# Viennot Shadows and Graded Module Structure in Colored Permutation Groups

Moxuan (Jasper) Liu  Department of Mathematics
University of California, San Diego
La Jolla, CA, 92093-0112, USA<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbf{x}_{n\times n}$ be a matrix of $n\times n$ variables, and let
$\mathbb{C}[\mathbf{x}_{n\times n}]$ be the polynomial ring on these
variables. Let $\mathfrak{S}_{n,r}$ be the group of colored permutations,
consisting of $n\times n$ complex matrices with exactly one nonzero entry in
each row and column, and each nonzero entry is a $r$-th root of unity. We
associate an ideal
$I_{\mathfrak{S}_{n,r}}\subseteq\mathbb{C}[\mathbf{x}_{n\times n}]$ with the
group $\mathfrak{S}_{n,r}$, and use orbit harmonics to give an ideal-theoretic
extension of the Viennot shadow line construction to $\mathfrak{S}_{n,r}$.
This extension gives a standard monomial basis of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$, and gives an
analogous definition of “longest increasing subsequence” to the group
$\mathfrak{S}_{n,r}$. We examine the extension of Chen’s conjecture to this
analogy. We also study the structure of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$ as a graded
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ module, which then induces a
graded $\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ module structure on the
$\mathbb{C}$-algebra $\mathbb{C}[\mathfrak{S}_{n,r}]$.

###### Key words and phrases:

Viennot’s shadow lines, orbit harmonics, ideals, graded modules

##  1\. Introduction

Longest increasing subsequences in the symmetric group $\mathfrak{S}_{n}$ have
been a well-studied object in combinatorics. In this paper, we will use orbit
harmonics and ideal-theoretical extensions of Viennot’s shadow line
construction to give an analogous definition of longest increasing subsequence
for the colored permutation group $\mathfrak{S}_{n,r}$.

Let $\mathbf{x}$ be a finite set of variables, and let $\mathbb{C}$ be the
field of complex numbers. Let $\mathbb{C}[\mathbf{x}]$ be the polynomial ring
with variable set $\mathbf{x}$. If $I$ is a graded ideal in
$\mathbb{C}[\mathbf{x}]$, the Hilbert series of $\mathbb{C}[\mathbf{x}]/I$ is

(1.1) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}]/I;q):=\sum_{d\geq
0}\dim_{\mathbb{C}}(\mathbb{C}[\mathbf{x}]/I)_{d}\cdot q^{d}.$

Let $\mathfrak{S}_{n}$ be the symmetric group on $[n]=\\{1,2,\dots,n\\}$.
Given $w\in\mathfrak{S}_{n}$, we write $w$ in its one line notation
$w=[w(1),w(2),\dots,w(n)]$. An increasing subsequence of length $l$ of $w$ is
a sequence of integers $1\leq i_{1}<i_{2}<\dots<i_{l}\leq n$ satisfying
$w(i_{1})<w(i_{2})<\dots<w(i_{l})$. We write

(1.2) $\mathrm{lis}(w)=\max\\{k:w\text{ has an increasing subsequence of
length }k\\}$

for the length of the longest increasing subsequence of $w$. Let $a_{n,k}$
denote the number of permutations $w\in\mathfrak{S}_{n}$ with
$\mathrm{lis}(w)=k$. For any positive integer $n$, Chen [2] conjectured that
the sequence $(a_{n,1},a_{n,2},\dots,a_{n,n})$ is log-concave, that is,

(1.3) $a_{n,i}\cdot a_{n,i+2}\leq a_{n,i+1}^{2}\text{ for }1\leq i\leq n-2.$

In [4], Rhoades studied the following ideal:

######  Definition 1.1.

Let $\mathbf{x}_{n\times n}=[x_{i,j}]_{1\leq i,j\leq n}$ be an $n\times n$
matrix of variables, and let $\mathbb{C}[\mathbf{x}_{n\times n}]$ be the
polynomial ring in these variables. The ideal $I_{n}$ is the ideal generated
by

  * •

the square $x_{i,j}^{2}$ for $1\leq i,j\leq n$ of any variable;

  * •

the product $x_{i,j}\cdot x_{i,j^{\prime}}$ for $1\leq i\leq n$ and $1\leq
j<j^{\prime}\leq n$ of two distinct variables in the same row;

  * •

the product $x_{i,j}\cdot x_{i^{\prime},j}$ for $1\leq i<i^{\prime}\leq n$ and
$1\leq j\leq n$ of two distinct variables in the same column;

  * •

the sum $\sum_{j=1}^{n}x_{i,j}$ for $1\leq i\leq n$ of variables in the same
row;

  * •

the sum $\sum_{i=1}^{n}x_{i,j}$ for $1\leq j\leq n$ of variables in the same
column.

As all the generators of $I_{n}$ are homogeneous,
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$ is a graded $\mathbb{C}$-algebra.
Let $\mathfrak{S}_{n}\times\mathfrak{S}_{n}$ act on $\mathbf{x}_{n\times n}$
by permuting rows and columns. This induces an action on
$\mathbb{C}[\mathbf{x}_{n\times n}]$ which stabilizes $I_{n}$, as the
collection of generators of $I_{n}$ is stable under this action. Thus,
$\mathbb{C}[\mathbf{x}_{n\times n}]$ has the structure of a graded
$\mathfrak{S}_{n}\times\mathfrak{S}_{n}$ module.

Rhoades proved [4] that $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$ has Hilbert
series

(1.4) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}/I_{n};q)=a_{n,n}+a_{n,n-1}\cdot q+a_{n,n-2}\cdot q^{2}+\dots+a_{n,1}\cdot
q^{n-1}$

with coefficients of $q^{k}$ being the number of permutations $w$ in
$\mathfrak{S}_{n}$ with $\mathrm{lis}(w)=n-k$. Using Viennot’s shadow line
construction, Rhoades attached a square-free monomial $\mathfrak{s}(w)$ for
each $w\in\mathfrak{S}_{n}$, with the property that

(1.5) $\deg(\mathfrak{s}(w))=n-\mathrm{lis}(w),$

and showed that (Theorem 3.7)

(1.6) $\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n}\\}$

is a spanning set of the vector space $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{n}$, and proved that this is actually the standard monomial basis with
respect to the Toeplitz order (see Definition 3.5). Rhoades also studied the
graded and ungraded module structure of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{n}$, giving the isomorphism

(1.7) $(\mathbb{C}[\mathbf{x}_{n\times
n}/I_{n})_{k}\cong\bigoplus_{\begin{subarray}{c}\lambda\,\vdash\,n\\\
\lambda_{1}\,=\,n-k\end{subarray}}\mathrm{End}_{\mathbb{C}}(V^{\lambda})$

and gave insights on the Novak-Rhoades conjecture [3], which is a
strengthening of Chen’s conjecture.

We generalize Rhoades’s work to colored permutation groups
$\mathfrak{S}_{n,r}$, which is the group of $n\times n$ matrices with exactly
one nonzero entry in each row or column, and all nonzero entries are $r$-th
roots of unity. We specifically emphasize on the case $r=2$, where
$\mathfrak{S}_{n,2}$ is denoted as $B_{n}$, the group of signed permutations.
In Section 2.4, we will give more detailed descriptions of the properties of
these groups. The following ideal is our object of study:

######  Definition 1.2.

$I_{\mathfrak{S}_{n,r}}$ is the ideal in $\mathbb{C}[\mathbf{x}_{n\times n}]$
with generators:

  * •

the $(r+1)$-th power $x_{i,j}^{r+1}$ for $1\leq i,j\leq n$ of any variable;

  * •

the product $x_{i,j}\cdot x_{i,j^{\prime}}$ for $1\leq i\leq n$ and $1\leq
j<j^{\prime}\leq n$ of two distinct variables in the same row;

  * •

the product $x_{i,j}\cdot x_{i^{\prime},j}$ for $1\leq i<i^{\prime}\leq n$ and
$1\leq j\leq n$ of two distinct variables in the same column;

  * •

the sum $\sum_{j=1}^{n}x_{i,j}^{r}$ for $1\leq i\leq n$ of $r$-th powers of
variables in the same row;

  * •

the sum $\sum_{i=1}^{n}x_{i,j}^{r}$ for $1\leq j\leq n$ of $r$-th powers of
variables in the same column.

In particular, when $r=2$, we write $I_{\mathfrak{S}_{n,2}}=I_{B_{n}}$.

Again, $I_{\mathfrak{S}_{n,r}}$ is a homogeneous ideal, so
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$ is a graded
$\mathbb{C}$-algebra. Let
$\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(C)$ act on
$\mathbb{C}[\mathbf{x}_{n\times n}]$, where the action is induced by the
following action on $\mathbf{x}_{n\times n}$: let
$A,A^{\prime}\in\mathrm{GL}_{n}(\mathbb{C})$,

(1.8) $(A,A^{\prime})\cdot\mathbf{x}_{n\times n}=A\mathbf{x}_{n\times
n}A^{\prime-1}.$

Then, treating $\mathfrak{S}_{n,r}$ as a subgroup of
$\mathrm{GL}_{n}(\mathbb{C})$, we obtain an action of
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ on
$\mathbb{C}[\mathbf{x}_{n\times n}]$. The ideal $I_{\mathfrak{S}_{n,r}}$ is
stable under this action, so $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$ has the structure of a graded
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ module. Using representation
theoretical tools including the Branching Rule (Equation (2.12)) and the
Generalized Murnaghan-Nakayama Rule (Theorem 2.3), we give the graded module
structure (Theorem 4.3)

(1.9) $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{k}\cong\bigoplus_{\begin{subarray}{c}\boldsymbol{\lambda}\vdash_{r}n\\\
r\lambda^{0}_{1}+\sum_{i=1}^{r-1}i|\lambda^{i}|=rn-k\end{subarray}}V^{\boldsymbol{\lambda}}\otimes
V^{\boldsymbol{\lambda}^{\prime}},$

where $V^{\boldsymbol{\lambda}}$ is the irreducible module of
$\mathfrak{S}_{n,r}$ labelled by the $r$-tuple of partitions
$\boldsymbol{\lambda}=(\lambda^{0},\lambda^{1},\dots,\lambda^{r-1})$ such that
$\sum_{i=0}^{r}|\lambda^{i}|=n$, and $V^{\boldsymbol{\lambda}^{\prime}}$ is
the dual module of $V^{\boldsymbol{\lambda}}$. See Section 2 for details of
relevant notations.

Focusing on the case of $B_{n}$, $b_{n,k}$ (first defined in Definition 3.12),
which is analogous to $a_{n,k}$, counts the number of signed permutations $w$
in $B_{n}$ with a property analogous to “having a longest increasing
subsequence of length $k$”. For $n\geq 2$, the Hilbert series of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$ is

(1.10) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}/I_{B_{n}};q)=b_{n,2n}+b_{n,2n-1}\cdot q+b_{n,2n-2}\cdot
q^{2}+\dots+b_{n,2}\cdot q^{2n-2}.$

By testing $n$ up tp $40$, we observe that the sequence $\\{b_{n,k}\\}_{2\leq
k\leq 2n}$ is “almost” log-concave: log-concavity holds for $n\leq 8$; and for
larger $n$’s, log-concavity breaks for some small $k$’s and then holds for the
remaining majority of the sequence.

We also extend this notion of “having longest increasing subsequence of length
$k$” to elements of $\mathfrak{S}_{n,r}$, and we define (Definition 3.21)
$c_{n,r,k}$ that counts number of elements in $\mathfrak{S}_{n,r}$ with this
property. We proved (Theorem 3.23) that the Hilbert series of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$ is

(1.11) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}};q)=\sum_{k=1}^{rn}c_{n,r,k}q^{rn-k},$

and also give the conjecture (Conjecture 5.1) that the sequence
$\\{c_{n,r,k}\\}$ is uni-modal, that is, there exists $k$ such that

(1.12) $c_{n,r,d}\leq c_{n,r,d+1}\text{ for }d<k\text{ and }c_{n,r,d}\geq
c_{n,r,d+1}\text{ for }d\geq k.$

The rest of the paper is structured as follows. In Section 2, we introduce the
background material on Gröbner theory, orbit harmonics, group structure of
$\mathfrak{S}_{n,r}$, the Schensted correspondence, and representation theory.
In Section 3, we generalize Viennot’s shadow line construction to the group
$\mathfrak{S}_{n,r}$, and use this construction to find a standard monomial
basis of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$. In
Section 4 we give the graded $\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$
module structure of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$. And in Section 5, we examine the generalization
of Chen’s conjecture, and close with possible directions of future research.

##  2\. Background

###  2.1. Gröbner Theory

Let $\mathbf{x}_{N}=(x_{1},x_{2},\dots,x_{N})$ be a finite list of variables,
and let $\mathbb{C}[\mathbf{x}_{N}]$ be the polynomial ring over these
variables. A total order $<$ on the set of monomials in
$\mathbb{C}[\mathbf{x}_{N}]$ is called a monomial order if

  * •

$1\leq m$ for any monomial $m$,

  * •

for any monomials $m_{1},m_{2},m_{3}$, we have that $m_{1}<m_{2}$ implies
$m_{1}m_{3}<m_{2}m_{3}$.

Given any polynomial $f$ in $\mathbb{C}[\mathbf{x}_{N}]$, the initial monomial
$\textrm{in}_{<}f$ of $f$ is defined to be the largest monomial term in $f$
with respect to $<$. And given $I$ an ideal in $\mathbb{C}[\mathbf{x}_{N}]$,
the initial ideal of $I$ is defined to be the ideal generated by the initial
monomials of all polynomials in $I$, i.e.

(2.1) $\textrm{in}_{<}I=\langle\textrm{in}_{<}f:f\in I\rangle$

A monomial in $\mathbb{C}[\mathbf{x}_{n\times n}]$ which is not an element of
$\textrm{in}_{<}I$ is called a standard monomial. It is known that

(2.2) $\\{m+I:m\text{ is a standard monomial}\\}$

forms a basis of the vector space $\mathbb{C}[\mathbf{x}_{N}]/I$, and this is
called the standard monomial basis.

###  2.2. Orbit Harmonics

Given $Z$ a finite collection of points in $\mathbb{C}^{N}$, the vanishing
ideal $\mathbf{I}(Z)\subseteq\mathbb{C}[\mathbf{x}_{N}]$ is the collection of
all polynomials $f$ satisfying $f(x)=0$ for all $x$ in $Z$. It is known that
we have the isomorphism of vector spaces

(2.3) $\mathbb{C}[\mathbf{x}_{N}]/\mathbf{I}(Z)\cong\mathbb{C}[Z],$

where we see $\mathbb{C}[Z]$ as the vector space of functions from $Z$ to
$\mathbb{C}$. Note that the vector space
$\mathbb{C}[\mathbf{x}_{N}]/\mathbf{I}(Z)$ is usually ungraded.

Given a polynomial $f$ in $\mathbb{C}[\mathbf{x}_{N}]$, we can write
$f=f_{0}+f_{1}+\dots+f_{d}$ where $f_{i}$ is homogeneous of degree $i$ and
$f_{d}$ is nonzero. Denote $\tau(f)$ as the highest degree homogeneous part of
$f$, i.e. $\tau(f)=f_{d}$. Given an ideal $I$, the associated graded ideal
$\mathrm{gr}I$ is defined to be

(2.4) $\mathrm{gr}I=\langle\tau(f):f\in I\rangle.$

The isomorphism (2.3) of ungraded vector spaces above can be further extended
to

(2.5)
$\mathbb{C}[\mathbf{x}_{N}]/\mathbf{I}(Z)\cong\mathbb{C}[Z]\cong\mathbb{C}[\mathbf{x}_{N}]/\mathrm{gr}\mathbf{I}(Z),$

and since $\tau(f)$ is homogeneous for all $f$,
$\mathbb{C}[\mathbf{x}_{N}]/\mathrm{gr}I$ has an additional structure as a
graded vector space.

###  2.3. The Schensted Correspondence

Let $n$ be any positive integer. A partition $\lambda$ of $n$ is a weakly
decreasing sequence of positive integers
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ such that
$\sum_{i=1}^{r}\lambda_{i}=n$. We use the notation $\lambda\vdash n$ to
indicate that $\lambda$ is a partition of $n$.

A Young diagram of shape $\lambda$ is a diagram of boxes with $\lambda_{i}$
boxes in the $i$-th row. A standard Young tableau of shape $\lambda$ is a
filling of $\\{1,2,\dots n\\}$ into the Young diagram of $\lambda$ such that
each number appears exactly once, and that the entries are increasing across
rows and down columns. For example, given $\lambda=(4,2,1)\vdash 7$, we give
the Young diagram of $\lambda$ and one standard Young tableau of shape
$\lambda$.

& 1 & 3 6 7 2 5 4

We write $\mathrm{SYT}(\lambda)$ to denote the collection of standard Young
tableaux of shape $\lambda$. The Schensted correspondence [6] is a bijection:

(2.6) $\mathfrak{S}_{n}\xrightarrow{\quad\sim\quad}\bigsqcup_{\lambda\vdash
n}\\{(P,Q):P,Q\in\mathrm{SYT}(\lambda)\\},$

that is, the Schensted correspondence associates to each
$w\in\mathfrak{S}_{n}$ an ordered pair $(P,Q)$ of standard Young tableaux of
the same shape. The explicit bijection is usually given by an insertion
algorithm, and details of the it can be found in [5].

###  2.4. Signed and Colored Permutation Groups

Recall that the group $\mathfrak{S}_{n,r}$ consists of all $n\times n$ complex
matrices with exactly one nonzero entry in each row and column, and each
nonzero entry is a $r$-th root of unity. We introduce a more combinatorial
interpretation of the the group in this section:

The $r$-colored permutation group $\mathfrak{S}_{n,r}$ is defined to be the
wreath product $(\mathbb{Z}/r\mathbb{Z})\wr\mathfrak{S}_{n}$. In particular,
when $r=2$, the group is called the signed permutation group, and we denote
the group as $B_{n}$. An element $w$ of $\mathfrak{S}_{n,r}$ can be viewed as
a pair $(\sigma,\kappa)$, where $\sigma$ is a permutation in
$\mathfrak{S}_{n}$, and $\kappa:[n]\rightarrow{0,1,\dots,r-1}$ is a coloring
of elements in $[n]$. Writing $[n]^{r}$ for the set
$\\{i^{j}:i\in[n],j\in{0,1,\dots,r-1}\\}$, we can see $w=(\sigma,\kappa)$ as a
function $[n]^{r}\rightarrow[n]^{r}$ via the rule

(2.7) $(w)(i^{j})=\sigma(i)^{\kappa(\sigma(i))+j},$

where addition is modulo $r$. An example is when $n=5,r=3,$,
$w=(\sigma,\kappa)$ where $\sigma=[4,2,5,3,1]=(1435)(2),\kappa=(2,1,1,2,0)$,
then we have
$w(1^{0})=4^{2},w(2^{0})=2^{1},w(3^{0})=5^{0},w(4^{0})=3^{2},w(5^{0})=1^{2}$.
In one-line notation, we denote $w=[4^{2},2^{1},5^{0},3^{2},1^{2}]$, and we
write the cycle notation of $w$ as $(1^{2},4^{2},3^{2},5^{0})(2^{1})$. For a
cycle in the cycle decomposition of $w$, we refer to the color of the cycle to
be the sum of colors of elements in it, reduced modulo $r$. For example, the
color of $(1^{2},4^{2},3^{2},5^{0})$ is $2+2+2=0\mod 3$, and the color of
$(2^{1})$ is $1$.

######  Definition 2.1.

Given integers $n\geq 0$, $r\geq 1$, a $r$-partition of $n$ is a collection of
$r$ partitions
$\boldsymbol{\lambda}=(\lambda^{0},\lambda^{1},\dots,\lambda^{r-1})$, such
that $\sum_{i=0}^{r-1}|\lambda^{i}|=n$. We write
$\boldsymbol{\lambda}\vdash_{r}n$ for $\boldsymbol{\lambda}$ being a
$r$-partition of $n$. In particular, when $r=2$, we call such $r$-partitions
bipartitions.

######  Definition 2.2.

The cycle type of an element $w=(\sigma,\kappa)\in\mathfrak{S}(n,r)$ is the
$r$-partition
$\boldsymbol{\lambda}=(\lambda^{0},\lambda^{1},\dots,\lambda^{r-1})$, such
that $\lambda^{i}$ consists of lengths of cycles in $w$ of color $i$.

Two elements in $\mathfrak{S}_{n,r}$ are conjugates if and only if they have
the same cycle type. Thus, the conjugacy classes of $\mathfrak{S}_{n,r}$ are
labelled by $r$-partitions of $n$.

###  2.5. Representation Theory

####  2.5.1. $\mathfrak{S}_{n}$ Representations

Let $\mathfrak{S}_{n}$ be the symmetric group. As the conjugacy classes of
$\mathfrak{S}_{n}$ are labelled by partitions of $n$, there is a bijection
between partitions of $n$ and irreducible representations of
$\mathfrak{S}_{n}$ over $\mathbb{C}$. For $\lambda\vdash n$, we denote
$V^{\lambda}$ as the irreducible representation of $\mathfrak{S}_{n}$
associated to $\lambda$. The dimension of $V^{\lambda}$ is given by
$|\mathrm{SYT}(\lambda)|$, the number of standard Young tableau of shape
$\lambda$.

####  2.5.2. Representations of Signed and Colored Permutations

Let
$\boldsymbol{\lambda}=(\lambda^{0},\lambda^{2},\dots,\lambda^{r-1})\vdash_{r}n$.
A standard Young tableau of shape $\boldsymbol{\lambda}$ is a bijection from
$[n]$ to the collection of boxes in the disjoint union of Young diagrams of
$\lambda^{0}$, $\lambda^{1}$, …,$\lambda^{r-1}$, such that the entries in each
diagram are increasing across rows and down columns. Below is an example of a
standard Young tableau for bipartition $((2,1),(4,2))\vdash 9$.

1 & 5 7 2 & 4 8 9 3 6

Specht constructed [8] a bijection between the irreducible representations of
$\mathfrak{S}_{n,r}$ and $r$-partitions of $n$, such that the dimension of the
irreducible representation labelled by $\boldsymbol{\lambda}$ is the number of
standard Young tableaux of shape $\boldsymbol{\lambda}$.

###  2.6. Branching Rule and Muranghan-Nakayama Rule

Let $\lambda\vdash n_{1}$ and $\mu\vdash n_{2}$, with the property that
$\lambda_{i}\geq\mu_{i}$ for all $i$, and let $n=n_{1}-n_{2}$. The skew
partition $\lambda/\mu$ has diagram obtained by deleting the diagram of $\mu$
from the diagram of $\lambda$. For example, when $\lambda=(4,3,2)\vdash 9$ and
$\mu=(2,1)\vdash 3$, the skew partition $\lambda/\mu$ has diagram drawn below.

&

The $\mathfrak{S}_{n}$ character $\chi^{\lambda/\mu}$ is defined to be

(2.8) $\chi^{\lambda/\mu}=\sum_{|\nu|=n}c^{\lambda}_{\mu,\nu}\chi^{\nu}$

where the sum is over all partitions $\nu$ of $n=n_{1}-n_{2}$ and
$c^{\lambda}_{\mu,\nu}$ is the Littlewoord-Richardson coefficient.

More generally, if
$\boldsymbol{\lambda}=(\lambda^{0},\lambda^{1},\dots,\lambda^{r-1})$ and
$\boldsymbol{\mu}=(\mu^{0},\mu^{1},\dots,\mu^{r-1})$ are $r$-partitions, such
that $\lambda^{i}/\mu^{i}$ are all well-defined skew partitions, the $r$-skew
partition $\boldsymbol{\lambda}/\boldsymbol{\mu}$ is then

(2.9)
$\boldsymbol{\lambda}/\boldsymbol{\mu}=(\lambda^{0}/\mu^{0},\lambda^{1}/\mu^{1},\dots,\lambda^{r-1}/\mu^{r-1}).$

The size of $\boldsymbol{\lambda}/\boldsymbol{\mu}$ is
$|\boldsymbol{\lambda}/\boldsymbol{\mu}|=\sum_{i=0}^{r-1}|\lambda^{i}|-|\mu^{i}|$.

Given a $r$-skew partition $\boldsymbol{\lambda}/\boldsymbol{\mu}$ such that
$|\boldsymbol{\lambda}/\boldsymbol{\mu}|=n$, we also have a character
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}$ of $\mathfrak{S}_{n,r}$
associated to it. It is defined to be

(2.10)
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}=\sum_{\boldsymbol{\nu}\vdash_{r}n}c^{\boldsymbol{\lambda}}_{\boldsymbol{\mu},\boldsymbol{\nu}}\chi^{\boldsymbol{\nu}}$

where the sum is over all $r$-partitions
$\boldsymbol{\nu}=(\nu^{1},\nu^{2},\dots,\nu^{r})\vdash_{r}n$, and
$c^{\boldsymbol{\lambda}}_{\boldsymbol{\mu},\boldsymbol{\nu}}$ is the
generalized Littlewood-Richardson coefficient, defined as

(2.11)
$c^{\boldsymbol{\lambda}}_{\boldsymbol{\mu},\boldsymbol{\nu}}=\prod_{i=1}^{r}c^{\lambda^{i}}_{\mu^{i},\nu^{i}},$

which is the product of Littlewood-Richardson coefficients.

####  2.6.1. Branching Rule

The characters labelled by $r$-skew partitions play an important role in the
branching rule of $\mathfrak{S}_{n,r}$ representations. Let
$\boldsymbol{\lambda}$ be a $r$-partition of $\mathfrak{S}_{n,r}$. For $k<n$,
$\mathfrak{S}_{k,r}\times\mathfrak{S}_{n-k,r}$ is a natural subgroup of
$\mathfrak{S}_{n,r}$, and we have the branching rule

(2.12)
$\mathrm{Res}^{\mathfrak{S}_{n,r}}_{\mathfrak{S}_{k,r}\times\mathfrak{S}_{n-k,r}}\chi^{\boldsymbol{\lambda}}=\sum_{\boldsymbol{\mu}}\chi^{\boldsymbol{\mu}}\times\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}$

where the sum is over all $r$-partitions $\boldsymbol{\mu}\vdash_{r}k$ such
that $\boldsymbol{\lambda}/\boldsymbol{\mu}$ is well defined. The proof of the
branching rule can be found in [9].

####  2.6.2. Generalized Murnaghan-Nakayama Rule

We then introduce how to evaluate
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}$ on $\mathfrak{S}_{n,r}$. A
ribbon is a skew partition with a rookwise connected diagram that does not
contain a $2\times 2$ square as a sub-diagram, and the height of a ribbon is
defined as the number of rows in it minus $1$. The diagram at the beginning of
the section is an example of a ribbon of height $2$. We say that a $r$-skew
partition $\boldsymbol{\lambda}/\boldsymbol{\mu}$ is a $r$-ribbon if only one
of the skew partitions $\lambda^{i}/\mu^{i}$ is non-empty the non-empty skew
partition is a ribbon. If $\boldsymbol{\lambda}/\boldsymbol{\mu}$ is a
$r$-ribbon with $\lambda^{j}/\mu^{j}$ non-empty, we say $\theta$ is the
character of $\boldsymbol{\lambda}/\boldsymbol{\mu}$, where $\theta$ is the
irreducible character of $\mathbb{Z}/r\mathbb{Z}$

(2.13)
$\theta:\mathbb{Z}/r\mathbb{Z}\rightarrow\mathrm{GL}_{1}(\mathbb{C})\quad\quad\theta(z)=e^{2j\pi
iz/r}.$

A decomposition of $\boldsymbol{\lambda}/\boldsymbol{\mu}$ into $r$-ribbons is
a nested sequence of $r$-partitions

(2.14)
$\boldsymbol{\mu}=\boldsymbol{\lambda}_{0}\subseteq\boldsymbol{\lambda}_{1}\subseteq\dots\subseteq\boldsymbol{\lambda}_{l}=\boldsymbol{\lambda}$

where $\boldsymbol{\lambda}_{i}/\boldsymbol{\lambda}_{i-1}$ is a $r$-ribbon
for $1\leq i\leq l$. Let
$\beta_{i}=|\boldsymbol{\lambda}_{i}/\boldsymbol{\lambda}_{i-1}|$ and
$\beta=(\beta_{1},\beta_{2},\dots,\beta_{l})$, we call the decomposition in
(2.14) a $\beta$-decomposition. With this definition, we introduce the
generalzied Murnaghan-Nakayama rule.

######  Theorem 2.3.

Let $w$ be an element of $\mathfrak{S}_{n,r}$, and suppose the cycles of $x$
have length $\beta_{1},\beta_{2},\dots,\beta_{l}$, and the cycles have colors
$c_{1},c_{2},\dots,c_{l}$. Then, we have the evaluation of
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}$ on $w$:

(2.15)
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}(w)=\sum_{\boldsymbol{\lambda}^{\prime}}\prod_{i=1}^{l}(-1)^{h_{i}}\theta_{i}(c_{i})$

where the sum is over $\boldsymbol{\lambda}^{\prime}$ of
$\beta$-decompositions of $\boldsymbol{\lambda}/\boldsymbol{\mu}$, $h_{i}$ is
the height of the $r$-ribbon in the $i$-th step of the $\beta$-decomposition,
and $\theta_{i}$ is the character of the $r$-ribbon in the $i$-th step as
defined above.

The proof of Theorem 2.3 can also be found in [9].

##  3\. Standard Monomial Basis

###  3.1. Viennot Shadow

Given a permutation $w\in\mathfrak{S}_{n}$, we associate to it a diagram on a
$n\times n$ grid, labelling all points $(i,w(i))$ on the grid. Below is the
diagram of $w=[5,1,3,6,7,2,4]\in\mathfrak{S}_{7}$.

Viennot proved [10] that the diagram can be used to construct the pair of
standard Young tableau in the Schensted correspondence. Imagine a light source
at the bottom left corner which shines northeast. Each point $(i,w(i))$ in the
diagram blocks light to its northeast. Consider the boundary of the shadow
region, and call it the first shadow line. The first shadow line for the
permutation $w$ above is depicted in the leftmost diagram below. Then, remove
all points on the first shadow line and iterate the process, we get the second
shadow line, the third shadow line, and so on. The diagram below in the middle
depicts all shadow lines of permutation $w=[5,1,3,6,7,2,4]$.

Suppose the shadow lines of $w$ are $L_{1},L_{2},\dots,L_{k}$, and we have
$w\rightarrow(P(w),Q(w))$ under the Schensted correspondence, Viennot proved
[10] the $y$ coordinates of the infinite horizontal rays of $L_{1},L_{2},\dots
L_{k}$ form the first row of $P(w)$, and the $x$ coordinates of the infinite
vertical rays of $L_{1},L_{2},\dots L_{k}$ for the first row of $Q(w)$. In the
example above, we have the first row of $P(w)$ is $1,2,4,7$; and the first row
of $Q(w)$ is $1,3,4,5$.

######  Definition 3.1.

Given the permutation $w\in\mathfrak{S}_{n}$, the shadow set $\mathcal{S}(w)$
of $w$ is the collection of points that lie on the northeast corners of the
shadow lines.

######  Remark 3.2.

From the construction of shadow lines, we can see that the shadow set of $w$
will contain no two points in the same row or column, which will induce the
following definition.

######  Definition 3.3.

A subset $\mathcal{R}$ of points in the $n\times n$ diagram is a (non-
attacking) rook placement if it contains at most one point in each row or
column.

As mentioned in the remark, the shadow set of any permutation is a rook
placement. However, not every rook placement is the shadow set of a
permutation. An algorithm for identifying whether a rook placement is the
shadow set of a permutation is specified in [4].

After obtaining the shadow set of a permutation, we can iterate the shadow
line construction on $\mathcal{S}(w)$, as depicted in the diagram below on the
left. Viennot proved [10] that the $y$ coordinates of the infinite horizontal
rays of the new shadow lines correspond to the second row of $P(w)$, and the
$x$ coordinates of the the infinite vertical rays correspond to the second row
of $Q(w)$. Keep iterating this process on the new shadow set, and we will get
the third row of the pair of tableau. The process ends when the new shadow set
is empty, and we would have the complete pair of tableaux $(P(w),Q(w))$.

Given the connections between the shadow sets and the Schensted
correspondence, the following lemma is immediate:

######  Lemma 3.4.

Let $w\in\mathfrak{S}_{n}$, the size of the shadow set is
$|\mathcal{S}(w)|=n-\mathrm{lis}(w)$, where $\mathrm{lis}(w)$ is the length of
the longest increasing subsequence of $w$, as defined in the introduction.

###  3.2. Shadow monomials

Given a collection of points $S\in[n]\times[n]$, we denote $m(S)$ to be the
square-free monomial $\prod_{(i,j)\in S}x_{i,j}$. Let $I_{n}$ be defined as in
the introduction. It is straightforward that

$\\{m(\mathcal{R}):\mathcal{R}\text{ is a rook placement}\\}$

is a spanning set of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$, since the
product of any two variables in the same row or column is in $I_{n}$.

Recall that given a set of variables $y_{1},y_{2},\dots,y_{N}$, the
lexicographical order is defined such that $y_{1}>y_{2}>\dots>y_{N}$ and
$y_{1}^{a_{1}}y_{2}^{a_{2}}\dots y_{N}^{a_{n}}<y_{1}^{b_{1}}y_{2}^{b_{2}}\dots
y_{N}^{b_{N}}$ if and only if there exists $j\leq n$ such that $a_{i}=b_{i}$
for all $i<j$, and $a_{j}<b_{j}$. The standard monomial basis of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$ was computed in [4] with respect to
the monomial order defined as follows.

######  Definition 3.5.

The Toeplitz order, denoted by $<_{\mathrm{Top}}$, is the lexicographical
order with respect the following ordering on the variables:

(3.1)
$x_{1,1}>x_{2,1}>x_{1,2}>x_{3,1}>x_{2,2}>x_{1,3}>\dots>x_{n,n-1}>x_{n-1,n}>x_{n,n}.$

Rhoades defined the shadow monomials as follows:

######  Definition 3.6.

Given a permutation $w\in\mathfrak{S}_{n}$, the shadow monomial of $w$, is the
square-free monomial

(3.2) $\mathfrak{s}(w)=m(\mathcal{S}(w)).$

Rhoades then proved the theorem:

######  Theorem 3.7.

[4] The monomials $\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n}\\}$ span the vector
space $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$.

Then, identifying $\mathfrak{S}_{n}$ as the group of $n\times n$ permutation
matrices (i.e. matrices such that all entries are $0$ and $1$, with exactly
one $1$ in each row and column), we have that the vanishing ideal of
$\mathfrak{S}_{n}$ is generated by

  * •

$x_{i,j}^{2}-x_{i,j}$ for $1\leq i,j\leq n$;

  * •

$x_{i,j}\cdot x_{i,j^{\prime}}$ for $1\leq i\leq n$, $1\leq j<j^{\prime}\leq
n$;

  * •

$x_{i,j}\cdot x_{i^{\prime},j}$ for $1\leq j\leq n$, $1\leq i<i^{\prime}\leq
n$;

  * •

$x_{i,1}+x_{i,2}+\dots+x_{i,n}$ for $1\leq i\leq n$;

  * •

$x_{1,j}+x_{2,j}+\dots+x_{n,j}$ for $1\leq j\leq n$.

A quick observation is that $I_{n}\subseteq\mathrm{gr}\mathbf{I}(P_{n})$,
since $I_{n}$ is generated by the highest degree homogeneous parts of the
generators of $\mathbf{I}_{P_{n}}$. The orbit harmonics (2.5) gives

(3.3)
$\mathbb{C}[\mathfrak{S}_{n}]\cong\mathbb{C}[P_{n}]\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}(P_{n})\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(P_{n}).$

Using a argument on the dimensions of the vector spaces, Rhoades showed [4]
that the ideals $I_{n}$ and $\mathrm{gr}\mathbf{I}(P_{n})$ are actually the
same, and showed that

$\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n}\\}$

is the standard monomial basis of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$
with respect to the Toeplitz order. Thus, let $a_{n,k}$ be the number of
permutations $w$ in $\mathfrak{S}_{n}$ with $\mathrm{lis}(w)=k$, the Hilbert
series of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$ is

(3.4) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{n};q)=a_{n,n}+a_{n,n-1}\cdot q+a_{n,n-2}\cdot q^{2}+\dots+a_{n,1}\cdot
q^{n-1}.$

###  3.3. Generalization to Signed Permutations

####  3.3.1. Spanning Set

Given a signed permutation $w=(\sigma,\kappa)$, we define its diagram on the
$n\times n$ grid. A point $(i,j)$ on the grid is colored red if $\sigma(i)=j$
and $\kappa(j)=1$. It is colored blue if $\sigma(i)=j$ and $\kappa(j)=0$. For
example, when $n=6$ and the one-line notation of $w$ is
$[2^{1},5^{0},3^{0},1^{0},6^{0},4^{1}]$, the diagram is below on the left:

We define the negative set of $w$ as

(3.5) $\mathrm{neg}(w)=\\{(i,j):\sigma(i)=j\text{ and }\kappa(j)=1\ \\}.$

or equivalently, it’s the collection of points colored red in the diagram.
Similarly, we define the positive set of $w$ to be

(3.6) $\mathrm{pos}(w)=\\{(i,j):\sigma(i)=j\text{ and }\kappa(j)=0\\}.$

######  Remark 3.8.

It is immediate result that $\mathrm{neg}(w)$ is a rook placement, and for any
rook placement $\mathcal{R}$, there is $w\in B_{n}$ such that
$\mathrm{neg}(w)=\mathcal{R}$.

Then, we do the shadow line construction on the positive set of $w$, and
obtain the shadow set $\mathcal{S}(\mathrm{pos}(w))$, as depicted in the above
on the right. The shadow lines are drawn in brown, and the points in the
shadow set are labelled in green. With this construction, we associate to the
signed permutation $w$ a monomial

(3.7) $\mathfrak{s}(w)=m(\mathrm{neg}(w))\cdot
m(\mathcal{S}(\mathrm{pos}(w)))^{2}.$

In the example above, the monomial is $x_{1,2}x_{6,4}x_{3,5}^{2}x_{4,3}^{2}$,
and we can introduce one of the main results:

######  Theorem 3.9.

The monomials $\\{\mathfrak{s}(w):w\in B_{n}\\}$ form a spanning set of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$.

Before presenting the proof of Theorem 3.9, we first introduce the following
lemma:

######  Lemma 3.10.

The following set descends to a spanning set of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$:

(3.8)
$\\{m(\mathcal{R}_{1})m(\mathcal{R}_{2})^{2}:\mathcal{R}_{1},\mathcal{R}_{2}\in[n]\times[n],\mathcal{R}=\mathcal{R}_{1}\cup\mathcal{R}_{2}\text{
is a rook placement}\\}.$

###### Proof.

Since the ideal $I_{B_{n}}$ contains the cubes of all variables, as well as
the products of any two variables in the same row or column, the result is
immediate. ∎

With the lemma, we present the proof of Theorem 3.9.

###### Proof.

(of Theorem 3.9) According to Lemma 3.10, we can fix a rook-placement
$\mathcal{R}_{1}$. By Remark 3.8, the set of $w\in B_{n}$ with
$\mathrm{neg}(w)=\mathcal{R}_{1}$ is nonempty, so it suffices to show

(3.9)
$\\{m(\mathcal{R}_{1})m(\mathcal{R}_{2})^{2}:\mathcal{R}=\mathcal{R}_{1}\cup\mathcal{R}_{2}\text{
is a rook placement}\\}$

lies, modulo the ideal $I_{B_{n}}$, in the linear span of

(3.10) $\\{\mathfrak{s}(w):w\in B_{n},\mathrm{neg}(w)=\mathcal{R}_{1}\\}.$

Let

(3.11) $I=\\{i:\text{there exists }j\text{ such that
}(i,j)\in\mathcal{R}_{1}\\},$

and

(3.12) $J=\\{j:\text{there exists }i\text{ such that
}(i,j)\in\mathcal{R}_{1}\\}.$

Fixing $i\in[n]\setminus I$, since the sum of squares of all variables in the
same row is in $B_{n}$, we have

(3.13) $\sum_{j\in[n]\setminus J}x_{i,j}^{2}\equiv-\sum_{j\in
J}x_{i,j}^{2}\mod I_{B_{n}}.$

Similarly, fixing $j\in[n]\setminus J$, we also have the equation

(3.14) $\sum_{i\in[n]\setminus I}x_{i,j}^{2}\equiv-\sum_{i\in
I}x_{i,j}^{2}\mod I_{B_{n}}.$

Thus, since $I_{B_{n}}$ contains the products of any two variables in the same
row or column, we obtain the following relations:

  * •

$m(\mathcal{R}_{1})\cdot\sum_{j\in[n]\setminus J}x_{i,j}^{2}\equiv 0\mod
I_{B_{n}}$

  * •

$m(\mathcal{R}_{1})\cdot\sum_{i\in[n]\setminus I}x_{i,j}^{2}\equiv 0\mod
I_{B_{n}}$

We also have the following relations directly from the definition of
$I_{B_{n}}$:

  * •

$(x_{i,j}^{2})^{2}\equiv 0\mod I_{B_{n}}$ for $i\in[n]\setminus I$,
$j\in[n]\setminus J$;

  * •

$x_{i,j}^{2}\cdot x_{i,j^{\prime}}^{2}\equiv 0\mod I_{B_{n}}$ for
$i\in[n]\setminus I$, $j,j^{\prime}\in[n]\setminus J$;

  * •

$x_{i,j}^{2}\cdot x_{i^{\prime},j}^{2}\equiv 0\mod I_{B_{n}}$ for
$i,i^{\prime}\in[n]\setminus I$, $j\in[n]\setminus J$.

Then, if we look at the $(n-|\mathcal{R}_{1}|)\times(n-|\mathcal{R}_{1}|)$
matrix of variables, where each variable is $x_{i,j}^{2}$ for
$i\in[n]\setminus I,j\in[n]\setminus J$, the collection of relations are
exactly the ones we used to define $I_{n}$ in Definition 1.1. For a signed
permutation $w$ with $\mathrm{neg}(w)=\mathcal{R}_{1}$, if we delete the rows
and columns that $\mathcal{R}_{1}$ appears in, $\mathrm{pos}(w)$ will form a
permutation in the remaining $(n-|I|)\times(n-|J|)$ grid. And since

(3.15) $\\{\mathrm{pos}(w):w\in B_{n},\mathrm{neg}(w)=\mathcal{R}_{1}\\}$

ranges over all possible permutations in the grid labelled by $([n]\setminus
I)\times([n]\setminus J)$, according to Theorem 3.7, the proof is complete. ∎

####  3.3.2. Standard Monomial Basis

Viewing $B_{n}$ as $n\times n$ matrices with exactly one nonzero entry in each
row or column, where the nonzero entries can be $\pm 1$, we have the vanishing
ideal $\mathbf{I}(B_{n})$ being generated by

  * •

$x_{i,j}^{3}-x_{i,j}$ for $1\leq i,j\leq n$;

  * •

$x_{i,j}\cdot x_{i,j^{\prime}}$ for $1\leq i\leq n$, $1\leq j<j^{\prime}\leq
n$;

  * •

$x_{i,j}\cdot x_{i^{\prime},j}$ for $1\leq j\leq n$, $1\leq i<i^{\prime}\leq
n$;

  * •

$x_{i,1}^{2}+x_{i,2}^{2}+\dots+x_{i,n}^{2}$ for $1\leq i\leq n$;

  * •

$x_{1,j}^{2}+x_{2,j}^{2}+\dots+x_{n,j}^{2}$ for $1\leq j\leq n$.

Comparing these generators with the generators of $I_{B_{n}}$, it immediately
follows that $I_{B_{n}}\subseteq\mathrm{gr}\mathbf{I}(B_{n})$. We further
improve this containment to an equality.

######  Theorem 3.11.

We have the equality of ideals $I_{B_{n}}=\mathrm{gr}\mathbf{I}(B_{n})$.
Moreover, the collection of monomials $\\{\mathfrak{s}(w):w\in B_{n}\\}$
descends to a basis of the vector space $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}}$. This is the standard monomial basis with respect to the
Toeplitz order.

###### Proof.

By the results from orbit harmonics, we have the following sequence of
isomorphisms

(3.16) $\mathbb{C}[B_{n}]\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}(B_{n})\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(B_{n}).$

With $I_{B_{n}}\subseteq\mathrm{gr}\mathbf{I}(B_{n})$, we have

(3.17) $\dim(\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(B_{n}))=2^{n}n!\leq\dim(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}}).$

since there is a unique pair $(\mathrm{neg}(w),\mathcal{S}(pos(w))$ for each
signed permutation $w$ , the spanning set $\\{\mathfrak{s}(w):w\in B_{n}\\}$
in Theorem 3.9 has size $2^{n}n!$. This forces the last inequality in (3.17)
to be an equality. This proves that $\\{\mathfrak{s}(w):w\in B_{n}\\}$
descends to a basis of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$.

To prove that this is actually the standard monomial basis with respect to the
Toeplitz order, let $\mathcal{R}_{1}$ be any rook placement, and let $f$ be a
monomial with square-free part $m(\mathcal{R}_{1})$. We claim that if $f$ is
in the standard monomial basis then $f$ is of the form $\mathfrak{s}(w)$ for
some $w\in B_{n}$. To see this, recall that $f$ is in the standard monomial
basis means that $f$ is not the initial monomial of any polynomial in
$I_{B_{n}}$. Then, using the notation in the end of the proof of Theorem 3.9,
the perfect square part $f/m(\mathcal{R}_{1})$ of $f$ must not be the initial
monomial of any polynomial in $I_{n-|\mathcal{R}_{1}|}$, where the variables
are $x_{i,j}^{2}$ with $i\in[n]\setminus I$ and $j\in[n]\setminus J$. Then,
according to the result on the standard monomial basis of $I_{n}$ by Rhoades
in [4], the claim is proved, and since the vector space
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$ has dimension $2^{n}n!$, all
monomials of the form $\mathfrak{s}(w)$ for $w\in B_{n}$ must be in the
standard monomial basis. ∎

With the following two definitions, we will be able to find the Hilbert series
of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$.

######  Definition 3.12.

Given $w\in B_{n}$, we define $\mathrm{lis}(\mathrm{pos}(w))$ to be the the
size of the largest subset $L$ of $\mathrm{pos}(w)$, such that given $(i,j)$
and $(i^{\prime},j^{\prime})$ in $L$, $i<i^{\prime}$ implies $j<j^{\prime}$.

######  Example 3.13.

Let $w\in B_{7}$ with one-line notation
$w=[3^{0},1^{1},6^{0},4^{0},7^{0},2^{1},5^{1}]$, we have
$\mathrm{pos}(w)=\\{(1,3),(3,6),(4,4),(5,7)\\}$, and
$\mathrm{lis}(\mathrm{pos}(w))=3$.

######  Definition 3.14.

For any pair of integers $n,k$, we define

(3.18) $b_{n,k}=|\\{w\in
B_{n}:2\mathrm{lis}(\mathrm{pos}(w))+|\mathrm{neg}(w)|=k\\}|$

From the definition, it is immediate that $b_{n,1}=0$ for $n\geq 2$, as
$|\mathrm{neg}(w)|\geq 2$ or $\mathrm{lis}(\mathrm{pos}(w))\geq 1$. Then, we
have the following corollary.

######  Corollary 3.15.

We have the Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$:

(3.19) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}};q)=b_{n,2n}+b_{n,2n-1}\cdot q+\cdots+b_{n,1}\cdot q^{2n-1}.$

###### Proof.

Follows directly from Theorem 3.11. ∎

######  Example 3.16.

When $n=3$, the Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{3}}$ is

(3.20) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{3}};q)=1+9q+22q^{2}+9q^{3}+q^{4}.$

###  3.4. Generalization to $\mathfrak{S}_{n,r}$

Given a $r$-colored permutation $w=(\sigma,\kappa)$, we also draw it on the
$n\times n$ grid. A point $(i,j)$ is labelled in color $c_{l}$ if
$\sigma(i)=j$ and $\kappa(j)=l$, where $l$ ranges over $0,1,\dots,r-1$. We
define the sets $C_{l}(w)$ to be the collection of points colored in $c_{l}$.
Then, we do the shadow set construction on the set $C_{0}(w)$ and obtain the
shadow set $\mathcal{S}(C_{0}(w))$, and we associate a monomial
$\mathfrak{s}(w)$ to $w$, defined as

(3.21) $\mathfrak{s}(w)=m(C_{1}(w))\cdot m(C_{2}(w))^{2}\cdot\dots\cdot
m(C_{r-1}(w))^{r-1}\cdot m(\mathcal{S}(C_{0}(w)))^{r}.$

With this construction, we give the following theorem:

######  Theorem 3.17.

The monomials $\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n,r}\\}$ descends to a
spanning set of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$.

The proof of Theorem 3.17 is similar to Theorem 3.9, and we also need the
following lemma which is similar to Lemma 3.10.

######  Lemma 3.18.

The following set of monomials descends to a spanning set of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$:

(3.22)
$~{}\left\\{\prod_{l=1}^{r}m(\mathcal{R}_{l})^{l}:\mathcal{R}_{l}\subset[n]\times[n],\mathcal{R}=\bigcup_{l=1}^{r}\mathcal{R}_{l}\text{
is a rook placement}\right\\}$

###### Proof.

Since the ideal $I_{\mathfrak{S}_{n,r}}$ contains the $r+1$-th power of all
variables, as well as the products of any two variables in the same row or
column, the result is immediate. ∎

With Lemma 3.18, we give the proof of Theorem 3.17.

###### Proof.

(of Theorem 3.17) According to Lemma 3.18, we can fix a collection of rook
placements $\mathcal{R}_{1},\mathcal{R}_{2},\dots,\mathcal{R}_{r-1}$, such
that

(3.23) $\bigcup_{l=1}^{r-1}\mathcal{R}_{l}$

is also a rook placement. Let $f$ be the monomial

(3.24) $f=\prod_{l=1}^{r-1}m(\mathcal{R}_{l})^{l},$

it suffices to show

(3.25) $\\{f\cdot
m(\mathcal{R}_{r})^{r}:\bigcup_{l=1}^{r}\mathcal{R}_{l}\text{ is a rook
placement}\\}$

lies, modulo $I_{\mathfrak{S}_{n,r}}$, in the linear span of

(3.26)
$\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n,r},C_{l}(w)=\mathcal{R}_{l}\text{ for
}1\leq l\leq r-1\\}.$

Let

(3.27) $I=\\{i:\text{there exists }j,l\text{ such that
}(i,j)\in\mathcal{R}_{l}\\},$

and

(3.28) $J=\\{j:\text{there exists }i,l\text{ such that
}(i,j)\in\mathcal{R}_{l}\\}.$

Note that we have

(3.29) $|I|=|J|=|\bigcup_{l=1}^{r-1}\mathcal{R}_{l}|.$

Fixing $i\in[n]\setminus I$, since the sum of $r$-th powers of all variables
in the same row is in $I_{\mathfrak{S}_{n,r}}$, we have

(3.30) $\sum_{j\in[n]\setminus J}x_{i,j}^{r}\equiv-\sum_{j\in
J}x_{i,j}^{r}\mod I_{\mathfrak{S}_{n,r}}.$

Similarly, fixing $j\in[n]\setminus J$, we also have the equation

(3.31) $\sum_{i\in[n]\setminus I}x_{i,j}^{r}\equiv-\sum_{i\in
I}x_{i,j}^{r}\mod I_{B_{n}}.$

Thus, since $I_{\mathfrak{S}_{n,r}}$ contains the products of any two
variables in the same row or column, we obtain the following relations:

  * •

$f\cdot\sum_{j\in[n]\setminus J}x_{i,j}^{r}\equiv 0\mod
I_{\mathfrak{S}_{n,r}}$;

  * •

$f\cdot\sum_{i\in[n]\setminus I}x_{i,j}^{r}\equiv 0\mod
I_{\mathfrak{S}_{n,r}}$.

We also have the following relations directly from the definition of
$I_{\mathfrak{S}_{n,r}}$:

  * •

$(x_{i,j}^{r})^{r}\equiv 0\mod I_{\mathfrak{S}_{n,r}}$ for $i\in[n]\setminus
I$, $j\in[n]\setminus J$;

  * •

$x_{i,j}^{r}\cdot x_{i,j^{\prime}}^{r}\equiv 0\mod I_{\mathfrak{S}_{n,r}}$ for
$i\in[n]\setminus I$, $j,j^{\prime}\in[n]\setminus J$;

  * •

$x_{i,j}^{r}\cdot x_{i^{\prime},j}^{r}\equiv 0\mod I_{\mathfrak{S}_{n,r}}$ for
$i,i^{\prime}\in[n]\setminus I$, $j\in[n]\setminus J$.

If we look at the $(n-|I|)\times(n-|J|)$ matrix of variables, where each
variable is $x_{i,j}^{r}$ for $i\in[n]\setminus I,j\in[n]\setminus J$, the
collection of relations are exactly the ones we used to define $I_{n}$ in
Definition 1.1. For a colored permutation $w$ with $C_{l}(w)=\mathcal{R}_{l}$
for $1\leq l\leq r-1$, if we delete the rows and columns with labels $I$ and
$J$ respectively, $C_{0}(w)$ will form a permutation in the remaining
$(n-|I|)\times(n-|J|)$ grid. And since

(3.32) $\\{C_{0}(w):w\in B_{n};C_{l}(w)=\mathcal{R}_{l}\\}$

ranges over all possible permutations in the grid labelled by $([n]\setminus
I)\times([n]\setminus J)$, according to Theorem 3.7, the proof is complete. ∎

Viewing $\mathfrak{S}_{n,r}$ as $n\times n$ matrices with exactly one nonzero
entry in each row and column, and every nonzero entry is a $r$-th root of
unity, we have that the vanishing ideal $\mathbf{I}(\mathfrak{S}_{n,r})$ being
generated by

  * •

$x_{i,j}^{r+1}-x_{i,j}$ for $1\leq i,j\leq n$;

  * •

$x_{i,j}\cdot x_{i,j^{\prime}}$ for $1\leq i\leq n$, $1\leq j<j^{\prime}\leq
n$;

  * •

$x_{i,j}\cdot x_{i^{\prime},j}$ for $1\leq j\leq n$, $1\leq i<i^{\prime}\leq
n$;

  * •

$x_{i,1}^{r}+x_{i,2}^{r}+\dots+x_{i,n}^{r}$ for $1\leq i\leq n$;

  * •

$x_{1,j}^{r}+x_{2,j}^{r}+\dots+x_{n,j}^{r}$ for $1\leq j\leq n$.

Thus, we have a theorem that is an analogy to Theorem 3.11.

######  Theorem 3.19.

We have the equality of ideals
$I_{\mathfrak{S}_{n,r}}=\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r})$. Moreover,
the collection of monomials $\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n,r}\\}$
descends to a basis of the vector space $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$. This is the standard monomial basis with respect
to the Toeplitz order.

###### Proof.

By the results from orbit harmonics, we have the following sequence of
isomorphisms

(3.33) $\mathbb{C}[\mathfrak{S}_{n,r}]\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}(\mathfrak{S}_{n,r})\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r}).$

With
$I_{\mathfrak{S}_{n,r}}\subseteq\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r})$, we
have

(3.34) $\dim(\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r}))=r^{n}n!\leq\dim(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}).$

since there is a unique tuple $(C_{1}(w),\dots
C_{r-1}(w),\mathcal{S}(C_{r}(w))$ for each colored permutation $w$, the
spanning set $\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n,r}\\}$ in Theorem 3.17
has size $r^{n}n!$. This forces the last inequality in (3.34) to be an
equality. This proves that $\\{\mathfrak{s}(w):w\in\mathfrak{S}_{n,r}\\}$
descends to a basis of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$. The proof that this is actually the standard
monomial basis with respect to the Toeplitz order is very similar to the proof
of Theorem 3.11. ∎

We also generalize the notion of longest increasing subsequence to the colored
permutation group. We first give the following two definitions:

######  Definition 3.20.

Given $w\in\mathfrak{S}_{n,r}$, we define $\mathrm{lis}(C_{0}(w))$ to be the
size of the largest subset $L$ of $C_{0}(w)$, such that given $(i,j)$ and
$(i^{\prime},j^{\prime})$ in $w$, $i<i^{\prime}$ implies $j<j^{\prime}$.

######  Definition 3.21.

For integers $n,r,k$, we define

(3.35)
$c_{n,r,k}=\left|\left\\{w\in\mathfrak{S}_{n,r}:r\cdot\mathrm{lis}(C_{0}(w))+\sum_{i=1}^{r-1}i\cdot|C_{i}(w)|=k\right\\}\right|$

######  Example 3.22.

Let $n=6$, $r=4$, and $w=[4^{2},1^{0},5^{2},6^{3},2^{1},3^{0}]$. We have
$C_{0}(w)=\\{(2,1),(5,3)\\}$, so $\mathrm{lis}(C_{0}(w))=2$, and we have

(3.36)
$r\cdot\mathrm{lis}(C_{0}(w))+|C_{1}(w)|+2|C_{1}(w)|+3|C_{3}(w)|=4\times
2+1+2\times 2+3\times 1=16,$

so $w$ would contribute to $c_{6,4,16}$.

With these two definitions, we give the Hilbert series of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$:

######  Theorem 3.23.

The Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$ is

(3.37) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}};q)=\sum_{k=1}^{rn}c_{n,r,k}q^{rn-k}$

###### Proof.

The result is immediate from the proof of Theorem 3.17 and Theorem 3.19. ∎

##  4\. Module Structure

As mentioned in the Introduction, treating $\mathfrak{S}_{n,r}$ as the group
of $n\times n$ matrices with exactly one nonzero entry in each row and column,
we can let $\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ act on
$\mathbb{C}[\mathbf{x}_{n\times n}]$, induced by its action on the matrix of
variables

(4.1) $(A,A^{\prime})\cdot\mathbf{x}_{n\times n}=A\mathbf{x}_{n\times
n}A^{\prime-1}.$

Since the collection of generators of $I_{\mathfrak{S}_{n,r}}$ are stable
under this action, we have that $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$ is a $\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$
module. By Theorem 3.19, we have an isomorphism and an equality of ideals

(4.2) $\mathbb{C}[\mathfrak{S}_{n,r}]\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r})=\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}},$

and since $\mathbb{C}[\mathfrak{S}_{n,r}]$ is semi-simple, (4.2) upgrades to
an isomorphism and an equality of ungraded
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ modules.

To investigate the graded structure of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$, we first introduce a lemma, which is a crucial
part in orbit harmonics. Let $\mathbf{x}$ be a collection of variables and
$\mathbb{C}[\mathbf{x}]$ be the polynomial ring. Given a graded
$\mathbb{C}$-algebra $A=\bigoplus_{i\geq 0}A_{i}$, let $A_{\leq
d}=\bigoplus_{i=0}^{d}A_{i}$ be the direct sum of degree less than or equal to
$d$ parts of it.

######  Lemma 4.1.

Let $I$ be an ideal in $\mathbb{C}[\mathbf{x}]$ and $\mathrm{gr}I$ be the
associated graded ideal of it. Let
$\mathcal{B}\subseteq\mathbb{C}[\mathbf{x}]_{\leq d}$ be a collection of
homogeneous polynomials with degree at most $d$. If $\mathcal{B}$ descends to
a basis of the vector space $(\mathbb{C}[\mathbf{x}]/grI)_{\leq d}$, then
$\mathcal{B}$ descends to a basis of the vector space
$\mathbb{C}[\mathbf{x}]_{\leq d}/(I\cap\mathbb{C}[\mathbf{x}]_{\leq d})$.

###### Proof.

The proof can be found in [4], Lemma 3.15. ∎

###  4.1. Graded Module Structure of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}}$

We first work on the case when $r=2$. In this case, the irreducible
$B_{n}\times B_{n}$ modules are of form $V^{(\lambda,\mu)}\otimes
V^{(\tilde{\lambda},\tilde{\mu})}$, where $(\lambda,\mu)$ and
$(\tilde{\lambda},\tilde{\mu})$ are bipartitions of $n$. Equation (4.2) now
reads:

(4.3) $\mathbb{C}[B_{n}]/\cong\mathbb{C}[\mathbf{x}_{n\times
n}]_{\mathrm{gr}}\mathbf{I}(B_{n})=\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}}.$

We give the main theorem of this section:

######  Theorem 4.2.

For any $k\geq 0$, the degree $k$ piece $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}})_{k}$ has graded $B_{n}\times B_{n}$ structure

(4.4) $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}})_{k}\cong\bigoplus_{\begin{subarray}{c}(\lambda,\mu)\vdash_{2}n\\\
2\lambda_{1}+|\mu|=2n-k\end{subarray}}V^{(\lambda,\mu)}\otimes
V^{(\lambda,\mu)}$

###### Proof.

If $W$ be any $B_{n}\times B_{n}$ module, $\mathrm{End}_{\mathbb{C}}(W)$ is
also a $B_{n}\times B_{n}$ module with the action

(4.5) $((u,v)\cdot\varphi)(w)=u\cdot\varphi(v^{-1}\cdot w)\text{ for all
}u,v\in B_{n}\text{, }\varphi\in\mathrm{End}_{\mathbb{C}}(W),w\in W.$

We also have that $\mathrm{End}_{\mathbb{C}}(W)$ is isomorphic to $W\otimes
W^{*}$ as $B_{n}\times B_{n}$ modules, where $W^{*}$ is the dual module of
$W$, via the isomorphism

(4.6) $W\otimes W^{*}\rightarrow\mathrm{End}_{\mathbb{C}}(W),\quad u\otimes
x\mapsto\varphi,\text{ with }\varphi(w)=x(w)\cdot u,$

where $u,w\in W$ and $x\in W^{*}$. Then, since all matrix representations of
$B_{n}$ can be realized over the real numbers, we have that $W$ is isomorphic
to $W^{*}$. So we have isomorphism of $B_{n}\times B_{n}$ modules

(4.7) $\mathrm{End}_{\mathbb{C}}(W)\cong W\otimes W.$

Since the algebra $\mathbb{C}[B_{N}]$ is semi-simple, the Artin-Wedderburn
Theorem gives the isomorphism of $\mathbb{C}$-algebras

(4.8)
$\Psi:\mathbb{C}[B_{n}]\xrightarrow{\,\,\sim\,\,}\bigoplus_{(\lambda,\mu)\vdash_{2}n}\mathrm{End}_{\mathbb{C}}(V^{(\lambda,\mu)}),$

where the $(\lambda,\mu)$-piece of $\Psi(a)$ maps $v$ to $a\cdot v$ for
$a\in\mathbb{C}[B_{n}]$, $v\in V^{(\lambda,\mu)}$. Since $\mathbb{C}[B_{n}]$
is a $B_{n}\times B_{n}$ module via left and right action, (4.7) and (4.8)
imply that we have the isomorphism of ungraded $B_{n}\times B_{n}$ modules

(4.9)
$\mathbb{C}[B_{n}]\cong\bigoplus_{(\lambda,\mu)\vdash_{2}n}\mathrm{End}_{\mathbb{C}}(V^{(\lambda,\mu)})\cong\bigoplus_{(\lambda,\mu)\vdash_{2}n}V^{(\lambda,\mu)}\otimes
V^{(\lambda,\mu)}.$

Returning to the statement of the theorem, by an inductive argument, it
suffices to prove that for any $k$, we have the isomorphism of ungraded
$B_{n}\times B_{n}$ modules:

(4.10) $(\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}})_{\leq
k}\cong\bigoplus_{\begin{subarray}{c}(\lambda,\mu)\vdash_{2}n\\\
2\lambda_{1}+|\mu|\geq
2n-k\end{subarray}}\mathrm{End}_{\mathbb{C}}(V^{(\lambda,\mu)}).$

With Theorem 3.11, we have a chain of isomorphisms and an equality of ungraded
$B_{n}\times B_{n}$ modules:

(4.11) $\mathbb{C}[B_{n}]\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}(B_{n})\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(B_{n})=\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}}.$

Then, consider the image of $\mathbb{C}[\mathbf{x}_{n\times n}]_{\leq k}$ in
$\mathbb{C}[\mathbf{x}_{n\times n}]/\mathbf{I}(B_{n})$, denoted by $L_{k}$,
i.e.

(4.12) $L_{k}=\mathrm{Image}(\mathbb{C}[\mathbf{x}_{n\times n}]_{\leq
k}\hookrightarrow\mathbb{C}[\mathbf{x}_{n\times
n}]\twoheadrightarrow\mathbb{C}[\mathbf{x}_{n\times n}]/\mathbf{I}(B_{n}))$

By Lemma 3.10 and Lemma 4.1, we have that

(4.13)
$L_{k}=\mathrm{span}_{\mathbb{C}}\\{m(\mathcal{R}_{1})m(\mathcal{R}_{2})^{2}+\mathbf{I}(B_{n}):\mathcal{R}_{1}\cup\mathcal{R}_{2}\text{
is a rook placement and }|\mathcal{R}_{1}|+2|\mathcal{R}_{2}|\leq k\\}.$

Then, Lemma 4.1 implies that we have the isomorphism of $B_{n}\times B_{n}$
modules

(4.14) $L_{k}\cong(\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}})_{\leq k}.$

So we are reduced to showing that

(4.15) $L_{k}\cong\bigoplus_{\begin{subarray}{c}(\lambda,\mu)\vdash_{2}n\\\
2\lambda_{1}+|\mu|\geq
2n-k\end{subarray}}\mathrm{End}_{\mathbb{C}}(V^{(\lambda,\mu)}).$

For any $j\leq k$, consider the collection of monomials

(4.16) $M^{j}=\left\\{m_{i}^{j}=x_{1,1}^{2}x_{2,2}^{2}\cdots
x_{i,i}^{2}x_{i+1,i+1}x_{i+2,i+2}\cdots x_{j-i,j-i}:0\leq i\leq\lfloor
j/2\rfloor\right\\}.$

For example, when $j=4$, the collection of monomials is

(4.17)
$M^{j}=\left\\{x_{1,1}^{2}x_{2,2}^{2},x_{1,1}^{2}x_{2,2}x_{3,3},x_{1,1}x_{2,2}x_{3,3}x_{4,4}\right\\}.$

Since nonzero monomials in $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$ are
cube-free, the degree $j$ monomials in $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}}$ are generated by the action of $B_{n}\times B_{n}$ on $M^{j}$.
Thus, $L_{k}$ is generated by the action of $B_{n}\times B_{n}$ on
$\bigcup_{j\leq k}M^{j}$.

Then, fix $0\leq j\leq k$, for all $i$ satisfying $0\leq i\leq\lfloor
j/2\rfloor$, embed $B_{n-j+i}\subseteq B_{n}$ by letting it act on the last
$n-j+i$ letters. Let $\epsilon_{i}^{j}$ be the group algebra element

(4.18)
$\epsilon_{i}^{j}=(e+t_{1,-1})(e+t_{2,-2})\cdots(e+t_{i,-i})(e-t_{i+1,-i-1})\cdots(e-t_{j-i,-j+i})\left(\sum_{w\in
B_{n-j+i}}w\right),$

where $e$ denotes the identity element and $t_{l,-l}$ denotes the
transposition in $B_{n}$ that interchanges $l$ and $-l$, and fixing all other
elements. Identifying $\mathbb{C}[\mathbf{x}_{n\times n}]$ with the collection
of functions from $B_{n}$ to $\mathbb{C}$, via the rule

(4.19) $\sum_{w\in B_{n}}a_{w}w\mapsto f:f(w)=a_{w},$

and since $\mathbb{C}[\mathbf{x}_{n\times n}]/\mathbf{I}(B_{n})$ is naturally
identified with functions from $B_{n}$ to $\mathbb{C}$, we can identify
$\epsilon_{i}^{j}$ with $m_{i}^{j}$. Let $\Sigma^{j}=\\{\epsilon_{i}^{j}:0\leq
i\leq\lfloor j/2\rfloor\\}$, and let $J_{k}$ be the two-sided ideal generated
by $\bigcup_{0\leq j\leq k}\Sigma^{j}$. We have the identification

(4.20) $J_{k}=L_{k}.$

So we are now reduced to find the image of $J_{k}$ under the Artin-Wedderburn
map in (4.8). Note that the image of $J_{k}$ would be direct sum of
endomorphisms of irreducible modules such that not all elements of $J_{k}$ act
as the $0$ operator. So fix $j\leq k$, we need to find the irreducible modules
such that not all elements of $\Sigma_{j}$ act as $0$.

Since $\epsilon_{i}^{j}$ is naturally an element of the group algebra
$\mathbb{C}[B_{j-i}\times B_{n-j+i}]$, given $V^{(\lambda,\mu)}$ an
irreducible character of $B_{n}$ labelled by bipartition $(\lambda,\mu)$, it
suffices to analyze the character of

(4.21) $\mathrm{Res}^{B_{n}}_{B_{j-i}\times B_{n-j+i}}V^{(\lambda,\mu)},$

evaluated on $\epsilon_{i}^{j}$. By the Branching Rule (2.12),

(4.22) $\mathrm{Res}^{B_{n}}_{B_{j-i}\times
B_{n-j+i}}\chi^{(\lambda,\mu)}=\sum_{(\tilde{\lambda},\tilde{\mu})}\chi^{(\tilde{\lambda},\tilde{\mu})}\times\chi^{(\lambda/\tilde{\lambda},\mu/\tilde{\mu})},$

where the sum is over all $(\tilde{\lambda},\tilde{\mu})$ that are bipartiions
of $j-i$ and that $(\lambda/\tilde{\lambda},\mu/\tilde{\lambda})$ is well-
defined. Then, denote

(4.23)
$\eta_{i}^{j}=(e+t_{1,-1})(e+t_{2,-2})\dots(e+t_{i,-i})(e-t_{i+1,-i-1})\dots(e-t_{j-i,-j+i}),$

that is, $\epsilon_{i}^{j}=\eta_{i}^{j}\sum_{w\in B_{n-k+i}}w$, we need to
analyze

(4.24)
$\sum_{(\tilde{\lambda},\tilde{\mu})}\chi^{(\tilde{\lambda},\tilde{\mu})}(\eta_{i}^{j})\chi^{(\lambda/\tilde{\lambda},\mu/\tilde{\mu})}\left(\sum_{w\in
B_{n-j+i}}w\right).$

Let $\theta$ be an irreducible character of $\mathbb{Z}/2\mathbb{Z}\cong
B_{1}$, we have that $\theta(e+t_{l,-l})\neq 0$ if and only if $\theta$ is the
trivial character; on the other hand, $\theta(e-t_{l,-l})\neq 0$ if and only
if $\theta$ is the signed character, i.e. $\theta(t_{l,l})=-1$. By the
Generalized Murnaghan-Nakayama Rule (2.15), we have that
$\chi^{(\tilde{\lambda},\tilde{\mu})}(\eta_{i}^{j})$ is nonzero if and only if
$|\tilde{\lambda}|=i$ and $|\tilde{\mu}|=j-2i$. On the other hand,

(4.25) $\chi^{(\lambda/\tilde{\lambda},\mu/\tilde{\mu})}\left(\sum_{w\in
B_{n-j+i}}w\right)$

is nonzero if and only if the trivial representation
$\chi^{((n-j+i),\emptyset)}$ has non-zero multiplicity in
$\chi^{(\lambda/\tilde{\lambda},\mu/\tilde{\mu})}$. This forces
$\mu/\tilde{\mu}=\emptyset$, and the diagram of $\lambda/\tilde{\lambda}$ has
no two boxes in the same row. Combining this, we have that $\epsilon_{i}^{j}$
acts as a nonzero operator on $V^{(\lambda,\mu)}$ if and only if the following
holds:

  * •

$|\mu|=j-2i$, thus $|\lambda|=n-j+2i$;

  * •

there exists $\tilde{\lambda}\vdash i$, such that the diagram of
$\lambda/\tilde{\lambda}$ has no two boxes in the same column.

Note that the second condition is equivalent to

  * •

the diagram of $\lambda$ has less than $i$ boxes beyond the first row, i.e.
$\lambda_{1}\geq|\lambda|-i=n-j+i$.

Since the first condition is an equality, the two condition can be combined to

(4.26) $2\lambda_{1}+|\mu|\geq 2n-j.$

Since $j\leq k$ is arbitrary, we can conclude that elements of $J_{k}$ act as
nonzero operators on all $V^{(\lambda,\mu)}$ satisfying

(4.27) $2\lambda_{1}+|\mu|\geq 2n-k.$

This gives the isomorphism of $B_{n}\times B_{n}$ modules

(4.28) $J_{k}\cong\bigoplus_{\begin{subarray}{c}(\lambda,\mu)\vdash_{2}n\\\
2\lambda_{1}+|\mu|\geq
2n-k\end{subarray}}\mathrm{End}_{\mathbb{C}}(V^{(\lambda,\mu)}),$

which completes the proof. ∎

###  4.2. Graded Module Structure of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$

We now investigate the graded module structure of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$ for general $r$.
Given $\boldsymbol{\lambda}=(\lambda^{0},\lambda^{1},\dots,\lambda^{r-1})$ a
$r$-partition of $n$, let
$\boldsymbol{\lambda}=(\lambda^{0},\lambda^{r-1},\lambda^{r-2},\dots\lambda^{1})$
be the dual $r$-partition of $\boldsymbol{\lambda}$, obtained by fixing
$\lambda^{0}$ and interchanging $\lambda^{i}$ with $\lambda^{r-i}$. As the
dual module of a unitary matrix representation is its conjugate, then by the
general Murnaghan-Nakayama (2.15), the dual module of
$V^{\boldsymbol{\lambda}}$ is then $V^{\boldsymbol{\lambda}^{\prime}}$, and we
have the following theorem:

######  Theorem 4.3.

For any $k\geq 0$, the degree $k$ piece $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{k}$ has graded
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ structure

(4.29) $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{k}\cong\bigoplus_{\begin{subarray}{c}\boldsymbol{\lambda}\vdash_{r}n\\\
r\lambda^{0}_{1}+\sum_{i=1}^{r-1}i|\lambda^{i}|=rn-k\end{subarray}}V^{\boldsymbol{\lambda}}\otimes
V^{\boldsymbol{\lambda}^{\prime}}$

###### Proof.

Similar to the proof of Theorem 4.2, we have that for any
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ module $W$,

(4.30) $\mathrm{End}_{\mathbb{C}}(W)\cong W\otimes W^{*}.$

Then, similar to (4.9), we have the isomorphisms of
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ modules

(4.31)
$\mathbb{C}[\mathfrak{S}_{n,r}]\cong\bigoplus_{\boldsymbol{\lambda}\vdash_{r}n}\mathrm{End}_{\mathbb{C}}(V^{\boldsymbol{\lambda}})\cong\bigoplus_{\boldsymbol{\lambda}\vdash_{r}n}V^{\boldsymbol{\lambda}}\otimes(V^{\boldsymbol{\lambda}})^{*}\cong\bigoplus_{\boldsymbol{\lambda}\vdash_{r}n}V^{\boldsymbol{\lambda}}\otimes
V^{\boldsymbol{\lambda}^{\prime}}.$

By Theorem 3.19, we have a chain of isomorphisms and an equality of ungraded
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ modules:

(4.32) $\mathbb{C}[\mathfrak{S}_{n,r}]\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}(\mathfrak{S}_{n,r})\cong\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r})=\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}.$

Again, let $L_{k}$ be the image of $\mathbb{C}[\mathbf{x}_{n\times n}]_{\leq
k}$ to $\mathbb{C}[\mathbf{x}_{n\times n}]/\mathbf{I}(\mathfrak{S}_{n,r})$,
that is,

(4.33) $L_{k}=\mathrm{Image}(\mathbb{C}[\mathbf{x}_{n\times n}]_{\leq
k}\hookrightarrow\mathbb{C}[\mathbf{x}_{n\times
n}]\twoheadrightarrow\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}(\mathfrak{S}_{n,r})).$

By Lemma 4.1, we have that

(4.34)
$~{}L_{k}=\mathrm{span}_{\mathbb{C}}\left\\{\left(\prod_{i=1}^{r}m(\mathcal{R}_{i})^{i}\right)+\mathbf{I}(\mathfrak{S}_{n,r}):\bigcup_{i=1}^{r}\mathcal{R}_{i}\text{
is a rook placement and }\sum_{i=1}^{r}i\cdot|\mathcal{R}_{i}|\leq
k\right\\}.$

Applying Lemma 4.1 again, we have the isomorphism of
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$ modules

(4.35) $L_{k}\cong(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{\leq k}.$

And we are reduced to showing that

(4.36)
$L_{k}\cong\bigoplus_{\begin{subarray}{c}\boldsymbol{\lambda}\vdash_{r}n\\\
r\lambda^{0}_{1}+\sum_{i=1}^{r-1}i|\lambda^{i}|\geq
rn-k\end{subarray}}\mathrm{End}_{\mathbb{C}}(V^{\boldsymbol{\lambda}}).$

Let $M_{k}$ be the subset of the spanning set in (4.34), containing all
monomials

(4.37) $\prod_{i=1}^{r}m(\mathcal{R}_{i})^{i}$

where the rook placements $\\{\mathcal{R}_{i}\\}_{i=1}^{r}$ satisfy the
criteria in (4.34), and for all $j\leq r$,

(4.38) $~{}\bigcup_{i=1}^{j}\mathcal{R}_{i}=\\{(a,a):1\leq
a\leq\sum_{i=1}^{j}|\mathcal{R}_{i}|\\}.$

In other words, in the $n\times n$ grid, the rook placements
$\\{\mathcal{R}_{i}\\}$ are arranged on the diagonals, from bottom left to top
right. For example, when $k=5$ and $r=3$, we have

(4.39)
$M_{k}=\\{x_{1,1}x_{2,2}x_{3,3}x_{4,4}x_{5,5},x_{1,1}x_{2,2}x_{3,3}x_{4,4}^{2},x_{1,1}x_{2,2}x_{3,3}^{3},x_{1,1}x_{2,2}^{2}x_{3,3}^{2},x_{1,1}^{2}x_{2,2}^{3}\\}.$

Monomials in $L_{k}$ are generated by monomials in $M_{k}$ by two sided action
of $\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$.

Again, identifying $\mathbb{C}[\mathfrak{S}_{n,r}]$ and
$\mathbb{C}[\mathbf{x}_{n\times n}]/\mathbf{I}_{\mathfrak{S}_{n,r}}$ with
functions from $\mathfrak{S}_{n,r}$ to $\mathbb{C}$, we can identify
$\mathbf{x}_{i,i}^{j}$ in $\mathbb{C}[\mathbf{x}_{n\times
n}]/\mathbf{I}_{\mathfrak{S}_{n,r}}$ with the group algebra element

(4.40)
$(e+\omega^{j}(i^{0},i^{1})+\omega^{2j}(i^{0},i^{2})+\cdots+\omega^{(r-1)j}(i^{0},i^{r-1}))\sum_{w\in\mathfrak{S}_{n-1,r}}w,$

where $\mathfrak{S}_{n-1,r}$ is embedded in $\mathfrak{S}_{n,r}$ by acting on
the last $n-1$ letters, $e$ denotes the identity in $\mathfrak{S}_{n,r}$,
$\omega$ is the primitive $r$-th root of unity in $\mathbb{C}$, and
$(i^{0},i^{l})$ denotes the element in $\mathfrak{S}_{n,r}$ that maps $i^{0}$
to $i^{l}$, while fixing all other elements. Let $\Sigma_{k}$ be the
collection of group algebra elements identified with monomials in $M_{k}$, and
let $J_{k}$ be the two-sided ideal generated by $J_{k}$. We have
identification

(4.41) $L_{k}=J_{k}.$

So we are reduced to finding $r$-partitions $\boldsymbol{\lambda}$ such that
not all elements of $J_{k}$ acts as zero operator. Denote $\epsilon_{i}^{j}$
as

(4.42)
$\epsilon_{i}^{j}=e+\omega^{j}(i^{0},i^{1})+\omega^{2j}(i^{0},i^{2})+\cdots+\omega^{(r-1)j}(i^{0},i^{r-1}),$

For rook placements $\\{\mathcal{R}_{j}\\}_{j=1}^{r}$ satisfying the criteria
in (4.34) and (4.38), let $\mathcal{R}=\bigcup_{j=1}^{r}\mathcal{R}_{j}$, we
have that the group algebra element that is identified with
$\prod_{j=1}^{r}m(\mathcal{R}_{j})^{i}$ is

(4.43)
$\epsilon=\left(\prod_{j=1}^{r}\prod_{(i,i)\in\mathcal{R}_{j}}\epsilon_{i}^{j}\right)\sum_{w\in\mathfrak{S}_{n-|\mathcal{R}|,r}}w.$

Let $\eta=\prod_{j=1}^{r}\prod_{(i,i)\in\mathcal{R}_{j}}\epsilon_{i}^{j}$, we
can embed $\mathfrak{S}_{n-|\mathcal{R}|,r}$ in $\mathfrak{S}_{n,r}$ by
letting it act on the last $n-|\mathcal{R}|$ letters. Then, by the Branching
Rule (2.12), we have

(4.44)
$\sum_{\boldsymbol{\mu}\vdash_{r}|\mathcal{R}|}\chi^{\boldsymbol{\mu}}(\eta)\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}\left(\sum_{w\in\mathfrak{S}_{n-|\mathcal{R}|,r}}w\right).$

Let $\theta$ be any irreducible $\mathbb{C}$-character of
$\mathfrak{S}_{1,r}\cong\mathbb{Z}/r\mathbb{Z}$, note that we have
$\theta(\epsilon^{j}_{i})\neq 0$ if and only if
$\theta((i^{0},i^{1}))=\omega^{-j}$. Thus, by the generalized Murnaghan-
Nakayama Rule (2.15), we have that for
$\boldsymbol{\mu}=(\mu^{0},\mu^{1},\dots\mu^{r-1})\vdash_{r}|\mathcal{R}|$,
$\chi^{\boldsymbol{\mu}}(\eta)$ is nonzero if and only if
$|\mu^{j}|=|\mathcal{R}_{r-j}|$. On the other hand,

(4.45)
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}\left(\sum_{w\in\mathfrak{S}_{n-|\mathcal{R}|,r}}w\right)$

is nonzero if and only if the trivial character has nonzero multiplicity in
$\chi^{\boldsymbol{\lambda}/\boldsymbol{\mu}}$, which indicates that
$|\mu^{j}|=|\lambda^{j}|$ for $1\leq j\leq r-1$, and there are no two boxes in
the same column in the diagram of $\lambda^{0}/\mu^{0}$, that is, there are
less than $|\mu^{0}|=|\mathcal{R}_{r}|$ boxes beyond the first row in the
diagram of $\lambda^{0}$.

Let $d=\sum_{j=1}^{r}j\cdot|\mathcal{R}_{j}|$ be the degree of the monomial
$\prod_{j=1}^{r}m(\mathcal{R}_{j})^{i}$. combining these conditions, we have
that $\epsilon$ acts as a nonzero operator on $V^{\boldsymbol{\lambda}}$ if
there exists $\boldsymbol{\mu}=(\mu^{0},\mu^{1},\dots,\mu^{r-1})$ such that
the following holds:

  1. (1)

$r|\mu^{0}|+(r-1)|\mu^{1}|+\dots+|\mu^{r-1}|=d$;

  2. (2)

$\lambda^{j}=\mu^{j}$ for $j\neq 0$;

  3. (3)

diagram of $\lambda^{0}/\mu^{0}$ has no two boxes in the same column.

Note that the third condition is equivalent to

(4.46)
$\lambda^{0}_{1}\geq|\lambda^{0}|-|\mu^{0}|=n-|\lambda^{1}|-|\lambda^{2}|-\dots-|\lambda^{r-1}|-|\mu^{0}|.$

Multiplying both side by $r$, we get

(4.47) $r\lambda^{1}_{0}\geq
rn-r|\lambda^{1}|-r|\lambda^{2}|-\dots-r|\lambda^{r-1}|-r|\mu^{0}|.$

Then, the first and second condition combined gives that

(4.48)
$r|\mu^{0}|=d-(r-1)|\lambda^{1}|-(r-2)|\lambda^{2}|-\dots-|\lambda^{r-1}|.$

Substituting this into the inequality (4.47), we have

(4.49)
$r\lambda^{0}_{1}+|\lambda^{1}|+2|\lambda^{2}|+\dots+(r-1)|\lambda^{r-1}|\geq
rn-d.$

This shows the isomorphism of $\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$
modules

(4.50)
$J_{k}\cong\bigoplus_{\begin{subarray}{c}\boldsymbol{\lambda}\vdash_{r}n\\\
r\lambda^{0}_{1}+\sum_{i=1}^{r-1}i|\lambda^{i}|\geq
rn-k\end{subarray}}\mathrm{End}_{\mathbb{C}}(V^{\boldsymbol{\lambda}}),$

which completes the proof.

∎

##  5\. Conclusions and Conjectures

Given a (possibly infinite) sequence ${a_{i}}$ of positive real numbers, we
say that the sequence is log-concave if for all positive integer $i$, we have

(5.1) $a_{i}\cdot a_{i+2}\leq a_{i+1}^{2};$

and we say the sequence is uni-modal if there exists a positive integer $k$
such that

(5.2) $a_{i}\leq a+_{i+1}\text{ for }i<k;\quad a_{i}\geq a_{i+1}\text{ for
}i\geq k.$

In the case of Hilbert series, we say a Hilbert series is log-concave or uni-
modal if its sequence of coefficients is log-concave or uni-modal. Recall that
the Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n}$ is

(5.3) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}/I_{n};q)=a_{n,n}+a_{n,n-1}\cdot q+a_{n,n-2}\cdot q^{2}+\dots+a_{n,1}\cdot
q^{n-1},$

where $a_{n},k$ counts the number of permutations $w\in\mathfrak{S}_{n}$ with
$\mathrm{lis}(w)=k$. Chen conjectured [2] that the sequence $\\{a_{n,k}\\}$ is
log-concave, that is, fixing an integer $n$, for all $k$ such that $2\leq
k\leq n-1$,

(5.4) $a_{n,k}^{2}\geq a_{n,k-1}\cdot a_{n,k+1}.$

We found some interesting patterns when we tried to generalize this to the
Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$. Recall that
we showed in Corollary 3.15 that the Hilbert series of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$ is

(5.5) ${\mathrm{Hilb}}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{B_{n}};q)=b_{n,2n}+b_{n,2n-1}\cdot q+\cdots+b_{n,1}\cdot q^{2n-1},$

where $\\{b_{n,k}\\}_{k=1}^{2n-2}$ is defined in Definition 3.14. By testing
$n$ up to $40$, we found that

  * •

for $n\leq 8$, the sequence $\\{b_{n,k}\\}$ is log-concave;

  * •

for $9\leq n\leq 17$, we have $b_{n,k}^{2}<b_{n,k-1}\cdot b_{n+1}$ when $k=3$,
and log-concavity holds for the remaining of the sequence;

  * •

for $18\leq n\leq 27$, we have $b_{n,k}^{2}<b_{n,k-1}\cdot b_{n+1}$ when
$k=3\text{ and }5$, and log-concavity holds for the remaining of the sequence;

  * •

for $27\leq n\leq 40$, we have $b_{n,k}^{2}<b_{n,k-1}\cdot b_{n+1}$ when
$k=3,5,\text{ and }7$, and log-concavity holds for the remaining of the
sequence.

From this observation, we conclude that the Hilbert series of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{B_{n}}$ is “almost log-concave”: log-
concavity breaks for some small $k$’s, but holds for the majority of the
sequence.

In [1], Baik, Deift, and Johansson proved that as $n\rightarrow\infty$, the
distribution function for $\mathrm{lis}$ on $\mathfrak{S}_{n}$, with proper
rescaling and recentering, converges to the Tracey-Widom distribution of the
largest eigenvalue of a random GUE matrix. That is, as $n\rightarrow\infty$,
the histogram of $\\{a_{n,k}\\}$ for a fixed $n$ will converge in distribution
(with rescaling and recentering) to the Tracey-Widom distribution. In Figure
1, we give the histogram of $\\{a_{n,k}\\}$ when $=65$. The horizontal axis
corresponds to values of $k$, and the height of each bar represents the
corresponding $a_{n,k}$.

Figure 1. $a_{n,k}$ when $n=65$

One possible direction for future studies is to study the limit of the
probability distributions corresponding to $\\{b_{n,k}\\}$ as
$n\rightarrow\infty$. In analogy with the case of $\\{a_{n,k}\\}$, one could
hope that we have convergence to a natural probability distribution. In Figure
2, we give the histogram of $\\{b_{n,k}\\}$ when $n=40$. The horizontal axis
corresponds to values of $k$, and the height of each bar represents the
corresponding $b_{n,k}$.

Figure 2. $b_{n,k}$ when $n=40$

Our testing on unimodality also yields a very nice pattern, and we give the
following conjecture:

######  Conjecture 5.1.

For any integers $n,r$, the Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$ is unimodal, that is, there exists $k$ such that

(5.6) $c_{n,r,d}\leq c_{n,r,d+1}\text{ for }d<k\text{, and }c_{n,r,d}\geq
c_{n,r,d+1}\text{ for }d\geq k.$

Conjecture 5.1 was tested to be true for $r\leq 4$ and $n\leq 20$.

As mentioned in [4], orbit harmonics played a key tool in our analysis. We
treated the colored permutation group $\mathfrak{S}_{n,r}$ as a collection of
$n\times n$ matrices, and proved that
$I_{\mathfrak{S}_{n,r}}=\mathrm{gr}\mathbf{I}(\mathfrak{S}_{n,r})$. We provide
one possible direction for future studies below:

A complex reflection on a finite-dimensional vector space $V$ is an element of
finite order in $\mathrm{GL}(V)$ which fixes a hyperplane pointwise. A complex
reflection group $W$ is a finite group generated by complex reflections.
Shepherd and Todd showed in [7] that any complex reflection group is either
one of the $34$ exceptional cases, or in the form $G(r,p,n)$, where $r,p,n$
are integers with $p$ divides r, and the group $G(r,p,n)$ consists of $n\times
n$ matrices with

  * •

exactly one nonzero entry in each row or column;

  * •

each nonzero entry is a $r$-th root of unity;

  * •

the product of all the nonzero-entries is a $r/p$-th root of unity.

When $p=1$, $G(r,1,n)$ is the colored permutation group $\mathfrak{S}_{n,r}$,
and we calculated $\mathrm{gr}\mathbf{I}(G(r,1,n))=I_{\mathfrak{S}_{n,r}}$ in
this paper. It may be interesting to study $\mathrm{gr}\mathbf{I}(G(r,p,n))$
for different $p$’s that divides $r$. For example, $G(2,2,n)$ coincides with
the Coexter group of type $D$, and it would be a good object for future study.

Another possible direction is on the equivariant log-concavity of
$\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$. In [4], Rhoades
conjectured that with $\mathfrak{S}_{n}\times\mathfrak{S}_{n}$ acting on
$(\mathbb{C}[\mathbf{x}_{n\times n}])_{d}$ by independent row and column
operation, there exists an injection

(5.7) $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{n})_{d+1}\otimes(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{n})_{d-1}\xhookrightarrow{\phi}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{n})_{d}\otimes(\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n})_{d}$

such that given $(w,v)\in\mathfrak{S}_{n}\times\mathfrak{S}_{n}$,
$f\in(\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n})_{d+1}$,
$g\in(\mathbb{C}[\mathbf{x}_{n\times n}]/I_{n})_{d-1}$,

(5.8) $\phi\left((w,v)\cdot(f\otimes g)\right)=(w,v)\cdot\left(\phi(f\otimes
g)\right).$

This conjecture would imply both Chen’s Conjecture [2] and the Novak-Rhoades
Conjecture [3]. As the Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}}$ is not always log-concave, it would be interesting
to study given $\mathfrak{S}_{n,r}$, for which $d$’s we can have an injection

(5.9) $(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{d+1}\otimes(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{d-1}\xhookrightarrow{}(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{d}\otimes(\mathbb{C}[\mathbf{x}_{n\times
n}]/I_{\mathfrak{S}_{n,r}})_{d}$

that commutes with the diagonal action of
$\mathfrak{S}_{n,r}\times\mathfrak{S}_{n,r}$. Our calculation shows that when
$n=1$, such an injection exists for any $r,d$.

##  6\. Acknowledgements

The author is very grateful to Brendon Rhoades for proof-reading this paper
and providing a lot of useful suggestions. The author is very grateful to
Brendon Rhoades for helpful conversations on Viennot shadow and orbit
harmonics.

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