)/\mathcal{T}(i-1)\to\mathcal{T}^{\prime}(i)/\mathcal{T}^{\prime}(i-1)$
is t-exact.

Given a perverse equivalence $F$, and carrying the notation from the
definition, we obtain by Condition (2) abelian equivalences
$\mathcal{A}(i)/\mathcal{A}(i-1)\to\mathcal{A}^{\prime}(i)/\mathcal{A}^{\prime}(i-1)$
for every $i$. Since abelian equivalences must map simple objects to simple
objects, these yield bijections

$\mathbb{B}(i)\setminus\mathbb{B}(i-1)\leftrightarrow\mathbb{B}^{\prime}(i)\setminus\mathbb{B}^{\prime}(i-1).$

Patching these together we obtain a bijection
$\varphi_{F}:\mathbb{B}\to\mathbb{B}$.

The functor $F$ induces an endomorphism
$f:[\mathcal{A}]_{\mathbb{C}}\to[\mathcal{A}]_{\mathbb{C}}$. For
$x,y\in\mathbb{B}$, write $x\leq y$ if
$x\in\mathbb{B}(i)\setminus\mathbb{B}(i-1)$ and
$y\in\mathbb{B}(j)\setminus\mathbb{B}(j-1)$ for $i\leq j$.

######  Lemma 3.4.

The operator $f$ acts on $([\mathcal{A}]_{\mathbb{C}},\mathbb{B},\leq)$ by
$\varphi_{F}$ up to lower-order terms.

###### Proof.

Suppose $x\in\mathbb{B}(i)\setminus\mathbb{B}(i-1)$. Then in
$[\mathcal{A}]_{\mathbb{C}}$ we have an equality

$f(x)=\pm\varphi_{F}(x)+\sum_{x^{\prime}\in\mathbb{B}^{\prime}(i-1)}a_{x^{\prime}}x^{\prime},$

where $a_{x^{\prime}}\in\mathbb{Z}$. Note that
$\varphi_{F}(x)\in\mathbb{B}^{\prime}(i)\setminus\mathbb{B}^{\prime}(i-1)$,
and since by construction
$\varphi_{F}(\mathbb{B}(i-1))=\mathbb{B}^{\prime}(i-1)$, we can rewrite this
as

$f(x)=\pm\varphi_{F}(x)+\sum_{y\in\mathbb{B}(i-1)}a_{y}\varphi_{F}(y).$

∎

###  3.2. Prior results

Let $\mathcal{C}$ be a categorical representation of $\mathfrak{g}$ with
associated $\mathfrak{g}$-crystal $\SS$ (so the underlying set of $\SS$ is the
isomorphism classes of simple objects in $\mathcal{C}$). We recall two results
of Halacheva, Licata, Losev and Yacobi:

######  Theorem 3.5 (Corollary 6.7 in [20]).

Suppose $\mathcal{C}$ is an isotypic categorical representation of type
$\lambda$. Then for any weight $\mu\in X$, the derived equivalence
$\Theta_{w_{I}}\mathsf{1}_{\mu}[-\mathrm{ht}(w_{I}(\lambda)-\mu)]:D^{b}(\mathcal{C}_{\mu})\to
D^{b}(\mathcal{C}_{w_{I}(\mu)})$ is t-exact.

Let $\mathcal{C}(1)\subset\cdots\subset\mathcal{C}(s)=\mathcal{C}$ be an
$I$-filtration of $\mathcal{C}$ such that $\mathcal{C}(i)/\mathcal{C}(i-1)$
categorifies an isotypic representation of type $\lambda_{i}$. For $\mu\in X$,
let $\mathcal{C}_{\mu}(i)=\mathcal{C}_{\mu}\cap\mathcal{C}(i)$. Recall that
the Schützenberger involution $\xi_{I}:\SS\to\SS$ satisfies the property that
$\xi_{I}(\SS_{\mu})=\SS_{w_{I}(\mu)}$.

######  Theorem 3.6 (Theorem 6.8 in [20]).

The autoequivalence
$\Theta_{w_{I}}\mathsf{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to
D^{b}(\mathcal{C}_{w_{I}(\mu)})$ is a perverse equivalence with respect to
$(\mathcal{C}_{\mu}(\bullet),\mathcal{C}_{w_{I}(\mu)}(\bullet),p)$, where
$p(i)=\mathrm{ht}(w_{I}(\lambda_{i})-\mu)$. Moreover, the bijection between
$\SS_{\mu}$ and $\SS_{w_{I}(\mu)}$ induced by $\Theta_{w_{I}}\mathsf{1}_{\mu}$
is equal to $\xi_{I}$.

Fix $J\subseteq I$, and let $\mathcal{D}$ denote the restriction of
$\mathcal{C}$ to $\mathfrak{g}_{J}$. That is, $\mathcal{D}=\mathcal{C}$ as
categories, but $\mathcal{D}$ is regarded as a categorical representation of
$U(\mathfrak{g}_{J})$. The weight subcategories are $\mathcal{D}_{\gamma}$ for
$\gamma\in X_{J}$, where
$\mathcal{D}_{\gamma}=\bigoplus_{\mu}\mathcal{C}_{\mu}$ and the sum ranges
over all $\mu\in X$ such that $\mu|_{\mathfrak{h}_{J}}=\gamma$.

Suppose $\mathcal{D}(1)\subset\cdots\subset\mathcal{D}(s)=\mathcal{D}$ is a
$J$-filtration of $\mathcal{D}$ such that $\mathcal{D}(j)/\mathcal{D}(j-1)$
categorifies an isotypic representation of type $\kappa_{j}$. For $\gamma\in
X_{J}$ define $\mathcal{D}_{\gamma}(j)$ as above, and for $\mu\in X$ let
$\mathcal{C}_{\mu}^{J}(j)=\mathcal{D}(j)\cap\mathcal{C}_{\mu}$. Recall that
$w_{J}$ is the longest element of $W_{J}\subseteq W$.

######  Corollary 3.7.

The autoequivalence
$\Theta_{w_{J}}\mathsf{1}_{\mu}:D^{b}(\mathcal{C}_{\mu})\to
D^{b}(\mathcal{C}_{w_{J}(\mu)})$ is a perverse equivalence with respect to
$(\mathcal{C}_{\mu}^{J}(\bullet),\mathcal{C}_{w_{J}(\mu)}^{J}(\bullet),p)$,
where $p(j)=\mathrm{ht}(w_{J}(\kappa_{j})-\gamma)$ and
$\gamma=\mu|_{\mathfrak{h}_{J}}\in X_{J}$. Moreover, the associated bijection
is equal to $\xi_{J}$.

###### Proof.

We know that $\Theta_{w_{J}}\mathsf{1}_{\mu}$ is an equivalence between
$D^{b}(\mathcal{C}_{\mu})$ and $D^{b}(\mathcal{C}_{w_{J}(\mu)})$, and we must
show it is a perverse equivalence. By Theorem 3.6, we have that
$\Theta_{w_{J}}\mathsf{1}_{\gamma}:D^{b}(\mathcal{D}_{\gamma})\to
D^{b}(\mathcal{D}_{w_{J}(\gamma)})$ is a perverse equivalence with respect to
$(\mathcal{D}_{\gamma}(\bullet),\mathcal{D}_{w_{J}(\gamma)}(\bullet),q)$,
where $p(j)=\mathrm{ht}(w_{J}(\kappa_{j})-\gamma)$, and the associated
bijection is
$\xi_{J}:\operatorname{res}_{J}(\SS)_{\gamma}\to\operatorname{res}_{J}(\SS)_{w_{J}(\gamma)}$.

Now, note that

$\displaystyle D^{b}_{\mathcal{D}_{\gamma}(j)}(\mathcal{D}_{\gamma})\cap
D^{b}(\mathcal{C}_{\mu})=D^{b}_{\mathcal{C}_{\mu}^{J}(j)}(\mathcal{C}_{\mu}).$

This follows from an easy fact: if $\mathcal{A},\mathcal{B},\mathcal{C}$ are
abelian categories with $\mathcal{C}\subseteq\mathcal{B}$, then
$D^{b}_{\mathcal{A}\cap\mathcal{B}}(\mathcal{A})\cap
D^{b}(\mathcal{C})=D^{b}_{\mathcal{A}\cap\mathcal{C}}(\mathcal{C})$. Therefore
the data defining the perverse equivalence $\Theta_{w_{J}}\mathsf{1}_{\gamma}$
restricts to define perversity data for $\Theta_{w_{J}}\mathsf{1}_{\mu}$ (with
the same perversity function) as claimed. Furthermore, the associated
bijection is the same as $\xi_{J}$ above, but the domain and codomain are
suitably shrunk to $\SS_{\mu}$ and $\SS_{w_{J}(\mu)}$. ∎

The case where $\mu=0$ is of most interest to us, so henceforth we set
$\mathcal{A}=\mathcal{C}_{0}$, $\mathbb{B}=\SS_{0}$ and
$\Theta_{w}:=\Theta_{w}\mathsf{1}_{0}$. For the longest element $w_{I}$ we set
$\Theta_{I}=\Theta_{w_{I}}$, and more generally $\Theta_{J}=\Theta_{w_{J}}$
for any $J\subseteq I$.

Note that in the set up of Theorem 3.6, $\Theta_{I}$ is perverse with respect
to $(\mathcal{A}(\bullet),\mathcal{A}(\bullet),p)$, i.e. the filtrations in
the domain and codomain agree. Moreover the perversity function in this case
simplifies to $p(i)=\mathrm{ht}(\lambda_{i})$. The analogous fact holds for
any $J\subseteq I$.

###  3.3. Perversity of $\Theta_{w}$ for separable elements

#### 3.3.1.

In this section we prove some technical results in preparation for the study
of $\Theta_{w}$ for separable elements $w\in W$.

We consider the following set-up: $\mathcal{A}$ is an abelian category and

$\mathcal{D}\subset\mathcal{C}\subset\mathcal{B}\subset\mathcal{A}$

is a chain of Serre subcategories. We suppose we have an equivalence
$F:D^{b}(\mathcal{A})\to D^{b}(\mathcal{A})$ which satisfies perversity-like
conditions with respect to the filtration
$\mathcal{D}\subset\mathcal{B}\subset\mathcal{A}$. Specifically, we have:

  1. (1)

$F(D^{b}_{\mathcal{B}}(\mathcal{A}))=D^{b}_{\mathcal{B}}(\mathcal{A})$,

  2. (2)

$F(D^{b}_{\mathcal{D}}(\mathcal{A}))=D^{b}_{\mathcal{D}}(\mathcal{A})$, and

  3. (3)

the induced autoequivalence of
$D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{D}}(\mathcal{A})$ (which we
denote by $\overline{F}$) is t-exact up to shift $b\in\mathbb{Z}$.

Notice that $F$ is not actually perverse since we don’t require any exactness
on $D^{b}_{\mathcal{D}}(\mathcal{A})$ nor on the quotient
$D^{b}(\mathcal{A})/D^{b}_{\mathcal{B}}(\mathcal{A})$.

Let $\mathbb{B}_{\mathcal{B}}$ (respectively
$\mathbb{B}_{\mathcal{C}},\mathbb{B}_{\mathcal{D}}$) be the set of simple
objects up to equivalence of $\mathcal{B}$ (respectively
$\mathcal{C},\mathcal{D}$). Let
$\varphi:\mathbb{B}_{\mathcal{B}}\setminus\mathbb{B}_{\mathcal{D}}\to\mathbb{B}_{\mathcal{B}}\setminus\mathbb{B}_{\mathcal{D}}$
be the bijection induced by (3). Define

$\mathcal{C}^{\prime}:=\big{<}\mathcal{D},\varphi(L)\;|\;L\in\mathbb{B}_{\mathcal{C}}\setminus\mathbb{B}_{\mathcal{D}}\big{>}.$

Let $Q:D^{b}_{\mathcal{B}}(\mathcal{A})\to
D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{D}}(\mathcal{A})$ be the
natural quotient functor. Note that the restriction of $Q$ to $\mathcal{B}$ is
the natural quotient functor $\mathcal{B}\to\mathcal{B}/\mathcal{D}$.

######  Lemma 3.8.

The category $\mathcal{C}^{\prime}$ is the full subcategory of $\mathcal{B}$
with objects

$\\{B\in\mathcal{B}\;|\;Q(B)\in\overline{F}[b](\mathcal{C}/\mathcal{D})\\}.$

###### Proof.

Denote by $\mathcal{C}^{\prime\prime}$ the full subcategory of $\mathcal{B}$
defined in the statement of the lemma. We first show that
$\mathcal{C}^{\prime\prime}$ is a Serre subcategory. Consider an exact
sequence $A_{1}\to A_{2}\to A_{3}$ where
$A_{1},A_{3}\in\mathcal{C}^{\prime\prime}$. Applying $Q$ we obtain
$Q(A_{1})\to Q(A_{2})\to Q(A_{3})$. Since $\overline{F}[b]$ restricts to an
autoequivalence of $\mathcal{B}/\mathcal{D}$, we have that
$\overline{F}[b](\mathcal{C}/\mathcal{D})$ is a Serre subcategory of
$\mathcal{B}/\mathcal{D}$. By hypothesis
$Q(A_{1}),Q(A_{3})\in\overline{F}[b](\mathcal{C}/\mathcal{D})$, and therefore
so is $Q(A_{2})$. This shows that $A_{2}\in\mathcal{C}^{\prime\prime}$.

Since both $\mathcal{C}^{\prime}$ and $\mathcal{C}^{\prime\prime}$ are Serre
subcategories, they are determined by the simple objects they contain.
Therefore to prove the lemma it suffices to show that they contain the same
simple objects. Clearly they both contain $\mathbb{B}_{\mathcal{D}}$. If $L$
is a simple object in $\mathcal{C}^{\prime}$ not in $\mathcal{D}$, then
$L=\varphi(L_{1})$ for some
$L_{1}\in\mathbb{B}_{\mathcal{C}}\setminus\mathbb{B}_{\mathcal{D}}$. Then
$Q(L)\cong\overline{F}[b](Q(L_{1}))$ and so $L\in\mathcal{C}^{\prime\prime}$.
Conversely, if $L$ is a simple object in $\mathcal{C}^{\prime\prime}$ not in
$\mathcal{D}$ then $Q(L)\cong\overline{F}[b](Q(L_{1}))$ for some simple object
in $\mathcal{C}/\mathcal{D}$, and so
$L=\varphi(L_{1})\in\mathcal{C}^{\prime}$.

∎

######  Proposition 3.9.

In the above set-up,
$F(D^{b}_{\mathcal{C}}(\mathcal{A}))=D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$.

###### Proof.

It suffices to show that $F(D^{b}_{\mathcal{C}}(\mathcal{A}))\subseteq
D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$, since the same argument will also
show that $F^{-1}(D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A}))\subseteq
D^{b}_{\mathcal{C}}(\mathcal{A})$.

We first show that if $X\in\mathcal{C}$ then $F(X)\in
D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$. Fix some $i\in\mathbb{Z}$ and
consider the functor
$D^{b}_{\mathcal{B}}(\mathcal{A})\to\mathcal{B}/\mathcal{C}^{\prime}$ given by
$R\circ H^{i}$, where $R:\mathcal{B}\to\mathcal{B}/\mathcal{C}^{\prime}$ is
the natural quotient functor and $H^{i}$ is the standard cohomology functor.
Note that $R\circ H^{i}(Y)=0$ for any $Y\in
D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$. Therefore, by the universal
property of Verdier quotients, there is a functor
$G:D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})\to\mathcal{B}/\mathcal{C}^{\prime}$
making the following commute:

${D^{b}_{\mathcal{B}}(\mathcal{A})}$${D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})}$${\mathcal{B}/\mathcal{C}^{\prime}}$$\scriptstyle{R\circ
H^{i}}$$\scriptstyle{G}$

(In fact, $G$ is given by the $i$-th cohomology with respect to the
t-structure which is induced on
$D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$,
but we do not require this fact.)

Note that by (1), $F(X)\in D^{b}_{\mathcal{B}}(\mathcal{A})$. Applying the
quotient functor to
$D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{D}}(\mathcal{A})$, we have
that $Q(F(X)[b])\cong\overline{F}[b](Q(X))$. Then by Lemma 3.8 we have
$Q(F(X)[b])\in\mathcal{C}^{\prime}$. Since the quotient functor
$D^{b}_{\mathcal{B}}(\mathcal{A})\to
D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$
factors through
$D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{D}}(\mathcal{A})$, the image
of $F(X)$ under the quotient functor $D^{b}_{\mathcal{B}}(\mathcal{A})\to
D^{b}_{\mathcal{B}}(\mathcal{A})/D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$ is
zero. Hence, by the commutativity of the above diagram we get that
$R(H^{i}(F(X)))=0$, i.e. $H^{i}(F(X))\in\mathcal{C}^{\prime}$ as desired.

For $X\in\mathcal{A}$ let $\ell(X)$ be the Jordan-Hölder length of $X$, i.e.
the length of its Jordan-Hölder filtration. For $X\in D^{b}(\mathcal{A})$ let

$\ell(X)=\sum_{i}\ell(H^{i}(X)).$

Now suppose $X\in D^{b}_{\mathcal{C}}(\mathcal{A})$. We will show by induction
on $\ell:=\ell(X)$ that $F(X)\in D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$.

If $\ell=1$ then, up to isomorphism, $X$ is concentrated in one degree and the
claim follows from the previous paragraphs. Now suppose $\ell>1$. Consider
first the case where there exists $r$ such that $\ell(\tau^{\leq
r}(X)),\ell(\tau^{\geq r+1}(X))<\ell$. Then applying $F$ to the exact triangle

$\tau^{\leq r}(X)\to X\to\tau^{\geq r+1}(X)\to$

we obtain

$F(\tau^{\leq r}(X))\to F(X)\to F(\tau^{\geq r+1}(X))\to$

and from the long exact sequence in cohomology we get exact sequences

$H^{i}(F(\tau^{\leq r}(X)))\to H^{i}(F(X))\to H^{i}(F(\tau^{\leq r}(X)))$

for every $i$. Now note that if $H^{i}(\tau^{\leq r}(X))$ is nonzero then it
is isomorphic to $H^{i}(X)$, and similarly for $H^{i}(\tau^{\geq r+1}(X))$.
Therefore $H^{i}(\tau^{\leq r}(X)),H^{i}(\tau^{\geq r+1}(X))\in\mathcal{C}$,
i.e. $\tau^{\leq r}(X),\tau^{\geq r+1}(X)\in
D^{b}_{\mathcal{C}}(\mathcal{A})$. By inductive hypothesis therefore we have
that $H^{i}(F(\tau^{\leq r}(X))),H^{i}(F(\tau^{\geq
r+1}(X)))\in\mathcal{C}^{\prime}$. Since $\mathcal{C}^{\prime}$ is a Serre
subcategory we obtain from the above exact sequence that
$H^{i}(F(X))\in\mathcal{C}^{\prime}$ as well, i.e. that $F(X)\in
D^{b}_{\mathcal{C}^{\prime}}(\mathcal{A})$.

Finally if there is no $r$ such that $\ell(\tau^{\leq r}(X)),\ell(\tau^{\geq
r+1}(X))<\ell$, then $X$ is a two-term complex with zero differential. Hence
it is isomorphic to a direct sum of two one-term complexes, and the result
follows from the work above. ∎

#### 3.3.2.

Consider a chain of diagrams $\mathcal{Z}=(I_{1},\ldots,I_{r})$ where
$I\supseteq I_{1}\supseteq\cdots\supseteq I_{r}$. Let $w\in W$ be the
corresponding separable element: $w=w_{\mathcal{Z}}:=w_{I_{1}}\cdots
w_{I_{r}}$. In this section we will prove that $\Theta_{w}$ is a perverse
equivalence. For this we need a filtration of $\mathcal{C}$ which is
compatible with $\mathcal{Z}$.

We set up up some notation for working with filtrations. Let
$[0,n]=\\{0,\ldots,n\\}$ and denote by $\mathcal{C}_{[0,n]}$ a filtration of
$\mathcal{C}$ with terms indexed by $[0,n]$:

$0=\mathcal{C}(0)\subset\mathcal{C}(1)\subset\cdots\subset\mathcal{C}(n)=\mathcal{C}.$

For a subset $A:=\\{i_{0},\dots,i_{\ell}\\}$ of indices with
$0=i_{0}<i_{1}<\dots<i_{\ell}=n$, we denote by $\mathcal{C}_{A}$ the
coarsening of $\mathcal{C}_{[0,n]}$ given by

$0=\mathcal{C}(i_{0})\subset\mathcal{C}(i_{1})\subset\cdots\subset\mathcal{C}(i_{\ell})=\mathcal{C}.$

Let $\mathcal{C}_{A}(k):=\mathcal{C}(i_{k})$ denote the $k$-th term of
$\mathcal{C}_{A}$. There is an induced filtration on each subquotient
$\mathcal{C}_{A}(k)/\mathcal{C}_{A}(k-1)$ by

$0=\mathcal{C}(i_{k-1})/\mathcal{C}(i_{k-1})\subset\mathcal{C}(i_{k-1}+1)/\mathcal{C}(i_{k-1})\subset\cdots\subset\mathcal{C}(i_{k})/\mathcal{C}(i_{k-1})=\mathcal{C}_{A}(k)/\mathcal{C}_{A}(k-1).$

We give a recursive definition for when $\mathcal{C}_{[0,n]}$ is an (ordered)
$\mathcal{Z}$-filtration.

######  Definition 3.10.

A filtration $\mathcal{C}_{[0,n]}$ of $\mathcal{C}$ is a
$\mathcal{Z}$-filtration if it can be coarsened to an $I_{1}$-filtration
$\mathcal{C}_{A}$, and if $r>1$, the induced filtration on each
$\mathcal{C}_{A}(k)/\mathcal{C}_{A}(k-1)$ is an
$(I_{2},\dots,I_{r})$-filtration.

Moreover, $\mathcal{C}_{[0,n]}$ is an ordered $\mathcal{Z}$-filtration if
$\mathcal{C}_{A}$ is an ordered $I_{1}$-filtration, and the induced filtration
on each $\mathcal{C}_{A}(k)/\mathcal{C}_{A}(k-1)$ is an ordered
$(I_{2},\dots,I_{r})$-filtration.

A filtration of $\mathcal{A}$ (the zero weight subcategory of $\mathcal{C}$)
induced by an (ordered) $\mathcal{Z}$-filtration is an (ordered)
$\mathcal{Z}$-filtration of $\mathcal{A}$.

Observe that $\mathcal{C}_{[0,n]}$ is a $\mathcal{Z}$-filtration if and only
if it is an $I_{r}$-filtration which can be coarsened to an
$I_{r-1}$-filtration, which in turn can be coarsened to an
$I_{r-2}$-filtration and so on.

The notion of an ordered $\mathcal{Z}$-filtration is more subtle, and we
elucidate this with an example when $r=2$. Consider the following
$(I,J)$-filtration of $\mathcal{C}$ for some $J\subset I$.

12345

In this picture, the rectangle represents the entire category $\mathcal{C}$,
and the vertical lines represent a filtration $\mathcal{C}_{[0,5]}$. We set
$A:=\\{3,5\\}$ so that $\mathcal{C}_{A}$ is the coarsening of
$\mathcal{C}_{[0,5]}$ given by the thick black lines.

If $\mathcal{C}_{[0,5]}$ is a $\mathcal{Z}$-filtration, then
$\mathcal{C}_{A}(1)=\mathcal{C}(3)$ and
$\mathcal{C}_{A}(2)/\mathcal{C}_{A}(1)=\mathcal{C}(5)/\mathcal{C}(3)$
categorify isotypic representations of $\mathfrak{g}_{I}$, and
$\mathcal{C}(i)/\mathcal{C}(i-1)$ categorifies an isotypic representation of
$\mathfrak{g}_{J}$ for every $1\leq i\leq 5$. This is an ordered
$\mathcal{Z}$-filtration if and only if:

  * •

The weights of $\mathcal{C}_{A}(1)$ and
$\mathcal{C}_{A}(2)/\mathcal{C}_{A}(1)$ are distinct and partially ordered
with respect to $\preceq_{I}$ (in particular they may be incomparable),

  * •

The weights of $\mathcal{C}(i)/\mathcal{C}(i-1)$ are distinct and partially
ordered with respect to $\preceq_{J}$ for $i=1,2,3$,

  * •

The weights of $\mathcal{C}(i)/\mathcal{C}(i-1)$ are distinct and partially
ordered with respect to $\preceq_{J}$ for $i=4,5$.

For example, it may be the case that $\mathcal{C}(4)/\mathcal{C}(3)$ has a
strictly lower weight than $\mathcal{C}(3)/\mathcal{C}(2)$ under
$\preceq_{J}$, but this is allowable in an ordered $\mathcal{Z}$-filtration
since they belong to different subquotients of the $I$-filtration.

######  Lemma 3.11.

Let $\mathcal{C}$ be a categorical representation of $U(\mathfrak{g})$, and
let $\mathcal{Z}=(I_{1},\ldots,I_{r})$ be a chain of diagrams. Then an ordered
$\mathcal{Z}$-filtration of $\mathcal{C}$ exists.

###### Proof.

We construct an ordered $\mathcal{Z}$-filtration inductively as follows: If
$r=1$ then an ordered $\mathcal{Z}$-filtration is given in [20, Remark 3.8].
Let $r>1$ and suppose we’ve constructed an ordered
$\mathcal{Z}^{\prime}$-filtration of any categorical representation of
$U(\mathfrak{g})$, where $\mathcal{Z}^{\prime}$ is a chain of length $r-1$.
Let

$0=\mathcal{C}(0)\subseteq\mathcal{C}(1)\subseteq\cdots\subseteq\mathcal{C}(s)=\mathcal{C}$

be an ordered $I_{1}$-filtration of $\mathcal{C}$. Then for any $i$ we have
that $\mathcal{C}(i)/\mathcal{C}(i-1)$ is a categorical representation of
$\mathfrak{g}_{I_{1}}$. Let $\mathcal{Z}^{\prime}:=(I_{2},\ldots,I_{r})$, and
consider $\mathcal{C}(i)/\mathcal{C}(i-1)$ as a categorical representation of
$\mathfrak{g}_{I_{2}}$ by restriction. By hypothesis we have an ordered
$\mathcal{Z}^{\prime}$-filtration of $\mathcal{C}(i)/\mathcal{C}(i-1)$. This
induces a chain of categories

$\mathcal{C}(i-1)=\mathcal{C}(i-1,0)\subseteq\mathcal{C}(i-1,1)\subseteq\cdots\subseteq\mathcal{C}(i-1,d_{i})=\mathcal{C}(i).$

Concatenating these chains together yields an ordered $\mathcal{Z}$-filtration
of $\mathcal{C}$. ∎

All the pieces are now in place to define the perversity data for
$\Theta_{w}$. Let $\mathcal{C}_{[0,n]}$ be a $\mathcal{Z}$-filtration of
$\mathcal{C}$ (and hence of $\mathcal{A}$). Fix $1\leq a\leq r$ and find
$A=\\{i_{0},\dots,i_{\ell}\\}$ so that the coarsening $\mathcal{C}_{A}$ is an
isotypic $I_{a}$-filtration. By Corollary 3.7,
$\Theta_{w_{I_{a}}}:D^{b}(\mathcal{A})\to D^{b}(\mathcal{A})$ is perverse with
respect to the filtration
$0=\mathcal{A}(i_{0})\subseteq\mathcal{A}(i_{1})\subseteq\cdots\subseteq\mathcal{A}(i_{\ell})=\mathcal{A}$
on the domain and codomain.

By construction each $\mathcal{A}(i)$ is a Serre subcategory of $\mathcal{A}$.
Therefore we can define $\mathbb{B}(i)\subseteq\mathbb{B}$ by
$\mathcal{A}(i):=\big{<}\mathbb{B}(i)\big{>}$. Now define

$\mathbb{B}^{\prime}(i)=\big{<}\xi_{I_{1}}\circ\cdots\circ\xi_{I_{r}}\big{(}\mathbb{B}(i)\big{)}\big{>},$

and let $\mathcal{A}^{\prime}(i):=\big{<}\mathbb{B}^{\prime}(i)\big{>}$.

Finally, to the subquotient $\mathcal{A}(i)/\mathcal{A}(i-1)$ in the
$\mathcal{Z}$-filtration we can associate a sequence of dominant weights
$\lambda^{i}_{a}\in X^{+}_{I_{a}}$ ($a=1,\ldots,r$) determined by:

$\displaystyle\big{[}\mathcal{A}(i)/\mathcal{A}(i-1)\big{]}_{\mathbb{C}}\subseteq\mathsf{Iso}_{\lambda_{a}^{i}}(\mathsf{Res}_{\mathfrak{g}_{I_{a}}}(V)).$

Set $p(i):=\sum_{a=1}^{r}\mathrm{ht}(\lambda^{i}_{a})$.

######  Theorem 3.12.

The autoequivalence $\Theta_{w}$ of $D^{b}(\mathcal{A})$ is perverse with
respect to $(\mathcal{A}(\bullet),\mathcal{A}^{\prime}(\bullet),p)$. Moreover,
the bijection $\varphi_{\Theta_{w}}$ is equal to
$\xi_{\mathcal{Z}}=\xi_{I_{1}}\circ\cdots\circ\xi_{I_{r}}$.

###### Proof.

We will prove the theorem by induction on $r$. If $r=1$ and $w=w_{I_{1}}$,
then by Corollary 3.7, $\Theta_{w}$ is a perverse equivalence with respect to
the $I_{1}$-filtration of $\mathcal{A}$ on both the domain and codomain.
Moreover, the perversity function is given by the height of the highest weight
corresponding to each subquotient. This agrees with the perversity data in the
statement of the theorem. Note that Corollary 3.7 also gives that the
associated bijection is $\xi_{I_{1}}$, completing the base case.

For $r>1$, let $\mathcal{T}(i)$ and $\mathcal{T}^{\prime}(i)$ be the
triangulated categories defined from $\mathcal{A}(i)$ and
$\mathcal{A}^{\prime}(i)$ as in Section 3.1. Write $w=w_{I_{1}}u$ where
$u=w_{I_{2}}\cdots w_{I_{r}}$. Set $\mathcal{Z}^{u}=(I_{2},\ldots,I_{r})$.
Note that the $\mathcal{Z}$-filtration of $\mathcal{A}$ is in particular also
a $\mathcal{Z}^{u}$-filtration. Let
$\mathbb{B}^{u}(i):=\big{<}\xi_{I_{2}}\circ\cdots\circ\xi_{I_{r}}\big{(}\mathbb{B}(i)\big{)}\big{>}$,
and define the associated categories $\mathcal{A}^{u}(i),\mathcal{T}^{u}(i)$
and the perversity function
$p^{u}(i):=\sum_{a=2}^{r}\mathrm{ht}(\lambda^{i}_{a})$. By induction we have
that $\Theta_{u}$ is perverse with respect to
$(\mathcal{A}(\bullet),\mathcal{A}^{u}(\bullet),p^{u})$, and
$\varphi_{\Theta_{u}}=\xi_{I_{2}}\circ\cdots\circ\xi_{I_{r}}$.

Suppose

$0=\mathcal{A}(i_{0})\subseteq\mathcal{A}(i_{1})\subseteq\cdots\subseteq\mathcal{A}(i_{t})=\mathcal{A}$

is a coarsening of the $\mathcal{Z}$-filtration to a $I_{1}$-filtration. First
observe that $\mathcal{A}^{u}(i_{\ell})=\mathcal{A}(i_{\ell})$ for
$1\leq\ell\leq t$. Indeed, for any $1\leq a\leq r$ we have that
$\varphi_{I_{a}}(\mathbb{B}(i_{\ell}))=\mathbb{B}(i_{\ell})$. Therefore
$\Theta_{u}(\mathcal{T}(i_{\ell}))=\mathcal{T}(i_{\ell})$. Since we also have
that $\Theta_{I_{1}}(\mathcal{T}(i_{\ell}))=\mathcal{T}(i_{\ell})$, it follows
that the same is true for $\Theta_{w}$.

Now take $i_{\ell-1}<j<i_{\ell}$ and consider the chain
$\mathcal{A}(i_{\ell-1})\subset\mathcal{A}^{u}(j)\subset\mathcal{A}(i_{\ell})\subset\mathcal{A}$.
We are in the setting of Section 3.3.1, where $F=\Theta_{I_{1}}$. By
Proposition 3.9, we deduce that
$\Theta_{I_{1}}(\mathcal{T}^{u}(j))=\mathcal{T}^{\prime}(j)$. Since we know by
hypothesis that $\Theta_{u}(\mathcal{T}(j))=\mathcal{T}^{u}(j)$, we conclude
that $\Theta_{w}(\mathcal{T}(j))=\mathcal{T}^{\prime}(j)$. This shows that
$\Theta_{w}$ preserves the filtrations in the perversity data.

We next show that $\Theta_{w}$ is t-exact up to shifts (prescribed by $p$) on
the subquotients of the filtration. Take $j$ as in the previous paragraph. By
inductive hypothesis the functor

$\Theta_{u}:\mathcal{T}(j)/\mathcal{T}(j-1)\to\mathcal{T}^{u}(j)/\mathcal{T}^{u}(j-1)$

is t-exact up to shift $-p^{u}(j)$. Moreover, by Proposition 3.6 the functor

$\Theta_{I_{1}}:\mathcal{T}(i_{\ell})/\mathcal{T}(i_{\ell}-1)\to\mathcal{T}(i_{\ell})/\mathcal{T}(i_{\ell}-1)$

is t-exact up to shift $-\mathrm{ht}(\lambda^{j}_{1})$. The restriction of
this functor to $\mathcal{T}^{u}(j)/\mathcal{T}^{u}(j-1)$ induces an
equivalence
$\mathcal{T}^{u}(j)/\mathcal{T}^{u}(j-1)\to\mathcal{T}^{\prime}(j)/\mathcal{T}^{\prime}(j-1)$
which is also t-exact up to shift $-\mathrm{ht}(\lambda^{j}_{1})$. Therefore
the composition $\Theta_{w}=\Theta_{I_{1}}\circ\Theta_{u}$ induces an
equivalence
$\mathcal{T}(j)/\mathcal{T}(j-1)\to\mathcal{T}^{\prime}(j)/\mathcal{T}^{\prime}(j-1)$
which is t-exact up shift $-p(j)=-\mathrm{ht}(\lambda^{j}_{1})-p^{u}(j)$.

Notice that this also shows that the induced bijection
$\mathbb{B}(j)\setminus\mathbb{B}(j-1)\to\mathbb{B}^{\prime}(j)\setminus\mathbb{B}^{\prime}(j-1)$
is given by $\xi_{\mathcal{Z}}$, since in the composition of t-exact functors
(up to shift) $\Theta_{I_{1}}\circ\Theta_{u}$, $\Theta_{u}$ gives rise to
$\xi_{I_{2}}\circ\cdots\circ\xi_{I_{r}}$ and $\Theta_{I_{1}}$ gives rise to
$\xi_{I_{1}}$. ∎

##  4\. Combinatorial Applications

###  4.1. The general set-up

We are now in position to deduce combinatorial consequences from Theorem 3.12.
First we describe a general schema for extracting combinatorial information
about the action of separable elements on canonical bases. We will then apply
this to study two situations: the Kazhdan–Lusztig basis of Specht modules in
Section 5, and the canonical bases in the zero weight space of tensor product
representations in Section 6.

To describe the general schema, let $U$ be a representation of $W$ equipped
with a basis $\mathbb{B}\subset U$. We suppose that the pair $(U,\mathbb{B})$
arises via categorification in the following sense: there is a categorical
representation $\mathcal{C}$ of $U(\mathfrak{g})$, and an isomorphism of
$W$-modules

$\displaystyle U\cong[\mathcal{C}_{0}]_{\mathbb{C}},$ (4.1)

such that the set $\mathbb{B}$ is identified with the set of isomorphism-
classes of simple objects in $\mathcal{C}_{0}$.

######  Remark 4.2.

It is possible to develop our theory also for other weight spaces, and deduce
results about the action of a cover of the Weyl group (the “Tits group”) on
canonical basis elements in nonzero weight spaces. We’ve opted to focus on the
zero weight space for simplicity.

###  4.2. The action of a separable element on $\mathbb{B}$

Let $\SS$ be the $I$-crystal arising from the categorical representation
$\mathcal{C}$. Recall that the crystals which arise in this manner are
“normal”, i.e. they are isomorphic to a direct sum of irreducible crystals
$\mathbb{L}(\lambda)$ (for $\lambda\in X_{I}^{+}$). For $b\in\SS$, we let
$\SS[b]$ be the connected component of $\SS$ containing $b$ and let
$\tau_{I}(b)\in X_{I}^{+}$ be the highest weight of $\SS[b]$, i.e.
$b\in\mathsf{Iso}_{\lambda}(\SS)$ if and only if $\tau_{I}(b)=\lambda$.

We begin by defining a preorder $\leq_{I}$ on $\SS$ (and hence on
$\mathbb{B}=\SS_{0}$), whose linearisations will correspond to ordered
$I$-filtrations. We then extend this to chains $\mathcal{Z}$.

######  Definition 4.3.

For $x,y\in\SS$, let $x\leq_{I}y$ if and only if
$\tau_{I}(x)\preceq_{I}\tau_{I}(y)$. We call this the $I$-preorder on $\SS$.

Note that two elements are equivalent in this preorder if and only if they lie
in the same maximal isotypic component of $\SS$, and we write $x\sim_{I}y$.
For $J\subseteq I$ we define $\leq_{J}$ on $\SS$ by checking the relation in
$\mathsf{Res}_{J}\SS$.

Fix a chain $\mathcal{Z}=(I_{1},\dots,I_{r})$ with $I\supseteq
I_{1}\supseteq\cdots\supseteq I_{r}$.

######  Definition 4.4.

If $r=1$, define $\leq_{\mathcal{Z}}$ equivalently to $\leq_{I_{1}}$.
Otherwise, for $x,y\in\SS$, set $x\leq_{\mathcal{Z}}y$ if and only if
$x<_{I_{1}}y$, or $x\sim_{I_{1}}y$ and $x\leq_{(I_{2},\dots,I_{r})}y$.

Note that $x<_{\mathcal{Z}}y$ if and only if $x<_{I_{a}}y$ for the minimal $a$
such that $x\not\sim_{I_{a}}y$. If $x\sim_{I_{a}}y$ for all $1\leq a\leq r$,
then $x$ and $y$ are equivalent under $\leq_{\mathcal{Z}}$ and we write
$x\sim_{\mathcal{Z}}y$.

The $\sim_{\mathcal{Z}}$ equivalence classes are labelled by sequences
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{r})\in
X_{I_{1}}^{+}\times\cdots\times X_{I_{r}}^{+}$, where $b\in\SS$ is in the
equivalence class labelled by $(\tau_{I_{1}}(b),\ldots,\tau_{I_{r}}(b))$. Let
$\SS(\boldsymbol{\lambda})$ denote the $\boldsymbol{\lambda}$-equivalence
class in $\SS$, and let

$\displaystyle\SS=\SS(\boldsymbol{\lambda}_{1})\sqcup\cdots\sqcup\SS(\boldsymbol{\lambda}_{n})$
(4.5)

be the decomposition into equivalence classes.

For $\boldsymbol{\lambda}$ as above, set
$\boldsymbol{\lambda}|_{a}=(\lambda_{1},\ldots,\lambda_{a})$.

######  Definition 4.6.

The list of $\sim_{\mathcal{Z}}$ equivalence classes
$(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})$ is normally
ordered if

  1. (1)

$\boldsymbol{\lambda}_{i}<_{\mathcal{Z}}\boldsymbol{\lambda}_{j}\Longrightarrow
i<j$, and

  2. (2)

if $i<j$ and $(\boldsymbol{\lambda}_{i})|_{a}=(\boldsymbol{\lambda}_{j})|_{a}$
then $(\boldsymbol{\lambda}_{i})|_{a}=(\boldsymbol{\lambda}_{k})|_{a}$ for all
$i\leq k\leq j$.

A normal ordering can be constructed by restricting the crystal $\SS$ along
$\mathcal{Z}$, and arranging the isotypic components along the way. More
precisely, first consider the isotypic decomposition of $\SS$ regarded as an
$I_{1}$-crystal
$\SS=\mathsf{Iso}_{\lambda_{1}}(\SS)\sqcup\cdots\sqcup\mathsf{Iso}_{\lambda_{N}}(\SS)$,
with the dominant weight arranged so that if
$\lambda_{i}\prec_{I_{1}}\lambda_{j}$ then $i<j$. Then for each component,
consider its decomposition as an $I_{2}$-crystal:

$\displaystyle\mathsf{Iso}_{\lambda_{i}}(\SS)=\mathsf{Iso}_{\lambda_{i},\mu_{1}}(\SS)\sqcup\cdots\sqcup\mathsf{Iso}_{\lambda_{i},\mu_{N_{i}}}(\SS),$

where
$\mathsf{Iso}_{\lambda_{i},\mu_{k}}(\SS):=\mathsf{Iso}_{\mu_{k}}(\mathsf{Iso}_{\lambda_{i}}(\SS))$
and the weights arranged so that if $\mu_{k}\prec_{I_{2}}\mu_{\ell}$ then
$k<\ell$. Continuing in this way down the chain $\mathcal{Z}$, at the final
step we obtain a normally ordered list of the $\sim_{\mathcal{Z}}$ equivalence
classes.

Let $\mathcal{C}(i)$ be the Serre subcategory of $\mathcal{C}$ generated by
$\SS(\boldsymbol{\lambda}_{1}),\ldots,\SS(\boldsymbol{\lambda}_{i})$, and let
$\mathcal{C}_{[0,n]}$ denote the associated filtration.

######  Lemma 4.7.

Let $\mathcal{C}$ be a categorical representation of $U(\mathfrak{g})$ with
crystal $\SS$. Let
$(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})$ be a normally
ordered list of the $\sim_{\mathcal{Z}}$ equivalence classes of $\SS$, and let
$\mathcal{C}_{[0,n]}$ be the resulting filtration. Then $\mathcal{C}_{[0,n]}$
is an ordered $\mathcal{Z}$-filtration.

###### Proof.

We prove this by induction on $r$. If $r=1$ then by the discussion in Section
2.3, $\mathcal{C}_{[0,n]}$ is an ordered $I_{1}$-filtration. Suppose now
$r>1$. We can break up the list (4.5) into segments:

$\displaystyle\SS(\boldsymbol{\lambda}_{1}),\ldots,\SS(\boldsymbol{\lambda}_{i_{1}})$
$\displaystyle\SS(\boldsymbol{\lambda}_{i_{1}+1}),\ldots,\SS(\boldsymbol{\lambda}_{i_{2}})$
$\displaystyle\hskip 42.67912pt\vdots$
$\displaystyle\SS(\boldsymbol{\lambda}_{i_{\ell}+1}),\ldots,\SS(\boldsymbol{\lambda}_{n})$

where
$(\boldsymbol{\lambda}_{1})|_{1}=(\boldsymbol{\lambda}_{2})|_{1}=\cdots=(\boldsymbol{\lambda}_{i_{1}})|_{1}$,
$(\boldsymbol{\lambda}_{i_{1}+1})|_{1}=\cdots=(\boldsymbol{\lambda}_{i_{2}})|_{1}$,
etc. Let $A=\\{i_{1},\ldots,i_{\ell}\\}$. Then by the same discussion in
Section 2.3, $\mathcal{C}_{A}$ is an ordered $I_{1}$-filtration.

Consider a subquotient $\mathcal{C}_{A}(k)/\mathcal{C}_{A}(k-1)$ of
$\mathcal{C}_{A}$ for some $1\leq k\leq\ell$. This is an isotypic
categorification of highest weight
$\lambda:=\boldsymbol{\lambda}_{i_{k}}|_{1}\in X_{I_{1}}^{+}$. Let
$\mathcal{Z}^{\prime}=(I_{2},\ldots,I_{r})$. The induced filtration on this
subquotient is then obtained from a normal ordering of the
$\sim_{\mathcal{Z}^{\prime}}$ equivalence classes of the crystal for
$\mathcal{C}_{A}(k)/\mathcal{C}_{A}(k-1)$. By induction this is an ordered
$\mathcal{Z}^{\prime}$-filtration, so by definition $\mathcal{C}_{[0,n]}$ is
an ordered $\mathcal{Z}$-filtration.

∎

We return to the general set-up described in Section 4.1: we have a based
representation $(U,\mathbb{B})$ of $W$ which arises from the categorical
representation $\mathcal{C}$ of $U(\mathfrak{g})$.

######  Theorem 4.8.

Let $w\in W^{\mathrm{sep}}$ and $x\in\mathbb{B}$. Then $w$ acts on
$(U,\mathbb{B},\leq_{\mathcal{Z}})$ by $\xi_{\mathcal{Z}}$ up to l.o.t., where
$\mathcal{Z}=(I_{1},\ldots,I_{r})$ is a chain such that $w:=w_{I_{1}}\cdots
w_{I_{r}}$.

###### Proof.

We consider a normal ordering of $\SS$, and the resulting filtration
$\mathscr{C}$ as defined above Lemma 4.7. By the Lemma this is an ordered
$\mathcal{Z}$-filtration. Let $\mathcal{A}(\bullet)$ be the associated
$\mathcal{Z}$-filtration of $\mathcal{A}=\mathcal{C}_{0}$. By Theorem 3.12,
$\Theta_{w}:D^{b}(\mathcal{A})\to D^{b}(\mathcal{A})$ is a perverse
equivalence with respect to $\mathcal{A}(\bullet)$ (on both the domain and
codomain), and the associated bijection is $\xi_{\mathcal{Z}}$. The result
follows from Lemma 3.4. ∎

######  Remark 4.9.

The bijections which arise in this manner are closley related to the cactus
group $C_{I}$ [21]. This is a group generated by $c_{J}$, where $J\subseteq I$
is a connected subdiagram, satisfying the following relations:

  1. (1)

$c_{J}^{2}=1$, for all $J\subseteq I$,

  2. (2)

$c_{J}c_{K}=c_{K}c_{J}$ if $J\cap K=\varnothing$ and there is no edge
connecting any $j\in J$ to any $k\in K$,

  3. (3)

$c_{J}c_{K}=c_{K}c_{\theta_{K}(J)}$ if $J\subseteq K$.

The assignment $c_{J}\mapsto\xi_{J}$ defines an action of $C_{I}$ on
$\mathbb{B}$ [20, Theorem 7.7].

###  4.3. The action of separable elements on the dual basis

In this section we describe how to obtain an analogue of Theorem 4.8 for a
basis dual to $\mathbb{B}$. Indeed, in our set-up (4.1) the basis $\mathbb{B}$
is often Lusztig’s “dual canonical basis”. Moreover, the representation $U$ is
equipped with a non-degenerate symmetric form $(\cdot,\cdot):U\times
U\to\mathbb{C}$, and the basis $\mathbb{B}^{\prime}$ dual to $\mathbb{B}$ with
respect to $(\cdot,\cdot)$ is the “canonical basis” (cf. Section 6).

For example, consider the case where $U$ is the zero weight space of an
irreducible representation $L(\lambda)$ with a fixed highest weight vector
$v_{\lambda}$. The category $\mathcal{C}_{0}$ is equivalent to
$A\mathrm{-mod}$ (the category of finite dimensional $A$-modules) for an
appropriate algebra $A$ (cf. Section 5.2). Then $\mathbb{B}$ corresponds to
simple $A$-modules, and is Lusztig’s dual canonical basis.

On the other hand, $U$ is also categorified by $A\mathrm{-pmod}$ (the category
of finitely generated projective $A$-modules) and in this model the projective
indecomposable modules give rise to a different basis $\mathbb{B}^{\prime}$
(the canonical basis). These bases are orthogonal with respect to the
symmetric bilinear form descended from the hom-pairing on $\mathcal{C}_{0}$:
for $X\in A\mathrm{-pmod},Y\in A\mathrm{-mod}$,
$([X],[Y])=\dim\operatorname{Hom}(X,Y)$. Note that this is precisely the
Shapavalov form and satisfies the properties:

  1. (1)

$(v_{\lambda},v_{\lambda})=1$,

  2. (2)

$(e_{i}\cdot u,v)=(u,f_{i}\cdot v)$ for all $i\in I$ and any $u,v\in U$.

To state our result, let $V=[\mathcal{C}]_{\mathbb{C}}$ be the representation
of $U(\mathfrak{g})$ categorified by $\mathcal{C}$ (so that $U=V_{0}$). Note
that we don’t require $V$ to be irreducible (e.g. $V$ could be a tensor
product of irreducible representations, cf. Section 6). Suppose that $V$ is
equipped with a non-degenerate symmetric form satisfying Property (2) above.
We let $(\cdot,\cdot)$ denote the restriction of this form to $U$.

Write $\mathbb{B}=\\{x_{\mathbf{t}}\mid\mathbf{t}\in\mathcal{X}\\}$ for the
basis of $U$ arising from simple objects under (4.1), indexed by a set
$\mathcal{X}$. Let
$\mathbb{B}^{\prime}=\\{x_{\mathbf{t}}^{\prime}\mid\mathbf{t}\in\mathcal{X}\\}$
be the dual basis with respect to $(\cdot,\cdot)$.

Given a chain of diagrams $\mathcal{Z}=(I_{1},\ldots,I_{r})$, define:
$\xi_{\mathcal{Z}}(\mathbf{t})=\mathbf{s}$ if and only if
$\xi_{\mathcal{Z}}(x_{\mathbf{t}})=x_{\mathbf{s}}$, and
$\mathbf{t}\leq_{\mathcal{Z}}\mathbf{s}$ if and only if
$x_{\mathbf{t}}\leq_{\mathcal{Z}}x_{\mathbf{s}}$.

Let $w:=w_{I_{1}}\cdots w_{I_{r}}$ be the separable element corresponding to
$\mathcal{Z}$, and write $\xi=\xi_{\mathcal{Z}}$.

######  Corollary 4.10.

Let $\mathbf{t}\in\mathcal{X}$. Then there exist $b_{\mathbf{r}}\in\mathbb{Z}$
such that

$w\cdot x_{\mathbf{t}}^{\prime}=\pm
x^{\prime}_{\xi^{-1}(\mathbf{t})}+\sum_{\xi^{-1}(\mathbf{r})\;>_{\mathcal{Z}}\;\xi^{-1}(\mathbf{t})}b_{\mathbf{r}}x^{\prime}_{\xi^{-1}(\mathbf{r})}.$

######  Lemma 4.11.

Any $w\in W$ is self-adjoint with respect to $(\cdot,\cdot):U\times
U\to\mathbb{C}$.

###### Proof.

We need to show that for any $u,v\in U$ we have that $(w\cdot u,v)=(u,w\cdot
v)$, and it suffices to prove this for simple generators $s_{i}\in W$. The
action of $s_{i}$ on $U$ can be expressed in terms of the Lie algebra action
as follows:

$s_{i}=\sum_{k\geq 0}(-1)^{k}e_{i}^{(k)}f_{i}^{(k)},$

where $x^{(k)}=x^{k}/k!$ is the divided-power. Therefore self-adjointness
follows immediately from Property (2). ∎

###### Proof of Corollary 4.10.

By Theorem 4.8 we can write for any $\mathbf{s}\in\mathcal{X}$:

$w\cdot
x_{\mathbf{s}}=\sum_{\mathbf{r}}a_{\mathbf{r}\mathbf{s}}x_{\xi(\mathbf{r})},$

where $a_{\mathbf{s}\mathbf{s}}=\pm 1$ and $a_{\mathbf{r}\mathbf{s}}=0$ if
$\mathbf{r}>_{\mathcal{Z}}\mathbf{s}$. Define
$b_{\mathbf{r}\mathbf{t}}\in\mathbb{Z}$ by

$w\cdot
x^{\prime}_{\mathbf{t}}=\sum_{\mathbf{r}}b_{\mathbf{r}\mathbf{t}}x^{\prime}_{\xi^{-1}(\mathbf{r})}.$

Then

$\displaystyle b_{\mathbf{r}\mathbf{t}}=(x_{\xi^{-1}(\mathbf{r})},w\cdot
x_{\mathbf{t}}^{\prime})=(w\cdot
x_{\xi^{-1}(\mathbf{r})},x_{\mathbf{t}}^{\prime})=a_{\xi^{-1}(\mathbf{t})\xi^{-1}(\mathbf{r})}$

Now, $a_{\xi^{-1}(\mathbf{t})\xi^{-1}(\mathbf{r})}=\pm 1$ if
$\mathbf{r}=\mathbf{t}$ and $a_{\xi^{-1}(\mathbf{t})\xi^{-1}(\mathbf{r})}=0$
if $\xi^{-1}(\mathbf{r})<_{\mathcal{Z}}\xi^{-1}(\mathbf{t})$. Therefore

$\displaystyle w\cdot x_{\mathbf{t}}$ $\displaystyle=\pm
x^{\prime}_{\xi^{-1}(\mathbf{t})}+\sum_{\xi^{-1}(\mathbf{r})\;>_{\mathcal{Z}}\;\xi^{-1}(\mathbf{t})}b_{\mathbf{r}\mathbf{t}}x^{\prime}_{\xi^{-1}(\mathbf{r})}.$

∎

##  5\. The Kazhdan–Lusztig Basis for the Specht Module

In this section, we construct the Specht module in two ways. For
$\lambda\vdash n$, let $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$ be the
irreducible representation for the symmetric group $S_{n}$ indexed by the
partition $\lambda$ equipped with its Kazhdan–Lusztig basis. The classical
realisation of this based representation comes from cell modules (cf. Section
B.1) in type A. We recall below that this basis also appears as the dual
canonical basis of $L(\lambda)_{0}$, where $L(\lambda)$ is the irreducible
$\mathfrak{sl}_{n}$-representation of highest weight $\lambda$.

We present both of these constructions, and then use Theorem 4.8 to deduce the
action of separable permutations on $\mathbb{K}\mathbb{L}_{\lambda}$, and
prove Theorem 1.4. Using the combinatorics of type A crystals, we determine
explicit descriptions for the bijection $\xi_{\mathcal{Z}}$ and preorder
$\leq_{\mathcal{Z}}$ on $\mathbb{K}\mathbb{L}_{\lambda}$ in terms of standard
tableaux.

###  5.1. The Specht module as a cell module

Following [3], we realise each Specht module as a cell module in type A. Fix
$n\geq 1$ and set $I=\mathrm{A}_{n-1}$, so that $W=S_{n}$ is the symmetric
group. See Appendix B for the general construction of cell modules. Let
$\mathbb{K}\mathbb{L}=\\{C_{w}\mid w\in S_{n}\\}$ denote the Kazhdan–Lusztig
basis for $\mathbb{C}[S_{n}]$ and let $\leq^{\mathrm{L}}$ denote the preorder
$\leq_{I}^{\mathrm{L}}$ (Definition B.2) on $S_{n}$, and $\sim^{\mathrm{L}}$
its induced equivalence relation. The equivalence classes under this relation
are known as left cells. Each left cell $\mathcal{C}$ has an associated based
$S_{n}$-representation $(V(\mathcal{C}),\mathbb{K}\mathbb{L}_{\mathcal{C}})$
which we call a cell module of $S_{n}$. We call
$\mathbb{K}\mathbb{L}_{\mathcal{C}}=\\{[C_{x}]\mid x\in\mathcal{C}\\}$ the KL
basis for $V(\mathcal{C})$.

We first give a combinatorial description of the left cells. Recall that the
RSK correspondence gives a bijection

$x\mapsto(P(x),Q(x))$

between elements of $S_{n}$ and pairs of standard Young tableaux of the same
shape and with $n$ boxes each. For the following two results, see Section 6.5
in [3].

######  Lemma 5.1.

Permutations $x,y\in S_{n}$ are equivalent under $\sim^{\mathrm{L}}$ if and
only if $Q(x)=Q(y)$.

Hence, the left cells of $S_{n}$ are naturally indexed by standard tableaux.
For a partition $\lambda\vdash n$ and $Q\in\mathrm{SYT}(\lambda)$, we define
the left cell

$\mathcal{C}_{Q}:=\\{w\in S_{n}\mid Q(w)=Q\\}.$

Write $[C_{P,Q}]$ for the unique KL basis element
$[C_{x}]\in\mathbb{K}\mathbb{L}_{\mathcal{C}_{Q}}$ satisfying $P(x)=P$. It
turns out that each cell module $V(\mathcal{C}_{Q})$ is an irreducible
$S_{n}$-representation:

######  Lemma 5.2.

Fix $\lambda\vdash n$ and $Q,Q^{\prime}\in\mathrm{SYT}(\lambda)$. The linear
extension of the map $[C_{P,Q}]\mapsto[C_{P,Q^{\prime}}]$ for every
$P\in\mathrm{SYT}(\lambda)$ is an isomorphism of $S_{n}$-representations
$V(\mathcal{C}_{Q})\xrightarrow{\sim}V(\mathcal{C}_{Q^{\prime}})$. Moreover,
$V(\mathcal{C}_{Q})$ is isomorphic to the Specht module $S^{\lambda}$.

Hence, we fix $Q\in\mathrm{SYT}(\lambda)$ and write $C_{P}$ for $C_{[P,Q]}$.
Then $S^{\lambda}$ has a basis

$\mathbb{K}\mathbb{L}_{\lambda}:=\\{C_{P}\mid P\in\mathrm{SYT}(\lambda)\\}$

which we call the KL basis, and this construction is independent of the choice
of $Q$.

###  5.2. The Specht module as the dual canonical basis

Consider the partition $\lambda$ as a highest weight for $\mathfrak{sl}_{n}$
and let $L(\lambda)$ denote the irreducible representation of
$\mathfrak{sl}_{n}$ with this highest weight. Recall that the Weyl group of
$\mathfrak{sl}_{n}$ is $S_{n}$, and hence $S_{n}$ acts on the zero weight
space $L(\lambda)_{0}$. By Schur–Weyl duality we have that
$L(\lambda)_{0}\cong S^{\lambda}$.

To apply Theorem 4.8 we require a categorification of $L(\lambda)$. This is
provided by the cyclotomic KLR algebras, as introduced by Khovanov–Lauda and
Rouquier. These algebras can be defined either by generators and relations
[25], or via the geometry of Lusztig quiver varieties [37]. We will not need
the precise definition of these algebras.

We let $\mathcal{L}(\lambda)$ denote the category of their finite dimensional
modules. Specifically, in the notation of [25], we consider the category

$\mathcal{L}(\lambda)=\bigoplus_{\nu}R^{\lambda}(\nu)\mathrm{-mod},$

where $\nu$ runs over elements of the root lattice of $\mathfrak{sl}_{n}$. We
need the following theorems:

######  Theorem 5.3.

Fix $\lambda\vdash n$.

  1. (1)

There is a categorical $\mathfrak{sl}_{n}$-action on $\mathcal{L}(\lambda)$
categorifying $L(\lambda)$ [25].

  2. (2)

Under the isomorphism $[\mathcal{L}(\lambda)]_{\mathbb{C}}\cong L(\lambda)$,
the equivalence classes of simple modules are identified with the dual
canonical basis of $L(\lambda)$ [41].

To complete the realisation of $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$
via categorification we recall the following folkloric result:

######  Proposition 5.4.

Under the isomorphism of $S_{n}$-modules $L(\lambda)_{0}\cong S^{\lambda}$,
the dual canonical basis corresponds to the Kazhdan–Lusztig basis.

###### Proof.

There are several ways to prove this; we take an explicit approach and use
Skandera’s construction of the dual canonical basis via immanents [39]. More
precisely, by Equation (4.3) of [34] the action of simple transpositions on
the dual canonical basis elements in $L(\lambda)_{0}$ agrees with their action
on KL basis elements. It follows that these bases agree up to a global scalar.
∎

We apply Theorem 4.8 to this situation:

######  Theorem 5.5.

Fix $\lambda\vdash n$. For $w\in S_{n}^{\mathrm{sep}}$, choose a chain
$\mathcal{Z}=(I_{1},\ldots,I_{r})$ in $\mathrm{A}_{n-1}$ such that
$w=w_{\mathcal{Z}}:=w_{I_{1}}\cdots w_{I_{r}}$. Then $w$ acts on
$(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda},\leq_{\mathcal{Z}})$ by
$\xi_{\mathcal{Z}}$ up to lower-order terms.

###  5.3. Crystal combinatorics

We make Theorem 5.5 more explicit by giving combinatorial descriptions of
$\leq_{\mathcal{Z}}$ and $\xi_{\mathcal{Z}}$ using standard tableaux. We
consider the case of a single parabolic subgroup $I^{\prime}\subseteq I$, and
generalise to chains via induction. These results will follow from the
construction of type A crystals, and the reader can find more information in
[7].

####  5.3.1. Irreducible Crystals

If $1\leq a<b\leq n$, the subset $I_{[a,b]}:=\\{a,a+1,\dots,b-1\\}\subseteq I$
generates the standard parabolic subgroup $S_{[a,b]}\leq S_{n}$ permuting the
elements $a,\dots,b$. Every nontrivial standard parabolic subgroup of $S_{n}$
splits into irreducibles as $I_{[a_{1},b_{1}]}\sqcup\cdots\sqcup
I_{[a_{m},b_{m}]}$ for $1\leq a_{1}<b_{1}<\cdots<a_{m}<b_{m}\leq n$.

For a partition $\lambda$, denote by $\mathbb{L}_{n}(\lambda)$ the crystal of
$L(\lambda)$. The objects in this crystal are identified with
$\mathrm{SSYT}_{n}(\lambda)$, the set of semistandard tableaux of shape
$\lambda$ containing entries from $\\{1,\dots,n\\}$. The weight map is given
by $\mathsf{wt}(T)=(\eta_{1},\dots,\eta_{n})$, where $\eta_{i}$ counts the
number of occurences of boxes with label $i$ in the tableau $T$. When
$\lambda\vdash n$, the zero-weight subspace
$\mathbb{B}(\lambda):=\mathbb{L}_{n}(\lambda)_{0}$ is indexed by the standard
tableaux $\mathrm{SYT}(\lambda)$.

The raising (respectively, lowering) operators $e_{i}$ (respectively, $f_{i}$)
for $1\leq i\leq n-1$ are described by replacing a single box $i$ with $i+1$
(respectively, $i+1$ with $i$) according to a bracketing rule applied to the
row-reading word of the tableau. We demonstrate this with the following
example:

######  Example 5.6.

Let $\lambda=(5,3,2)$ and set

$T=\young(11223,334,45)\in\mathbb{L}_{5}(\lambda).$

The row-reading word is $4533411223$. To calculate the action of $f_{3}$, we
place a closed bracket beneath every occurence of $3$, and an open bracket
beneath every occurence of $4$.

$\begin{matrix}4&5&3&3&4&1&1&2&2&3\\\ (&&)&)&(&&&&&)\end{matrix}$

We increment the rightmost unpaired $3$. Hence:

$\young(11223,334,45)\xrightarrow{\quad
f_{3}\quad}\young(11223,3{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4}}4,45)$

The Schützenberger involution $\xi_{I}$ acts on $\mathbb{L}_{n}(\lambda)$ by
the evacuation operator for tableaux. To calculate this, we rotate the tableau
by $180^{\circ}$, replace each entry $i$ with $n-i+1$, and rectify (that is,
remove all empty inside boxes using jeu-de-taquin slides). Again, in lieu of a
precise definition we provide an example:

######  Example 5.7.

We calculate the evacuation of $T$ from Example 5.6.

$\young(11223,334,45)\xrightarrow{\mathrm{rotate}}\young(:::54,::433,32211)\xrightarrow{\mathrm{replace}}\young(:::12,::233,34455)\xrightarrow{\mathrm{rectify}}\young(12355,234,34)$

Figure 1. The crystal graph for $\mathbb{L}_{4}(2,2)$ with arrow labels
$\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color<EMAIL_ADDRESS>$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\xrightarrow{2}}$
and
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\xrightarrow{3}}$.
If $J=I_{[2,4]}=\\{2,3\\}$, then $\mathsf{Res}_{J}\mathbb{L}_{4}(2,2)$ has
three connected components. The standard tableaux are separately coloured, and
are fixed by $\xi_{I}$.
${\young(11,22)}$${\young(11,23)}$${\young(12,23)}$${\young(11,33)}$${\young(11,24)}$${\young(12,33)}$${\young(12,24)}$${\young(11,34)}$${\young(22,33)}$${{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\young(11,11)}}$${{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\young(12,34)}}$${{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\young(13,24)}}$${\young(11,44)}$${\young(22,34)}$${\young(13,34)}$${\young(12,44)}$${\young(23,34)}$${\young(22,44)}$${\young(13,44)}$${\young(23,44)}$${\young(33,44)}$

####  5.3.2. Skew Crystals

It is useful to extend our definition of $\mathbb{L}_{n}(\lambda)$ to allow
skew shapes. If $\mu$ is a skew partition, let $\mathbb{L}_{n}(\mu)$ be the
crystal with objects indexed by $\mathrm{SSYT}_{n}(\mu)$, the skew
semistandard tableaux of shape $\mu$ on labels $\\{1,\dots,n\\}$. The weight
map and action of the operators $e_{i}$ and $f_{i}$ are defined as above. We
note that $\mathbb{L}_{n}(\mu)$ is a normal $I$-crystal, and so is isomorphic
to a disjoint union of crystals each of the form $\mathbb{L}_{n}(\lambda)$ for
some partitions $\lambda$.

We recall a number of definitions from [19]. Given a skew semistandard tableau
$X$, the rectification of $X$ is the semistandard tableaux $\mathsf{rect}(X)$
obtained by removing all inside empty boxes via jeu-de-taquin slides. Two skew
tableaux $X$ and $Y$ are called dual equivalent if and only if they have the
same shape after performing any sequence of slides to the same outside or
inside boxes of both tableaux, and we write $X\approx Y$. Note that $X\approx
Y$ implies $\operatorname{sh}(X)=\operatorname{sh}(Y)$, where
$\operatorname{sh}$ denotes the shape of a (skew) tableaux.

For $X\in\mathbb{L}_{n}(\mu)$, let $\mathbb{L}_{n}(\mu)[X]$ denote the
connected component of $\mathbb{L}_{n}(\mu)$ containing $X$.

######  Proposition 5.8.

The objects of $\mathbb{L}_{n}(\mu)[X]$ are identified with the dual
equivalence class of $X$. Moreover, $\mathsf{rect}(-)$ induces a crystal
isomorphism $\mathbb{L}_{n}(\mu)[X]\xrightarrow{\sim}\mathbb{L}_{n}(\nu)$,
where $\nu:=\operatorname{sh}(\mathsf{rect}(X))$.

###### Proof.

For any $1\leq i\leq n$, the operator $f_{i}$ on tableaux commutes with jeu-
de-taquin slides [28, Theorem 3.3.1]. Hence, $\mathsf{rect}(f_{i}\cdot
X)=f_{i}\cdot\mathsf{rect}(X)$ since $\mathsf{rect}(-)$ can be described as a
composition of slides. The same holds for $e_{i}$. Moreover, it is easy to see
that rectification preserves the weight. Finally, this map is an isomorphism
since $\mathsf{rect}(-)$ is a bijection between the dual equivalence class of
$X$ and $\mathrm{SSYT}_{n}(\nu)$ [19, Theorem 2.13]. ∎

######  Corollary 5.9.

The crystal $\mathbb{L}_{n}(\mu)[X]$ has highest weight
$\operatorname{sh}(\mathsf{rect}(X))$, and $\xi_{I}$ acts on
$\mathbb{L}_{n}(\mu)$ by sending $X$ to the unique tableau $Y$ such that
$Y\approx X$ and $\mathsf{rect}(Y)=\mathsf{ev}(\mathsf{rect}(X))$.

For a semistandard (skew) tableau $X$, set $\mathsf{ev}(X):=\xi_{I}\cdot X$.
We can calculate $\mathsf{ev}(-)$ by rectifying, evacuating, and then
‘unrectifying’ backward along the same path to preserve dual equivalence. We
continue to call this map evacuation for skew tableaux, although the term
_reversal_ is also used in the literature [1].

####  5.3.3. Levi Branching

Fix $I^{\prime}:=I_{[a,b]}\subseteq I$ for $1\leq a<b\leq n$ and set
$\mathbb{L}_{n}^{\prime}(\lambda):=\mathsf{Res}_{I^{\prime}}\mathbb{L}_{n}(\lambda)$.
We wish to calculate the highest-weight and action of $\xi_{I^{\prime}}$ on
$\mathbb{L}_{n}^{\prime}(\lambda)$, from which we can describe
$\leq_{I^{\prime}}$ and $\xi_{I^{\prime}}$ on $\mathbb{L}_{n}(\lambda)$.

For a (skew) tableau $T$, let $T_{[a,b]}$ denote the tableau obtained by
restricting $T$ to the boxes with labels in $\\{a,\dots,b\\}$, and subtracting
$a-1$ from each value. For example:

$T=\young(11223,334,45)\implies T_{[2,4]}=\young(::112,223,3)$

Observe that acting by $f_{i}$ on $T$ for $i\in I^{\prime}$ only affects boxes
with labels in $\\{a,\dots,b\\}$. In particular, the restriction map
$(-)_{[a,b]}$ gives a crystals isomorphism

$\mathbb{L}_{n}^{\prime}(\lambda)[T]\xrightarrow{\sim}\mathbb{L}_{b-a+1}(\mu)[T_{[a,b]}],$

where $\mu:=\operatorname{sh}(T_{[a,b]})$. Applying Corollary 5.9 to the
right-hand side of this crystal isomorphism gives the following:

  * •

The highest-weight element of $\mathbb{L}_{n}^{\prime}(\lambda)[T]$ has weight
$\tau_{I^{\prime}}(T)=\operatorname{sh}(\mathsf{rect}(T_{[a,b]}))$.

  * •

For $R,T\in\mathbb{L}_{n}(\lambda)$, $R\leq_{I^{\prime}}T$ if and only if
$\operatorname{sh}(\mathsf{rect}(R_{[a,b]}))\preceq\operatorname{sh}(\mathsf{rect}(T_{[a,b]}))$
in the dominance order on partitions.

  * •

For $T\in\mathbb{L}_{n}(\lambda)$, $\xi_{I^{\prime}}\cdot T$ is the unique
tableau $R$ such that $R\sim_{I^{\prime}}T$ and
$R_{[a,b]}=\mathsf{ev}(T_{[a,b]})$.

Set $\mathsf{ev}_{I^{\prime}}(T):=\xi_{I^{\prime}}\cdot T$ for every
$T\in\mathbb{L}_{n}(\lambda)$. We observe that $\mathsf{ev}_{I^{\prime}}(T)$
can be calculated by replacing $T_{[a,b]}$ with its evacuation (up to shifting
by $a-1$), and preserving the rest of the tableau. This fact can also be
proved using growth diagrams [10]. We give an example below.

######  Example 5.10.

Set $n=5$ and $I^{\prime}=I_{[2,4]}$. We have

$\young(::112,223,3)\quad\xrightarrow{\quad\mathsf{ev}\quad}\quad\young(::122,133,2)$

and hence

$\young(11223,334,45)\quad\xrightarrow{\,\,\,\,\,\mathsf{ev}_{I^{\prime}}\,\,\,\,\,}\quad\young(11233,244,35)$

To extend this to subdiagrams with multiple components, we first define the
external products of crystals. Given a normal $I_{1}$-crystal $\SS_{1}$ and a
normal $I_{2}$-crystal $\SS_{2}$, let $\SS_{1}\boxtimes\SS_{2}$ be the normal
$(I_{1}\sqcup I_{2})$-crystal with objects $x_{1}\boxtimes x_{2}$ for
$x_{1}\in\SS_{1}$ and $x_{2}\in\SS_{2}$. The crystal operator $f_{i}$ acts by

$f_{i}(x_{1}\boxtimes x_{2})=\begin{cases}f_{i}(x_{1})\boxtimes x_{2}&i\in
I_{1}\\\ x_{1}\boxtimes f_{i}(x_{2})&i\in I_{2}\end{cases}$

for any $i\in I_{1}\sqcup I_{2}$ and similarly for $e_{i}$. Moreover,
$\mathsf{wt}(x_{1}\boxtimes
x_{2})\mapsto(\mathsf{wt}(x_{1}),\mathsf{wt}(x_{2}))\in X_{I_{1}}\times
X_{I_{2}}$. From this, we observe that $\tau_{I_{1}\sqcup
I_{2}}(x_{1}\boxtimes x_{2})=(\tau_{I_{1}}(x_{1}),\tau_{I_{2}}(x_{2}))$, and
the Schützenberger operator $\xi_{I_{1}\sqcup I_{2}}$ acts by $x_{1}\boxtimes
x_{2}\mapsto\xi_{I_{1}}(x_{1})\boxtimes\xi_{I_{2}}(x_{2})$. The external
product is associative and we extend the definition to finite products
$\SS_{1}\boxtimes\cdots\boxtimes\SS_{m}$.

We now consider a general subset
$I^{\prime}=I_{[a_{1},b_{1}]}\sqcup\cdots\sqcup I_{[a_{m},b_{m}]}\subseteq I$.
We observe that
$\mathbb{L}_{n}^{\prime}(\lambda):=\mathsf{Res}_{I^{\prime}}\mathbb{L}_{n}(\lambda)$
has a natural decomposition into products of skew crystals.

######  Lemma 5.11.

Fix $I^{\prime}\subseteq I$ as above and $T\in\mathbb{L}_{n}(\lambda)$. Set
$\mu_{k}:=\operatorname{sh}(T_{[a_{k},b_{k}]})$ and $n_{k}:=b_{k}-a_{k}+1$ for
every $1\leq k\leq m$. The map $R\mapsto
R_{[a_{1},b_{1}]}\boxtimes\cdots\boxtimes R_{[a_{m},b_{m}]}$ gives an
isomorphism of $I^{\prime}$-crystals

$\mathbb{L}_{n}^{\prime}(\lambda)[T]\xrightarrow{\sim}\mathbb{L}_{n_{1}}(\mu_{1})[T_{[a_{1},b_{1}]}]\boxtimes\cdots\boxtimes\mathbb{L}_{n_{m}}(\mu_{m})[T_{[a_{m},b_{m}]}].$

###### Proof.

We treat $\mathbb{L}_{n}^{\prime}(\lambda)$ as a crystal with Dynkin type
$\mathrm{A}_{n_{1}-1}\sqcup\cdots\sqcup\mathrm{A}_{n_{m}-1}$, where
$a_{k}+i\in I^{\prime}$ for $1\leq i<n_{k}$ corresponds to
$i\in\mathrm{A}_{n_{k}-1}$. It is clear that this crystal map preserves
weights. We are required to show that acting by $f_{a_{k}+i}$ on
$\mathbb{L}_{n}^{\prime}(\lambda)$ is equivalent to acting by $f_{i}$ on the
component $\mathbb{L}_{n_{k}}(\mu_{k})$ under the map.

Let $\rho_{k}$ denote the restriction map $R\mapsto R_{[a_{k},b_{k}]}$ for any
$R\in\mathbb{L}_{n}(\lambda)$. Note that acting by $f_{a_{k}+i}\in
I_{[a_{k},b_{k}]}$ for $1\leq i<n_{i}$ only affects tableaux on the labels
$a_{k},\dots,b_{k}$. Hence:

$\rho_{l}f_{a_{k}+i}=\begin{cases}\rho_{l}&l\neq k\\\
f_{i}\rho_{k}&l=k\end{cases}$

as required. ∎

We again apply Corollary 5.9, this time on each component of the external
product from the decomposition of $\mathbb{L}_{n}^{\prime}(\lambda)[T]$ in the
above Lemma.

######  Proposition 5.12.

Fix $T\in\mathbb{L}_{n}(\lambda)$ and $I^{\prime}\subseteq I$ as above.

  1. (1)

The connected component of $\mathbb{L}_{n}^{\prime}(\lambda)[T]$ consists of
$R\in\mathbb{L}_{n}(\lambda)$ satisfying $R_{[a_{k},b_{k}]}\approx
T_{[a_{k},b_{k}]}$ for every $1\leq k\leq m$.

  2. (2)

This component has highest-weight $(\nu_{1},\dots,\nu_{m})$, where
$\nu_{k}:=\operatorname{sh}(\mathsf{rect}(T_{[a_{k},b_{k}]}))$.

  3. (3)

The Schützenberger involution $\xi_{I^{\prime}}$ acts on this component by
$\mathsf{ev}_{I_{[a_{1},b_{1}]}}\cdots\mathsf{ev}_{I_{[a_{m},b_{m}]}}$.

We set $\mathsf{ev}_{I^{\prime}}(T):=\xi_{I^{\prime}}(T)$ for any
$T\in\mathbb{L}_{n}(\lambda)$ as before. This can be calculated by evacuating
$T_{[a_{1},b_{1}]}$ (up to shift by $a_{1}-1$), evacuating $T_{[a_{2},b_{2}]}$
(up to shift by $a_{2}-1$), and so on. From these results we immediately
deduce our description of $\leq_{I^{\prime}}$ and $\xi_{I^{\prime}}$ for any
$I^{\prime}\subseteq I$, and combine this with Theorem 5.5.

For the following result, fix $n\geq 1$ and
$I^{\prime}=I_{[a_{1},b_{1}]}\sqcup\cdots\sqcup I_{[a_{m},b_{m}]}\subseteq I$
as before. For $\lambda\vdash n$ and $R,T\in\mathrm{SYT}(\lambda)$, set
$R\leq_{I^{\prime}}T$ if and only if
$\operatorname{sh}(\mathsf{rect}(R_{[a_{k},b_{k}]}))\preceq\operatorname{sh}(\mathsf{rect}(T_{[a_{k},b_{k}]}))$
for every $1\leq k\leq m$.

######  Corollary 5.13.

Let $w_{I^{\prime}}$ be the permutation reversing $a_{k},\dots,b_{k}$ for
every $1\leq k\leq m$. For any $\lambda\vdash n$, associate
$\mathrm{SYT}(\lambda)$ with $\mathbb{K}\mathbb{L}_{\lambda}$ by
$T\leftrightarrow C_{T}$. Then $w_{I^{\prime}}$ acts on
$(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda},\leq_{I^{\prime}})$ by
$\mathsf{ev}_{I^{\prime}}$ up to l.o.t.

###### Proof.

The element $w_{I^{\prime}}$ is the longest element of the parabolic subgroup
of $S_{n}$ induced by $I^{\prime}$. Under the association of
$\mathbb{K}\mathbb{L}_{\lambda}$ with the dual canonical basis for
$L(\lambda)_{0}$, we have shown that $\xi_{I^{\prime}}$ acts by $C_{T}\mapsto
C_{\mathsf{ev}_{I^{\prime}}(T)}$ and $C_{R}\leq_{I^{\prime}}C_{T}$ (in the
crystal ordering) if and only if $R\leq_{I^{\prime}}T$. Then apply Theorem
5.5. ∎

If $\mathcal{Z}=(I_{1},\dots,I_{r})$ is a descending chain, we can extend the
above result to all separable permutations $w_{\mathcal{Z}}:=w_{I_{1}}\cdots
w_{I_{r}}$ by replacing $\mathsf{ev}_{I^{\prime}}$ with
$\mathsf{ev}_{\mathcal{Z}}:=\mathsf{ev}_{I_{1}}\cdots\mathsf{ev}_{I_{r}}$ and
$\leq_{I^{\prime}}$ with $\leq_{\mathcal{Z}}$ using Definition 4.4.

###  5.4. Generalisation of Rhoades’ Theorem and other examples

We consider a number of special cases of Corollary 5.13. First, we study the
cases for which $\leq_{\mathcal{Z}}$ is trivial on
$\mathbb{K}\mathbb{L}_{\lambda}$.

When $I^{\prime}=I$, $w_{I}$ is the longest element $w_{0}$,
$\mathsf{ev}_{I}=\mathsf{ev}$ and $\leq_{I}$ is the trivial preorder. Hence,
$w_{0}\cdot C_{T}=\pm C_{\mathsf{ev}(T)}$ for all $T\in\mathrm{SYT}(\lambda)$
with constant sign, recovering the theorem of Berenstein–Zelevinsky [2] and
Stembridge [40].

Next, set $J:=I_{[1,n-1]}$ and $\mathcal{Z}:=(I,J)$. Then $w_{\mathcal{Z}}$ is
the long cycle $c_{n}=(1,2,\ldots,n)$, or equivalently, the Coxeter element
$s_{1}\cdots s_{n-1}$. It was observed by Rush [38] that
$\mathsf{ev}_{\mathcal{Z}}=\mathsf{ev}_{I}\mathsf{ev}_{J}$ is the promotion
map $\mathsf{pr}$. The preorder $\leq_{\mathcal{Z}}$ is equivalent to
$\leq_{J}$, and $R\leq_{J}T$ if and only if the $n$-box occurs in a higher row
of $R$ than it occurs in $T$. When $\lambda$ is a rectangular shape, the
$n$-box always occurs in the same position and the order $\leq_{\mathcal{Z}}$
is trivial. Hence, we have $c_{n}\cdot C_{T}=\pm C_{\mathsf{pr}(T)}$ for every
$T\in\mathrm{SYT}(\lambda)$ with constant sign, obtaining Rhoades’ Theorem
[34]. By Corollary 5.13 we can now generalise this formula to arbitrary
shapes.

######  Proposition 5.14.

Fix $\lambda\vdash n$ and $T\in\mathrm{SYT}(\lambda)$. The Coxeter element
$c_{n}$ acts on $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda},\leq)$ by
$\mathsf{pr}$ up to l.o.t., where $\leq$ is the ordering given by comparing
the position of the $n$-box.

######  Remark 5.15.

This result was first announced (with a combinatorial proof) by the authors at
FPSAC 2022 [17].

We note that every power of the long cycle $c_{n}^{k}$ is a separable
permutation, and so it acts on $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$
by some bijection up to lower-order terms. On rectangular tableaux, this
bijection will be $\mathsf{pr}^{k}$ by Rhoades’ Theorem. However, this is not
true for general shapes.

Consider the chain $\mathcal{Z}=(I_{n},I_{n-1},\ldots,I_{1})$ where
$I_{r}:=I_{[1,r]}$. This corresponds to the standard tower of symmetric groups
which appears e.g. in constructions of Gelfand-Tsetlin bases of $S^{\lambda}$.
The corresponding separable permutation $g_{n}:=w_{I_{n}}\cdots w_{I_{1}}$ is
given by (in one-line notation):

$\displaystyle n\;n-2\;\cdots\;2\;1\;3\;\cdots\text{ if }n\text{ is even,}$
$\displaystyle n\;n-2\;\cdots\;1\;2\;4\;\cdots\text{ if }n\text{ is odd.}$

For example, $g_{5}=53124$ and $g_{6}=642135$.

The preorder $\leq_{\mathcal{Z}}$ on $\mathrm{SYT}(\lambda)$ becomes a linear
order, determined as follows: to each tableau $T$ assign the sequence
$(r_{n},\dots,r_{1})$ with $r_{i}$ the row of $T$ containing $i$, and order
these sequences lexicographically. For example, if $\lambda=(3,2)$ the
standard Young tableaux from smallest to largest are:

(135,24)(125,34)(134,25)(124,35)(123,45)

Note that this order linearises the dominance order on tableaux. The bijection
$\xi_{\mathcal{Z}}$ is given by an operation $\mathsf{nev}$ we’ll call “nested
evacuation”, where $\mathsf{nev}(T)$ is obtained from $T$ as follows: first
evacuate the smallest box, then evacuate the smallest two boxes, and so on
until we evacuate the entire tableaux.

######  Remark 5.16.

Unlike the previous examples, where the bijection $\xi_{\mathcal{Z}}$
recovered well-studied operations in algebraic combinatorics (namely the
evacuation and promotion operators), we are not aware of nested evacuation
appearing elsewhere in the combinatorics literature.

By Corollary 5.13 we obtain:

######  Proposition 5.17.

Fix $\lambda\vdash n$ and $T\in\mathrm{SYT}(\lambda)$. The permutation $g_{n}$
acts on $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda},\leq)$ by $\mathsf{nev}$
up to l.o.t., where $<$ is the ordering given by comparing the position of the
$n$-box, then comparing the position of the $(n-1)$-box, and so on.

Finally, we consider the action of a single generator. Every irreducible
normal crystal of type $\mathrm{A}_{1}$ is isomorphic to
$\mathbb{L}_{2}(\lambda)$ for some partition $\lambda=(n)$ with $n\geq 0$
(when $n=0$ we get the crystal with a single object of weight zero). The
weight-zero subset of this crystal is empty when $\lambda$ has an odd number
of boxes, and contains a single element otherwise.

We return briefly to the general setup of Section 4.1. Let $W$ be a simply-
laced Weyl group with Dynkin diagram $I$ and $(U,\mathbb{B})$ a based
$W$-representation arising from categorification.

######  Proposition 5.18.

For $i\in I$, the generator $s_{i}$ acts on $(U,\mathbb{B})$ by the identity
up to l.o.t.

###### Proof.

Let $J\subset I$ be the subdiagram $J=\\{i\\}$ which has Dynkin type
$\mathrm{A}_{1}$. Then $s_{i}$ is the longest element of $W_{J}$. Suppose
$\SS$ is the underlying crystal for $(U,\mathbb{B})$ such that
$\SS_{0}=\mathbb{B}$. Then $\operatorname{res}_{J}\SS$ is a normal crystal of
type $\mathrm{A}_{1}$, hence $\xi_{J}$ acts by the identity on $\mathbb{B}$.
Then apply Theorem 4.8. ∎

For $(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$ when $\lambda\vdash n$ and
$J=\\{i\\}\subseteq\mathrm{A}_{n-1}$, we can explicitly determine the order
$\leq_{J}$. For a tableau $T\in\mathrm{SYT}(\lambda)$, write
$i\in\operatorname{Des}(T)$ if and only if $i$ is a descent of $T$, i.e. the
box containing $i+1$ occurs in a strictly lower row of $T$ than the box
containing $i$. By considering $\mathsf{rect}(T_{[i,i+1]})$, one can observe
that:

$\tau_{J}(T)=\begin{cases}(1,1)&i\in\operatorname{Des}(T),\\\
(2,0)&i\not\in\operatorname{Des}(T).\end{cases}$

Applying Proposition 5.18, we recover a well-known fact about the action of
each generator $s_{i}$ on $\mathbb{K}\mathbb{L}_{\lambda}$ [3, Equation 6.20].
For $R,T\in\mathrm{SYT}(\lambda)$, $C_{T}$ always appears in $s_{i}\cdot
C_{T}$ with coefficient $\pm 1$ (depending on whether $i\in D(T)$), and if any
other element $C_{R}$ appears with nonzero coefficient, then
$i\in\operatorname{Des}(R)\setminus\operatorname{Des}(T)$.

We conclude this section with an example showing that non-separable
permutations may not act by a bijection up to lower-order terms on
$\mathbb{K}\mathbb{L}_{\lambda}$.

######  Example 5.19.

Set $\lambda=(3,1)$ and fix the ordered basis:

$\mathbb{K}\mathbb{L}_{\lambda}=\left(\,\,\young(123,4)\,,\,\,\young(124,3)\,,\,\,\young(134,2)\,\,\right)$

Then the permutation $w=2413\in S_{4}\setminus S_{4}^{\mathrm{sep}}$ acts on
$\mathbb{K}\mathbb{L}_{\lambda}$ by

$[w]_{\mathbb{K}\mathbb{L}_{\lambda}}=\begin{bmatrix}0&-1&1\\\ 1&-1&1\\\
1&-1&0\end{bmatrix}.$

Since there are only two zero entries in this matrix, $w$ cannot act by a
bijection up to lower-order terms for any ordering of
$\mathbb{K}\mathbb{L}_{\lambda}$ (Corollary A.3).

######  Conjecture 5.20.

Choose $\lambda\vdash n$ with $\dim S^{\lambda}\geq 3$. Then $w$ acts on
$(S^{\lambda},\mathbb{K}\mathbb{L}_{\lambda})$ by a bijection up to l.o.t. if
and only if $w\in W^{\mathrm{sep}}$.

##  6\. Canonical bases in tensor product representations

### 6.1.

Let $I$ be a finite type simply-laced Dynkin diagram, and let $W=W_{I}$ be the
associated Weyl group. In this section we apply Theorem 4.8 to study the
action of separable elements of $W$ on the dual canonical basis elements (of
weight zero) in tensor product representations of
$\mathfrak{g}=\mathfrak{g}_{I}$.

Recall that $X^{+}$ denotes the set of integral dominant weights of
$\mathfrak{g}$, and for $\lambda\in X^{+}$, $L(\lambda)$ denotes the
irreducible representation of $\mathfrak{g}$ of highest weight $\lambda$.
Given a sequence $\underline{\lambda}=(\lambda_{1},\ldots,\lambda_{n})$ of
dominant weights, we define
$L(\underline{\lambda}):=L(\lambda_{1})\otimes\cdots\otimes L(\lambda_{n})$.

These tensor product representations possess (dual) canonical bases
constructed in [31, 32]. These bases have remarkable properties such as
positivity and compatibility with isotypic filtrations, and important
connections to the BGG category $\mathcal{O}$ and quantum topology.

Both the canonical and dual canonical basis are indexed by the crystal of the
representation. More precisely, the underlying set of $\SS(\lambda)$ indexes
two bases of $L(\lambda)$: the dual canonical basis $\\{x_{b}\mid
b\in\SS(\lambda)\\}$ and the canonical basis $\\{x^{\prime}_{b}\mid
b\in\SS(\lambda)\\}$. This follows from [18], and also follows from the
categorical point of view.

Indeed, as mentioned in Section 4.3, elements of the dual canonical basis
correspond to classes of simple modules in a categorication of $L(\lambda)$,
which can be constructed via the category of finite-dimensional modules over
cyclotomic KLR algebras. If one instead considers the category of finitely-
generated projective modules over the same algebras, this produces another
categorification of $L(\lambda)$ but now the classes of indecomposable
projective modules correspond to the canonical basis. This theory is well-
explained in [41].

Consequently, the dual canonical and canonical bases of
$L(\underline{\lambda})$ are indexed by the crystal
$\SS(\underline{\lambda}):=\SS(\lambda_{1})\otimes\cdots\otimes\SS(\lambda_{n})$,
and we denote these $\\{x_{b}\mid b\in\SS(\underline{\lambda})\\}$ and
$\\{x^{\prime}_{b}\mid b\in\SS(\underline{\lambda})\\}$ as above. Note that
these are not simply the tensor product of bases, e.g. in $L(\lambda)\otimes
L(\lambda^{\prime})$, it is not necessarily true that $x_{b\otimes
b^{\prime}}=x_{b}\otimes x_{b^{\prime}}$ for $b\in\SS(\lambda)$ and
$b^{\prime}\in\SS(\lambda^{\prime})$. However, we do have that $x_{b\otimes
b^{\prime}}=x_{b}\otimes x_{b^{\prime}}+\text{ lower-order terms}$, where the
order on tensor products of bases is defined via the partial order on weights
[32, Section 27.3].

### 6.2.

Consider the zero weight space
$U(\underline{\lambda}):=L(\underline{\lambda})_{0}$, which as usual we view
as a $W$-module. We let $\mathbb{B}(\underline{\lambda})\subset
U(\underline{\lambda})$ denote the set of dual canonical basis elements of
$L(\underline{\lambda})_{0}$.

######  Theorem 6.1 (Webster’s Theorem).

The pair $(U(\underline{\lambda}),\mathbb{B}(\underline{\lambda}))$ arises via
categorification.

###### Proof.

Webster constructs KLRW algberas $T^{\underline{\lambda}}_{\nu}$, which are
generalisations of KLR algebras, in order to categorify the representations
$L(\underline{\lambda})$. Specifically, the category

$\mathcal{L}(\underline{\lambda})=\bigoplus_{\nu}T^{\underline{\lambda}}_{\nu}\mathrm{-mod},$

carries an action of $\mathfrak{g}$ (and the corresponding quantum group), and
this representation categorifies $L(\underline{\lambda})$. By [41, Corollary
8.10], if we instead consider the additive categorification given by taking
the categories of projective modules above, then the classes of indecomposable
projectives correspond to the canonical basis of $L(\underline{\lambda})$.
Then, by [41, Corollary 2.4], we obtain that the classes of simple modules in
$\mathcal{L}(\underline{\lambda})$ correspond to the dual canonical basis
elements of $L(\underline{\lambda})$. ∎

We are in the position to apply Theorem 4.8. Consider a chain $\mathcal{Z}$ of
subdiagrams of $I$, and let $w=w_{\mathcal{Z}}\in W$ be the corresponding
separable element, $\leq_{\mathcal{Z}}$ the partial order on
$\mathbb{B}(\underline{\lambda})$, and $\xi_{\mathcal{Z}}$ the bijection of
$\mathbb{B}(\underline{\lambda})$. Then $w$ acts on
$(U(\underline{\lambda}),\mathbb{B}(\underline{\lambda}),\leq_{\mathcal{Z}})$
by $\xi_{\mathcal{Z}}$ up to l.o.t.

### 6.3.

We investigate this result through some examples, and begin by considering the
longest element $w=w_{I}$. In the simplest case
$\underline{\lambda}=(\lambda)$, i.e. when $w$ acts on the zero weight space
of the irreducible representation $L(\lambda)$, we recover, and slightly
refine, a classical result of Lusztig [30, Corollary 5.9]: for any
$x\in\mathbb{B}(\lambda)$ we have that $w_{I}\cdot x=\pm\xi_{I}(x)$, where
$\xi_{I}:\mathbb{B}(\lambda)\to\mathbb{B}(\lambda)$ is the Schützenberger
involution. Note that Lusztig’s result predates the theory of crystals, and
does not explicitly construct the involution.

Next, take $I=A_{n-1}$ and let $w_{I}$ act on tensor product representations
of $\mathfrak{sl}_{n}$. For instance, consider
$\mathbb{C}^{n}\otimes\mathbb{C}^{n}\cong L(2\varpi_{1})\oplus L(\varpi_{2})$,
where $\varpi_{i}$ denotes the $i$-th fundamental weight. Write
$\mathbb{B}:=\big{(}\SS(\varpi_{1})\otimes\SS(\varpi_{1})\big{)}_{0}=\mathbb{B}_{1}\oplus\mathbb{B}_{2}$,
where $\mathbb{B}_{1}\cong\SS(2\varpi_{1})_{0}$ and
$\mathbb{B}_{2}\cong\SS(\varpi_{2})_{0}$, and let
$\xi:\mathbb{B}\to\mathbb{B}$ be the Schützenberger involution. Since
$\varpi_{2}<_{I}2\varpi_{1}$, we have:

  1. (1)

For $b\in\mathbb{B}_{2}$, $w\cdot x_{b}=\pm x_{\xi(b)}$,

  2. (2)

For $b\in\mathbb{B}_{1}$, $w\cdot x_{b}=\pm x_{\xi(b)}+v_{b}$ where
$v_{b}\in\operatorname{span}_{\mathbb{Z}}(\mathbb{B}_{2})$.

Note that in this small example we already exhibit new combinatorial
information which hasn’t been apparent via classical methods.

More generally, consider the $d$-fold tensor product of $\mathbb{C}^{n}$, and
for simplicity suppose $n\geq d$. By Schur–Weyl duality,
$(\mathbb{C}^{n})^{\otimes d}\cong\bigoplus_{\lambda\vdash d}L(\lambda)\otimes
S^{\lambda}$. We write

$\mathbb{B}:=\big{(}\SS(\varpi_{1})^{\otimes
d}\big{)}_{0}=\bigoplus_{\lambda\vdash d}\mathbb{B}_{\lambda},$

where $\mathbb{B}_{\lambda}\cong\big{(}\SS(\lambda)^{\oplus
d_{\lambda}}\big{)}_{0}$ and $d_{\lambda}=\dim S^{\lambda}$. Let
$\xi:\mathbb{B}\to\mathbb{B}$ be the Schützenberger involution. Since the
dominance order $\prec$ on partitions is equivalent to the weight order
$<_{I}$, we have for every $b\in\mathbb{B}_{\lambda}$:

$w\cdot x_{b}=\pm x_{\xi(b)}+v_{b},\;\text{where
}v_{b}\in\operatorname{span}_{\mathbb{Z}}(\\{\mathbb{B}_{\mu}\mid\mu\prec\lambda\\}).$

Finally, we consider an example of our results that shows how other separable
permutations act on bases. Set $I=A_{2}$ and let $L(\lambda)=\mathfrak{g}$ be
the adjoint representation of $\mathfrak{g}=\mathfrak{sl}_{3}$, which has
highest weight $\lambda=(2,1,0)$. Also, let $J=A_{1}\subset I$ be the
embedding of $A_{1}$ as the first vertex.

We’ll compare the action of the separable permutations $w=(1,3)$ and
$c=(1,2,3)$ on the canonical and dual canonical basis of $(L(\lambda)\otimes
L(\lambda))_{0}$. The crystal of $L(\lambda)\otimes L(\lambda)$ is defined on
the set $\mathrm{SYT}(\lambda)\times\mathrm{SYT}(\lambda)$, and there are ten
weight-zero elements:

$\displaystyle
b_{0}=\tiny\young(23,3)\otimes\tiny\young(11,2),\;b_{1}=\tiny\young(22,3)\otimes\tiny\young(11,3),\;b_{2}=\tiny\young(13,2)\otimes\tiny\young(13,3),\;b_{3}=\tiny\young(12,3)\otimes\tiny\young(12,3),\;b_{4}=\tiny\young(13,3)\otimes\tiny\young(12,2)$
$\displaystyle
b_{5}=\tiny\young(13,2)\otimes\tiny\young(12,3),\;b_{6}=\tiny\young(12,3)\otimes\tiny\young(13,2),\;b_{7}=\tiny\young(12,2)\otimes\tiny\young(13,3),\;b_{8}=\tiny\young(11,2)\otimes\tiny\young(23,3),\;b_{9}=\tiny\young(11,3)\otimes\tiny\young(22,3)$

Below we list their images under $\tau_{I}$ and $\tau_{J}$ (cf. Section 4.2).

| $b_{0}$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | $b_{4}$ | $b_{5}$ | $b_{6}$ | $b_{7}$ | $b_{8}$ | $b_{9}$
---|---|---|---|---|---|---|---|---|---|---
$\tau_{I}$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{1}$ | $\lambda_{1}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{4}$ | $\lambda_{4}$
$\tau_{J}$ | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 4

where $\lambda_{0}=(0,0,0)$, $\lambda_{1}=(2,1,0)$, $\lambda_{2}=(3,3,0)$,
$\lambda_{3}=(4,1,1)$ and $\lambda_{4}=(4,2,0)$. The Hasse diagram of these
weights is given by:

${\lambda_{4}}$${\lambda_{3}}$${\lambda_{2}}$${\lambda_{1}}$${\lambda_{0}}$

Finally, we note that $\xi_{I}=(0,7)(1,3)(2,4)$ as a permutation of the
$b_{i}$ and $\xi_{J}$ is trivial.

Let $x_{i}$ be the dual canonical basis element corresponding to each $b_{i}$.
As an example, we consider the action of $w$ on $x_{3}$. Since $b_{0}$ is the
only element smaller than $b_{3}$ under $<_{I}$,

$w\cdot x_{3}=\pm x_{\xi(3)}+ax_{\xi(0)}=\pm x_{1}+ax_{7}$

for some $a\in\mathbb{Z}$. To compute the action of $c$, note that
$x_{i}<_{(I,J)}x_{3}$ for $i=0,1,2$ and so

$c\cdot x_{3}=\pm
x_{\xi(3)}+a_{0}x_{\xi(0)}+a_{1}x_{\xi(1)}+a_{2}x_{\xi(2)}=\pm
x_{1}+a_{0}x_{7}+a_{1}x_{3}+a_{2}x_{4}$

for $a_{0},a_{1},a_{2}\in\mathbb{Z}$. We can make similar computations to
determine the actions on the canonical basis using Corollary 4.10.

##  Appendix A Acting by Bijections up to Lower-Order Terms

In this section we discuss a number of general results concerning linear
operators which act by bijections up to lower-order terms, as introduced in
Definition 1.2.

For the following, let $U$ be a vector space over $\mathbb{C}$ with a fixed
basis $\mathbb{B}$, and let $w:U\to U$ be a linear operator. Our first result
is that if $w$ acts on $(U,\mathbb{B})$ by a bijection up to lower-order
terms, this bijection is unique.

######  Lemma A.1.

For $i=1,2$, let $\xi_{i}:\mathbb{B}\to\mathbb{B}$ be bijections and let
$\leq_{i}$ be preorders on $\mathbb{B}$ such that $w$ acts on
$(U,\mathbb{B},\leq_{i})$ by $\xi_{i}$ up to lower-order terms. Then
$\xi_{1}=\xi_{2}$.

###### Proof.

Suppose for a contradiction that $\xi_{1}\neq\xi_{2}$, and let
$x\in\mathbb{B}$ be minimal under $\leq_{1}$ such that
$\xi_{1}(x)\neq\xi_{2}(x)$. By definition, we have:

$w(x)=\pm\xi_{1}(x)+\sum_{y<_{1}x}a_{y}\xi_{1}(y)$

and we also know that $\xi_{2}(x)$ appears with coefficient $\pm 1$ in $w\cdot
x$. Since $\xi_{1}(x)\neq\xi_{2}(x)$ by assumption, we must have
$\xi_{2}(x)=\xi_{1}(y)$ for some $y<_{1}x$. Since $x$ was minimal, we conclude
that $\xi_{1}(y)=\xi_{2}(y)$ and hence $y=x$, a contradiction. ∎

Next, we interpret bijections up to lower-order terms through the QR
decomposition of a matrix. Recall that a square matrix $A$ with real
coefficients has a unique decomposition into a product of square matrices
$A=QR$, where $Q$ is orthogonal and $R$ is upper-triangular with positive
diagonal entries.

Suppose $(U,\mathbb{B})$ is a finite-dimensional based vector space. Given a
preorder $\leq$ on $\mathbb{B}$, let $\preceq$ be a linearisation of this
order, which is a total order satisfying $x\prec y$ for $x,y\in\mathbb{B}$
whenever $x<y$. For an operator $w:U\to U$, let $[w]_{(\mathbb{B},\preceq)}$
be the matrix of $w$ with respect to the ordered basis $(\mathbb{B},\preceq)$.

######  Lemma A.2.

Suppose that $w$ acts on $(U,\mathbb{B},\leq)$ by $\xi$ up to lower-order
terms. Then $[w]_{(\mathbb{B},\preceq)}$ has a unique $QR$ decomposition, and
the orthogonal part $Q$ is a generalised permutation matrix for $\xi$.

###### Proof.

Since $[w]_{(\mathbb{B},\preceq)}$ is a square matrix with integer entries, it
has a unique QR decomposition. For every $x\in\mathbb{B}$ we can find
$a_{y}\in\mathbb{Z}$ such that

$w(x)=\pm\xi(x)+\sum_{y<x}a_{y}\xi(y).$

Define maps $\xi^{\prime}:U\to U$ and $w^{\prime}:U\to U$ by the linear
extensions of $\xi^{\prime}(x):=\pm\xi(x)$ and

$w^{\prime}(x):=x\pm\sum_{y<x}a_{y}y$

for every $x\in\mathbb{B}$, where the sign in both cases matches the sign of
$\xi(x)$ in $w(x)$. Then $w=\xi^{\prime}w^{\prime}$,
$[\xi^{\prime}]_{(\mathbb{B},\preceq)}$ is a generalised permutation matrix
for $\xi$ with entries $\pm 1$, and $[w^{\prime}]_{(\mathbb{B},\preceq)}$ is
an upper-triangular matrix whose diagonal entries are all $1$, since the order
$<$ implies $\prec$ by construction. Permutation matrices with nonzero entries
$\pm 1$ are orthogonal, so

$[w]_{(\mathbb{B},\preceq)}=[\xi^{\prime}]_{(\mathbb{B},\preceq)}[w^{\prime}]_{(\mathbb{B},\preceq)}$

is the QR decomposition as required. ∎

We note that a decomposition of this form is extremely rare in general, since
permutation matrices with entries $\pm 1$ form a finite subset of the space of
orthogonal matrices.

Given some based vector space $(U,\mathbb{B})$ and an operator $w:U\to U$, we
would like to determine whether $w$ acts by _some_ bijection up to lower-order
terms. Suppose $A:=[w]_{(\mathbb{B},\preceq)}$, where $\preceq$ is an
arbitrary ordering on $\mathbb{B}$.

######  Corollary A.3.

The operator $w:U\to U$ acts by some bijection on $(U,\mathbb{B})$ up to
lower-order terms if and only if there exist permutation matrices
$P_{1},P_{2}$ such that $P_{1}AP_{2}$ is an upper-triangular integer matrix
with diagonal entries $\pm 1$.

###### Proof.

By the Lemma, $w$ acts by a bijection on $(U,\mathbb{B})$ by a bijection up to
l.o.t. if and only if there is some ordering $(\mathbb{B},\preceq^{\prime})$
of the basis such that $[w]_{(\mathbb{B},\preceq^{\prime})}=PR$, where $P$ is
a permutation matrix and $R$ is upper-triangular with integer entries and
diagonal entries $\pm 1$. Since reordering the basis corresponds to
conjugating by a permutation matrix, this is equivalent to the existence of a
permutation matrix $P_{0}$ such that

$P_{0}AP_{0}^{-1}=PR.$

Setting $P_{1}:=P^{-1}P_{0}$, $P_{2}:=P_{0}^{-1}$ and multiplying by $P^{-1}$
on both sides completes the proof. ∎

This is the problem of ‘upper-triangularisation’: asking whether the rows and
columns of a matrix $A$ can be permuted to achieve upper-triangularity
(although we also impose some additional constraints on the entries). When $A$
is a binary matrix, the computational complexity of this problem has been
shown to be NP-complete [14].

##  Appendix B The Kazhdan–Lusztig Basis for Arbitrary Coxeter Groups

In this section, we prove that separable elements of a Coxeter group act on
the Kazhdan-Lusztig basis of the left regular representation by bijections up
to lower-order terms. Our argument is combinatorial, and builds on work of
Mathas [33] and Roichman [34]. We note that a version of these results for
representations of Hecke algebras (including those with unequal parameters)
can be similarly developed using recent work by Bonnafé [4], although this
involves a number of technicalities which we choose to avoid for the sake of
exposition.

For the general theory of Coxeter groups and their Kazhdan–Lusztig
representations, see [3]. We begin by making the necessary definitions to
state our main result.

Let $W:=(W,I)$ be a Coxeter system. That is, $I$ is a finite set, which is the
vertex set of a simple, undirected graph with edges labelled
$\\{3,4,\dots,\infty\\}$. Given such an $I$, the associated Coxeter group $W$
is generated by $s_{i}$ for $i\in I$ satisfying $s_{i}^{2}=1$ and
$(s_{i}s_{j})^{m_{ij}}=1$, where $m_{ij}$ is the weight of the edge between
$i$ and $j$. If no such edge exists, we set $m_{ij}:=2$ so that $s_{i}$ and
$s_{j}$ commute. If $m_{ij}=\infty$, no relation is imposed. Note that every
Weyl group can be realised as a finite Coxeter group.

An expression for $x\in W$ is a word $(i_{1},\dots,i_{r})$ with $i_{j}\in I$
such that $x=s_{i_{1}}\cdots s_{i_{r}}$. We call this expression reduced if it
is of minimal length among all expressions for $x$. The length of such an
expression is called the length of $x$.

For $J\subseteq I$, we let $(W_{J},J)$ be the induced Coxeter system. Then
$W_{J}\leq W$ is the subgroup generated by $\\{s_{i}\mid i\in J\\}$. When
$W_{J}$ is a finite group, it contains a unique longest element which we
denote $w_{J}$. It turns out that this element is always an involution. We use
this to extend the definition of separable elements of Weyl groups from
Section 1.3 to all Coxeter groups by

$W^{\mathrm{sep}}:=\\{w_{I_{1}}\cdots w_{I_{r}}\in W\mid I\supseteq
I_{1}\supseteq\cdots\supseteq I_{r},\,W_{I_{1}}\text{ finite}\\}.$

In the following section, we construct a canonical basis for the regular
representation $\mathbb{C}[W]$, which we call the Kazhdan–Lusztig (KL) basis.
Our main result will be as follows:

######  Theorem B.1.

Let $W$ be a Coxeter group and $\mathbb{K}\mathbb{L}$ the KL basis of
$\mathbb{C}[W]$. Every $w\in W^{\mathrm{sep}}$ acts on
$(\mathbb{C}[W],\mathbb{K}\mathbb{L})$ by a bijection up to lower-order terms.

###  B.1. Setup

In this section we construct the KL basis $\mathbb{C}[W]$ following [3]. As
before, let $W:=(W,I)$ be an arbitrary Coxeter system.

Denote by $\mathcal{H}:=\mathcal{H}(W;q)$ the Hecke algebra, which is the
$\mathbb{Z}[q^{\pm 1/2}]$-algebra generated by $\\{T_{i}\mid i\in I\\}$
satisfying the relations $(T_{i}T_{j})^{m_{ij}}=1$, and the quadratic relation
$T_{i}^{2}=(q-1)T_{i}+q$. If $w=s_{i_{1}}\cdots s_{i_{r}}$ is a reduced
expression, define $T_{w}:=T_{i_{1}}\cdots T_{i_{r}}$ (this is independent of
the chosen reduced expression). Observe that $\mathcal{H}$ is spanned over
$\mathbb{Z}[q^{\pm 1/2}]$ by $\\{T_{w}\mid w\in W\\}$, and each $T_{w}$ is
invertible.

There is a ring involution on $\mathcal{H}$ given by the $\mathbb{Z}$-linear
extension of $q^{1/2}\mapsto q^{-1/2}$ and $T_{w}\mapsto T_{w^{-1}}^{-1}$ for
every $w\in W$. By work of Kazhdan and Lusztig [24], there is a unique basis
$\\{C^{\prime}_{w}(q)\mid w\in W\\}$ for $\mathcal{H}$ which is invariant
under this involution, and satisfies some additional properties which we won’t
explicitly need.

To obtain representations of $W$, we specialise this construction to $q\mapsto
1$. Observe that there is a ring isomorphism

$\mathcal{H}(W;1)\xrightarrow{\sim}\mathbb{Z}[W]$

sending $T_{w}$ to $w$. We let $C_{w}$ denote the image of $C^{\prime}_{w}(1)$
under this isomorphism, and write $\mathbb{K}\mathbb{L}=\\{C_{w}\mid w\in
W\\}$ for the induced basis of the regular representation
$\mathbb{C}[W]=\mathbb{C}\otimes_{\mathbb{Z}}\mathbb{Z}[W]$, which we call the
Kazhdan–Lusztig (KL) basis.

The construction of a fixed basis for $\mathbb{C}[W]$ gives a natural
filtration of subrepresentations. We generalise this to determining a
filtration of
$\mathsf{Res}_{J}\mathbb{C}[W]:=\mathsf{Res}_{W_{J}}\mathbb{C}[W]$ for every
$J\subseteq I$, whose subquotients can be given a KL basis in a natural way.
When $W_{J}$ is finite, we will see that the longest element $w_{J}\in W_{J}$
acts by bijection (up to sign) on the KL basis of these subquotients.

For $x,y\in W$ and $i\in I$, write $x\xleftarrow{i}y$ if $C_{x}$ appears with
nonzero coefficient in the expansion of $s_{i}\cdot C_{y}$ with respect to the
KL basis.

######  Definition B.2.

Fix $J\subseteq I$. For $x,y\in W$, set $x\leq_{J}^{\mathrm{L}}y$ if and only
if there exists a chain

$x=x_{0}\xleftarrow{i_{1}}x_{1}\xleftarrow{i_{2}}x_{2}\xleftarrow{i_{3}}\cdots\xleftarrow{i_{r}}x_{r}=y$

for $x_{0},x_{1},\dots,x_{r}\in W$ and $i_{1},\dots,i_{r}\in J$. Write
$\sim_{J}^{L}$ for the equivalence relation induced by this preorder.

Note that $C_{x}$ appears with nonzero coefficient in $w\cdot C_{x}$ for some
$w\in W_{J}$ only if $x\leq_{J}^{\mathrm{L}}y$, and that
$\leq_{J^{\prime}}^{\mathrm{L}}$ implies $\leq_{J}^{\mathrm{L}}$ whenever
$J^{\prime}\subseteq J\subseteq I$. Let $\mathcal{C}$ be an equivalence class
of $W$ under $\leq_{J}^{\mathrm{L}}$, which we call a $J$-relative left cell.
For $x\in W$, write $x\leq_{J}^{\mathrm{L}}\mathcal{C}$ if
$x\leq_{J}^{\mathrm{L}}y$ for some (equivalently, for all) $y\in\mathcal{C}$.
Define the subspace

$\mathcal{C}^{\leq}:=\operatorname{span}_{\mathbb{C}}(\\{C_{x}\mid
x\leq_{J}^{\mathrm{L}}\mathcal{C}\\})\subseteq\mathbb{C}[W]$

which is a subrepresentation of $\mathsf{Res}_{J}\mathbb{C}[W]$. We define
$x<_{J}^{\mathrm{L}}\mathcal{C}$ and $\mathcal{C}^{<}$ similarly. Observe that
$\mathcal{C}^{<}$ is a $\mathsf{Res}_{J}\mathbb{C}[W]$-subrepresentation of
$\mathcal{C}^{\leq}$, and so we can define the quotient representation

$V(\mathcal{C}):=\mathcal{C}^{\leq}/\mathcal{C}^{<}.$

This is a $W_{J}$-representation, and we call such representations
$J$-relative cell modules of $W$. For $x\in\mathcal{C}$, let $[C_{x}]_{J}$ be
the image of the KL basis element $C_{x}\in\mathcal{C}^{\leq}$ under the
natural projection map $\mathcal{C}^{\leq}\twoheadrightarrow V(\mathcal{C})$.
Observe that $\mathbb{K}\mathbb{L}_{\mathcal{C}}:=\\{[C_{x}]_{J}\mid
x\in\mathcal{C}\\}$ is a basis for $V(\mathcal{C})$, which we call the KL
basis.

When $J=I$, we write $\leq^{\mathrm{L}}$ (respectively $\sim^{\mathrm{L}}$)
instead of $\leq_{I}^{\mathrm{L}}$ (respectively $\sim_{I}^{\mathrm{L}}$), and
$[C_{x}]$ instead of $[C_{x}]_{I}$. We often drop the prefix ‘$I$-relative’
for cells and modules and refer simply to left cells and cell modules of $W$.

######  Remark B.3.

When $W=S_{n}$ is the symmetric group, the cell modules are irreducible (see
Section 5.1). However, cell modules of general Coxeter groups will be neither
irreducible nor finite-dimensional in general.

###  B.2. Actions on cell modules

Let $(W,I)$ be an arbitrary Coxeter system. Our goal is to show that for any
$J\subseteq I$ with $W_{J}$ finite, the element $w_{J}$ acts on
$\mathbb{C}[W]$ by an involution on the KL basis.

We begin with the action of the longest element on cell modules for finite
Coxeter groups, which forms the basis for the remainder of our calculations.

######  Proposition B.4 (Theorem 3.1 in [33]).

Suppose $W$ is finite and $\mathcal{C}$ is a left cell of $W$. Then there is
an involution $\psi_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}$ such that the
longest element $w_{I}\in W$ acts on the KL basis of $V(\mathcal{C})$ by

$w_{I}\cdot[C_{x}]=\pm[C_{\psi_{\mathcal{C}}(x)}]$

for every $x\in\mathcal{C}$, where the sign is constant.

######  Remark B.5.

In [33], Mathas states this result only for Weyl groups, since the general
case relies on the so-called ‘positivity conjecture’ to hold for each Coxeter
group. However, this conjecture has since been proven [13].

To extend this to parabolic subgroups, we need to compare the $J$-relative
left cells of $W$ to the left cells of $W_{J}$, which is possible due to a
number of observations of Roichman [35].

Fix $(W,I)$ an arbitrary Coxeter system and $J\subseteq I$ a subdiagram. Every
$x\in W$ has a unique decomposition $x=ua$, where $u\in W_{J}$ and $a$ is the
minimal-length element of $W_{J}x\subseteq W$. We define special notation for
this decomposition by setting $\mathsf{Res}_{J}(x):=u$ and $\pi_{J}(x):=a$.

######  Lemma B.6 (Theorem 5.2 in [35]).

Fix $w\in W_{J}$, $\mathcal{C}$ a $J$-relative left cell and
$x,y\in\mathcal{C}$. Then $[C_{x}]_{J}\in V(\mathcal{C})$ appears in
$w\cdot[C_{y}]_{J}$ with nonzero coefficient only if $\pi_{J}(x)=\pi_{J}(y)$.
If this is the case, the coefficient with which it appears matches the
coefficient of $C_{\operatorname{res}_{J}(x)}\in\mathbb{C}[W_{J}]$ in the
action of $w\cdot C_{\operatorname{res}_{J}(y)}$.

In particular, for every $J$-relative left cell $\mathcal{C}$ of $W$, there is
a left cell $\mathcal{C}^{\prime}$ of $W_{J}$ such that
$\operatorname{res}_{J}(-):\mathcal{C}\to\mathcal{C}^{\prime}$ is a bijection,
and the map $[C_{x}]_{J}\mapsto[C_{\operatorname{res}_{J}(x)}]$ for every
$x\in\mathcal{C}$ linearly extends to an isomorphism $V(\mathcal{C})\to
V(\mathcal{C}^{\prime})$ of $W_{J}$-represetations.

By this Lemma, we are able to deduce the action of $w_{J}$ on the KL basis of
$\mathbb{C}[W]$ up to lower-order terms. We first define the bijection that