commutators $[x_{i},x_{j}]_{+}$. Consequently, $J_{2}=I_{2}^{\perp}$ lies in
the perp space of the span of all such commutators, which is the
$\mathbf{k}$-span of all anti-commutators $[y_{i},y_{j}]_{-}$, allowing $i=j$
here.

Claim: every anti-commutator $[y_{i},y_{j}]_{-}$ is primitive, meaning
$\Delta[y_{i},y_{j}]_{-}=1\otimes[y_{i},y_{j}]_{-}+[y_{i},y_{j}]_{-}\otimes
1$.

Assuming the claim, every $j\in J_{2}$ is also primitive, so $\Delta
j=1\otimes j+j\otimes 1\in H\otimes J+J\otimes H$, as desired. Checking the
claim is a standard calculation: when $a=y_{i},$ and $b=y_{j}$ are both
primitive, and of odd degree, then their anti-commutator is also primitive:

$\displaystyle\Delta[a,b]_{-}=\Delta(ab+ba)$ $\displaystyle=(1\otimes
a+a\otimes 1)(1\otimes b+b\otimes 1)+(1\otimes b+b\otimes 1)(1\otimes
a+a\otimes 1)$ $\displaystyle=1\otimes ab-b\otimes a+a\otimes b+ab\otimes
1+1\otimes ba-a\otimes b+b\otimes a+ba\otimes 1$ $\displaystyle=1\otimes
ab+ab\otimes 1+1\otimes ba+ba\otimes 1$
$\displaystyle=1\otimes(ab+ba)+(ab+ba)\otimes
1=1\otimes[a,b]_{-}+[a,b]_{-}\otimes 1.\qed$

######  Remark 8.2.

If a quadratic algebra $A$ is neither commutative nor anti-commutative, then
$A^{!}$ might not inherit a Hopf algebra structure from the tensor algebra.
Consider the quadratic algebra

$A^{!}=\mathbb{k}[x,y]/(x^{2}+y^{2})=\mathbf{k}\langle
x,y\rangle/(x^{2}+y^{2},xy-yx).$

This is the quadratic dual of $A=\mathbf{k}\langle
x,y\rangle/(xy+yx,y^{2}-x^{2})$, which is neither commutative nor anti-
commutative. Notice that if the characteristic of $\mathbf{k}$ is not equal to
$2$, the ideal $J=(x^{2}+y^{2},xy-yx)$ is not a co-ideal for the coproduct
$\Delta$ on the tensor algebra $H=\mathbf{k}\langle x,y\rangle$:

$\Delta(x^{2}+y^{2})=(x^{2}+y^{2})\otimes 1+2(x\otimes x+y\otimes
y)+1\otimes(x^{2}+y^{2}),$

which one can check does not lie in $H\otimes J+J\otimes H$.

When $A=\bigoplus_{d=0}^{\infty}A_{d}$ is an associative standard graded
$\mathbf{k}$-algebra, so generated by $A_{1}$, and is either commutative or
anti-commutative, the Yoneda algebra
$\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k})$ has a natural
coproduct giving it the structure of a graded Hopf algebra. Therefore,
$\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k})$ is also the
universal enveloping algebra of a graded Lie (super-)algebra. See Avramov
[Avramov, §10.1] for a discussion when $A$ is commutative, and Denham and
Suciu [DenhamSuciu, §1] for the case where $A$ is anti-commutative.

######  Definition 8.3.

In the above setting, the homotopy Lie algebra $\pi_{A}$ is the graded Lie
algebra or Lie superalgebra of primitive elements in the Yoneda algebra
$\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k})$ of $A$, that is,

$\mathcal{U}(\pi_{A})\cong\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k}).$

###  8.1. The holonomy Lie algebra

Let $A=\bigoplus_{d=0}^{\infty}A_{d}$ be an associative graded
$\mathbf{k}$-algebra, with $\mathbb{k}$-basis $x_{1},\dots,x_{n}$ for
$V=A_{1}$; for the moment we do not assume that $A$ is generated by $A_{1}$.
Then the decomposable elements of $A_{2}$ are defined to be those in the image
of the multiplication map

$\phi:A_{1}\otimes A_{1}\to A_{2}.$

Letting $V=A_{1}^{*}$ have dual basis $y_{1},\ldots,y_{n}$, if one considers
the dual of this multiplication map

$\phi^{\ast}:A_{2}^{\ast}\to(A_{1}\otimes A_{1})^{\ast}\cong
A_{1}^{\ast}\otimes A_{1}^{\ast},$

then one has these identifications:

$\displaystyle\operatorname{im}(\phi^{\ast})$
$\displaystyle\cong\left(\frac{A_{1}\otimes
A_{1}}{\ker\phi}\right)^{\ast}\cong\\{f\in A_{1}^{\ast}\otimes
A_{1}^{\ast}:f(\ker\phi)=0\\},$ (68)
$\displaystyle\cong\left(\frac{T^{2}(V)}{I_{2}}\right)^{\ast}\cong
I_{2}^{\perp}=:J_{2}.$

Here we consider $J_{2}=I_{2}^{\perp}$ as a subspace of
$T^{2}(V^{*})=V^{*}\otimes V^{*}$, with pairing $T^{2}(V^{*})\times
T^{2}(V)\rightarrow\mathbf{k}$ just as in (8). Now just as the proof of
Proposition 8.1, if one further assumes that $A$ is commutative (resp. anti-
commutative), then $I_{2}$ contains the $\mathbf{k}$-span of all commutators
$[x_{i},x_{j}]_{+}$ (resp. anti-commutators $[x_{i},x_{j}]_{-}$).
Consequently, $J_{2}=I_{2}^{\perp}$ lies in the perp space of the span of all
such commutators or anti-commutators, which is the $\mathbf{k}$-span of all
anti-commutators $[y_{i},y_{j}]_{-}$, allowing $i=j$ (resp. all commutators
$[y_{i},y_{j}]_{+})$. In other words, $\operatorname{im}(\phi^{*})=J_{2}$ is
identified with a subspace of $[A_{1}^{*},A_{1}^{*}]_{-}$ or
$[A_{1}^{*},A_{1}^{*}]_{+}$ inside

$\mathrm{Lie}(V^{*})=\mathrm{Lie}(A_{1}^{*})=\mathrm{Lie}(y_{1},\ldots,y_{n})\subset
T^{*}(A_{1}^{*})$

where $\mathrm{Lie}(y_{1},\ldots,y_{n})$ denotes the free Lie algebra (resp.
free Lie superalgebra) on $y_{1},\ldots,y_{n}$ when $A$ is anti-commutative
(resp. commutative).

######  Definition 8.4.

In the above context of an associative graded $\mathbf{k}$-algebra $A$ which
is either commutative or anti-commutative, define the holonomy Lie algebra
$\mathfrak{h}_{A}$ via the quotient

$\mathfrak{h}_{A}=\textrm{Lie}(A_{1}^{\ast})/\langle\operatorname{im}(\phi^{\ast})\rangle=\textrm{Lie}(y_{1},\ldots,y_{n})/\langle
J_{2}\rangle.$ (69)

Here $\mathrm{Lie}(A_{1}^{\ast})$ is the free Lie algebra (resp. free Lie
superalgebra) on the $\mathbf{k}$-basis $y_{1},\dots,y_{n}$ for $V^{*}$ if $A$
is anti-commutative (resp. commutative), and $\langle J_{2}\rangle=\langle
I_{2}^{\perp}\rangle$ is the Lie ideal generated by $J_{2}=I_{2}^{\perp}$.

The following result of Löfwall connects the holonomy and homotopy Lie
algebras.

######  Lemma 8.5.

[lofwall86]*Theorem 1.1 The universal enveloping algebra
$\mathcal{U}(\mathfrak{h}_{A})$ of the holonomy Lie algebra $\mathfrak{h}_{A}$
equals the linear strand of the Yoneda algebra
$\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k})$. That is,

$\displaystyle\mathcal{U}(\mathfrak{h}_{A})$
$\displaystyle\cong\bigoplus_{i\geq
0}\operatorname{Ext}_{A}^{i}(\mathbf{k},\mathbf{k})_{i}$
$\displaystyle\subset\bigoplus_{i,j\geq
0}\operatorname{Ext}_{A}^{i}(\mathbf{k},\mathbf{k})_{j}=\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k}).$

In particular, if $A$ is a Koszul algebra, so
$A^{!}=\operatorname{Ext}^{\bullet}_{A}(\mathbf{k},\mathbf{k})$ is equal to
its own linear strand, one has

$A^{!}=\mathcal{U}(\mathcal{L})\text{ where
}\mathcal{L}={\mathfrak{h}}_{A}=\pi_{A}.$

We next give a simple presentation for the holonomy Lie algebra
${\mathfrak{h}}_{A}$ when $A=\mathrm{OS}(M)$ or $A=\mathrm{VG}(\mathcal{M})$.
In the case of $\mathrm{OS}(M)$, this is a well-known result of Kohno
[kohno1983holonomy], but as far as the authors are aware, for
$\mathrm{VG}(\mathcal{M})$ the presentation is new.

######  Theorem 8.6.

The holonomy Lie algebra of $\mathrm{OS}(M)$ (resp.
$\mathrm{VG}(\mathcal{M})$) for any simple (oriented) matroid $M$ (resp.
$\mathcal{M}$) is generated by the relations (31) (resp. (32).

###### Proof.

We give a proof for $\mathrm{VG}(\mathcal{M})$ analogous to Löfwall’s proof
[lofwall-matroid] of Kohno’s result for $\mathrm{OS}(M)$. By Equation 68, we
can identify $\operatorname{im}(\phi^{\ast})$ with
$J_{2}:=I_{2}^{\perp}\subset\mathbf{k}\langle\mathbf{y}\rangle$, where
$A=\mathbf{k}\langle x_{1},\ldots,x_{n}\rangle/I$. There are three families of
quadratic relations in the ideal $I$ presenting $\mathrm{VG}(\mathcal{M})$ to
consider:

$\displaystyle x_{k}^{2}$ $\displaystyle\text{ for }k=1,2,\ldots,n,$ (70)
$\displaystyle x_{k}x_{\ell}-x_{\ell}x_{k}$ $\displaystyle\text{ for }1\leq
k\leq\ell\leq n,$ (71)
$\displaystyle\partial^{\pm}(C):={\mathrm{sgn}_{{C},{m}}}x_{k}x_{\ell}+{\mathrm{sgn}_{{C},{\ell}}}x_{k}x_{m}+{\mathrm{sgn}_{{C},{k}}}x_{\ell}x_{m}$
$\displaystyle\text{ for circuits }C=\\{k,\ell,m\\}\text{ of size three.}$
(72)

Recall from (25) in 5.5 that the pairing $T^{2}(V^{*})\otimes
T^{2}(V)\rightarrow\mathbf{k}$ makes $\mathbf{k}\langle\mathbf{x}\rangle_{F}$
and $\mathbf{k}\langle\mathbf{y}\rangle_{F^{\prime}}$ pair to zero unless
$F=F^{\prime}$, so that one can compute $J_{2}=I_{2}^{\perp}$ flat-by-flat,
obtaining

$[J_{2}\cap\mathbf{k}\langle\mathbf{y}\rangle]_{F}=[I_{2}\cap\mathbf{k}\langle\mathbf{x}\rangle]_{F}^{\perp}$

for all rank $2$ flats $F$ in $\mathcal{F}$. From this one sees that it
suffices to prove the result assuming $\mathcal{M}=\mathcal{M}|_{F}\cong
U_{2,n}$ a uniform rank $2$ matroid on $E=\\{1,2,\ldots,n\\}$ with one rank 2
flat $F=E$.

The quadratic part $I_{2}$ contains these
$n+\binom{n}{2}+\binom{n-1}{2}=n^{2}-n+1$ elements among (70), (71), (72):

  * •

$n$ of the form $x_{i}^{2}$,

  * •

$\binom{n}{2}$ of the form $x_{i}x_{j}-x_{j}x_{i}$ for $i<j$, and

  * •

$\binom{n-1}{2}$ of the form $\partial^{\pm}(C)$ for circuits $C=\\{1,i,j\\}$
with $1<i<j\leq n$.

One can also easily check that they are $\mathbf{k}$-linearly independent
inside $T^{2}(V)$. Consequently one has

$\dim_{\mathbf{k}}J_{2}=\dim_{\mathbf{k}}I_{2}^{\perp}=\dim_{\mathbf{k}}T^{2}(V^{*})-\dim_{\mathbf{k}}I_{2}\leq
n^{2}-(n^{2}-n+1)=n-1.$

On the other hand, the proof of Theorem 5.18 showed each of these elements
from (32) lies in $I_{2}^{\perp}$:

$r^{\pm}(j,E):=\sum_{\begin{subarray}{c}1\leq k\leq n:\\\ k\neq
j\end{subarray}}\chi_{\mathcal{M}|_{F}}(j,k)\cdot[y_{j},y_{k}]_{-}=\sum_{\begin{subarray}{c}1\leq
k\leq n:\\\ k\neq
j\end{subarray}}\chi_{\mathcal{M}|_{F}}(j,k)\cdot(y_{j}y_{k}+y_{k}y_{j}),$

But the subset $\\{r^{\pm}(j,E)\\}_{1\leq j\leq n-1}$ gives $n-1$ such
elements which are linearly independent in $T^{2}(V^{*})$, and therefore they
span $J_{2}=I_{2}^{\perp}$. ∎

###  8.2. PBW decomposition

When the Koszul algebra $A$ is commutative or anti-commutative, we can use
variants of the Poincaré-Birkhoff-Witt (PBW) Theorem to relate the
($G$-equivariant) Hilbert series for $A^{!}=\mathcal{U}(\mathcal{L})$ and that
of the graded Lie algebra
$\mathcal{L}=\bigoplus_{d=0}^{\infty}\mathcal{L}_{d}$.

######  Remark 8.7.

We state the results in this section assuming that the characteristic of
$\mathbf{k}$ is zero; however, these results can be extended to arbitrary
characteristic by replacing the symmetric algebra $\operatorname{Sym}(V)$ or
symmetric powers $\operatorname{Sym}^{k}(V)$ with the divided power algebra
$D(V)$ or a divided power $D^{k}(V)$ in every place it appears.

####  8.2.1. The anti-commutative case

When $A$ is anti-commutative, and $\mathbf{k}$ has characteristic zero, the
PBW Theorem gives a graded $\mathbf{k}$-vector space isomorphism between
$A^{!}=\mathcal{U}(\mathcal{L})\cong\operatorname{Sym}(\mathcal{L})$.
Therefore, we have the Hilbert series relation

$\operatorname{Hilb}(A^{!},t)=\operatorname{Hilb}(\mathcal{U}(\mathcal{L}),t)=\operatorname{Hilb}(\operatorname{Sym}(\mathcal{L}),t)=\prod_{d=1}^{\infty}\frac{1}{(1-t^{d})^{\varphi_{d}}},$
(73)

where $\varphi_{d}=\dim_{\mathbf{k}}\mathcal{L}_{d}$; see
[PolishchukPositselski, §2.2 Example 2].

######  Remark 8.8.

The lower central series (LCS) of a finitely-generated group $G$ is a chain of
normal subgroups $G=G_{1}\geq G_{2}\geq\dots$ defined recursively by
$G_{k}=[G_{k-1},G]$. Kohno [Kohno] used the topological interpretation of the
Orlik-Solomon algebra that we will discuss in Section 9 to investigate the LCS
of the homotopy group of the complement of a complex hyperplane arrangement.
By studying the holonomy Lie algebra of the Orlik-Solomon algebra of the braid
arrangement, Kohno proved that the ranks $\varphi_{d}$ of the successive
quotients in the lower central series of the homotopy group of the complement
of the braid arrangement satisfy Equation 73. Falk and Randell [falk-randell-
LCS] later showed that this formula also holds for supersolvable arrangements,
and Shelton and Yuzvinsky [shelton1997koszul] proved that an LCS formula of
the form in Equation 73 holds if and only if the Orlik-Solomon algebra of the
arrangement is Koszul. Peeva [Peeva] gave another proof that the LCS formula
of this form holds for supersolvable arrangements using the fact that they
have a quadratic Gröbner basis.

Any group $G$ of graded $\mathbf{k}$-algebra symmetries of $A$, and therefore
of $A^{!}$, will also act as graded Lie algebra symmetries of $\mathcal{L}$.
The PBW Theorem then gives these equalities in $R_{\mathbf{k}}(G)[[t]]$:

$\displaystyle\operatorname{Hilb}_{\mathrm{eq}}(A^{!},t)=\operatorname{Hilb}_{\mathrm{eq}}(\mathcal{U}(\mathcal{L}),t)$
$\displaystyle=\operatorname{Hilb}_{\mathrm{eq}}(\operatorname{Sym}(\mathcal{L}),t)$
(74)
$\displaystyle=\operatorname{Hilb}_{\mathrm{eq}}\left(\operatorname{Sym}\left(\bigoplus_{d=0}^{\infty}\mathcal{L}_{d}\right),t\right)$
$\displaystyle=\sum_{\lambda=(1^{m_{1}}2^{m_{2}}\cdots)}t^{|\lambda|}\prod_{j\geq
1}\left[\operatorname{Sym}^{m_{j}}\mathcal{L}_{j}\right].$

In Section 10, we will use this description to investigate representation
stability for $\mathcal{L}$ in the setting where $A$ is anti-commutative.

####  8.2.2. The commutative case

Similarly, when $A$ is commutative, Polishchuk and Positselski discuss in
[PolishchukPositselski, §1.2, Example 4] how $A^{!}=\mathcal{U}(\mathcal{L})$
for the (graded) Lie superalgebra
$\mathcal{L}=\bigoplus_{d=}^{\infty}\mathcal{L}_{d}$ over $\mathbf{k}$, in
which the parity is induced by the grading, that is,

$\displaystyle\mathcal{L}_{\mathrm{even}}$ $\displaystyle=\bigoplus_{d\equiv
0\bmod{2}}\mathcal{L}_{d},$ (75) $\displaystyle\mathcal{L}_{\mathrm{odd}}$
$\displaystyle=\bigoplus_{d\equiv 1\bmod{2}}\mathcal{L}_{d}.$ (76)

The graded version of the PBW Theorem (see Milnor and Moore [MilnorMoore, Thm.
5.15], Ross [Ross, Thm. 2.1], Scheunert [Scheunert, §2.3 Thm. 1]), asserts
that when $\mathbf{k}$ is a field of characteristic zero, one has a graded
$\mathbf{k}$-vector space isomorphism

$A^{!}=\mathcal{U}(\mathcal{L})\cong\operatorname{Sym}_{\pm}(\mathcal{L}):=\operatorname{Sym}(\mathcal{L}_{\textrm{even}})\otimes\wedge(\mathcal{L}_{\textrm{odd}})$

and hence a Hilbert series relation

$\operatorname{Hilb}(A^{!},t)=\operatorname{Hilb}(\mathcal{U}(\mathcal{L}),t)=\operatorname{Hilb}(\operatorname{Sym}_{\pm}(\mathcal{L}),t)=\frac{\prod_{d\text{
odd}}(1+t^{d})^{\varphi_{d}}}{\prod_{d\text{ even},d\geq
2}(1-t^{d})^{\varphi_{d}}}$

where $\varphi_{d}=\dim_{\mathbf{k}}\mathcal{L}_{d}$.

######  Remark 8.9.

For any formal power series $P(t)=1+\sum_{j\geq 1}b_{j}t^{j}$ with
$b_{j}\in{\mathbb{Z}}$, there exist uniquely defined $\varphi_{d}$ such that

$P(t)=\frac{\prod_{d\text{ odd}}(1+t^{d})^{\varphi_{d}}}{\prod_{d\text{
even},d\geq 2}(1-t^{d})^{\varphi_{d}}}.$

If
$P(t)=\sum_{d=0}^{\infty}\dim_{\mathbf{k}}\text{Tor}_{j}^{R}(\mathbf{k},\mathbf{k})$
is the Poincaré series of a Noetherian commutative ring $R$ in either

  * •

the local setting, where $(R,\mathfrak{m})$ is a local ring with residue field
$\mathbf{k}=R/\mathbf{m}$, or

  * •

the graded setting, where $R=\bigoplus_{d=0}^{\infty}$ is an
${\mathbb{N}}$-graded commutative $\mathbf{k}$-algebra with
$R_{0}=\mathbf{k}$,

the exponent $\varphi_{d}$ is called the $d^{th}$ deviation of the ring $R$.
This is because the nonvanishing of the $\varphi_{d}$ measure whether $R$
“deviates” from being a regular ring or a complete intersection in precise
senses:

  * •

$R$ is regular if and only if $\varphi_{2}=\varphi_{3}=\dots=0$; see [Avramov,
7.3.2]

  * •

$R$ is a complete intersection if and only if
$\varphi_{3}=\varphi_{3}=\dots=0$; see [Avramov, 7.3.3].

Moreover, in the local setting one can always resolve $\mathbf{k}$ over $R$
via an acyclic closure; this was first proven in [gulliksenLevin69]. See
[Avramov, §6.3, §7, §10.2] for an in-depth discussion in the local setting;
analogous results hold for commutative Noetherian graded
$\mathbf{k}$-algebras. Informally, an acyclic closure is built by recursively
adjoining formal variables to represent boundaries of any cycles that appear
while computing an $R$-free resolution of $\mathbf{k}$. The number of formal
variables that one must adjoin in homological degree $d$ is exactly
$\varphi_{d}$, which predicts the dimension of the $d^{th}$ graded component
of the indecomposables within $\text{Tor}^{R}(\mathbf{k},\mathbf{k})$. Since
the graded dual of $\text{Tor}^{R}(\mathbf{k},\mathbf{k})$ is exactly
$\operatorname{Ext}_{R}(\mathbf{k},\mathbf{k})$, the space of indecomposables
of $\text{Tor}^{R}(\mathbf{k},\mathbf{k})$ is the graded dual to the space of
primitives in $\operatorname{Ext}_{R}(\mathbf{k},\mathbf{k})$, so that
$\varphi_{d}=\dim_{\mathbf{k}}\mathcal{L}_{d}$.

Again, in the presence of a group $G$ of graded $\mathbf{k}$-algebra
symmetries acting on $A,A^{!}$, one also has these equalities in
$R_{\mathbf{k}}(G)[[t]]$:

$\displaystyle\operatorname{Hilb}_{\mathrm{eq}}(A^{!},t)$
$\displaystyle=\operatorname{Hilb}_{\mathrm{eq}}(\mathcal{U}(\mathcal{L}),t)$
(77)
$\displaystyle=\operatorname{Hilb}_{\mathrm{eq}}(\operatorname{Sym}_{\pm}(\mathcal{L}),t)$
$\displaystyle=\operatorname{Hilb}_{\mathrm{eq}}\left(\wedge\left(\mathcal{L}_{\mathrm{odd}}\right)\otimes\operatorname{Sym}\left(\mathcal{L}_{\mathrm{even}}\right),t\right)$
$\displaystyle=\sum_{\lambda=(1^{m_{1}}2^{m_{2}}\cdots)}t^{|\lambda|}\prod_{j\text{
odd}}\left[\wedge^{m_{j}}\mathcal{L}_{j}\right]\cdot\prod_{j\text{ even},j\geq
2}\left[\operatorname{Sym}^{m_{j}}\mathcal{L}_{j}\right].$

We will use this description in Section 10 to investigate representation
stability for $\mathcal{L}$ in the setting where $A$ is commutative.

######  Example 8.10.

Let us return to the Boolean matroid $U_{n,n}$ discussed in Example 4.5 and
Section 6.1, but now considered as an oriented matroid represented by the
standard basis vectors $v_{1},\ldots,v_{n}$ in ${\mathbb{R}}^{n}$. Since the
$\\{v_{i}\\}$ are linearly independent, there are no circuits, and the graded
Varchenko-Gel’fand ring $A=\mathrm{VG}(U_{n,n})$ and its Koszul dual $A^{!}$
have these descriptions:

$\displaystyle A$
$\displaystyle=\mathbf{k}[x_{1},\ldots,x_{n}]/(x_{1}^{2},\ldots,x_{n}^{2})$
$\displaystyle=\mathbf{k}\langle
x_{1},\ldots,x_{n}\rangle/(x_{k}^{2},x_{i}x_{j}-x_{j}x_{i})_{1\leq k\leq
n,1\leq<j\leq n}$ $\displaystyle A^{!}$ $\displaystyle=\mathbf{k}\langle
y_{1},\ldots,y_{n}\rangle/(y_{i}y_{j}+y_{j}y_{i})_{1\leq<j\leq n}$

The oriented matroid automorphisms $\text{Aut}(\mathcal{M})$ are the full
hyperoctahedral group $\mathfrak{S}_{n}^{\pm}$, in which a signed permutation
$w$ with $w(v_{i})=\pm v_{j}$ acts on the variables via $w(x_{i})=\pm
x_{j},w(y_{i})=\pm y_{j}$.

We next analyze the graded $\mathbf{k}\mathfrak{S}_{n}^{\pm}$-modules
$A,A^{!}$ when $\mathbf{k}$ has characteristic zero. To do this, first recall
(e.g., from Geissinger and Kinch [Geissinger], Macdonald [Macdonald, Chap. 1.
App. B]) that irreducible $\mathbf{k}\mathfrak{S}_{n}^{\pm}$-modules
$\mathcal{S}^{(\lambda^{+},\lambda^{-})}$ are indexed by ordered pairs of
partitions $(\lambda^{+},\lambda^{-})$ where
$|\lambda^{+}|=n_{+},|\lambda|=n_{-}$ with $n_{+}+n_{-}=n$. One can construct
$\mathcal{S}^{(\lambda^{+},\lambda^{-})}$ using the irreducible
$\mathbf{k}\mathfrak{S}_{n}$-modules $\\{\mathcal{S}^{\mu}\\}$ as building
blocks as follows. Introduce the operation of inflation $U\longmapsto
U\Uparrow$ of a $\mathbf{k}\mathfrak{S}_{n}$-module $U$ to a
$\mathbf{k}\mathfrak{S}_{n}^{\pm}$-module by precomposing with the group
surjection $\pi:\mathfrak{S}_{n}^{\pm}\longrightarrow\mathfrak{S}_{n}$ that
ignores the $\pm$ signs in a signed permutation. Also introduce the one-
dimensional character $\chi_{\pm}:\mathfrak{S}_{n}^{\pm}\rightarrow\\{\pm
1\\}$ sending a signed permutation $w$ to the product of its $\pm 1$ signs,
that is, $\chi_{\pm}(w):=\det(w)/\det(\pi(w))$. Then starting with irreducible
$\mathbf{k}\mathfrak{S}_{n}$-modules $\mathcal{S}^{\lambda}$, one builds
$\mathcal{S}^{(\lambda^{+},\lambda^{-})}$ as follows:

$\mathcal{S}^{(\lambda^{+},\lambda^{-})}:=\left(\mathcal{S}^{\lambda^{+}}\Uparrow\quad\otimes\quad\left(\chi_{\pm}\otimes(\mathcal{S}^{\lambda^{-}}\Uparrow)\right)\right)\big{\uparrow}_{\mathfrak{S}_{n_{+}}^{\pm}\times\mathfrak{S}_{n_{-}}^{\pm}}^{\mathfrak{S}_{n}^{\pm}}.$

For example, this identifies the graded component $A_{i}$ of
$A=\mathbf{k}[x_{1},\ldots,x_{n}]/(x_{1}^{2},\ldots,x_{n}^{2})$ as the
irreducible $\mathbf{k}\mathfrak{S}_{n}^{\pm}$-module
$\mathcal{S}^{((n-i),(i))}$. This is because it is a direct sum of the
$\binom{n}{i}$ lines which are the $\mathfrak{S}_{n}^{\pm}$-images of the line
$L:=\mathbf{k}\cdot x_{1}x_{2}\cdots x_{i}$. This line $L$ is stabilized
setwise by the subgroup
$\mathfrak{S}_{n-i}^{\pm}\times\mathfrak{S}_{i}^{\pm}$, with the
$\mathfrak{S}_{n-i}^{\pm}$ factor acting trivially, and the
$\mathfrak{S}_{i}^{\pm}$ factor acting via $\chi_{\pm}$. Hence one has

$\operatorname{Hilb}_{\mathrm{eq}}(A,t)=\sum_{i=0}^{n}t^{i}\cdot[\mathcal{S}^{((n-i),(i))}].$

We next analyze $A^{!}$ as a $\mathbf{k}\mathfrak{S}_{n}^{\pm}$-module. Since
$(x_{1}^{2},\ldots,x_{n}^{2})$ is a regular sequence in
$\mathbf{k}[\mathbf{x}]$, the quotient $A$ is a complete intersection, and
$\mathcal{L}_{1}=V^{*}=\mathrm{span}_{\mathbf{k}}\\{y_{1},\ldots,y_{n}\\}$ and
$\mathcal{L}_{2}=\mathrm{span}_{\mathbf{k}}\\{y_{1}^{2},\ldots,y_{n}^{2}\\}$.
This gives a $\mathbf{k}\mathfrak{S}_{n}^{\pm}$-module isomorphism

$\displaystyle A^{!}$
$\displaystyle\cong\wedge_{\mathbf{k}}\mathcal{L}_{1}\otimes_{\mathbf{k}}\operatorname{Sym}\mathcal{L}_{2}$
$\displaystyle=\wedge_{\mathbf{k}}(y_{1},\ldots,y_{n})\otimes_{\mathbf{k}}\mathbf{k}[y_{1}^{2},\ldots,y_{n}^{2}].$
(78)

One can analyze each tensor factor in (78) separately. An analysis similar to
the one for $A_{i}$ gives an $\mathbf{k}\mathfrak{S}_{n}^{\pm}$-module
isomorphism

$\wedge^{i}_{\mathbf{k}}(y_{1},\ldots,y_{n})\cong\mathcal{S}^{((n-i),(1^{i}))}.$

In the other tensor factor of (78), the action of $\mathfrak{S}_{n}^{\pm}$ on
$\mathbf{k}[y_{1}^{2},\ldots,y_{n}^{2}]$ is inflated through the surjection
$\pi:\mathfrak{S}_{n}^{\pm}\longrightarrow\mathfrak{S}_{n}$, letting one
compute its $\mathfrak{S}_{n}^{\pm}$-equivariant Hilbert series from the one
for $\mathfrak{S}_{n}$ on $\mathbf{k}[y_{1},\ldots,y_{n}]$ given in (50), and
doubling the grading. The upshot is this equivariant Hilbert series:

$\displaystyle\operatorname{Hilb}_{\mathrm{eq}}(A^{!},t)$
$\displaystyle=\operatorname{Hilb}_{\mathrm{eq}}(\mathbf{k}[y_{1}^{2},\ldots,y_{n}^{2}],t)\cdot\operatorname{Hilb}_{\mathrm{eq}}(\wedge\\{y_{1},\ldots,y_{n}\\},t)$
$\displaystyle=\frac{1}{(1-t^{2})(1-t^{4})\cdots(1-t^{2n})}\left(\sum_{Q}t^{2\mathrm{maj}(Q)}\cdot[\mathcal{S}^{(\lambda(Q)),\varnothing)}]\right)\left(\sum_{i=0}^{n}t^{i}\cdot[\mathcal{S}^{((n-i),(1^{i}))}]\right)$
$\displaystyle=\frac{\sum_{i=0}^{n}\sum_{Q}t^{2\mathrm{maj}(Q)+i}\cdot[\mathcal{S}^{(\lambda(Q)),\varnothing)}]\cdot[\mathcal{S}^{((n-i),(1^{i}))}]}{(1-t^{2})(1-t^{4})\cdots(1-t^{2n})}$
(79)

where in the sums above, $Q$ ranges over all standard Young tableaux with $n$
cells.

##  9\. Topological interpretations of
$\mathrm{OS}(M),\mathrm{VG}(\mathcal{M})$ and Koszul duality

Orlik-Solomon algebras $\mathrm{OS}(M)$ have their origins in the following
result.

######  Theorem 9.1.

[OrlikSolomon] For an arrangement $\mathcal{A}=\\{H_{1},\ldots,H_{n}\\}$ of
linear hyperplanes in ${\mathbb{C}}^{r}$ with normal vectors
$v_{1},\ldots,v_{n}$ representing a matroid $M$, the cohomology ring of their
complement $X:={\mathbb{C}}^{r}\setminus\bigcup_{H\in\mathcal{A}}H$ has
presentation (using any coefficient ring $\mathbf{k}$) as

$H^{*}(X,\mathbf{k})\cong\mathrm{OS}(M).$

An analogue for $\mathrm{VG}(\mathcal{M})$ was given by de Longueville and
Schultz [DelonguevilleSchultz] and later Moseley [Moseley].

######  Theorem 9.2.

[DelonguevilleSchultz, Cor. 5.6], Moseley [Moseley] For an arrangement
$\mathcal{A}=\\{H_{1},\ldots,H_{n}\\}$ of linear hyperplanes in
${\mathbb{R}}^{r}$ with normal vectors $v_{1},\ldots,v_{n}$ representing an
oriented matroid $\mathcal{M}$, the cohomology ring of their
“${\mathbb{R}}^{3}$-thickened” complement
$X_{{\mathbb{R}}^{3}}:=\left({\mathbb{R}}^{r}\otimes_{\mathbb{R}}{\mathbb{R}}^{3}\right)\setminus\bigcup_{H\in\mathcal{A}}\left(H\otimes_{\mathbb{R}}{\mathbb{R}}^{3}\right)$
has presentation (using any coefficient ring $\mathbf{k}$) as

$H^{*}(X_{{\mathbb{R}}^{3}},\mathbf{k})\cong\mathrm{VG}(\mathcal{M}),$

with the cohomology concentrated in even degrees, so that the isomorphism
halves the grading.

######  Remark 9.3.

The result of de Longueville and Schultz [DelonguevilleSchultz, Cor. 5.6]
proves more generally that, for any $d\geq 2$, under the same assumptions on
$\mathcal{A}\subset{\mathbb{R}}^{r}$, the “${\mathbb{R}}^{d}$-thickened”
complement
$X_{{\mathbb{R}}^{d}}:=\left({\mathbb{R}}^{r}\otimes_{\mathbb{R}}{\mathbb{R}}^{d}\right)\setminus\bigcup_{H\in\mathcal{A}}\left(H\otimes_{\mathbb{R}}{\mathbb{R}}^{d}\right)$
has presentation (using any coefficient ring $\mathbf{k}$) as

$H^{*}(X_{{\mathbb{R}}^{d}},\mathbf{k})\cong\begin{cases}\mathrm{OS}(M)&\text{
for }d=2,4,6,\ldots,\text{ even},\\\ \mathrm{VG}(\mathcal{M})&\text{ for
}d=3,5,7,\ldots,\text{ odd},\end{cases}$

with cohomology concentrated in degrees divisible by $d_{1}$, so the
isomorphism divides the grading by $d-1$. Here $M,\mathcal{M}$ are the
matroid, oriented matroid associated to the normal vectors
$v_{1},\ldots,v_{n}$.

######  Remark 9.4.

The type $A_{n}$ and $B_{n}$ reflection arrangements are both supersolvable
and realizable over ${\mathbb{R}}$ (and therefore ${\mathbb{C}}$) and
therefore we can apply this topological interpretation of the Orlik-Solomon
and Varchenko-Gel’fand rings of these families of arrangements. In fact, for
the type $A$ reflection arrangements, we can also view the Orlik-Solomon and
Varchenko-Gel’fand rings as cohomology rings of configuration spaces of points
in ${\mathbb{R}}^{d}$; this perspective will be discussed further in Section
11.1.

In general, not every arrangement realizable over ${\mathbb{C}}$ is realizable
over ${\mathbb{R}}$, and there exist matroids (including supersolvable ones)
that are represented only in positive characteristics, and some not
representable over any field. See for example, some of the matroids discussed
in Sections 12.1 and 12.2. One may visualize some of the implications as
follows:

$\begin{array}[]{ccc}\text{matroid }M&\Leftarrow&\text{oriented matroid
}\mathcal{M}\\\ \Uparrow&&\Uparrow\\\ \mathcal{A}\text{ realized over
}{\mathbb{C}}&\Leftarrow&\mathcal{A}\text{ realized over
}{\mathbb{R}}\end{array}$

If the cohomology ring $H^{\ast}(X,\mathbf{k})$ of a simply connected
topological space $X$ is a Koszul $\mathbf{k}$-algebra (as in the case of the
Orlik-Solomon and Varchenko-Gel’fand rings for supersolvable arrangements),
then the Koszul dual $H^{\ast}(X,\mathbf{k})^{!}$ can be interpreted as the
homology ring $H_{\ast}(\Omega X,\mathbf{k})$ of the based loop space $\Omega
X$.

######  Proposition 9.5 (See [Berglund, berglund2017free]).

Let $X$ be a simply connected topological space such that its cohomology ring
$A:=H^{\ast}(X,\mathbf{k})$ is a Koszul $\mathbf{k}$-algebra over a field
$\mathbf{k}$. Then

$A^{!}=H^{\ast}(X,\mathbf{k})^{!}=H_{\ast}(\Omega X,\mathbf{k})$

where $\Omega X$ is the basepointed loop space of $X$.

###### Proof.

The authors thank Craig Westerland for communicating the following proof to
them. Observe that these spaces participate in the path-loop fibration

$\Omega X\rightarrow PX\rightarrow X,$

where $PX\coloneqq\\{f:I\to X:f(0)=\ast\text{ and }f\text{ continuous}\\}$ is
the space of based maps from an interval into $X$; note that $PX$ is
contractible. In general, for a Serre fibration $F\rightarrow E\rightarrow B$
having $B$ simply connected, the Eilenberg-Moore spectral sequence for
cohomology is

$E^{*,*}_{2}=\text{Tor}_{H^{*}(B)}(\mathbf{k},H^{*}(E))\Rightarrow H^{*}(F)$

where $H^{*}(E)$ is a $H^{*}(B)$-module by the map in the fibration, and
$\mathbf{k}$ is our base field (or ring, if everything is suitably flat over
$\mathbf{k}$). In the case of the path-loop fibration, since $PX$ is
contractible,

$\text{Tor}_{H^{*}(X)}(\mathbf{k},\mathbf{k})\Rightarrow H^{*}(\Omega X).$

If $\mathbf{k}$ is a field, or if everything is free over $\mathbf{k}$, we can
dualize everything to get

$\operatorname{Ext}_{H^{*}(X)}(\mathbf{k},\mathbf{k})\Rightarrow H_{*}(\Omega
X).$

If $H^{*}(X)$ is Koszul, then
$\operatorname{Ext}_{H^{*}(X)}(\mathbf{k},\mathbf{k})\cong H^{*}(X)^{!}$.
Further, as $\operatorname{Ext}_{H^{*}(X)}(\mathbf{k},\mathbf{k})$ is
concentrated in diagonal bidegrees, its differentials are zero, so the
spectral sequence collapses at $E_{2}$, giving

$H_{*}(\Omega X,\mathbf{k})=H^{*}(X,\mathbf{k})^{!}.\qed$

######  Remark 9.6.

Under the hypotheses of Proposition 9.5, the terminology from Definition 8.3
of homotopy Lie algebra for the $\mathbf{k}$-subspace of primitives
$\mathcal{L}\subset
A^{!}=\operatorname{Ext}_{A}^{\bullet}(\mathbf{k},\mathbf{k})$ is consistent
with the same terminology in rational homotopy theory, where a simply
connected space $X$ has homotopy Lie algebra defined as the
$\mathbf{k}$-subspace of primitives $\mathcal{L}\cong\pi_{*}(\Omega
X)\otimes\mathbf{k}$ inside the Hopf algebra $H_{*}(\Omega X,\mathbf{k})$;
see, e.g., Félix, Halperin and Thomas [FelixHalperinThomas, §21(d), Thm.
21.5].

##  10\. Representation stability and Koszul algebras

We wish to show how sequences of Koszul algebras $\\{A(n)\\}_{n\geq 1}$ with
$\mathfrak{S}_{n}$-actions that satisfy representation stability in the sense
of Church and Farb [ChurchFarb] lead to representation stability for their
Koszul duals $\\{A(n)^{!}\\}_{n\geq 1}$, and for the primitive parts
$\\{\mathcal{L}(n)\\}_{n\geq 1}$ of the duals. Useful references on
representation stability are Church and Farb [ChurchFarb], Church, Ellenberg
and Farb [CEF] and Matherne, Miyata, Proudfoot and Ramos
[MatherneMiyataProudfootRamos].

In this section, $\mathbf{k}$ is a field of characteristic zero. Recall this
definition from [ChurchFarb].

######  Definition 10.1.

For a partition $\mu$ of $k$, recall $\mathcal{S}^{\mu}$ denotes the
irreducible $\mathbf{k}\mathfrak{S}_{k}$-module indexed by $\nu$. Given a
partition $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})$ and
$n\geq|\lambda|+\lambda_{1}$, define a partition of $n$ by

$\lambda[n]:=(n-|\lambda|,\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell}).$

Say that a sequence $\\{V_{n}\\}_{n\geq 1}$ of
$\mathbf{k}\mathfrak{S}_{n}$-modules777Such a sequence of
$\mathbf{k}\mathfrak{S}_{n}$-modules is equivalent to what is called an
$FB$-module in [MatherneMiyataProudfootRamos]. is representation stable if
there is a list of (not necessarily distinct) partitions
$\\{\lambda^{(i)}\\}_{i=1}^{t}$ and an integer $N$ such that for $n\geq N$,
one has

$V_{n}=\bigoplus_{i=1}^{t}\mathcal{S}^{\lambda^{(i)}[n]}.$

Say that $\\{V_{n}\\}_{n\geq 1}$ is representation stable past $N$ when the
above equality holds for $n\geq N$.

The following easy observations are recorded in [MatherneMiyataProudfootRamos,
§3].

######  Proposition 10.2.

When $\\{V_{n}\\},\\{W_{n}\\}$ are representation stable sequences, then so is
$\\{V_{n}\oplus W_{n}\\}$. On the other hand, if the virtual modules
$[U_{n}]=[V_{n}]-[W_{n}]$ come from genuine
$\mathbf{k}\mathfrak{S}_{n}$-modules $\\{U_{n}\\}$, then $\\{U_{n}\\}$ is also
a representation stable sequence.

It is less obvious what happens for tensor products. The following precise
version of a result of Murnaghan was proven by Briand, Orellana, Rosas
[BriandOrellanaRosas, Thm. 1.2].

######  Theorem 10.3.

The sequence $\\{\mathcal{S}^{\alpha[n]}\otimes\mathcal{S}^{\beta[n]}\\}$ is
representation stable past $|\alpha|+|\beta|+\alpha_{1}+\beta_{1}$.

This consequence was noted by Matherne, Miyata, Proudfoot and Ramos
[MatherneMiyataProudfootRamos, Thm. 3.3].

######  Lemma 10.4.

If the $\\{V_{n}\\},\\{W_{n}\\}$ are representation stable past $A,B$,
respectively, then $\\{V_{n}\otimes W_{n}\\}$ is representation stable past
$A+B$.

The following observation is occasionally useful for pinpointing the onset of
representation stability.

######  Lemma 10.5.

[HershReiner, Lemma 2.2] Suppose $\\{V_{n}\\}_{n\geq N}$ are
$\mathbf{k}\mathfrak{S}_{n}$-modules defined via

$V_{n}\cong\bigoplus_{\mu}\left(\mathcal{S}^{\mu}\uparrow_{\mathfrak{S}_{|\mu|}\times\mathfrak{S}_{n-|\mu|}}^{\mathfrak{S}_{n}}\right)^{\oplus
c_{\mu}}$

for $\mu$ with $|\mu|\geq N$, and integers $c_{\mu}\geq 1$. Then $\\{V_{n}\\}$
is representation stable, stabilizing exactly at

$n=\max_{\mu}\\{|\mu|+\mu_{1}\\}.$

We next use some of the foregoing observations to show how representation
stability of families of Koszul algebras $\\{A(n)\\}$ passes to their Koszul
duals $\\{A(n)^{!}\\}$.

######  Corollary 10.6.

Let $\\{A(n)\\}_{n\geq 1}$ be a sequence of Koszul algebras, with Koszul duals
$\\{A(n)^{!}\\}$.

  1. (i)

If for each fixed $i\geq 0$, the sequence $\\{A(n)_{i}\\}$ is representation
stable, then so is each $\\{A(n)^{!}_{i}\\}$.

  2. (ii)

If furthermore there exists a constant $c$ (independent of $i$) such that each
sequence $\\{A(n)_{i}\\}$ is representation stable past $ci$, then each
$\\{A(n)^{!}_{i}\\}$ is also representation stable past $ci$.

###### Proof.

We prove (ii); the proof for (i) is the same, ignoring the bounds involving
$ci$. But (ii) is immediate using (16) and Lemma 10.4, since each factor
$A_{\alpha_{p}}=A_{\alpha_{p}}(n)$ in the right-hand side of (16), now in
characteristic zero, has the sequence $\\{A_{\alpha_{p}}(n)\\}$ representation
stable past $c{\alpha_{p}}$. ∎

######  Example 10.7.

Rank two matroids $M=U_{2,n}$ were discussed in Example 4.6 and Section 6.3.
Their group of matroid automorphisms $\text{Aut}(U_{2,n})=\mathfrak{S}_{n}$,
and (57) showed that as $\mathbf{k}\mathfrak{S}_{n}$-modules, one has

$\displaystyle[\mathrm{OS}(U_{2,n})_{0}]$
$\displaystyle\cong[\mathcal{S}^{(n)}],$
$\displaystyle[\mathrm{OS}(U_{2,n})_{1}]$
$\displaystyle\cong[\mathcal{S}^{(n)}]+[\mathcal{S}^{(n-1,1)}],$
$\displaystyle[\mathrm{OS}(U_{2,n})_{2}]$
$\displaystyle\cong[\mathcal{S}^{(n-1,1)}],$

which are representation stable past $n=2$. Consequently, applying Corollary
10.6(ii) with $c=2$ implies that the Koszul duals
$\\{\mathrm{OS}(U_{2,n})^{!}_{i}\\}$, which are the
$\mathfrak{S}_{n}$-permutation representations discussed in Proposition 6.1,
should be representation stable past $n=2i$. In fact, one has the following.

######  Proposition 10.8.

For $i\geq 0$, representation stability of
$\\{\mathrm{OS}(U_{2,n})^{!}_{i}\\}$ starts exactly at $n=2i$.

###### Proof.

Recall that Proposition 6.1 expresses $[\mathrm{OS}(U_{2,n})^{!}_{i}]$ as a
nonnegative combination of classes $\varphi_{(n-d,1^{d})}$ for various $d$ in
the range $2\leq d\leq i$, where $\varphi_{\lambda}$ is the class of the coset
representation $\mathbf{k}[\mathfrak{S}_{n}/\mathfrak{S}_{\lambda}]$.
Furthermore, the coefficient on $\varphi_{(n-i,1^{i})}$ is $1$. Since one can
write

$\varphi_{(n-d,1^{d})}=\left[\left(\mathbf{k}\mathfrak{S}_{d}\right)\uparrow_{\mathfrak{S}_{d}\times\mathfrak{S}_{n-d}}^{\mathfrak{S}_{n}}\right]\quad\text{
where
}\quad\mathbf{k}\mathfrak{S}_{d}\cong\bigoplus_{\mu:|\mu|=d}(\mathcal{S}^{\mu})^{\oplus\dim\mathcal{S}^{\mu}},$

Lemma 10.5 shows $\\{\varphi_{(n-d,1^{d})}\\}_{n}$ stabilizes exactly at
$n=2d$, and $\\{[\mathrm{OS}(U_{2,n})^{!}_{i}]\\}$ exactly at $n=2i$. ∎

Recall from Section 8 that when $A$ is Koszul and commutative or anti-
commutative, then $A^{!}=\mathcal{U}(\mathcal{L})$ is the universal enveloping
algebra for a graded Lie algebra or superalgebra $\mathcal{L}$. We next show
how representation stability of families of Koszul algebras $\\{A(n)\\}$
passes to the Lie (super-)algebras $\\{\mathcal{L}(n)\\}$. The following lemma
will be useful for this purpose.

######  Lemma 10.9.

For any representation stable sequence $\\{V_{n}\\}$ and any partition $\mu$
giving rise to a Schur functor ${\mathbb{S}}^{\mu}(-)$, the sequence
$\\{{\mathbb{S}}^{\mu}(V_{n})\\}$ is also representation stable. In
particular, for each fixed $m=0,1,2,\ldots$, the sequences
$\\{\wedge^{m}(V_{n})\\},\\{\operatorname{Sym}^{m}(V_{n})\\}$ are
representation stable.

###### Proof.

Express the representation stable sequence $\\{V_{n}\\}$ for $n\gg 0$ as
$V_{n}=\bigoplus_{i=1}^{t}\mathcal{S}^{\lambda^{(i)}[n]},$ and proceed by
induction on $t$ to show $\\{{\mathbb{S}}^{\mu}(V_{n})\\}$ is representation
stable for all $\mu$.

The case $t=1$ was proven by Church, Ellenberg, Farb [CEF, Prop. 3.4.5] who
showed that for any partitions $\lambda,\mu$, the sequence
$\\{{\mathbb{S}}^{\mu}(\mathcal{S}^{\lambda[n]})\\}$ is representation stable.
In the inductive step, write $V_{n}=U_{n}\oplus\mathcal{S}^{\lambda^{(i)}[n]}$
for $n\gg 0$, where
$U_{n}:=\bigoplus_{i=1}^{t-1}\mathcal{S}^{\lambda^{(i)}[n]}$, so that
induction applies to the representation stable sequence $\\{U_{n}\\}$. Using
the general isomorphism (see, e.g., [AkinBuchsbaumWeyman, Thm. II.4.11])

${\mathbb{S}}(X\oplus
Y)\cong\bigoplus_{\nu\subseteq\mu}{\mathbb{S}}^{\nu}(X)\otimes{\mathbb{S}}^{\mu/\nu}(Y)$

one concludes that, for $n\gg 0$,

$\displaystyle{\mathbb{S}}^{\mu}(V_{n})={\mathbb{S}}^{\mu}(U_{n}\oplus\mathcal{S}^{\lambda[n]})\cong\bigoplus_{\nu\subseteq\mu}{\mathbb{S}}^{\nu}(U_{n})\otimes{\mathbb{S}}^{\mu/\nu}(\mathcal{S}^{\lambda[n]})\cong\bigoplus_{\nu,\mu,\rho}\left({\mathbb{S}}^{\nu}(U_{n})\otimes{\mathbb{S}}^{\rho}(\mathcal{S}^{\lambda[n]})\right)^{\oplus
c^{\mu/\nu}_{\rho}}$

for some nonnegative integer (Littlewood-Richardson) coefficients
$c^{\mu/\nu}_{\rho}$. By induction on $t$, the sequences
$\\{{\mathbb{S}}^{\nu}(U_{n})\\}$ are representation stable, and by the $t=1$
case, the same holds for
$\\{{\mathbb{S}}^{\rho}(\mathcal{S}^{\lambda[n]})\\}$. Hence by Theorem 10.3,
each summand
$\\{{\mathbb{S}}^{\nu}(U_{n})\otimes{\mathbb{S}}^{\rho}(\mathcal{S}^{\lambda[n]})\\}$
on the right side is a representation stable sequence, and the same holds for
the entire direct sum. ∎

We now apply this to the sequences of Lie (super-)algebras
$\\{\mathcal{L}(n)\\}$.

######  Corollary 10.10.

Let $\\{A(n)\\}$ be a family of Koszul algebras, all commutative (resp. anti-
commutative), with $\\{\mathcal{L}(n)\\}$ defined by
$A(n)^{!}=\mathcal{U}(\mathcal{L}(n))$. If for each fixed $i=0,1,2,\ldots$,
the sequence $\\{A(n)_{i}\\}$ is representation stable, then for each fixed
$i=1,2,\ldots$, the sequence $\\{\mathcal{L}(n)_{i}\\}$ is also representation
stable.

###### Proof.

In either case where $\\{A(n)\\}$ are commutative or anti-commutative, use
induction on $i$. In the base case $i=1$, one has this string of equalities,
justified below:

$[\mathcal{L}(n)_{1}]\overset{(a)}{=}[A(n)^{!}_{1}]\overset{(b)}{=}[\left(A(n)_{1}\right)^{*}]\overset{(c)}{=}[A(n)_{1}].$

Equality (a) comes from comparing coefficients of $t^{1}$ on either side of
(77) or (74), equality (b) comes from (15), and equality (c) comes from the
fact that $\mathbf{k}\mathfrak{S}_{n}$-modules are all self-contragredient.
Since the right sides $\\{A(n)_{1}\\}$ are representation stable, so are the
left sides $\\{\mathcal{L}(n)_{1}\\}$ .

In the induction step where $i\geq 2$, rewrite the equalities that come from
comparing the coefficient of $t^{i}$ on either side of (77) or (74), isolating
the summand $[\mathcal{L}(n)_{i}]$ on the right corresponding to
$\lambda=(i)$. For fixed $n$, this expresses $\mathcal{L}(n)_{i}$ recursively
in terms of $A(n)^{!}_{i}$ and
$\mathcal{L}(n)_{1},\mathcal{L}(n)_{2},\ldots,\mathcal{L}(n)_{i-1}$:

$[\mathcal{L}(n)_{i}]=[A(n)^{!}_{i}]-\begin{cases}\displaystyle\sum_{\begin{subarray}{c}\lambda\vdash
i:\\\ \lambda=(1^{m_{1}}2^{m_{2}}\cdots i^{m_{i}})\\\
\lambda\neq(i)\end{subarray}}\prod_{1\leq
j<i}[\operatorname{Sym}^{m_{j}}(\mathcal{L}(n)_{j})]\quad\quad\quad\quad\text{
for }A(n)\text{ anti-commutative}.\\\ &\\\
\displaystyle\sum_{\begin{subarray}{c}\lambda\vdash i:\\\
\lambda=(1^{m_{1}}2^{m_{2}}\cdots i^{m_{i}})\\\
\lambda\neq(i)\end{subarray}}\prod_{\begin{subarray}{c}1\leq j<i\\\ j\text{
odd
}\end{subarray}}[\wedge^{m_{j}}(\mathcal{L}(n)_{j})]\prod_{\begin{subarray}{c}2\leq
j<i\\\ j\text{ even
}\end{subarray}}[\operatorname{Sym}^{m_{j}}(\mathcal{L}(n)_{j})]\text{ for
}A(n)\text{ commutative}.\end{cases}$

Now use Corollary 10.6 asserting that each sequence $\\{A(n)^{!}_{i}\\}$ is
representation stable. Induction on $i$ shows each sequence
$\\{\mathcal{L}(n)_{j}\\}$ for $j\leq i-1$ appearing on the right is
representation stable. Lemma 10.9, then implies the same for all sequences
$\\{\wedge^{m_{j}}(\mathcal{L}(n)_{j}\\},\\{\operatorname{Sym}^{m_{j}}(\mathcal{L}(n)_{j}\\}$
appearing on the right. Then Theorem 10.3 gives the same for their tensor
products. Thus every summand on the right is a representation stable sequence
in $n$, and hence so is $\\{\mathcal{L}(n)_{i}\\}$. ∎

######  Remark 10.11.

The above proof shows the following statement for a sequence of graded Lie
(super-)algebras and $\mathbf{k}\mathfrak{S}_{n}$-modules $\mathcal{L}(n)$,
with universal enveloping algebras $\mathcal{U}(\mathcal{L}(n))$: one has for
all $i\geq 1$ that $\\{\mathcal{L}(n)_{i}\\}$ is representation stable if and
only if one has for all $i\geq 0$ that $\\{\mathcal{U}(\mathcal{L}(n))_{i}\\}$
is representation stable.

######  Remark 10.12.

Unlike Corollary 10.6, we have not seriously tried to bound the onset of
stability for the sequences $\\{\mathcal{L}(n)_{i}\\}$, in terms of a given
bound for the onset of stability in $\\{A(n)_{i}\\}$. However, Sage
computations up to $i=10$ suggest the following for uniform matroids $U_{2,n}$
of rank $2$.

######  Conjecture 10.13.

Defining $\mathcal{L}(n)_{i}$ by
$\mathrm{OS}(U_{2,n})^{!}=\mathcal{U}(\mathcal{L}(n))$, the sequence
$\\{\mathcal{L}(n)_{i}\\}$ is representation stable past $n=2i-1$ for fixed
$i\geq 3$.

######  Remark 10.14.

Although $[A(n)_{i}^{!}]$ is a permutation representation when
$A(n)=\mathrm{OS}(U_{2,n})$ by 6.1, the primitives $[\mathcal{L}(n)_{i}]$ are
generally not classes of permutation representations. This fails immediately
for $n=3$ and $i=2$, where $[\mathcal{L}(3)_{2}]$ is the sign representation.
Also for braid matroids, if $A(n)=\mathrm{OS}(\operatorname{Br}_{n})$, and
$A(n)^{!}=\mathrm{OS}(\operatorname{Br}_{n})^{!}=\mathcal{U}(\mathcal{L}(n))$,
one can check

$\mathcal{L}(n)_{2}=(\mathrm{sgn}_{\mathfrak{S}_{3}}\otimes\mathbb{1}_{\mathfrak{S}_{n-3}})\big{\uparrow}^{\mathfrak{S}_{n}}_{\mathfrak{S}_{3}\times\mathfrak{S}_{n-3}}$

which is again not a permutation module.

##  11\. The motivating example: braid matroids and Stirling representations

As mentioned prior to Example 4.7, our motivation came from the braid matroids
$M=\operatorname{Br}_{n}$, which are also known as the graphic matroids for
complete graphs $K_{n}$. They are also known as the matroids represented by
the vectors $\\{v_{ij}\\}_{1\leq i<j\leq n}$ with $v_{ij}:=e_{i}-e_{j}$ in
${\mathbb{R}}^{n}$ which are the (positive) roots in the type $A_{n-1}$ root
system, whose normal hyperplanes
$H_{ij}=\\{\mathbf{x}\in{\mathbb{R}}^{n}:x_{i}=x_{j}\\}$ are the reflecting
hyperplanes for the transposition $(i,j)$ in the symmetric group
$\mathfrak{S}_{n}$ when it acts on $V={\mathbb{R}}^{n}$ by permuting
coordinates. Thus $M=\operatorname{Br}_{n}$ is orientable, and abusing
notation slightly, we will also denote by $\mathcal{M}=\operatorname{Br}_{n}$
the oriented matroid on the ground set $E=\\{\\{i,j\\}:1\leq i<j\leq n\\}$
represented by these vectors $\\{v_{ij}\\}$.

###  11.1. Comparison with cohomology of configuration spaces

It turns out that the algebras
$\mathrm{OS}(\operatorname{Br}_{n}),\mathrm{VG}(\operatorname{Br}_{n})$ had
been studied historically earlier as the cohomology rings of certain
configuration spaces of $n$ ordered (labeled) points in a space $X$

$\mathrm{Conf}_{n}(X):=\\{(x_{1},\ldots,x_{n})\in X^{n}:x_{i}\neq x_{j}\text{
for }1\leq i<j\leq n\\}.$

The arrangement of hyperplanes $H_{ij}$ in $V={\mathbb{R}}^{n}$ described
above allow one to identify

$\mathrm{Conf}_{n}({\mathbb{R}}^{d})=V\otimes_{\mathbb{R}}{\mathbb{R}}^{d}\setminus\bigcup_{1\leq
i<j\leq n}H_{ij}\otimes_{\mathbb{R}}{\mathbb{R}}^{d},$

that is, as the complement of subspace arrangements coming from the reflection
hyperplane arrangements “thickened” by tensoring with ${\mathbb{R}}^{d}$ as in
Theorems 9.1, 9.2 and Remark 9.3. In fact, the special cases of those results
for the braid arrangements, along with quadratic presentations for the
associated algebras, were known to Arnol’d [Arnold] for
$\mathrm{OS}(\operatorname{Br}_{n})$ and Cohen [CohenLadaMay] for
$\mathrm{VG}(\operatorname{Br}_{n})$:

$\displaystyle\mathrm{OS}(\operatorname{Br}_{n})$
$\displaystyle\cong\wedge_{\mathbf{k}}(x_{ij})/(x_{ij}x_{ik}-x_{ij}x_{jk}+x_{ik}x_{jk})$
$\displaystyle\mathrm{VG}(\operatorname{Br}_{n})$
$\displaystyle\cong\mathbf{k}[x_{ij}]/\,\,(x_{ij}x_{ik}-x_{ij}x_{jk}+x_{ik}x_{jk},\,\,x_{ij}^{2}\,\,)$

Here permutations $\sigma$ in $\mathfrak{S}_{n}$ act on the variables by
permuting subscripts, that is, $\sigma(x_{ij})=x_{\sigma(i),\sigma(j)}$, but
with the convention that $x_{ji}=x_{ij}$ in
$\mathrm{OS}(\operatorname{Br}_{n})$, and $x_{ji}=-x_{ij}$ in
$\mathrm{VG}(\operatorname{Br}_{n})$.

Note that that these presentations are consistent with the general
presentation coming from supersolvable matroids in Corollary 5.14, using the
modular complete flag $\underline{F}$ of flats chosen in Example 4.7: one
checks that the corresponding decomposition
$\underline{E}=(E_{1},E_{2},\ldots,E_{n-1})$ of $E=\\{\\{i,j\\}\\}_{1\leq
i<j\leq n}$ has

$\displaystyle E_{1}$ $\displaystyle=\\{\\{1,2\\}\\},$ (80) $\displaystyle
E_{2}$ $\displaystyle=\\{\\{1,3\\},\\{2,3\\}\\},$ $\displaystyle E_{3}$
$\displaystyle=\\{\\{1,4\\},\\{2,4\\},\\{3,4\\}\\},$
$\displaystyle\quad\vdots$ $\displaystyle E_{n-1}$
$\displaystyle=\\{\\{1,n\\},\\{2,n\\},\ldots,\\{n-2,n\\},\\{n-1,n\\}\\}.$

and the subset of circuits
$\mathcal{C}_{\operatorname{BEZ}}(\underline{E})=\\{\\{\\{i,j\\},\\{i,k\\},\\{j,k\\}\\}_{1\leq
i<j<k\leq n}.$ Here the NBC monomial basis for either
$\mathrm{OS}(\operatorname{Br}_{n}),\mathrm{VG}(\operatorname{Br}_{n})$ are
the products of $x_{ij}$ that choose at most one $\\{i,j\\}$ from each set
$E_{p}$ with $p=1,2,\ldots,n-1$ above; Barcelo [Barcelo, Thm. 2.1] calls this
picking at most one finger $x_{ij}$ from each hand $E_{p}$. Since the
exponents $e_{p}=|E_{p}|$ here are $(e_{1},\ldots,e_{r})=(1,2,\ldots,n-1)$,
one has these Hilbert series

$\displaystyle\operatorname{Hilb}(\mathrm{OS}(\operatorname{Br}_{n}),t)$
$\displaystyle=\operatorname{Hilb}(\mathrm{VG}(\operatorname{Br}_{n}),t)$
$\displaystyle=(1+t)(1+2t)\cdots(1+(n-1)t)=\sum_{i=0}^{n-1}t^{i}\,c(n,n-i)$
$\displaystyle\operatorname{Hilb}(\mathrm{OS}(\operatorname{Br}_{n})^{!},t)$
$\displaystyle=\operatorname{Hilb}(\mathrm{VG}(\operatorname{Br}_{n})^{!},t)$
$\displaystyle=\frac{1}{(1-t)(1-2t)\cdots(1-(n-1)t)}=\sum_{i=0}^{\infty}t^{i}\,S((n-1)+i,n-1).$

where the coefficients $c(n,k),S(n,k)$ appearing here are the (signless)
Stirling numbers of the first kind $c(n,k)$, counting permutations in
$\mathfrak{S}_{n}$ with $k$ cycles, and the Stirling numbers of the 2nd kind
$S(n,k)$, counting partitions of the set $\\{1,2,\ldots,n\\}$ into $k$ blocks.
Comparing coefficients on powers of $t$, one has for either
$A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ or
$\mathrm{VG}(\operatorname{Br}_{n})$ that

$\displaystyle\dim_{\mathbf{k}}A(n)_{i}$ $\displaystyle=c(n,n-i),$
$\displaystyle c(n,k)$ $\displaystyle=\dim_{\mathbf{k}}A(n)_{n-k},$
$\displaystyle\dim_{\mathbf{k}}A(n)^{!}_{i}$ $\displaystyle=S((n-1)+i,n-1),$
$\displaystyle S(n,k)$ $\displaystyle=\dim_{k}A(k+1)^{!}_{n-k}.$

######  Definition 11.1.

(Stirling representations) For either
$A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ or
$A(n)=\mathrm{VG}(\operatorname{Br}_{n})$, call $A(n)_{i}$ the Stirling
representations of the first kind, and call $A(n)^{!}_{i}$ the Stirling
representations of the second kind. When emphasizing their dimensions as
representations, we will abbreviate them

$\displaystyle\mathfrak{c}_{\mathrm{OS}}(n,k)$
$\displaystyle:=\mathrm{OS}(\operatorname{Br}_{n})_{n-k},$
$\displaystyle\mathfrak{c}_{\mathrm{VG}}(n,k)$
$\displaystyle:=\mathrm{VG}(\operatorname{Br}_{n})_{n-k},\text{ both
}\mathbf{k}\mathfrak{S}_{n}\text{-modules},$
$\displaystyle\mathcal{S}_{\mathrm{OS}}(n,k)$
$\displaystyle:=\mathrm{OS}(\operatorname{Br}_{k+1})^{!}_{n-k},$
$\displaystyle\mathcal{S}_{\mathrm{VG}}(n,k)$
$\displaystyle:=\mathrm{VG}(\operatorname{Br}_{k+1})^{!}_{n-k},\text{ both
}\mathbf{k}\mathfrak{S}_{k+1}\text{-modules},.$

######  Remark 11.2.

The coincidence between $\dim_{\mathbf{k}}A(k+1)^{!}_{n-k}$ and $S(n,k)$,
counting set partitions of $\\{1,2,\ldots,n\\}$ into $k$ blocks, is closely
related to a well-known combinatorial encoding of set partitions via
restricted growth functions, as we explain here; see also Stanton and White
[StantonWhite, Sec. 1.5].

Given any $k$-block set partition $\pi=\\{B_{1},\ldots,B_{k}\\}$ of
$\\{1,2,\ldots,n\\}$, re-index the blocks so that $\min B_{1}<\min
B_{2}<\cdots<\min B_{k}$. Then the restricted growth function (rgf) encoding
$\pi$ is the sequence $(i_{1},i_{2},\ldots,i_{n})$ defined by $i_{j}=\ell$ if
$j\in B_{\ell}$ for $j=1,2,\ldots,n$. By definition, $i_{1}:=1$ and $i_{j}\leq
1+\max\\{i_{0},i_{1},\ldots,i_{j-1}\\}$; it is not hard to check that these
two properties characterize the rgf’s. For example, with $n=15$ and $k=3$,
this set partition

$\pi=\\{\underbrace{\\{\mathbf{1},2,3,5,8,10,15\\}}_{B_{1}},\underbrace{\\{\mathbf{4},6,7,12\\}}_{B_{2}},\underbrace{\\{\mathbf{9},11,13,14\\}}_{B_{3}}\\}$

corresponds to this rgf $(i_{1},i_{2},\ldots,i_{15})$:

$\begin{array}[]{rcccccccccccccccl}i_{1}&i_{2}&i_{3}&i_{4}&i_{5}&i_{6}&i_{7}&i_{8}&i_{9}&i_{10}&i_{11}&i_{12}&i_{13}&i_{14}&i_{15}\\\
1&1&1&2&1&2&2&1&3&1&3&2&3&3&1\end{array}$

We claim that these rgf’s correspond bijectively to the standard monomial
$\mathbf{k}$-basis for $A(k+1)^{!}_{n-k}$ given in Corollary 5.18. To explain
this bijection, underline the first (leftmost) occurrence of each value
$p=1,2,\ldots,k$ among the $i_{j}$, and append an extra (underlined)
$i_{n+1}:=k+1$ on the right, as a convention. One then associates to
$(i_{1},\ldots,i_{n})$ the product $m_{2}\cdot m_{3}\cdots m_{k}\cdot m_{k+1}$
where $m_{p}$ is the noncommutative monomial in variables
$\\{y_{ip}\\}_{i=1}^{p-1}$ obtained by replacing each non-underlined value
$i_{j}$ above with the variable $x_{i_{j},p}$ if $p$ is the next underlined
value to the right of $i_{j}$:

$\begin{array}[]{rcccccccccccccccl}\underline{1}&1&1&\underline{2}&1&2&2&1&\underline{3}&1&3&2&3&3&1&\underline{4}\\\
&y_{12}&y_{12}&\cdot&y_{13}&y_{23}&y_{23}&y_{13}&\cdot&y_{14}&y_{34}&y_{24}&y_{34}&y_{34}&y_{14}&\end{array}$

Since the number of non-underlined values $i_{j}$ is $n-k$, this is a standard
monomial in $A(k+1)^{!}_{n-k}$.

######  Remark 11.3.

The presentations of
$\mathrm{OS}(\operatorname{Br}_{n})^{!},\mathrm{VG}(\operatorname{Br}_{n})^{!}$
in Theorem 5.18 are equivalent to what Cohen and Gitler [CohenGitler] called
graded infinitesimal braid relations in their presentation of the loop space
homology algebra $H_{*}(\Omega X,\mathbf{k})$ where
$X=\mathrm{Conf}_{n}({\mathbb{R}}^{d})$; see also Berglund [Berglund, Example
5.5]. For the case of $\mathrm{OS}(\operatorname{Br}_{n})^{!}$, considered as
a universal enveloping algebra
$\mathrm{OS}(\operatorname{Br}_{n})^{!}=\mathcal{U}(\mathcal{L})$, see also
the discussion by Fresse [Fresse, Ch. 10], referring to $\mathcal{L}$ as the
Drinfeld-Kohno Lie algebra and $\mathcal{U}(\mathcal{L})$ as the algebra of
chord diagrams.

###  11.2. Stirling representations of the first kind: generating functions

The $\mathbf{k}\mathfrak{S}_{n}$-module structure for either
$A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ or
$\mathrm{VG}(\operatorname{Br}_{n})$ are well-studied. Explicit irreducible
decompositions are not known, but can be computed reasonably efficiently
through symmetric function formulas involving plethysm and generating
functions, given in work of Sundaram and Welker [SundaramWelker, Thm.
4.4(iii)] and reviewed here; see also the summary in Hersh and Reiner
[HershReiner, Sec. 2].

Let $\mathbf{k}$ be a field of characteristic zero. The Frobenius
characteristic isomorphism $R_{\mathbf{k}}(\mathfrak{S}_{n})\cong\Lambda_{n}$,
where $\Lambda_{n}$ are the degree $n$ homogeneous symmetric functions
$\Lambda(z_{1},z_{2},\ldots)_{n}$, mentioned in Section 6.3 above, can be
compiled for all $n$ to give a ring isomorphism

$\bigoplus_{n=0}^{\infty}R_{\mathbf{k}}(\mathfrak{S}_{n})\overset{\mathrm{ch}}{\longrightarrow}\bigoplus_{n=0}^{\infty}\Lambda_{n}=\Lambda.$

Here the product on the left is the external or induction product

$([U],[V])\longmapsto\left[(U\otimes_{\mathbf{k}}V)\uparrow_{\mathfrak{S}_{a}\times\mathfrak{S}_{b}}^{\mathfrak{S}_{a+b}}\right],$

while the product on the right simply multiplies symmetric functions. If one
defines the power sum symmetric function $p_{r}:=z_{1}^{r}+z_{2}^{r}+\cdots$,
and the $\mathbf{k}$-basis
$\\{p_{\lambda}:=p_{\lambda_{1}}p_{\lambda_{2}}\cdots\\}$ indexed by
partitions $\lambda$ of $n$ for $\Lambda_{n}$, then for each
$\mathbf{k}\mathfrak{S}_{n}$-module $U$, the Frobenius isomorphism maps
$[U]\overset{\mathrm{ch}}{\longmapsto}\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}_{n}}p_{\lambda(\sigma)}$
where $\lambda(\sigma)$ is the cycle type partition of $\sigma$.

Let $\mathrm{Lie}_{n}$ denote the $n^{th}$ Lie representation: the
$\mathfrak{S}_{n}$-representation on the multilinear component of the free Lie
algebra on $n$ variables. It has a formula due to Klyachko [Klyachko] as
$\mathrm{Lie}_{n}=\zeta\uparrow_{C_{n}}^{\mathfrak{S}_{n}},$ where $\zeta$ is
the one-dimensional representation of the cyclic group $C_{n}$ inside
$\mathfrak{S}_{n}$ generated by an $n$-cycle $c$ that sends $c\mapsto
e^{\frac{2\pi i}{n}}$. Defining symmetric functions

$\displaystyle\ell_{n}$ $\displaystyle:=\mathrm{ch}(\mathrm{Lie}_{n}),$
$\displaystyle\pi_{n}$
$\displaystyle:=\mathrm{ch}(\mathrm{sgn}_{n}\otimes\mathrm{Lie}_{n}),$

and letting $(f,g)\mapsto f[g]$ denote plethystic composition of symmetric
functions [Macdonald, §I.8], one has the following plethystic expressions and
product generating functions (see Sundaram [Sundaram, Thm. 1.8, and p.249],
Sundaram and Welker [SundaramWelker, Thm. 4.4(iii)] and Hersh and Reiner
[HershReiner, §2, Thm. 2.17]):

$\displaystyle
1+\sum_{n=1}^{\infty}u^{n}\sum_{k=1}^{n}\mathrm{ch}([\mathrm{VG}(\operatorname{Br}_{n})_{n-k}])t^{k}$
$\displaystyle=\sum_{\lambda=1^{m_{1}}2^{m_{2}}\cdots}u^{|\lambda|}t^{\ell(\lambda)}\prod_{j\geq
1}h_{m_{j}}[\ell_{j}]$ (81)
$\displaystyle=\prod_{m=1}^{\infty}(1-u^{m}p_{m})^{-a_{m}(t)},$ (82)
$\displaystyle
1+\sum_{n=1}^{\infty}u^{n}\sum_{k=1}^{n}\mathrm{ch}([\mathrm{OS}(\operatorname{Br}_{n})_{n-k}])t^{k}$
$\displaystyle=\sum_{\lambda=1^{m_{1}}2^{m_{2}}\cdots}u^{|\lambda|}t^{\ell(\lambda)}\prod_{j\text{odd
}}h_{m_{j}}[\pi_{j}]\prod_{\begin{subarray}{c}j\text{ even}\\\ j\geq
2\end{subarray}}e_{m_{j}}[\pi_{j}],$ (83)
$\displaystyle=\prod_{m=1}^{\infty}(1+(-u)^{m}p_{m})^{a_{m}(-t)},$ (84)

where here $a_{m}(t)\coloneqq\frac{1}{m}\sum_{d|m}\mu(d)t^{\frac{m}{d}},$ with
$\mu(d)$ the number-theoretic Möbius function. Equivalently, define for a
partition $\lambda=1^{m_{1}}2^{m_{2}}\cdots$ of $n$ (written $\lambda\vdash
n$) with $m_{i}$ parts equal to $i$, the $\mathfrak{S}_{n}$-representations
$\mathrm{OS}_{\lambda},\mathrm{VG}_{\lambda}$ whose Frobenius characteristics
are the products appearing above. Then

$\begin{array}[]{rclcccl}\mathrm{ch}\,\mathrm{VG}_{\lambda}&:=&\displaystyle\prod_{i}h_{m_{i}}[\ell_{i}],&\text{
so that
}&\mathrm{ch}(\mathrm{VG}(\operatorname{Br}_{n})_{n-k})&=&\displaystyle\sum_{\begin{subarray}{c}\lambda\vdash
n\\\ \ell(\lambda)=k\end{subarray}}\mathrm{VG}_{\lambda},\\\
\mathrm{ch}\,\mathrm{OS}_{\lambda}&:=&\displaystyle\prod_{i\
\mathrm{odd}}h_{m_{i}}[\pi_{i}]\prod_{i\
\mathrm{even}}e_{m_{i}}[\pi_{i}],&\text{ so that
}&\mathrm{ch}(\mathrm{OS}(\operatorname{Br}_{n})_{n-k})&=&\displaystyle\sum_{\begin{subarray}{c}\lambda\vdash
n\\\ \ell(\lambda)=k\end{subarray}}\mathrm{OS}_{\lambda}.\\\ \end{array}$ (85)

Thus $\mathrm{VG}_{(n)}$ is the Lie representation with
$\mathrm{ch}\mathrm{VG}_{(n)}=\ell_{n}$ mentioned above. Similarly,
$\\{\mathrm{VG}_{\lambda}\\}$ are called higher Lie characters; see Schocker
[Schocker]. Also, the last equality in (85) implies that
$\mathrm{OS}(\operatorname{Br}_{n})_{n-k}$ coincides with the
$\mathfrak{S}_{n}$-representation on the $(n-k)$th Whitney homology of the
partition lattice, $1\leq k\leq n$; see Lehrer and Solomon [LehrerSolomon,
Thm. 4.5], Sundaram [Sundaram, Thm. 1.8].

###  11.3. Data on Stirling representations of the second kind

In contrast to the above $\mathbf{k}\mathfrak{S}_{n}$-descriptions of
$A(n)_{i}$ when
$A(n)=\mathrm{OS}(\operatorname{Br}_{n}),\mathrm{VG}(\operatorname{Br}_{n})$,
for the Koszul duals $A(n)^{!}_{i}$, we currently lack formulas of this
nature, although we can tabulate $A(n)^{!}_{i}$ recursively from the
$A(n)_{i}$ using (15).

######  Question 11.4.

Are there formulas like (81), (82), (83), (84) for the duals
$\mathrm{VG}(\operatorname{Br}_{n})^{!},\mathrm{OS}(\operatorname{Br}_{n})^{!}$?

###  11.4. Branching rules for both kinds of Stirling representations

Stirling numbers of both kinds satisfy well-known recurrences, mentioned in
the Introduction:

$\displaystyle c(n,k)$ $\displaystyle=(n-1)\cdot c(n-1,k)+c(n-1,k-1)$ (86)
$\displaystyle S(n,k)$ $\displaystyle=k\cdot S(n-1,k)+S(n-1,k-1)$ (87)

Theorem 7.1 will allow us to lift these to branching rules for the Stirling
representations of both kinds. We consider here the action of
$G=\mathfrak{S}_{n}$ on the matroid and oriented matroid
$\operatorname{Br}_{n}$. In this case, the setwise
$\mathfrak{S}_{n}$-stabilizer for the modular coatom $F=E_{n-2}$ in (80) is
the subgroup $H=\mathfrak{S}_{n-1}$. Furthermore, the permutation action
$\mathcal{X}$ of $\mathfrak{S}_{n-1}$ on the set

$E_{n-1}=E\setminus F=\\{\\{1,n\\},\\{2,n\\},\ldots,\\{n-1,n\\}\\}$

and its signed permtuation action on the real vectors representing $E_{n-1}$
in the oriented matroid $\operatorname{Br}_{n}$

$\\{v_{1n},v_{2n},\ldots,v_{n-1,n}\\}=\\{e_{1}-e_{n},e_{2}-e_{n},\cdots,e_{n-1}-e_{n}\\},$

are both equivalent to the defining $\mathfrak{S}_{n-1}$-permutation
representation $\chi^{(n-1)}_{\mathrm{def}}$ via $(n-1)\times(n-1)$
permutation matrices. Translating Theorem 7.1 then immediately implies the
following.

######  Corollary 11.5.

For any field $\mathbf{k}$, the recurrences (86), (87) lift equivariantly as
follows.

  * (i)

Letting $\mathfrak{c}(n,k)$ denote either
$\mathrm{OS}(\operatorname{Br}_{n})_{n-k}$ or
$\mathrm{VG}(\operatorname{Br}_{n})_{n-k}$ as
$\mathbf{k}\mathfrak{S}_{n}$-module, the recurrence (86) lifts to two
recurrences in $R_{\mathbf{k}}(\mathfrak{S}_{n-1})$

$[\mathfrak{c}(n,k)\downarrow^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-1}}]=[\chi^{(n-1)}_{\mathrm{def}}]\cdot[\mathfrak{c}(n-1,k)]+[\mathfrak{c}(n-1,k-1)],$
(88)

reflecting two $\mathbf{k}\mathfrak{S}_{n-1}$-module exact sequences

$0\longrightarrow\mathfrak{c}(n-1,k-1)\longrightarrow\mathfrak{c}(n,k)\downarrow^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-1}}\longrightarrow\chi^{(n-1)}_{\mathrm{def}}\otimes\mathfrak{c}(n-1,k)\longrightarrow
0.$ (89)

  * (ii)

Letting $\mathcal{S}(n,k)$ denote either
$\mathrm{OS}(\operatorname{Br}_{k+1})^{!}_{n-k}$ or
$\mathrm{VG}(\operatorname{Br}_{k+1})^{!}_{n-k}$ as
$\mathbf{k}\mathfrak{S}_{k+1}$-module, the recurrence (87) lifts to two
relations in $R_{\mathbf{k}}(\mathfrak{S}_{k})$

$[\mathcal{S}(n,k)\downarrow^{\mathfrak{S}_{k+1}}_{\mathfrak{S}_{k}}]=[\chi^{(k)}_{\mathrm{def}}]\cdot[\mathcal{S}(n-1,k)\downarrow^{\mathfrak{S}_{k+1}}_{\mathfrak{S}_{k}}]+[\mathcal{S}(n-1,k-1)],$
(90)

reflecting two $\mathbf{k}\mathfrak{S}_{k}$-module exact sequences

$0\longrightarrow\chi^{(k)}_{\mathrm{def}}\otimes\mathcal{S}(n-1,k)\downarrow^{\mathfrak{S}_{k+1}}_{\mathfrak{S}_{k}}\longrightarrow\mathcal{S}(n,k)\downarrow^{\mathfrak{S}_{k+1}}_{\mathfrak{S}_{k}}\longrightarrow\mathcal{S}(n-1,k-1)\longrightarrow
0.$ (91)

######  Remark 11.6.

Proposition 2.16 implies the two versions of (88) are equivalent to those of
(90).

######  Remark 11.7.

All of the assertions Corollary 11.5 are new, as far as we know, when working
over fields $\mathbf{k}$ of positive characteristic, and (89),(90),(91) are
new even when $\mathbf{k}$ has characteristic zero. However, when $\mathbf{k}$
has characteristic zero, it turns out that (88) also follows from work of
Sundaram in [Sundaram, SuLiesup2]. For example, the relation (88) for
$\mathfrak{c}(n,k)=\mathrm{OS}(\operatorname{Br}_{n})_{n-k}$ can be deduced by
combining [Sundaram, Thm 2.2, Part (2) and Prop. 1.9]; we omit the details
here.

And it turns out that both cases of (88), when either
$\mathfrak{c}(n,k)=\mathrm{OS}(\operatorname{Br}_{n})_{n-k}$ or
$\mathrm{VG}(\operatorname{Br}_{n})_{n-k}$, follow from the symmetric function
branching result [SuLiesup2, Thm. 4.10]. In the notation there, choosing
$F=\sum_{n\geq 1}\ell_{n}$, one takes $G^{j}_{n}=h_{j}[F]|_{\deg n}$ to deduce
(88) for $\mathfrak{c}(n,k)=\mathrm{VG}(\operatorname{Br}_{n})_{n-k}$, and one
takes $G^{j}_{n}=e_{j}[F]|_{\deg n}$ to deduce (88) for
$\mathfrak{c}(n,k)=\mathrm{OS}(\operatorname{Br}_{n})_{n-k}$. We again omit
the details here.

###  11.5. Braid matroids and representation stability

Here we wish to apply the representation stability results of Section 10 to
the braid matroids $\operatorname{Br}_{n}$. A special case of the main result
of Church and Farb [ChurchFarb] shows in our language, that for each fixed
$i=0,1,2\ldots$, both sequences $\\{A(n)_{i}\\}$ where
$A(n)=\mathrm{OS}(\operatorname{Br}_{n}),\mathrm{VG}(\operatorname{Br}_{n})$
are representation stable. Hersh and Reiner [HershReiner, Thm. 1.1] pinned
down the onset of this representation stability.

######  Theorem 11.8.

For each fixed $i\geq 1$, both sequences $\\{A(n)_{i}\\}$ where
$A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ and
$\mathrm{VG}(\operatorname{Br}_{n})$ are representation stable, past $3i$ for
$\mathrm{VG}(\operatorname{Br}_{n})$ and past $3i+1$ for
$\mathrm{OS}(\operatorname{Br}_{n})$.

One then deduces the following representation stability for their Koszul
duals.

######  Corollary 11.9.

For each fixed $i\geq 1$, both sequences $\\{A(n)^{!}_{i}\\}$ where
$A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ and
$\mathrm{VG}(\operatorname{Br}_{n})$ are representation stable, past $3i$ for
$\\{\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}\\}$ and past $4i$ for
$\\{\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i}\\}$.

###### Proof.

Theorem 11.8 gives the necessary hypotheses to apply Corollary 10.6, using the
constant $c=3$ for $\\{\mathrm{VG}(\operatorname{Br}_{n})_{i}\\}$ and using
the constant $c=4$ (since $3i+1\leq 4i$) for
$\\{\mathrm{OS}(\operatorname{Br}_{n})_{i}\\}$. ∎

######  Remark 11.10.

The bounds in Corollary 11.9 happen to be tight for
$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i},\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}$
when $i=0,1,2$. To see this, one can apply Lemma 10.5 to Propositions 11.16,
11.17 and Remark 11.18 below (specifically, see equations (98), (100)) to
deduce that for $i=0,1,2$, the sequences
$\\{\mathrm{OS}(\operatorname{Br}_{n})_{i}^{!}\\}$ start to stabilize exactly
when $n\geq 4i$, and the sequences
$\\{\mathrm{VG}(\operatorname{Br}_{n})_{i}^{!}\\}$ start to stabilize exactly
when $n\geq 3i$. This suggests the following conjecture, confirmed by
Sage/Cocalc for $\mathrm{OS}(\operatorname{Br}_{n})_{i}^{!}$ up to $i=5$, and
for $\mathrm{VG}(\operatorname{Br}_{n})_{i}^{!}$ up to $i=7$.

######  Conjecture 11.11.

The bounds for onset of stability in Corollary 11.9 are tight: for $i\geq 0$,
the sequences $\\{\mathrm{OS}(\operatorname{Br}_{n})_{i}^{!}\\}$ and
$\\{\mathrm{VG}(\operatorname{Br}_{n})_{i}^{!}\\}$ start to stabilize exactly
when $n=4i$ and $n=3i$, respectively.

Since
$A(n)=\mathrm{OS}(\operatorname{Br}_{n}),\mathrm{VG}(\operatorname{Br}_{n})$
are anti-commutative and commutative, respectively, Corollary 10.10
immediately implies the following.

######  Corollary 11.12.

Letting
$A(n)=\mathrm{OS}(\operatorname{Br}_{n}),\mathrm{VG}(\operatorname{Br}_{n})$,
and defining $\mathcal{L}(n)_{i}$ by $A(n)^{!}=\mathcal{U}(\mathcal{L}(n))$,
for each fixed $i=1,2,\ldots$, the sequence $\\{\mathcal{L}(n)_{i}\\}$ is
representation stable.

######  Remark 11.13.

The case of Corollary 11.12 for $A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ also
follows from work of Church, Ellenberg and Farb [CEF, Thm. 7.3.4]. They
consider instead of $\mathcal{L}(n)$ the Malcev Lie algebra $\mathfrak{p}_{n}$
associated to the fundamental group $\pi_{1}(X)$ for the configuration space
$X=\mathrm{Conf}_{n}({\mathbb{R}}^{2})=\mathrm{Conf}_{n}({\mathbb{C}})$
considered in Section 11.1; alternatively, $X$ is the complement of the
complex braid arrangement $\mathcal{A}$ as in Theorem 9.1. These two Lie
algebras $\mathfrak{p}_{n}$ and $\mathcal{L}(n)$ coincide due to the
$1$-formality of complements of complex algebraic hypersurfaces; see, e.g.,
Suciu and Wang [SuciuWang, §6,7].

Computations in Sage/Cocalc through $i=8$ suggest the following conjecture.

######  Conjecture 11.14.

Defining $\\{\mathcal{L}(n)_{i}\\}$ by $A(n)^{!}=\mathcal{U}(\mathcal{L}(n))$,
its onset of representation stability is:

  * •

$n=2i$ for a fixed $i\geq 1$ when $A(n)=\mathrm{OS}(\operatorname{Br}_{n})$,

  * •

$n=2i$ for a fixed $i\geq 3$ when $A(n)=\mathrm{VG}(\operatorname{Br}_{n})$.

###  11.6. Near-boundary cases for Stirling representations of the second kind

Stirling numbers $S(n,k)$ of the second kind have more explicit formulas when
either $k$ or $n-k$ is small. We similarly present here more explicit
formulas, in the language of symmetric functions for the Stirling
representations

$\displaystyle\mathrm{OS}(\operatorname{Br}_{n})_{i}$
$\displaystyle=\mathcal{S}_{\mathrm{OS}}((n-1)+i,n-1),$
$\displaystyle\mathrm{VG}(\operatorname{Br}_{n})_{i}$
$\displaystyle=\mathcal{S}_{\mathrm{VG}}((n-1)+i,n-1),$

as $\mathbf{k}\mathfrak{S}_{n}$-modules, when either $i$ or $n$ is small.

Part of our motivation comes from the following observations about when
$\mathrm{OS}(M)_{i},\mathrm{VG}(\mathcal{M})_{i}$ and their Koszul duals
$\mathrm{OS}(M)^{!}_{i},\mathrm{VG}(\mathcal{M})^{!}_{i}$ turn out to be
permutation representations of their automorphism groups $G=\text{Aut}(M)$ or
$\text{Aut}(\mathcal{M})$. The discussion of Boolean matroids Section 6.1 and
low rank matroids in Section 6.2 and Proposition 6.1 showed that

  * •

$\mathrm{OS}(M)_{i},\mathrm{VG}(\mathcal{M})_{i}$ are rarely permutation
representations,

  * •

$\mathrm{VG}(\mathcal{M})^{!}_{i}$ is not always a permutation representation,

  * •

but $\mathrm{OS}(M)^{!}_{i}$ was always a $G$-permutation representation in
these previous examples.

For the braid matroids $\operatorname{Br}_{n}$, it is not always true that
$\mathrm{OS}(\operatorname{Br}_{n})_{i}$ is a permutation representation, but
the next result shows that it happens in many cases where $i$ or $n$ is small.

######  Theorem 11.15.

For $\mathbf{k}$ of characteristic zero, the
$\mathbf{k}\mathfrak{S}_{n}$-modules
$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i}=\mathcal{S}_{\mathrm{OS}}((n-1)+i,n-1)$

  * (i)

are permutation modules for $i=0,1$,

  * (ii)

are half-permutation modules for $i=2$, meaning that
$2\cdot[\mathrm{OS}(\operatorname{Br}_{n})^{!}_{2}]$ is the class of a
permutation module in $R_{\mathbf{k}}(\mathfrak{S}_{n})$,

  * (iii)

are permutation modules888And $[\mathrm{OS}(\operatorname{Br}_{n})_{i}]$ are
even $h$-positive permutation modules when $n=1,2,3$, overlapping with the
discussion in Section 6.3 on rank two matroids, as
$U_{2,3}=\operatorname{Br}_{3}$. for $n=1,2,3,4,5$.

However, both

$\displaystyle\mathcal{S}_{\mathrm{OS}}(10,5)$
$\displaystyle=\mathrm{OS}(\operatorname{Br}_{6})^{!}_{5}\text{ with }n=6,$
$\displaystyle\mathcal{S}_{\mathrm{OS}}(11,8)$
$\displaystyle=\mathrm{OS}(\operatorname{Br}_{9})^{!}_{3}\text{ with }i=3$

fail to be permutation modules, even after scaling them by positive
integers999Trevor Karn’s Burnside Solver further shows that
$\mathcal{S}_{\mathrm{OS}}(5+i,5)$ is a permutation module for $i\leq 4$ and
$i=6,7$; it fails to be one at $i=8,9,10$..

$i$$n$01234567823456789Permutation moduleNot a “fraction” of a permutation
moduleHalf-permutation module Table 1. When are
$[\mathrm{OS}(\operatorname{Br}_{n})_{i}]=\mathcal{S}_{\mathrm{OS}}((n-1)+i,n-1)$
permutation modules or “fractions” thereof?

Table 1 summarizes the results of Theorem 11.15. The proof of Theorem 11.15
and calculation of near-boundary cases for
$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i},\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}$
employs a brute-force strategy, which we outline here, giving only brief
sketches of the proofs.

Note that since all $\mathbf{k}\mathfrak{S}_{n}$-modules $U$ are self-
contragredient, one has $[U^{*}]=[U]$ in $R_{\mathbf{k}}(\mathfrak{S}_{n})$,
and so the defining recurrence (15) for Koszul modules simplifies to this:

$[A^{!}_{d}]=\sum_{i=1}^{d}(-1)^{i-1}[A_{i}]\cdot[(A^{!}_{d-i})]$ (92)

This means that if one defines

$\displaystyle g_{i}$ $\displaystyle=\mathrm{ch}\,\mathrm{OS}(n)_{i}\text{ for
}0\leq i\leq n-1,$ $\displaystyle f_{i}$
$\displaystyle:=\mathrm{ch}\,\mathrm{OS}(n)^{!}_{i}=\mathcal{S}_{\mathrm{OS}}(n-1+i,n-1)\text{
for }i=0,1,2,\ldots,$

then, with $*$ below denoting the internal (Kronecker) product of symmetric
functions, (85) lets one sometimes compute explicit formulas for the $g_{i}$
in terms of the homogeneous symmetric functions $\\{h_{\lambda}\\}$, and (92)
gives a recurrence for $f_{i}$ in terms of $f_{0},f_{1},\ldots,f_{i-1}$:

$f_{i}=\sum_{i=1}^{d}(-1)^{i-1}g_{i}*f_{d-i}$ (93)

In each of the cases below, we identify a small subset $T$ of partitions of
$n$ such that the linear span of $\\{h_{\lambda}:\lambda\in T\\}$ contains the
$f_{i}$. Further manipulation then gives the results described above, and the
precise formulas below.

###  11.7. Proof of Theorem 11.15 part (i).

Corresponding to the Stirling number formulas

$\displaystyle S(n-1,n-1)$ $\displaystyle=1,$ $\displaystyle S(n,n-1)$
$\displaystyle=\binom{n}{2},$

one has the following result, implying Theorem 11.15 part (i).

######  Proposition 11.16.

For the cases $i=0,1$, one has

$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(n-1,n-1)=\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(n-1,n-1)$
$\displaystyle=h_{n}\quad\text{ for }n\geq 1,$ (94)
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(n,n-1)$
$\displaystyle=h_{2}h_{n-2}\quad\text{ for }n\geq 2,$ (95)
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(n,n-1)$
$\displaystyle=e_{2}h_{n-2}\quad\text{ for }n\geq 2.$ (96)

###### Proof.

Equation (94) follows since
$\mathrm{OS}(\operatorname{Br}_{n})_{0}=\mathrm{VG}(\operatorname{Br}_{n})_{0}=\mathbf{k}$,
carrying the trivial $\mathfrak{S}_{n}$ representation in either case. For
(95), (96), note that (92) and (85) imply

$\displaystyle[\mathrm{OS}(\operatorname{Br}_{n})^{!}_{1}]$
$\displaystyle=[\mathrm{OS}(\operatorname{Br}_{n})_{1}]=[\mathrm{OS}(\operatorname{Br}_{n})_{(21^{n-2})}]=h_{1}[\pi_{2}]\cdot
h_{n-2}[\pi_{1}]=h_{2}h_{n-2},,$
$\displaystyle[\mathrm{VG}(\operatorname{Br}_{n})^{!}_{1}]$
$\displaystyle=[\mathrm{VG}(\operatorname{Br}_{n})_{1}]=[\mathrm{VG}(\operatorname{Br}_{n})_{(21^{n-2})}]=h_{1}[\ell_{2}]\cdot
h_{n-2}[\ell_{1}]=e_{2}h_{n-2}.\qed$

###  11.8. Proof of Theorem 11.15 part (ii).

Here we prove the curious fact that for $n\geq 7$,
$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{2}=\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$
is in fact _half_ of a permutation module.

For $n=7,8,9,10$, Sage computation with the Burnside ring shows that
$\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$ is NOT a permutation module. By running
his Burnside solver on the first formula in Proposition 11.17 below with
rational coefficients, Trevor Karn noticed positive half-integers in the data
and conjectured that two copies of $\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$
together constitute a permutation module.

######  Proposition 11.17.

One has the following decompositions as permutation modules for $n\leq 6$:

$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(3,1)$
$\displaystyle=h_{2},$
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(4,2)$
$\displaystyle=h_{3}+h_{1}^{3},$
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(5,3)$
$\displaystyle=h_{2}[h_{1}^{2}]+h_{1}^{2}h_{2}+h_{4},$
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(6,4)$
$\displaystyle=h_{1}\cdot h_{2}[h_{2}]+h_{2}(h_{3}+e_{3})+h_{2}^{2}h_{1},$
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(7,5)$
$\displaystyle=h_{2}[h_{2}h_{1}]+h_{3}^{2}+h_{4}h_{1}^{2},$

and then for $n\geq 4$ one has

$\displaystyle\mathrm{ch}\,\mathrm{OS}(\operatorname{Br}_{n})^{!}_{2}$
$\displaystyle=\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$
$\displaystyle=h_{n-2}h_{2}+h_{n-3}h_{1}^{3}+h_{n-3}h_{3}+h_{n-4}h_{2}^{2}+h_{n-4}h_{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-h_{n-3}h_{2}h_{1}-h_{n-4}h_{3}h_{1}}}$
(97)
$\displaystyle=h_{n-2}\,s_{(2)}+h_{n-3}\,(s_{(1^{3})}+s_{(2,1)}+s_{(3)})+h_{n-4}\,(s_{(2,2)}+s_{(4)})$
(98)
$\displaystyle=h_{n-2}h_{2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{1}{2}}h_{n-3}h_{1}^{3}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{1}{2}}h_{n-3}(h_{3}+e_{3})+h_{n-4}\cdot
h_{2}[h_{2}].$ (99)

So two copies of
$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{2}=\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$
together form a permutation module, with orbit stabilisers

$\\{\mathfrak{S}_{n-2}\times
S_{2},\,\,\mathfrak{S}_{n-3},\,\,\mathfrak{S}_{n-3}\times
C_{3},\,\,\mathfrak{S}_{n-4}\times I_{2}(4)\\}$

where $C_{n}$ is the cyclic group of order $n$ (generated by the $n$-cycle
$(1,2,\ldots,n)$ in $\mathfrak{S}_{n}$) and $I_{2}(n)$ is the dihedral group
of order $2n$ inside $\mathfrak{S}_{n}$ containing that same $n$-cycle.

###### Sketch of proof..

The expansion for $n=2$ is clear. For $n\geq 3$, writing $f_{i},g_{i}$ as in
(93), one finds that

$\displaystyle f_{1}$ $\displaystyle=g_{1},$ $\displaystyle f_{2}$
$\displaystyle=f_{1}*g_{1}-g_{2}.$

Using (85) and writing $\delta(S)\in\\{0,1\\}$ depending on whether statement
$S$ is false or true, one has

$\displaystyle f_{1}=g_{1}$
$\displaystyle=\mathrm{ch}\,A(n)_{1}=\mathrm{ch}\,\mathrm{OS}_{(2,1^{n-2})}=h_{n-2}\pi_{2}=h_{n-2}h_{2},$
$\displaystyle g_{2}$
$\displaystyle=h_{n-3}(h_{2}h_{1}-h_{3})\cdot\delta_{n\geq
3}+h_{n-4}(h_{3}h_{1}-h_{4})\cdot\delta_{n\geq 4}.$

Using the standard fact that
$U\otimes(V\big{\uparrow}_{H}^{G})\,\cong(U\big{\downarrow}_{H}\otimes
V)\big{\uparrow}_{H}^{G},$ for the Young subgroup
$H=\mathfrak{S}_{2}\times\mathfrak{S}_{n-2},$ and the skewing operators
$s_{(2)}^{\perp}$ and $s_{(1^{2})}^{\perp}$ as defined in Macdonald
[Macdonald, Ex. I.5.3], the expression (97) follows by routine manipulation.
Then to establish (99), we use these facts:

$\displaystyle h_{1}^{3}+2h_{3}-2h_{2}h_{1}=h_{3}+e_{3}$
$\displaystyle=\mathrm{ch}\,1\big{\uparrow}_{C_{3}}^{\mathfrak{S}_{3}},$
$\displaystyle h_{2}[h_{2}]=h_{4}+s_{(2,2)}=h_{4}+h_{2}^{2}-h_{3}h_{1}$
$\displaystyle=\mathrm{ch}\,1\big{\uparrow}_{I_{2}(4)}^{\mathfrak{S}_{4}}.\qed$

######  Remark 11.18.

A similar analysis gives the following for
$\mathrm{VG}(\operatorname{Br}_{n}^{!})_{2}$:

$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(n+1,n-1)$
$\displaystyle=h_{n-2}h_{2}+h_{n-3}(h_{3}-h_{2}h_{1}+h_{1}^{3})+h_{n-4}(h_{4}+h_{2}h_{1}^{2}-h_{3}h_{1}-h_{2}^{2}),\,n\geq
4$
$\displaystyle=h_{n-2}\,s_{(2)}+h_{n-3}\,(s_{(1^{3})}+s_{(2,1)}+s_{(3)})+h_{n-4}\,s_{(2,1,1)}.$
(100)

Also $\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(n+1,n-1)=h_{1}^{n}+h_{n}$ for
$n=3,4,$ and hence $\mathcal{S}_{\mathrm{VG}}(n+1,n-1)$ is a permutation
module for $n\leq 4$. However for $n=5,6,7,$ the Burnside solver shows that it
is not a permutation module (even after scaling), even though all character
values are nonnegative. At $n=8$ there are negative character values, so even
scaling will not result in a permutation module.

###  11.9. Proof of Theorem 11.15 part (iii).

For fixed small $k$, the general Stirling number formula

$S(n,k)={\frac{1}{k!}}\sum_{i=0}^{k}(-1)^{i}{\binom{k}{i}}(k-i)^{n}$ (101)

gives fairly simple explicit formulas for $S(n,k)$ as a function of $n$, e.g.,
for $k=1,2,3,4,5$:

$\displaystyle S(n,1)$ $\displaystyle=1$ (102) $\displaystyle S(n,2)$
$\displaystyle=\frac{1}{2}(2^{n}-2\cdot
1^{n})=2^{n-1}-1=1+2+2^{2}+\cdots+2^{n-2},$ (103) $\displaystyle S(n,3)$
$\displaystyle=\frac{1}{6}(3^{n}-3\cdot 2^{n}+3)$ (104) $\displaystyle S(n,4)$
$\displaystyle=\frac{1}{24}(4^{n}-4\cdot 3^{n}+6\cdot 2^{n}-4)$ (105)
$\displaystyle S(n,5)$ $\displaystyle=\frac{1}{120}(5^{n}-5\cdot 4^{n}+10\cdot
3^{n}-10\cdot 2^{n}+5)$ (106)

We give here analogous descriptions of the
$\mathbf{k}\mathfrak{S}_{n}$-modules
$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i},\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}$,
having dimension $S(n-1+i,n-1)$, starting101010There is little to say for
$n=1$, as $\mathfrak{S}_{1}$ is the trivial group, and
$\mathrm{OS}(\operatorname{Br}_{1})=\mathrm{VG}(\operatorname{Br}_{1})=\mathbf{k}=\mathrm{OS}(\operatorname{Br}_{1})^{!}=\mathrm{VG}(\operatorname{Br}_{1})^{!},$
and
$\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(0,0)=\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(0,0)=h_{1}.$
with $n=2,3$.

######  Proposition 11.19.

The Frobenius characteristics of the $\mathbf{k}\mathfrak{S}_{n}$-modules

$\displaystyle\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i}$
$\displaystyle=\mathcal{S}_{\mathrm{OS}}(n-1+i,n-1),$
$\displaystyle\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}$
$\displaystyle=\mathcal{S}_{\mathrm{VG}}(n-1+i,n-1)$

for $n=2,3$ have these formulas:

  1. $n=2:$

     * $\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(i+1,1)=h_{2},$

     * $\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(i+1,1)=\begin{cases}h_{2},&i\text{ even},\\\ e_{2},&i\text{ odd}.\end{cases}$

  2. $n=3:$

     * $\mathrm{ch}\,\mathcal{S}_{OS}(i+2,2)=\mathrm{ch}\,\mathcal{S}_{VG}(i+2,2)=\frac{2^{i}-1}{3}h_{1}^{3}+h_{3}$, if $i$ even,

     * $\mathrm{ch}\,\mathcal{S}_{OS}(i+2,2)=\frac{2(2^{i-1}-1)}{3}h_{1}^{3}+h_{1}h_{2}$, if $i$ odd,

     * $\mathrm{ch}\,\mathcal{S}_{VG}(i+2,2)=\frac{2(2^{i-1}-1)}{3}h_{1}^{3}+h_{1}e_{2},$ if $i$ odd.

In particular,

  * •

$\mathrm{OS}(\operatorname{Br}_{2})^{!}_{i},\mathrm{OS}(\operatorname{Br}_{3})^{!}_{i},$
are permutation modules, while

  * •

$\mathrm{VG}(\operatorname{Br}_{2})^{!}_{i},\mathrm{VG}(\operatorname{Br}_{3})^{!}_{i}$
are permutation modules for all _even_ $i$, and

  * •

When $n=2,3$ one has

$\mathrm{OS}(\operatorname{Br}_{n})^{!}_{i}\cong\begin{cases}\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}&\text{
if }i\text{ is even},\\\
\mathrm{sgn}_{n}\otimes\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i}&\text{ if
}i\text{ is odd}.\\\ \end{cases}$

###### Sketch of proof..

These all follow by induction on $i$ via the recurrences (92) and (85). ∎

######  Remark 11.20.

Note the expressions for $n=2$ are consistent with the formula $S(i+1,1)=1$
coming from (102). We claim that the expressions for $n=3$ are also consistent
with the formulas

$\displaystyle S(i+2,2)$ $\displaystyle=2^{i+1}-1$ (107)
$\displaystyle=1+2+2^{2}+\cdots+2^{i}$ (108)

coming from (103), which we illustrate here for
$\mathrm{OS}(\operatorname{Br}_{3})_{i}=\mathcal{S}(i+2,2)$. One can rewrite
it as

$\displaystyle\mathrm{ch}\,\,\mathrm{OS}(\operatorname{Br}_{3})_{i}=\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(i+2,2)$
$\displaystyle=\underbrace{\left(\frac{2^{i+1}+(-1)^{i}}{6}-\frac{1}{2}\right)}_{\text{call
this }c_{i}}\cdot h_{1}^{3}+\begin{cases}h_{3}&\text{ if }i\text{ is even},\\\
h_{2}h_{1}&\text{ if }i\text{ is odd}\end{cases}$
$\displaystyle=\frac{2^{i+1}}{6}\cdot
h_{1}^{3}+\begin{cases}h_{3}-\frac{1}{3}h_{1}^{3}&\text{ if }i\text{ is
even},\\\ h_{2}h_{1}-\frac{2}{3}h_{1}^{3}&\text{ if }i\text{ is
odd}.\end{cases}$

Since $h_{1}^{3},h_{2}h_{1},h_{3}$ correspond to
$\mathbf{k}\mathfrak{S}_{3}$-modules of dimensions $6,3,1$, one can check that
this last formula lifts (107). Interestingly, the number $c_{i}$ of copies of
the regular representation here (that is, the coefficient of $h_{1}^{3}$)
gives a sequence $0,0,1,2,5,10,21,42,85,\ldots$ for which every other term
$0,1,5,21,85,341,\ldots$ appears on the Online Encyclopedia of Integer
Sequences as OEIS A002450.

Expressions lifting (108) arise when one uses the recurrences (92) and (85),
without trying to rewrite things in terms of $h_{\lambda}$. Recalling that
$\mathcal{S}^{\lambda}$ is the irreducible $\mathbf{k}\mathfrak{S}_{n}$-module
indexed by $\lambda$, with $\mathrm{ch}\,\,\mathcal{S}^{\lambda}=s_{\lambda}$,
a Schur function. One can check that these recurrences give

$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(i+2,2)$
$\displaystyle=h_{3}+s_{(2,1)}+s_{(2,1)}^{*2}+\cdots+s_{(2,1)}^{*i}.$ (109)
$\displaystyle\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(i+2,2)$
$\displaystyle=\omega^{i}(h_{3})+s_{(2,1)}+s_{(2,1)}^{*2}+\cdots+s_{(2,1)}^{*i},$
(110)

where $\omega:\Lambda\rightarrow\Lambda$ is the involution on symmetric
functions swapping $h_{n}\leftrightarrow e_{n}$ for $n\geq 1$, corresponding
to tensoring $\mathbf{k}\mathfrak{S}_{n}$-modules by the sign character
$\mathrm{sgn}_{n}$. Bearing in mind that $h_{3},e_{3}$ correspond to
$1$-dimensional modules, while $s_{(2,1)}$ corresponds to the $2$-dimensional
reflection representation $\mathcal{S}^{(2,1)}$ of $\mathfrak{S}_{3}$, one
sees that (109), (110) lift (108). Note also that (109) is consistent with the
$n=3$ case of (59), since one has a matroid isomorphism
$\operatorname{Br}_{3}\cong U_{2,3}$.

###  11.10. The cases $\mathrm{OS}(\operatorname{Br}_{4})^{!}$ and
$\mathrm{VG}(\operatorname{Br}_{4})^{!}$

blank line

Here we show that the $\mathfrak{S}_{4}$-modules
$\mathcal{S}_{\mathrm{OS}}(n+3,3)$ are permutation modules. One observes a
periodicity in the initial expressions for
$f_{n}=\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(n+3,3)$ below.

$\begin{split}&f_{0}=h_{4}\\\ &f_{1}=h_{2}^{2}\\\
&f_{2}=h_{1}^{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-h_{2}h_{1}^{2}}}+2h_{2}^{2}+h_{4}\quad\quad=h_{2}[h_{1}^{2}]+h_{1}^{2}h_{2}+h_{4}\\\
&f_{3}=4h_{1}^{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-3h_{2}h_{1}^{2}}}+5h_{2}^{2}\quad\qquad\
=2h_{1}^{4}+2h_{2}[h_{1}^{2}]+h_{1}^{2}h_{2}+h_{2}^{2}\\\
&f_{4}=14h_{1}^{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-8h_{2}h_{1}^{2}}}+10h_{2}^{2}+h_{4}=10h_{1}^{4}+4h_{2}[h_{1}^{2}]+2h_{2}^{2}+h_{4}\\\
&f_{5}=44h_{1}^{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-18h_{2}h_{1}^{2}}}+21h_{2}^{2}\qquad=35h_{1}^{4}+9h_{2}[h_{1}^{2}]+3h_{2}^{2}\\\
&f_{6}=135h_{1}^{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-39h_{2}h_{1}^{2}}}+42h_{2}^{2}+h_{4}=115h_{1}^{4}+20h_{2}[h_{1}^{2}]+2h_{2}^{2}+h_{2}h_{1}^{2}+h_{4}\\\
&f_{7}=408h_{1}^{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-81h_{2}h_{1}^{2}}}+85h_{2}^{2}\qquad=367h_{1}^{4}+41h_{2}[h_{1}^{2}]+3h_{2}^{2}+h_{2}h_{1}^{2}\end{split}$
(111)

######  Proposition 11.21.

The Frobenius characteristic $\mathrm{ch}\,\mathcal{S}_{OS}(n+3,3)$ is an
integer combination of $\\{h_{1}^{4},h_{1}^{2}h_{2},h_{2}^{2},h_{4}\\}$.

Let
$\mathbf{\mathrm{ch}\,\mathcal{S}_{OS}(n+3,3)=a_{n}h_{1}^{4}+b_{n}h_{1}^{2}h_{2}+c^{\prime}_{n}h_{2}^{2}+d_{n}h_{4}},\,n\geq
0$. Let $G_{2}$ be the subgroup of order 2 generated by $(12)(34)$.Then
$\mathcal{S}_{OS}(n+3,3)$ is a permutation module with orbit stabilisers
consisting of the wreath product $\mathfrak{S}_{2}[G_{2}]$, as well as a
subset of the Young subgroups $\mathfrak{S}_{\lambda}$,
$\lambda\in\\{(1^{4}),(2,1^{2}),(2^{2}),(4)\\}$. We have, for
$a_{n},c^{\prime}_{n},d_{n}\geq 0$ and $b_{n}<0$,

$\displaystyle\mathrm{ch}\,\mathcal{S}_{OS}(n+3,3)$
$\displaystyle=(a_{n}+\frac{b_{n}}{2})h_{1}^{4}-\frac{b_{n}}{2}h_{2}[h_{1}^{2}]+(c^{\prime}_{n}+b_{n})h_{2}^{2}+d_{n}h_{4},\
n\equiv 0,1\bmod 4,$ (112)
$\displaystyle=(a_{n}+\frac{b_{n}-1}{2})h_{1}^{4}-\frac{b_{n}-1}{2}h_{2}[h_{1}^{2}]+h_{1}^{2}h_{2}+(c^{\prime}_{n}+b_{n}-1)h_{2}^{2}+d_{n}h_{4},\
n\equiv 2,3\bmod 4.$ (113)

The coefficients $a_{n},b_{n},c^{\prime}_{n},d_{n}$ are determined below.

  1. (1)

The coefficients of $h_{4}$ are the sequence
$d_{n}=\frac{1+(-1)^{n}}{2},\,n\geq 0.$

  2. (2)

The coefficients of $h_{2}^{2}$ are $\\{0,0,1,2,5,10,21,\ldots\\}$, i.e. the
numbers $c_{n}$ from Remark 11.20. More precisely,
$c^{\prime}_{n}=c_{n+3}=\frac{2^{n+2}-3+(-1)^{n+3}}{6},\,n\geq 0.$

  3. (3)

$b_{n}-b_{n-1}=-c_{n+2}=-\frac{2^{n+1}-3+(-1)^{n}}{6},n\geq 1,$ with
$b_{0}=0=b_{1}$. In particular, $b_{n}<0$ for $n\geq 2$. We have

$b_{n}=-\frac{2}{3}(2^{n}-1)+\frac{n}{2}+\frac{1}{12}(1-(-1)^{n})=-\frac{2^{n+1}}{3}+\frac{n}{2}+\frac{3}{4}-\frac{1}{12}(-1)^{n},\,n\geq
0.$

Thus $\mathcal{S}_{OS}(n+3,3)$ is NOT $h$-positive for $n\geq 2$.

  4. (4)

The coefficients $a_{n}$ are all nonnegative, and strictly positive if $n\geq
2$. We have $a_{0}=a_{1}=0$ and for $n\geq 2$,

$a_{n}=\frac{3^{n+1}}{16}-\frac{n}{4}-\frac{4-(-1)^{n}}{16}$

In particular, $a_{n}+\frac{b_{n}}{2}$, and $a_{n}+\frac{b_{n}-1}{2}$ are
positive integers for $n\geq 3$.

Also, $a_{n}$ is the multiplicity of the sign representation.

######  Remark 11.22.

Computing dimensions shows that the $h$-expansion of
$\mathrm{ch}\,\mathcal{S}(n+3,3)$ lifts the formula (104) for the Stirling
number $S(n+3,3)$.

The coefficient $a_{n}$ of $h_{1}^{4}$ gives the sequence
$0,0,1,4,14,44,135,408,1228,\ldots$, appearing as OEIS A097137. (One checks
that $a_{n}-a_{n-2}=(3^{n-1}-1)/2$.) Also the negative of the coefficient
$b_{n}$ of $h_{1}^{2}h_{2}$ gives $0,0,1,3,8,18,39,81,166,336,677,\ldots$,
which is OEIS A011377 or OEIS A178420.

A similar analysis for $\mathrm{VG}(\operatorname{Br}_{4})_{i}^{!}$ shows that
its Frobenius characteristic $\mathrm{ch}\,\mathcal{S}_{\mathrm{VG}}(3+i,3)$
is also an integer combination of
$\\{h_{1}^{4},h_{1}^{2}h_{2},h_{2}^{2},h_{4}\\}$, in fact of
$\\{h_{1}^{4},h_{2}e_{2},h_{4}\\}$.

Here is the data for $f_{i}=\mathrm{ch}\,\mathrm{VG}^{!}(4)_{i}$ with $0\leq
i\leq 11$:

$\displaystyle f_{0}$ $\displaystyle=h_{4}$ $\displaystyle f_{1}$
$\displaystyle=h_{2}e_{2}$ $\displaystyle f_{2}$
$\displaystyle=h_{1}^{4}+h_{4}$ $\displaystyle f_{3}$
$\displaystyle=4\,h_{1}^{4}-\,h_{2}e_{2}$ $\displaystyle f_{4}$
$\displaystyle=12\,h_{1}^{4}+2\,h_{2}e_{2}+h_{4}$ $\displaystyle f_{5}$
$\displaystyle=40\,h_{1}^{4}+\,h_{2}e_{2}$ $\displaystyle f_{6}$
$\displaystyle=127\,h_{1}^{4}-4\,h_{2}e_{2}+h_{4}$ $\displaystyle f_{7}$
$\displaystyle=388\,h_{1}^{4}+3\,h_{2}e_{2}$ $\displaystyle f_{8}$
$\displaystyle=1186\,h_{1}^{4}+6\,h_{2}e_{2}+h_{4}$ $\displaystyle f_{9}$
$\displaystyle=3608\,h_{1}^{4}-11\,h_{2}e_{2}$ $\displaystyle f_{10}$
$\displaystyle=10901\,h_{1}^{4}+h_{4}$ $\displaystyle f_{11}$
$\displaystyle=32868\,h_{1}^{4}+23\,h_{2}e_{2}$

Observe that the set $\\{h_{1}^{4},h_{2}e_{2},h_{4}\\}$ is linearly
independent. One then has the following more precise statement:

######  Proposition 11.23.

Write $f_{n}$ for $\mathrm{ch}\,\mathcal{S}_{VG}(n+3,3)$. Then
$f_{2n-1},f_{2n}-h_{4}\in{\mathbb{Z}}[h_{1}^{4},h_{2}e_{2}]$ and hence for
$n\geq 0$, both the representation $\mathcal{S}_{VG}(2n+2,3)$ and the quotient
representation $\mathcal{S}_{VG}(2n+3,3)/\mathbb{1}_{\mathfrak{S}_{4}}$ are
fixed under tensoring with the sign representation $\mathrm{sgn}$ of
$\mathfrak{S}_{4}$.

Let $f_{n}=a_{n}h_{1}^{4}+b_{n}h_{2}e_{2}+d_{n}h_{4}.$ Then, with initial
values $a_{0}=a_{1}=0,a_{2}=1,a_{3}=4$, $b_{0}=0,b_{1}=1,b_{2}=0,b_{3}=-1$,
one has that $d_{n}=\frac{1+(-1)^{n}}{2}$, $a_{n}\geq 0$ for all $n\geq 0$,
and for $n\geq 3$:

$\displaystyle a_{n}$
$\displaystyle=6a_{n-1}-11a_{n-2}+6a_{n-3}+2(b_{n-1}-b_{n-2}+b_{n-3})-d_{n-2},$
$\displaystyle b_{n}$ $\displaystyle=-b_{n-2}+2b_{n-3}.$

The sequence $\\{b_{n}\\}_{n\geq 0}$ appears in OEIS A077912, with generating
function $\frac{x}{1+x^{2}-2x^{3}}$.

Moreover $\mathcal{S}_{VG}(n+3,3)$ is a permutation module if and only if
$b_{n}=0$ or $b_{n}\leq-2$. Write $\ -b_{n}=2\alpha_{n}+3\beta_{n}$ for
nonnegative integers $\alpha_{n},\beta_{n}$. Then
$a_{n}-(\alpha_{n}+\beta_{n})$ is nonnegative and

$f_{n}=(a_{n}-(\alpha_{n}+\beta_{n}))h_{1}^{4}+\alpha_{n}\,\mathrm{ch}\,(1\big{\uparrow}_{G_{2}}^{\mathfrak{S}_{4}})+\beta_{n}\,\mathrm{ch}\,(1\big{\uparrow}_{V_{4}}^{\mathfrak{S}_{4}})+d_{n}h_{4}$

is the Frobenius characteristic of a permutation module, where the orbit
stabilisers are $\mathfrak{S}_{1}$, $\mathfrak{S}_{4}$ and the subgroups
$G_{2}=\langle(12)(34)\rangle$ and
$V_{4}=\\{(1),(12)(34),(13)(24),(14)(23)\\}$ of $\mathfrak{S}_{4}$.

###  11.11. The case $\mathrm{OS}(\operatorname{Br}_{5})^{!}$

blank line

In this section we show that the $\mathfrak{S}_{5}$-modules
$\mathcal{S}_{\mathrm{OS}}(n+4,4)$ are also permutation modules. We also show
that the $h$-expansions exhibit a curious periodicity modulo 4.

The initial expressions for
$f_{n}=\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(n+4,4)$ are as follows.

$\begin{split}f_{0}&=h_{5},\ f_{1}=h_{3}h_{2},\\\
f_{2}&=h_{4}h_{1}+h_{2}h_{1}^{3}+2h_{3}h_{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-h_{3}h_{1}^{2}}}=h_{1}h_{2}[h_{2}]+h_{2}(h_{3}+e_{3})+h_{2}^{2}h_{1},\\\
f_{3}&=2h_{1}^{5}+3h_{2}^{2}h_{1}+2h_{3}h_{2},\\\
f_{4}&=12h_{1}^{5}+8h_{2}^{2}h_{1}+2h_{3}h_{2}+h_{5}\\\
f_{5}&=60h_{1}^{5}+18h_{2}^{2}h_{1}+3h_{3}h_{2}\\\
f_{6}&=274h_{1}^{5}+38h_{2}^{2}h_{1}+h_{2}h_{1}^{3}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{-h_{3}h_{1}^{2}}}+4h_{3}h_{2}+h_{4}h_{1}\\\
&=(274h_{1}^{5}+38h_{2}h_{1}^{2}+2h_{3}h_{2})+f_{2}\\\
f_{7}&=1194h_{1}^{5}+81h_{2}^{2}h_{1}+4h_{3}h_{2}.\end{split}$ (114)

######  Proposition 11.24.

The $\mathfrak{S}_{5}$-module $\mathcal{S}_{\mathrm{OS}}(n+4,4)$ is a
permutation module for all $n\geq 0$, with orbit stabilisers given by

  * •

the Young subgroups $\mathfrak{S}_{\lambda}$ for
$\lambda\in\\{(1^{5}),(2^{2},1),(3,2)\\}$ if $\mathbf{n\equiv 1,3\bmod 4}.$

  * •

the Young subgroup $\mathfrak{S}_{(2^{2},1)}$, as well as the subgroups
$\mathfrak{S}_{1}\times I_{2}(4),A_{3}\times\mathfrak{S}_{2}$ if
$\mathbf{n\equiv 2\bmod 4}.$

Here $A_{3}\times\mathfrak{S}_{2}$ is the subgroup of the Young subgroup
$\mathfrak{S}_{3}\times\mathfrak{S}_{2}$, for the alternating subgroup $A_{3}$
of $\mathfrak{S}_{3}$.

  * •

the Young subgroups $\mathfrak{S}_{\lambda}$ for
$\lambda\in\\{(1^{5}),(2^{2},1),(3,2),(5)\\}$ if $\mathbf{n\equiv 0\bmod 4}.$

Let $J=\\{h_{1}^{5},h_{2}^{2}h_{1},h_{3}h_{2}\\}$. Let
$f_{n}=\mathrm{ch}\,\mathcal{S}_{OS}(n+4,4)=\mathrm{ch}\,A^{!}(5)_{n}$. Then

  1. (1)

$f_{n}$ is a nonnegative integer combination of the set $J$ if $n\equiv
1,3\bmod 4.$

  2. (2)

$f_{n}-f_{2}$ is a nonnegative integer combination of $J$ if $n\equiv 2\bmod
4.$

  3. (3)

$f_{n}-f_{0}$ is a nonnegative integer combination of $J$ if $n\equiv 0\bmod
4.$

The following explicit decomposition holds for $f_{n+4}-f_{n}$:

$f_{n+4}-f_{n}=a_{n}h_{1}^{5}+b_{n}h_{1}h_{2}^{2}+2h_{2}h_{3},\ n\geq 0,$
(115)

where $b_{0}=8$, $b_{n}=10(2^{n})-2,\,n\geq 1,$ and

$a_{n}=\frac{1}{3}(1+17\cdot 4^{n+1}-3\cdot 2^{n+1}-3^{n+3}).$ (116)

Let $0\leq i\leq 3$ and $k\geq 0$. Then

$\begin{split}&f_{4k+4+i}-f_{i}=\alpha_{k,i}h_{1}^{5}+\beta_{k,i}h_{2}^{2}h_{1}+2(k+1)h_{2}h_{3}\\\
&\text{where}\\\
&\alpha_{k,i}=\frac{k+1}{3}+4^{i+1}\frac{256^{k+1}-1}{45}-3^{i+2}\frac{81^{k+1}-1}{80}-2^{i+1}\frac{16^{k+1}-1}{15},\\\
&\beta_{k,i}=2^{i+1}\frac{16^{k+1}-1}{3}-2(k+1).\end{split}$ (117)

The multiplicity of the sign representation in $f_{n}$ is
$\begin{cases}\alpha_{k,i},&n=4(k+1)+i\text{ and }k\geq 0,\\\ 2,&n=3,\\\
0,&n<3.\end{cases}$

######  Remark 11.25.

(The restriction of $\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$ and
$\mathcal{S}_{\mathrm{VG}}(n+1,n-1)$ to $\mathfrak{S}_{n-1}$)

Observe that in each of the cases $\mathrm{OS}(\operatorname{Br}_{n})$, $1\leq
n\leq 5$, the restriction of the $\mathfrak{S}_{n}$-module to
$\mathfrak{S}_{n-1}$ is always an $h$-positive permutation module. The
restriction is not $h$-positive for $\mathcal{S}(n+1,n-1)$ when $n\geq 5$,
although the following formula shows that it is a permutation module.

$\mathrm{ch}\,\mathcal{S}(n+1,n-1)\downarrow^{\mathfrak{S}_{n}}_{\mathfrak{S}_{n-1}}=(h_{n-2}h_{1}+h_{n-3}h_{2}+2h_{n-3}h_{1}^{2}+h_{n-4}h_{1}^{3}+h_{n-4}h_{3})\delta_{n\geq
4}+h_{n-5}\,\mathrm{ch}\,\mathbb{1}\uparrow_{I_{2}(4)}^{\mathfrak{S}_{4}}\delta_{n\geq
5}.$

Here $I_{2}(4)$ is the dihedral group of order 8.

######  Remark 11.26.

(The restriction of $\mathcal{S}_{\mathrm{OS}}(n+3,3)$ and
$\mathcal{S}_{\mathrm{VG}}(n+3,3)$) With the coefficients defined in
Proposition 11.21, the restriction of $\mathcal{S}_{OS}(n+3,3)$ to
$\mathfrak{S}_{3}$ has Frobenius characteristic

$(4a_{n}+b_{n})h_{1}^{3}+2(b_{n}+c^{\prime}_{n})h_{1}h_{2}+d_{n}h_{3},$

and is thus $h$-positive. In particular
$\mathcal{S}_{\mathrm{OS}}(n+3,3)\big{\downarrow}_{\mathfrak{S}_{3}}$ is a
permutation module whose point stabilisers are Young subgroups.

Proposition 11.23 shows that a similar statement holds for
$\mathcal{S}_{\mathrm{VG}}(n+3,3)\big{\downarrow}_{\mathfrak{S}_{3}}$ ; here
the orbit stabilisers are $\mathfrak{S}_{1}$ and $\mathfrak{S}_{3}$.

######  Remark 11.27.

(The restriction of $\mathcal{S}_{\mathrm{OS}}(n+4,4)$) The restriction of
$f_{n}=\mathrm{ch}\,\mathcal{S}_{\mathrm{OS}}(n+4,4)$ to $\mathfrak{S}_{4}$ is
$h$-positive, supported on the set $\\{h_{1}^{4},h_{1}^{2}h_{2},h_{2}^{2}\\}$
if $n\equiv 1,3\bmod 4$, the set
$\\{h_{1}^{4},h_{1}^{2}h_{2},h_{2}^{2},h_{4}\\}$ if $n\equiv 0\bmod 4$, and
finally the set $\\{h_{1}^{4},h_{1}^{2}h_{2},h_{2}^{2},h_{3}h_{1},h_{4}\\}$ if
$n\equiv 2\bmod 4$. In particular
$\mathcal{S}_{\mathrm{OS}}(n+4,4)\big{\downarrow}_{\mathfrak{S}_{4}}$ is a
permutation module whose point stabilisers are Young subgroups.

######  Remark 11.28.

(The multiplicity of the trivial representation) Here we collect formulas for
the multiplicity of the trivial representation:

For $\mathcal{S}_{\mathrm{OS}}(n+1,n-1)$, the multiplicity of the trivial
representation is 3 for $n\geq 4$, and the multiplicity of the sign
representation is 0 for $n\neq 3,4$, and 1 otherwise.

For $\mathcal{S}_{OS}(n+2,2)$, the multiplicity is

$\frac{2^{n+1}}{6}+\frac{3+(-1)^{n}}{6}.$

For $\mathcal{S}_{\mathrm{OS}}(n+3,3)$, the multiplicity of the trivial
representation is

$\frac{3^{n+1}}{16}+\frac{n}{4}+\frac{8+5(-1)^{n}}{16},$

giving $\\{1,1,3,6,17,47,139,412,\ldots\\}$.

For $\mathcal{S}_{\mathrm{OS}}(n+4,4)$, (with definitions as in Proposition
11.24), the multiplicity of the trivial representation is

$\frac{k+1}{3}+4^{i+1}\frac{256^{k+1}-1}{45}-3^{i+2}\frac{81^{k+1}-1}{80}+2^{i+3}\frac{16^{k+1}-1}{5}+\langle
f_{i},h_{5}\rangle.$

##  12\. Further remarks and questions

We remark here on some further directions which could merit exploration.

###  12.1. Projective geometries over finite fields

The Boolean matroids $U_{n,n}$ discussed in Example 4.5 and Section 6.1 have a
well-studied “$q$-analogue”: the projective geometries $PG(n,q)$, associated
with the finite vector spaces ${\mathbb{F}}_{q}^{n}$. These $PG(n,q)$ are non-
orientable simple matroids whose ground set
$E={\mathbb{P}}({\mathbb{F}}_{q}^{n})={\mathbb{P}}^{n-1}_{{\mathbb{F}}_{q}}$
is the set of points in a finite projective space, so
$|E|=[n]_{q}:=1+q+q^{2}+\cdots+q^{n-1}$, with poset of flats $\mathcal{F}$
given by the lattice of all subspaces in ${\mathbb{F}}_{q}^{n}$; see Oxley
[Oxley, §6.1] and Orlik and Terao [OrlikTerao, Example 4.33]. The lattices
$\mathcal{F}$ are modular, meaning that every flat is a modular element, so
that every complete flag $\underline{F}$ is a modular complete flag. Hence the
matroids $PG(n,q)$ are supersolvable, with exponents
$(e_{1},\ldots,e_{n})=(1,q,q^{2},\ldots,q^{n-1})$. Consequently, the family of
$\mathbf{k}$-algebras $A(n):=\mathrm{OS}(PG(n,q))$ are Koszul, satisfying

$\displaystyle\operatorname{Hilb}(A(n),t)$
$\displaystyle=(1+t)(1+qt)(1+q^{2}t)\cdots(1+q^{n-1}t)\quad\text{ with
}\dim_{\mathbf{k}}A(n)^{!}_{i}=q^{\binom{i}{2}}\left[\genfrac{}{}{0.0pt}{}{n}{i}\right]_{q}$
$\displaystyle\operatorname{Hilb}(A(n)^{!},t)$
$\displaystyle=\frac{1}{(1-t)(1-qt)(1-q^{2}t)\cdots(1-q^{n-1}t)}\quad\text{
with
}\dim_{\mathbf{k}}A(n)^{!}_{i}=\left[\genfrac{}{}{0.0pt}{}{n+i-1}{i}\right]_{q},$

where
$\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]_{q}:=\frac{[n]!_{q}}{[k]!_{q}[n-k]!_{q}}$
with $[n]!_{q}:=[n]_{q}[n-1]_{q}\cdots[2]_{q}[1]_{q}$; see Macdonald
[Macdonald, Example I.2.2].

######  Problem 12.1.

Study $A(n)=\mathrm{OS}(PG(n,q))$ and $A(n)^{!}=\mathrm{OS}(PG(n,q))^{!}$ as
$GL_{n}({\mathbb{F}}_{q})$-representations.

For example, the $q$-Pascal recurrences for
$A(n)_{i}=\left[\genfrac{}{}{0.0pt}{}{n}{i}\right]_{q}$ and
$A(n)^{!}_{i}=\left[\genfrac{}{}{0.0pt}{}{n+i-1}{i}\right]_{q}$ will have
lifts to branching rules via Proposition 2.16 and Theorem 7.1. There is also
an appropriate analogue here of representation stability for
$GL_{n}({\mathbb{F}}_{q})$-representations developed by Putman and Sam
[PutmanSam].

###  12.2. Type $B$, wreath products, and Dowling geometries

As mentioned in Section 11, the braid matroids $\operatorname{Br}_{n}$ are
represented by the root systems of type $A_{n-1}$, accounting for the action
of the reflection group $\mathfrak{S}_{n}$ on them as symmetries.

There are other real and complex reflection groups giving rise to matroids
with large symmetry, but relatively few of these matroids are supersolvable;
see Hoge and Röhrle [HogeRohrle] for their classification. They include the
dihedral reflection groups giving rise to the rank two matroids already
discussed in Example 4.6 and Section 6.3. They also include the reflection
groups of type $B_{n}$ or $C_{n}$, isomorphic to the hyperoctahedral group or
signed permutation group $\mathfrak{S}^{\pm}_{n}$ that appeared in Section
5.2. Their root systems can be realized over ${\mathbb{R}}$, giving rise to an
oriented matroid from the positive roots

$\Phi_{B_{n}}^{+}:=\\{+e_{i}\pm e_{j}\\}_{1\leq i<j\leq
n}\,\,\sqcup\,\,\\{e_{i}\\}_{1\leq i\leq n}.$ (118)

More generally, one has the complex reflection groups
$\mathfrak{S}_{n}[{\mathbb{Z}}/m{\mathbb{Z}}]=({\mathbb{Z}}/m{\mathbb{Z}})\wr\mathfrak{S}_{n}$
for $m\geq 2$, also known as the groups $G(m,1,n)$ within Shephard and Todd’s
classification [ShephardTodd] of irreducible complex reflection groups.
Letting $\zeta_{m}:=e^{\frac{2\pi i}{m}}$, their associated matroids can be
represented by this list of vectors in ${\mathbb{C}}^{n}$:

$\\{e_{i}-\zeta^{k}e_{j}\\}_{\begin{subarray}{c}1\leq i<j\leq n\\\ 0\leq k\leq
m-1\end{subarray}}\,\,\sqcup\,\,\\{e_{i}\\}_{1\leq i\leq n}.$ (119)

These matroids are not realizable over ${\mathbb{R}}$ (and not orientable)
unless $m=2$ where they recover the type $B_{n}/C_{n}$ reflection groups.

Motivated by these examples, Dowling [Dowling] introduced a more general class
of matroids, now known as the Dowling geometries ${Q_{n}(G)}$; see Oxley
[Oxley, §6.10] for definitions and discussion. Here $G$ is any finite group,
and the matroid automorphisms of ${Q_{n}(G)}$ contain the wreath product
$\mathfrak{S}_{n}[G]=G\wr\mathfrak{S}_{n}$. Interestingly, Dowling proved that
the matroid ${Q_{n}(G)}$ is representable over a field ${\mathbb{F}}$ if and
only if the finite group $G$ is a subgroup of ${\mathbb{F}}^{\times}$; in
particular, this forces $G$ to be cyclic, as in the complex reflection groups
$\mathfrak{S}_{n}[{\mathbb{Z}}/m{\mathbb{Z}}]$ mentioned above.

Dowling also showed that ${Q_{n}(G)}$ is supersolvable for any finite group
$G$. Consequently, their Orlik-Solomon algebras $\mathrm{OS}({Q_{n}(G)})$ are
always Koszul, and when $|G|=2$, the same holds for the Varchenko-Gel’fand
ring $\mathrm{VG}(\mathcal{M}(B_{n}))$, e.g., if $\mathcal{M}(B_{n})$ is
realized by the vectors in (118) above.

######  Problem 12.2.

Study these families of Koszul algebras $A(n)=\mathrm{OS}({Q_{n}(G)})$ and
$\mathrm{VG}(\mathcal{M}(B_{n}))$, along with their Koszul duals $A(n)^{!}$,
as $\mathfrak{S}_{n}[G]$-representations.

If $m:=|G|$, then the exponents for the supersolvable matroids ${Q_{n}(G)}$
turn out to be

$(e_{1},e_{2},\ldots,e_{n})=(1,m+1,2m+1,\ldots,(n-1)m+1).$

Combining this with Dowling’s formulas [Dowling, §4], for the rank
sizes111111Also called the Whitney numbers of the second kind for the poset.
in the poset of flats of ${Q_{n}(G)}$, one encounters a similar coincidence to
the equality
$\dim_{\mathbf{k}}\mathrm{OS}(\operatorname{Br}_{n})_{i}=S((n-1)+i,n-1)$
discussed in Remark 11.2: the dimension of $\mathrm{OS}({Q_{n}(G)})^{!}_{i}$
is the size of the $(n-1)^{st}$ rank in the flat poset of ${Q_{(n-1)+i}(G)}$.
This again reflects a bijection between the standard monomial
$\mathbf{k}$-basis for $\mathrm{OS}({Q_{n}(G)})^{!}_{i}$ from Theorem 5.18 and
an encoding of flats in ${Q_{n}(G)}$ generalizing restricted growth functions,
similar to work of Komatsu, Bagno, and Garber [BagnoGarberKomatsu, §2.3]. We
omit the details here.

###  12.3. Equivariant degree one injections

Recall the following consequences of Theorem 5.21: By Part (ii) of Corollary
5.22, for the matroid automorphism group $G=\text{Aut}(M)$, there are
$G$-equivariant degree one injections

$[\mathrm{OS}(M)^{!}_{i}]\hookrightarrow[\mathrm{OS}(M)^{!}_{i+1}],\text{ for
all }i\geq 0$ (120)

while Part (iii) of Corollary 5.22 asserts that for the full oriented matroid
automorphism group $G=\text{Aut}(\mathcal{M})$, there are $G$-equivariant
degree _two_ injections

$[\mathrm{VG}(\mathcal{M})^{!}_{i}]\hookrightarrow[\mathrm{VG}(\mathcal{M})_{i+2}^{!}],\text{
for all }\ i\geq 0.$

The latter injections arise from right-multiplication by a degree two
$G$-invariant $\underline{E}$-generic power sum $p_{2}(\mathbf{y})$, such as
$p_{2}(\mathbf{y})=\sum_{i}y_{i}^{2}$. Unfortunately, for some oriented
matroids $\mathcal{M}$, there are no degree one $\underline{E}$-generic
element power sums $p_{1}(\mathbf{y})$ in $A^{!}_{1}$ that are also
$G$-invariant. For example, $p_{1}(\mathbf{y})=\sum_{i}y_{i}$ is not always
$G$-invariant. In fact, the calculation for rank one oriented matroids
$\mathcal{M}=U_{1,1}$ in (53) show that in that case, there are no
$G$-equivariant injections
$\mathrm{VG}(\mathcal{M})^{!}_{i}\hookrightarrow\mathrm{VG}(\mathcal{M})_{i+1}^{!}$
for any $i$.

Nonetheless, for the braid matroids $\mathcal{M}=\operatorname{Br}_{n}$, Sage
calculations for $n\leq 10$ and $1\leq i\leq 9$ support the following
conjecture.

######  Conjecture 12.3.

For the braid oriented matroid $\mathcal{M}=\operatorname{Br}_{n}$, there
exist equivariant injections

$\displaystyle[\mathcal{S}_{\mathrm{VG}}((n-1)+i,n-1)]=[{\mathrm{VG}}(\operatorname{Br}_{n})^{!}_{i}]$
$\displaystyle\quad\hookrightarrow[\mathcal{S}_{\mathrm{VG}}((n-1)+i+1,n-1)]=[\mathrm{VG}(\operatorname{Br}_{n})^{!}_{i+1}],\text{
for all }i\geq 1.$

Propositions 11.19 and 11.23 establish Conjecture 12.3 for $n\leq 4$ in
characteristic zero.

We close with some observations on a consequence of the $G$-equivariant
injections in (120): they imply that the following alternating sum in
$R_{\mathbf{k}}(G)$ is always the class of a genuine $\mathbf{k}G$-module:

$[\mathrm{OS}^{!}(M)_{i}]-[\mathrm{OS}^{!}(M)_{i-1}]+\ldots+(-1)^{i-1}[\mathrm{OS}^{!}(M)_{0}]$
(121)

######  Problem 12.4.

Investigate the genuine $\mathbf{k}G$-modules (121). Do they have interesting
descriptions?

For example, for the braid matroid $M=\operatorname{Br}_{n}$, the dimension of
the genuine module (121) is

$S(n\\!-\\!1\\!+\\!i,n\\!-\\!1)-S(n\\!-\\!2\\!+\\!i,n\\!-\\!1)+\cdots+(-1)^{i}S(n\\!-\\!1\\!,n\\!-\\!1).$
(122)

This has an interpretation via a result of Mansour and Munagi
[MansourMunagi2014, Corollary 11]: it is the number of set partitions of
$\\{1,2,\ldots,n+i\\}$ into $n$ blocks, where no block contains a pair $j,j+1$
modulo $n+i$ for $1\leq j\leq n+i$. We know of no accompanying
$\mathbf{k}\mathfrak{S}_{n}$-module built from these objects.

We remark that for any matroid $M$, the alternating sum analogous to (121) for
$\mathrm{OS}(M)$, namely

$[\mathrm{OS}(M)_{i}]-[\mathrm{OS}(M)_{i-1}]+\ldots+(-1)^{i-1}[\mathrm{OS}(M)_{0}]$
(123)

is always a genuine $\mathbf{k}G$-module for $G=\text{Aut}(M)$, isomorphic to
the top homology of a rank-selected subposet of the lattice of flats. We
quickly sketch how this follows from combining these two results:

  * •

[OrlikSolomon] exhibits a $\mathbf{k}G$-module isomorphism
$\mathrm{OS}(M)\cong\text{\bf Whit}(\mathcal{L}_{M})$, where $\text{\bf
Whit}(\mathcal{L}_{M})$ is the Whitney homology of the lattice of flats
$\mathcal{L}_{M}$ of $M$, and

  * •

[Sundaram] If $G=\text{Aut}(P)$ for a Cohen-Macaulay poset $P$, then the
alternating sum in ${\mathbb{R}}_{\mathbf{k}}(G)$

$[\text{\bf Whit}_{i}(P)]-[\text{\bf
Whit}_{i-1}(P)]+\ldots+(-1)^{i-1}[\text{\bf Whit}_{0}(P)]$

is $\mathbf{k}G$-isomorphic to the top homology of the rank-selected subposet
of $P$ consisting of the bottom $i$ nonzero ranks. The Hopf trace argument in
[Sundaram, Lemma 1.1], written for characteristic zero, can be replaced by
applying, for an arbitrary field $\mathbf{k}$, Proposition 2.11(ii) to
Baclawski’s complex. Similarly the arguments of Baclawski and Björner as cited
in [Sundaram, Theorem 1.2] can be adapted for any field $\mathbf{k}$, by
appealing to the isomorphism in [bjorner1992homology, p. 262, Theorem 7.9.6].
Finally, the equivariant isomorphism with $\mathrm{OS}(M)$ follows from
[bjorner1992homology, Theorem 7.10.2], extending the argument of
[OrlikSolomon, Theorem 4.3] to the whole Orlik-Solomon algebra.

## Acknowledgements

The authors thank Shiyue Li, Ivan Marin, Keller VandeBogert, Peter Webb, and
Craig Westerland for helpful conversations and references. They are grateful
to Trevor Karn for his wonderful Sage/Cocalc code that checks whether a
symmetric group representation is isomorphic to a permutation representation,
which helped us create Table 1. In particular, we are grateful to him for
experimentally discovering the half-integers in the decomposition of
Proposition 11.17. First and second authors received partial support from NSF
grants DMS-2053288 and DMS-1745638.

##  Appendix A Tables of characters for Stirling representations

This section consists of several tables of characters for $A(n)_{i}$ and its
primitives $\mathcal{L}(A(n))_{i}$ when
$A(n)=\mathrm{OS}(\operatorname{Br}_{n})$ or
$\mathrm{VG}(\operatorname{Br}_{n})$. For each table, the observed onset of
representation stability in each column is shaded in blue. All data was
generated using SAGE code which is publicly available at
[Almousa_StirlingRepresentations_2024].

$n$ $i$ 0 |  | 1 | 2 | 3
---|---|---|---|---
3 |  | $s_{2,1}+s_{3}$ | $s_{1,1,1}+2s_{2,1}+2s_{3}$ | $2s_{1,1,1}+5s_{2,1}+3s_{3}$
4 |  | $s_{2,2}+s_{3,1}+s_{4}$ | $\begin{gathered}s_{1,1,1,1}+2s_{2,1,1}\\\ +3s_{2,2}+3s_{3,1}+3s_{4}\end{gathered}$ | $\begin{gathered}4s_{1,1,1,1}+9s_{2,1,1}\\\ +10s_{2,2}+11s_{3,1}+6s_{4}\end{gathered}$
5 |  | $s_{3,2}+s_{4,1}+s_{5}$ | $\begin{gathered}s_{2,1,1,1}+2s_{2,2,1}+2s_{3,1,1}\\\ +4s_{3,2}+4s_{4,1}+3s_{5}\end{gathered}$ | $\begin{gathered}2s_{1,1,1,1,1}+8s_{2,1,1,1}+13s_{2,2,1}\\\ +15s_{3,1,1}+18s_{3,2}+16s_{4,1}+7s_{5}\end{gathered}$
6 |  | $s_{4,2}+s_{5,1}+s_{6}$ | $\begin{gathered}s_{2,2,2}+s_{3,1,1,1}\\\ +2s_{3,2,1}+s_{3,3}+2s_{4,1,1}\\\ +5s_{4,2}+4s_{5,1}+3s_{6}\end{gathered}$ | $\begin{gathered}3s_{2,1,1,1,1}+5s_{2,2,1,1}+7s_{2,2,2}\\\ +10s_{3,1,1,1}+21s_{3,2,1}+8s_{3,3}\\\ +17s_{4,1,1}+24s_{4,2}+17s_{5,1}+8s_{6}\end{gathered}$
7 |  | $s_{5,2}+s_{6,1}+s_{7}$ | $\begin{gathered}s_{3,2,2}+s_{4,1,1,1}\\\ +2s_{4,2,1}+2s_{4,3}+2s_{5,1,1}\\\ +5s_{5,2}+4s_{6,1}+3s_{7}\end{gathered}$ | $\begin{gathered}s_{2,2,1,1,1}+2s_{2,2,2,1}\\\ +3s_{3,1,1,1,1}+7s_{3,2,1,1}+9s_{3,2,2}\\\ +8s_{3,3,1}+10s_{4,1,1,1}+24s_{4,2,1}+14s_{4,3}\\\ +17s_{5,1,1}+25s_{5,2}+18s_{6,1}+8s_{7}\end{gathered}$
8 |  | $s_{6,2}+s_{7,1}+s_{8}$ |  $\begin{gathered}s_{4,2,2}+s_{4,4}+s_{5,1,1,1}\\\ +2s_{5,2,1}+2s_{5,3}+2s_{6,1,1}\\\ +5s_{6,2}+4s_{7,1}+3s_{8}\end{gathered}$ | $\begin{gathered}s_{2,2,2,2}+s_{3,2,1,1,1}+2s_{3,2,2,1}+2s_{3,3,1,1}\\\ +2s_{3,3,2}+3s_{4,1,1,1,1}+7s_{4,2,1,1}+10s_{4,2,2}\\\ +11s_{4,3,1}+6s_{4,4}+10s_{5,1,1,1}+24s_{5,2,1}\\\ +15s_{5,3}+17s_{6,1,1}+26s_{6,2}+18s_{7,1}+8s_{8}\end{gathered}$
9 |  | $s_{7,2}+s_{8,1}+s_{9}$ | $\begin{gathered}s_{5,2,2}+s_{5,4}+s_{6,1,1,1}\\\ +2s_{6,2,1}+2s_{6,3}+2s_{7,1,1}\\\ +5s_{7,2}+4s_{8,1}+3s_{9}\end{gathered}$ | $\begin{gathered}s_{3,2,2,2}+s_{4,2,1,1,1}+2s_{4,2,2,1}\\\ +2s_{4,3,1,1}+3s_{4,3,2}+3s_{4,4,1}\\\ +3s_{5,1,1,1,1}+7s_{5,2,1,1}+10s_{5,2,2}\\\ +11s_{5,3,1}+7s_{5,4}+10s_{6,1,1,1}+24s_{6,2,1}\\\ +16s_{6,3}+17s_{7,1,1}+26s_{7,2}+18s_{8,1}+8s_{9}\end{gathered}$
10 |  | $s_{8,2}+s_{9,1}+s_{10}$ | $\begin{gathered}s_{6,2,2}+s_{6,4}+s_{7,1,1,1}\\\ +2s_{7,2,1}+2s_{7,3}+2s_{8,1,1}\\\ +5s_{8,2}+4s_{9,1}+3s_{10}\end{gathered}$ | $\begin{gathered}s_{4,2,2,2}+s_{4,4,2}+s_{5,2,1,1,1}+2s_{5,2,2,1}\\\ +2s_{5,3,1,1}+3s_{5,3,2}+3s_{5,4,1}+s_{5,5}\\\ +3s_{6,1,1,1,1}+7s_{6,2,1,1}+10s_{6,2,2}+11s_{6,3,1}\\\ +8s_{6,4}+10s_{7,1,1,1}+24s_{7,2,1}+16s_{7,3}\\\ +17s_{8,1,1}+26s_{8,2}+18s_{9,1}+8s_{10}\end{gathered}$
11 |  | $s_{9,2}+s_{10,1}+s_{11}$ | $\begin{gathered}s_{7,2,2}+s_{7,4}+s_{8,1,1,1}\\\ +2s_{8,2,1}+2s_{8,3}+2s_{9,1,1}\\\ +5s_{9,2}+4s_{10,1}+3s_{11}\end{gathered}$ | $\begin{gathered}s_{5,2,2,2}+s_{5,4,2}+s_{6,2,1,1,1}+2s_{6,2,2,1}\\\ +2s_{6,3,1,1}+3s_{6,3,2}+3s_{6,4,1}+2s_{6,5}\\\ +3s_{7,1,1,1,1}+7s_{7,2,1,1}+10s_{7,2,2}+11s_{7,3,1}\\\ +8s_{7,4}+10s_{8,1,1,1}+24s_{8,2,1}+16s_{8,3}\\\ +17s_{9,1,1}+26s_{9,2}+18s_{10,1}+8s_{11}\end{gathered}$
12 |  | $s_{10,2}+s_{11,1}+s_{12}$ | $\begin{gathered}s_{8,2,2}+s_{8,4}+s_{9,1,1,1}\\\ +2s_{9,2,1}+2s_{9,3}+2s_{10,1,1}\\\ +5s_{10,2}+4s_{11,1}+3s_{12}\end{gathered}$ | $\begin{gathered}s_{6,2,2,2}+s_{6,4,2}+s_{6,6}+s_{7,2,1,1,1}\\\ +2s_{7,2,2,1}+2s_{7,3,1,1}+3s_{7,3,2}\\\ +3s_{7,4,1}+2s_{7,5}+3s_{8,1,1,1,1}\\\ +7s_{8,2,1,1}+10s_{8,2,2}+11s_{8,3,1}\\\ +8s_{8,4}+10s_{9,1,1,1}+24s_{9,2,1}+16s_{9,3}\\\ +17s_{10,1,1}+26s_{10,2}+18s_{11,1}+8s_{12}\end{gathered}$
Table 2. Characters for $[{\mathrm{OS}}(\operatorname{Br}_{n})^{!}_{i}]=[\mathcal{S}_{\mathrm{OS}}((n-1)+i,n-1)]$. Note that $\mathrm{ch}[\mathrm{OS}(\operatorname{Br}_{1})^{!}_{i}]=s_{2}$ for $i\geq 0$ and $\mathrm{ch}[\mathrm{OS}(\operatorname{Br}_{n})^{!}_{0}]=s_{n}$ for $n\geq 2$.  $n$ $i$ | 0 | 1 | 2 | 3
---|---|---|---|---
2 | $s_{2}$ | $s_{1,1}$ | $s_{2}$ | $s_{1,1}$
3 | $s_{3}$ | $s_{1,1,1}+s_{2,1}$ | $s_{1,1,1}+2s_{2,1}+2s_{3}$ | $3s_{1,1,1}+5s_{2,1}+2s_{3}$
4 | $s_{4}$ | $s_{2,1,1}+s_{3,1}$ | $\begin{gathered}s_{1,1,1,1}+3s_{2,1,1}\\\ +2s_{2,2}+3s_{3,1}+2s_{4}\end{gathered}$ | $\begin{gathered}4s_{1,1,1,1}+11s_{2,1,1}+8s_{2,2}\\\ +11s_{3,1}+4s_{4}\end{gathered}$
5 | $s_{5}$ | $s_{3,1,1}+s_{4,1}$ | $\begin{gathered}2s_{2,1,1,1}+2s_{2,2,1}+3s_{3,1,1}\\\ +3s_{3,2}+3s_{4,1}+2s_{5}\end{gathered}$ | $\begin{gathered}3s_{1,1,1,1,1}+10s_{2,1,1,1}+14s_{2,2,1}\\\ +17s_{3,1,1}+15s_{3,2}+14s_{4,1}+4s_{5}\end{gathered}$
6 | $s_{6}$ | $s_{4,1,1}+s_{5,1}$ | $\begin{gathered}s_{2,2,1,1}+2s_{3,1,1,1}\\\ +2s_{3,2,1}+s_{3,3}+3s_{4,1,1}\\\ +3s_{4,2}+3s_{5,1}+2s_{6}\end{gathered}$ | $\begin{gathered}s_{1,1,1,1,1,1}+5s_{2,1,1,1,1}+8s_{2,2,1,1}\\\ +7s_{2,2,2}+13s_{3,1,1,1}+21s_{3,2,1}+7s_{3,3}\\\ +18s_{4,1,1}+18s_{4,2}+14s_{5,1}+4s_{6}\end{gathered}$
7 | $s_{7}$ | $s_{5,1,1}+s_{6,1}$ | $\begin{gathered}s_{3,2,1,1}+2s_{4,1,1,1}\\\ +2s_{4,2,1}+s_{4,3}+3s_{5,1,1}\\\ +3s_{5,2}+3s_{6,1}+2s_{7}\end{gathered}$ | $\begin{gathered}s_{2,1,1,1,1,1}+2s_{2,2,1,1,1}+3s_{2,2,2,1}\\\ +6s_{3,1,1,1,1}+11s_{3,2,1,1}+8s_{3,2,2}\\\ +7s_{3,3,1}+13s_{4,1,1,1}+22s_{4,2,1}+10s_{4,3}\\\ +18s_{5,1,1}+18s_{5,2}+14s_{6,1}+4s_{7}\end{gathered}$
8 | $s_{8}$ | $s_{6,1,1}+s_{7,1}$ | $\begin{gathered}s_{4,2,1,1}+2s_{5,1,1,1}\\\ +2s_{5,2,1}+s_{5,3}\\\ +3s_{6,1,1}+3s_{6,2}\\\ +3s_{7,1}+2s_{8}\end{gathered}$ | $\begin{gathered}s_{2,2,2,2}+s_{3,1,1,1,1,1}+3s_{3,2,1,1,1}\\\ +3s_{3,2,2,1}+3s_{3,3,1,1}+s_{3,3,2}\\\ +6s_{4,1,1,1,1}+11s_{4,2,1,1}+8s_{4,2,2}\\\ +8s_{4,3,1}+3s_{4,4}+13s_{5,1,1,1}\\\ +22s_{5,2,1}+10s_{5,3}+18s_{6,1,1}\\\ +18s_{6,2}+14s_{7,1}+4s_{8}\end{gathered}$
9 | $s_{9}$ | $s_{7,1,1}+s_{8,1}$ | $\begin{gathered}s_{5,2,1,1}+2s_{6,1,1,1}\\\ +2s_{6,2,1}+s_{6,3}\\\ +3s_{7,1,1}+3s_{7,2}\\\ +3s_{8,1}+2s_{9}\end{gathered}$ | $\begin{gathered}s_{3,2,2,2}+s_{3,3,1,1,1}+s_{4,1,1,1,1,1}\\\ +3s_{4,2,1,1,1}+3s_{4,2,2,1}+3s_{4,3,1,1}\\\ +s_{4,3,2}+s_{4,4,1}+6s_{5,1,1,1,1}\\\ +11s_{5,2,1,1}+8s_{5,2,2}+8s_{5,3,1}\\\ +3s_{5,4}+13s_{6,1,1,1}+22s_{6,2,1}+10s_{6,3}\\\ +18s_{7,1,1}+18s_{7,2}+14s_{8,1}+4s_{9}\end{gathered}$
10 | $s_{10}$ | $s_{8,1,1}+s_{9,1}$ | $\begin{gathered}s_{6,2,1,1}+2s_{7,1,1,1}\\\ +2s_{7,2,1}+s_{7,3}\\\ +3s_{8,1,1}+3s_{8,2}\\\ +3s_{9,1}+2s_{10}\end{gathered}$ | $\begin{gathered}s_{4,2,2,2}+s_{4,3,1,1,1}+s_{5,1,1,1,1,1}\\\ +3s_{5,2,1,1,1}+3s_{5,2,2,1}+3s_{5,3,1,1}\\\ +s_{5,3,2}+s_{5,4,1}+6s_{6,1,1,1,1}\\\ +11s_{6,2,1,1}+8s_{6,2,2}+8s_{6,3,1}\\\ +3s_{6,4}+13s_{7,1,1,1}+22s_{7,2,1}+10s_{7,3}\\\ +18s_{8,1,1}+18s_{8,2}+14s_{9,1}+4s_{10}\end{gathered}$
Table 3. Characters for
$[{\mathrm{VG}}(\operatorname{Br}_{n})^{!}_{i}]=[\mathcal{S}_{\mathrm{VG}}((n-1)+i,n-1)]$.

$n$ $i$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
2 | $s_{2}$ | 0 | 0 | 0 | 0
3 | $s_{2,1}+s_{3}$ | $s_{1,1,1}$ | $s_{2,1}$ | $s_{1,1,1}+s_{2,1}$ | $s_{1,1,1}+2s_{2,1}+s_{3}$
4 | $s_{2,2}+s_{3,1}+s_{4}$ | $s_{1,1,1,1}+s_{2,1,1}$ | $s_{2,1,1}+2s_{2,2}+s_{3,1}$ | $2s_{1,1,1,1}+3s_{2,1,1}+2s_{2,2}+2s_{3,1}$ | $3s_{1,1,1,1}+6s_{2,1,1}+6s_{2,2}+6s_{3,1}+3s_{4}$
5 | $s_{3,2}+s_{4,1}+s_{5}$ | $s_{2,1,1,1}+s_{3,1,1}$ | $\begin{gathered}2s_{2,2,1}+s_{3,1,1}\\\ +2s_{3,2}+s_{4,1}\end{gathered}$ | $\begin{gathered}s_{1,1,1,1,1}+3s_{2,1,1,1}+3s_{2,2,1}\\\ +5s_{3,1,1}+3s_{3,2}+2s_{4,1}\end{gathered}$ | $\begin{gathered}2s_{1,1,1,1,1}+8s_{2,1,1,1}+11s_{2,2,1}\\\ +11s_{3,1,1}+12s_{3,2}+10s_{4,1}+3s_{5}\end{gathered}$
6 | $s_{4,2}+s_{5,1}+s_{6}$ | $s_{3,1,1,1}+s_{4,1,1}$ | $\begin{gathered}s_{2,2,2}+2s_{3,2,1}\\\ +s_{4,1,1}+2s_{4,2}+s_{5,1}\end{gathered}$ | $\begin{gathered}s_{2,1,1,1,1}+s_{2,2,1,1}+s_{2,2,2}\\\ +4s_{3,1,1,1}+5s_{3,2,1}+s_{3,3}\\\ +5s_{4,1,1}+3s_{4,2}+2s_{5,1}\end{gathered}$ | $\begin{gathered}4s_{2,1,1,1,1}+7s_{2,2,1,1}+7s_{2,2,2}\\\ +11s_{3,1,1,1}+18s_{3,2,1}+6s_{3,3}\\\ +13s_{4,1,1}+17s_{4,2}+10s_{5,1}+3s_{6}\end{gathered}$
7 | $s_{5,2}+s_{6,1}+s_{7}$ | $s_{4,1,1,1}+s_{5,1,1}$ | $\begin{gathered}s_{3,2,2}+2s_{4,2,1}\\\ +s_{5,1,1}+2s_{5,2}+s_{6,1}\end{gathered}$ | $\begin{gathered}s_{3,1,1,1,1}+2s_{3,2,1,1}+s_{3,2,2}\\\ +2s_{3,3,1}+4s_{4,1,1,1}+5s_{4,2,1}\\\ +s_{4,3}+5s_{5,1,1}+3s_{5,2}+2s_{6,1}\end{gathered}$ | $\begin{gathered}2s_{2,2,1,1,1}+3s_{2,2,2,1}+5s_{3,1,1,1,1}\\\ +10s_{3,2,1,1}+9s_{3,2,2}+7s_{3,3,1}\\\ +11s_{4,1,1,1}+21s_{4,2,1}+11s_{4,3}\\\ +13s_{5,1,1}+17s_{5,2}+10s_{6,1}+3s_{7}\end{gathered}$
8 | $s_{6,2}+s_{7,1}+s_{8}$ | $\begin{gathered}s_{5,1,1,1}+s_{6,1,1}\end{gathered}$ | $\begin{gathered}s_{4,2,2}+2s_{5,2,1}\\\ +s_{6,1,1}+2s_{6,2}+s_{7,1}\end{gathered}$ |  $\begin{gathered}s_{3,3,1,1}+s_{4,1,1,1,1}\\\ +2s_{4,2,1,1}+s_{4,2,2}+2s_{4,3,1}\\\ +4s_{5,1,1,1}+5s_{5,2,1}+s_{5,3}\\\ +5s_{6,1,1}+3s_{6,2}+2s_{7,1}\end{gathered}$ | $\begin{gathered}s_{2,2,2,2}+3s_{3,2,1,1,1}+3s_{3,2,2,1}\\\ +3s_{3,3,1,1}+2s_{3,3,2}+5s_{4,1,1,1,1}\\\ +10s_{4,2,1,1}+10s_{4,2,2}+10s_{4,3,1}\\\ +5s_{4,4}+11s_{5,1,1,1}+21s_{5,2,1}+11s_{5,3}\\\ +13s_{6,1,1}+17s_{6,2}+10s_{7,1}+3s_{8}\end{gathered}$
9 | $s_{7,2}+s_{8,1}+s_{9}$ | $s_{6,1,1,1}+s_{7,1,1}$ | $\begin{gathered}s_{5,2,2}+2s_{6,2,1}\\\ +s_{7,1,1}+2s_{7,2}+s_{8,1}\end{gathered}$ | $\begin{gathered}s_{4,3,1,1}+s_{5,1,1,1,1}\\\ +2s_{5,2,1,1}+s_{5,2,2}+2s_{5,3,1}\\\ +4s_{6,1,1,1}+5s_{6,2,1}+s_{6,3}\\\ +5s_{7,1,1}+3s_{7,2}+2s_{8,1}\end{gathered}$ | $\begin{gathered}s_{3,2,2,2}+s_{3,3,1,1,1}+3s_{4,2,1,1,1}\\\ +3s_{4,2,2,1}+3s_{4,3,1,1}+3s_{4,3,2}\\\ +3s_{4,4,1}+5s_{5,1,1,1,1}+10s_{5,2,1,1}\\\ +10s_{5,2,2}+10s_{5,3,1}+5s_{5,4}\\\ +11s_{6,1,1,1}+21s_{6,2,1}+11s_{6,3}\\\ +13s_{7,1,1}+17s_{7,2}+10s_{8,1}+3s_{9}\end{gathered}$
10 | $s_{8,2}+s_{9,1}+s_{10}$ | $s_{7,1,1,1}+s_{8,1,1}$ | $\begin{gathered}s_{6,2,2}+2s_{7,2,1}\\\ +s_{8,1,1}+2s_{8,2}+s_{9,1}\end{gathered}$ | $\begin{gathered}s_{5,3,1,1}+s_{6,1,1,1,1}\\\ +2s_{6,2,1,1}+s_{6,2,2}+2s_{6,3,1}\\\ +4s_{7,1,1,1}+5s_{7,2,1}+s_{7,3}\\\ +5s_{8,1,1}+3s_{8,2}+2s_{9,1}\end{gathered}$ |  $\begin{gathered}s_{4,2,2,2}+s_{4,3,1,1,1}+s_{4,4,2}\\\ +3s_{5,2,1,1,1}+3s_{5,2,2,1}+3s_{5,3,1,1}\\\ +3s_{5,3,2}+3s_{5,4,1}+5s_{6,1,1,1,1}\\\ +10s_{6,2,1,1}+10s_{6,2,2}+10s_{6,3,1}\\\ +5s_{6,4}+11s_{7,1,1,1}+21s_{7,2,1}+11s_{7,3}\\\ +13s_{8,1,1}+17s_{8,2}+10s_{9,1}+3s_{10}\end{gathered}$
Table 4. Characters for $[\mathcal{L}(n)_{i}]$ when
$A=\mathrm{OS}(\operatorname{Br}_{n})$.

$n$ $i$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
2 | $s_{1,1}$ | $s_{2}$ | 0 | 0 | 0
3 | $s_{1,1,1}+s_{2,1}$ | $s_{2,1}+2s_{3}$ | $s_{2,1}$ | $s_{1,1,1}+s_{2,1}$ | $s_{1,1,1}+2s_{2,1}+s_{3}$
4 | $s_{2,1,1}+s_{3,1}$ | $s_{2,2}+2s_{3,1}+2s_{4}$ | $s_{2,1,1}+2s_{2,2}+s_{3,1}$ | $\begin{gathered}s_{1,1,1,1}+4s_{2,1,1}+s_{2,2}+2s_{3,1}\end{gathered}$ | $\begin{gathered}2s_{1,1,1,1}+7s_{2,1,1}+4s_{2,2}+7s_{3,1}+2s_{4}\end{gathered}$
5 | $s_{3,1,1}+s_{4,1}$ | $2s_{3,2}+2s_{4,1}+2s_{5}$ | $\begin{gathered}2s_{2,2,1}+s_{3,1,1}\\\ +2s_{3,2}+s_{4,1}\end{gathered}$ | $\begin{gathered}3s_{2,1,1,1}+3s_{2,2,1}\\\ +6s_{3,1,1}+2s_{3,2}+2s_{4,1}\end{gathered}$ | $\begin{gathered}s_{1,1,1,1,1}+8s_{2,1,1,1}+10s_{2,2,1}\\\ +13s_{3,1,1}+11s_{3,2}+10s_{4,1}+2s_{5}\end{gathered}$
6 | $s_{4,1,1}+s_{5,1}$ | $\begin{gathered}s_{3,3}+2s_{4,2}\\\ +2s_{5,1}+2s_{6}\end{gathered}$ | $\begin{gathered}s_{2,2,2}+2s_{3,2,1}\\\ +s_{4,1,1}+2s_{4,2}+s_{5,1}\end{gathered}$ | $\begin{gathered}2s_{2,2,1,1}+4s_{3,1,1,1}\\\ +5s_{3,2,1}+s_{3,3}\\\ +6s_{4,1,1}+2s_{4,2}+2s_{5,1}\end{gathered}$ | $\begin{gathered}3s_{2,1,1,1,1}+9s_{2,2,1,1}+4s_{2,2,2}\\\ +10s_{3,1,1,1}+18s_{3,2,1}+8s_{3,3}\\\ +16s_{4,1,1}+14s_{4,2}+10s_{5,1}+2s_{6}\end{gathered}$
7 | $s_{5,1,1}+s_{6,1}$ | $\begin{gathered}s_{4,3}+2s_{5,2}\\\ +2s_{6,1}+2s_{7}\end{gathered}$ | $\begin{gathered}s_{3,2,2}+2s_{4,2,1}\\\ +s_{5,1,1}+2s_{5,2}+s_{6,1}\end{gathered}$ | $\begin{gathered}3s_{3,2,1,1}+2s_{3,3,1}\\\ +4s_{4,1,1,1}+5s_{4,2,1}+s_{4,3}\\\ +6s_{5,1,1}+2s_{5,2}+2s_{6,1}\end{gathered}$ | $\begin{gathered}3s_{2,2,1,1,1}+3s_{2,2,2,1}+3s_{3,1,1,1,1}\\\ +11s_{3,2,1,1}+6s_{3,2,2}+9s_{3,3,1}\\\ +11s_{4,1,1,1}+21s_{4,2,1}+11s_{4,3}\\\ +16s_{5,1,1}+14s_{5,2}+10s_{6,1}+2s_{7}\end{gathered}$
8 | $s_{6,1,1}+s_{7,1}$ | $\begin{gathered}s_{5,3}+2s_{6,2}\\\ +2s_{7,1}+2s_{8}\end{gathered}$ | $\begin{gathered}s_{4,2,2}+2s_{5,2,1}\\\ +s_{6,1,1}+2s_{6,2}+s_{7,1}\end{gathered}$ | $\begin{gathered}s_{3,3,1,1}+3s_{4,2,1,1}\\\ +2s_{4,3,1}+4s_{5,1,1,1}\\\ +5s_{5,2,1}+s_{5,3}\\\ +6s_{6,1,1}+2s_{6,2}+2s_{7,1}\end{gathered}$ | $\begin{gathered}s_{2,2,2,1,1}+3s_{3,2,1,1,1}+3s_{3,2,2,1}\\\ +2s_{3,3,1,1}+3s_{3,3,2}+3s_{4,1,1,1,1}\\\ +12s_{4,2,1,1}+6s_{4,2,2}+12s_{4,3,1}\\\ +3s_{4,4}+11s_{5,1,1,1}+21s_{5,2,1}+11s_{5,3}\\\ +16s_{6,1,1}+14s_{6,2}+10s_{7,1}+2s_{8}\end{gathered}$
9 | $s_{7,1,1}+s_{8,1}$ | $\begin{gathered}s_{6,3}+2s_{7,2}\\\ +2s_{8,1}+2s_{9}\end{gathered}$ | $\begin{gathered}s_{5,2,2}+2s_{6,2,1}\\\ +s_{7,1,1}+2s_{7,2}+s_{8,1}\end{gathered}$ | $\begin{gathered}s_{4,3,1,1}+3s_{5,2,1,1}\\\ +2s_{5,3,1}+4s_{6,1,1,1}\\\ +5s_{6,2,1}+s_{6,3}\\\ +6s_{7,1,1}+2s_{7,2}+2s_{8,1}\end{gathered}$ | $\begin{gathered}s_{3,2,2,1,1}+s_{3,3,3}+3s_{4,2,1,1,1}\\\ +3s_{4,2,2,1}+3s_{4,3,1,1}+3s_{4,3,2}\\\ +3s_{4,4,1}+3s_{5,1,1,1,1}+12s_{5,2,1,1}\\\ +6s_{5,2,2}+12s_{5,3,1}+3s_{5,4}\\\ +11s_{6,1,1,1}+21s_{6,2,1}+11s_{6,3}\\\ +16s_{7,1,1}+14s_{7,2}+10s_{8,1}+2s_{9}\end{gathered}$
10 | $s_{8,1,1}+s_{9,1}$ | $\begin{gathered}s_{7,3}+2s_{8,2}\\\ +2s_{9,1}+2s_{10}\end{gathered}$ | $\begin{gathered}s_{6,2,2}+2s_{7,2,1}\\\ +s_{8,1,1}+2s_{8,2}+s_{9,1}\end{gathered}$ | $\begin{gathered}s_{5,3,1,1}+3s_{6,2,1,1}\\\ +2s_{6,3,1}+4s_{7,1,1,1}\\\ +5s_{7,2,1}+s_{7,3}\\\ +6s_{8,1,1}+2s_{8,2}+2s_{9,1}\end{gathered}$ | $\begin{gathered}s_{4,2,2,1,1}+s_{4,3,3}+s_{4,4,1,1}\\\ +3s_{5,2,1,1,1}+3s_{5,2,2,1}+3s_{5,3,1,1}\\\ +3s_{5,3,2}+3s_{5,4,1}+3s_{6,1,1,1,1}\\\ +12s_{6,2,1,1}+6s_{6,2,2}+12s_{6,3,1}\\\ +3s_{6,4}+11s_{7,1,1,1}+21s_{7,2,1}+11s_{7,3}\\\ +16s_{8,1,1}+14s_{8,2}+10s_{9,1}+2s_{10}\end{gathered}$
Table 5. Characters for $[\mathcal{L}(n)_{i}]$ when
$A=\mathrm{VG}(\operatorname{Br}_{n})$.

## References