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%%%%% Auteur
\author{\firstname{Nicholas} \lastname{Proudfoot}}
\address{University of Oregon\\
Department of Mathematics\\
Eugene\\
OR 97403, USA}
\email{njp@uoregon.edu}
\thanks{The author is supported by NSF grant DMS-1565036.}
%%%%% Sujet
\subjclass{05B35, 20C33}
\keywords{Kazhdan--Lusztig polynomial, matroid, unipotent representation}
%%%%% Gestion
\DOI{10.5802/alco.59}
\datereceived{2018-09-01}
\daterevised{2019-01-18}
\dateaccepted{2019-01-14}
%%%%% Titre et résumé
\title[Equivariant Kazhdan--Lusztig polynomials of $q$-niform matroids]{Equivariant Kazhdan--Lusztig polynomials of $q$-niform matroids}
\begin{abstract}
We study $q$-analogues of uniform matroids, which we call $q$-niform matroids.
While uniform matroids admit actions of symmetric groups, $q$-niform matroids admit actions
of finite general linear groups. We show that the equivariant Kazhdan--Lusztig polynomial
of a $q$-niform matroid is the unipotent $q$-analogue of the equivariant Kazhdan--Lusztig polynomial
of the corresponding uniform matroid, thus providing evidence for the positivity conjecture for equivariant
Kazhdan--Lusztig polynomials.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
For any matroid $M$, the \emph{Kazhdan--Lusztig polynomial} $P_M(t) \in \Z[t]$ was introduced in~\cite{EPW}.
In the case where the matroid $M$ admits the action of a finite group $W$, one can define the \emph{equivariant Kazhdan--Lusztig polynomial}
$P_M^W(t)$~\cite{GPY}; this is a polynomial whose coefficients are virtual representations of $W$ (in characteristic zero)
with dimensions equal to the coefficients of $P_M(t)$.
Though these polynomials admit elementary recursive definitions, there are not many families of matroids for which explicit formulas
are known. Non-equivariant formulas exist for thagomizer matroids~\cite{thag} and fan, wheel, and whirl matroids~\cite{fan-wheel-whirl}.
Kazhdan--Lusztig polynomials of braid matroids have been studied extensively, both in the equivariant~\cite{fs-braid}
and non-equivariant~\cite{Karn-Wakefield} settings, though no simple formulas have been obtained.
The most interesting explicit formulas that we have are for uniform matroids. Let $U_{n,m}$ be the uniform matroid
of rank $n-m$ on a set of cardinality $n$, which admits an action of the symmetric group $S_n$.
For any partition $\la$ of $n$, let $V[\la]$ be the associated irreducible representation of $S_n$.
The following theorem was
proved in~\cite[Theorem~3.1]{GPY}; an independent proof of the non-equivariant statement was later given in
\cite[Theorem~1.2]{Chinese-uniform}.
\begin{theorem}\label{uniform}
Let $C_{n,m}^i$ be the coefficient of $t^i$ in the $S_n$-equivariant Kazhdan--Lusztig polynomial of $U_{n,m}$,
and let $c_{n,m}^i := \dim C_{n,m}^i$ be the corresponding non-equivariant coefficient.
\begin{itemize}
\item $C_{n,m}^0 = V[n]$, and for all $i>0$, $$C_{n,m}^i = \sum_{b=1}^{\min(m,n-m-2i)} V[n-2i-b+1,b+1,2^{i-1}].$$
\item $c_{n,m}^0 = 1$, and for all $i>0$, $c_{n,m}^i$ is equal to
$$\sum_{b=1}^{\min(m,n-m-2i)} \frac{(n - 2i - 2b + 1)n!}{(n\mk -\mk i\mk-\mk b)
(n\mk -\mk i\mk -\mk b\mk +\mk 1)(i\mk +\mk b)(i \mk +\mk b \mk - \mk 1)(n\mk -\mk 2i\mk -\mk b)!
(b\mk -\mk 1)!i!(i \mk - \mk 1)!}.$$
\end{itemize}
\end{theorem}
The purpose of this note is to obtain a $q$-analogue of Theorem~\ref{uniform}.
Let $q$ be a prime power, and let $U_{n,0}(q)$ be the rank $n$ matroid associated with
the collection of all hyperplanes in the vector space $\Fq^n$, which we regard as a $q$-analogue
of the Boolean matroid of rank $n$. For any natural number $m\leq n$,
let $U_{n,m}(q)$ be the truncation of $U_{n,0}(q)$ to rank $n-m$.
More concretely, a basis for $U_{n,m}(q)$ is a set of $n-m$ hyperplanes whose intersection has dimension $m$.
The matroid $U_{n,m}(q)$ is a $q$-analogue of the uniform matroid $U_{n,m}$,
and we will therefore refer to it as a \emph{$q$-niform matroid}.
This matroid was also studied in~\cite{HRS}, where the authors computed the Hilbert series of its Chow ring.
The $q$-niform matroid $U_{n,m}(q)$ admits a natural action of the group $\Gn$ of invertible $n\times n$
matrices with coefficients in $\Fq$, which is a $q$-analogue of $S_n$.
The representation theory of $\Gn$ is much more complicated than the representation theory of $S_n$.
However, there is a certain subset of irreducible representations of $\Gn$, known as \emph{irreducible unipotent representations},
that correspond bijectively to the irreducible representations of $S_n$.
For any partition $\la$ of $n$, let $V(q)[\la]$ be the associated irreducible unipotent representation of $\Gn$, which
we will refer to as the \emph{unipotent $q$-analogue} of $V[\la]$.
For any positive integer $k$, we use the standard notation
$$[k]_q := 1 + q + \cdots + q^{k-1}\andH [k]_q! := [k]_q[k-1]_q\cdots[1]_q.$$
The following theorem, which is our main result, says that the equivariant Kazhdan--Lusztig coefficients of $U_{n,m}(q)$
are precisely the unipotent $q$-analogues of the equivariant Kazhdan--Lusztig coefficients of $U_{n,m}$.
\begin{theorem}\label{q-niform}
Let $C_{n,m}^i(q)$ be the coefficient of $t^i$ in the $\Gn$-equivariant Kazhdan--Lusztig polynomial of $U_{n,m}(q)$,
and let $c_{n,m}^i(q) := \dim C_{n,m}^i(q)$ be the corresponding non-equivariant coefficient.
\begin{itemize}
\item $C_{n,m}^0(q) = V(q)[n]$, and for all $i>0$, $$C_{n,m}^i(q) = \sum_{b=1}^{\min(m,n-m-2i)} V(q)[n-2i-b+1,b+1,2^{i-1}].$$
\item $c_{n,m}^0(q) = 1$, and for all $i>0$, $c_{n,m}^i(q)$ is equal to
$$
\sum_{b=1}^{\min(m,n\Mk -\Mk m\Mk -\Mk 2i)}\Mk \Mk
\frac{q^{b-1+i(i+1)}\;[n - 2i - 2b + 1]_q[n]_q!}{[n\Mk -\Mk i\Mk -\Mk b]_q[n\Mk -\Mk i\Mk -\Mk b\Mk +\Mk 1]_q
[i\Mk +\Mk b]_q[i\Mk +\Mk b \Mk - \Mk 1]_q[n\Mk -\Mk 2i\Mk -\Mk b]_q![b\Mk - \Mk 1]_q![i]_q![i\Mk -\Mk 1]_q!}.
$$
\end{itemize}
\end{theorem}
\begin{rema}
For any matroid $M$, the coefficients of $P_M(t)$ are conjectured to be non-negative~\cite[Conjecture~2.3]{EPW}.
More generally, the coefficients of $P_M^W(t)$ are conjectured to be honest (rather than virtual) representations of $W$
\cite[Conjecture~2.13]{GPY}. These conjectures are proved when $M$ is realizable~\cite[Theorem~3.10]{EPW}
(respectively equivariantly realizable~\cite[Corollary~2.12]{GPY}), but no proof exists in the general case.
The matroid $U_{n,m}$ is always realizable, but it is not equivariantly realizable unless $m\in\{0,1,n-1,n\}$
(of these, only the $m=1$ case yields nontrivial Kazhdan--Lusztig coefficients).
Similarly, the matroid $U_{n,m}(q)$ is always realizable, but it is typically not equivariantly realizable.
Thus Theorems~\ref{uniform} and~\ref{q-niform} both provide significant evidence for the equivariant
non-negativity conjecture.
\end{rema}
\begin{rema}
\looseness-1
Theorem~\ref{uniform} implies that $\{C_{n,m}^i \mid n\geq m\}$ admits the structure of a finitely
generated FI-module~\cite[Theorem~1.13]{CEF},
while Theorem~\ref{q-niform} implies that $\{C_{n,m}^i(q) \mid n\geq m\}$ admits the structure of a finitely
generated VI-module~\cite[Theorem~1.6]{GanW}.
In order to define these structures in a natural way, we would need to be able to define $C_{n,m}^i$
and $C_{n,m}^i(q)$ as actual vector spaces rather than as isomorphism classes of vector spaces.
The matroid $U_{n,1}$ is equivariantly realizable, which means that we have a cohomological interpretation of $C_{n,1}^i$,
and we obtain a canonical $\operatorname{FI^{op}}$-module structure from~\cite[Theorem~3.3(1)]{fs-braid};
dualizing then gives a canonical finitely generated FI-module.
In joint work with Braden, Huh, Matherne, and Wang, the author is working to construct a canonical vector space
isomorphic to the coefficient of $t^i$ in $P_M(t)$ for any matroid $M$. When this goal is achieved, we believe
that this construction will induce a canonical $\operatorname{FI^{op}}$-module structure on $\{C_{n,m}^i \mid n\geq m\}$
and a canonical $\operatorname{VI^{op}}$-module structure on $\{C_{n,m}^i(q) \mid n\geq m\}$, each with finitely generated duals.
\end{rema}
Our proof of Theorem~\ref{q-niform} relies heavily on Theorem~\ref{uniform} along with the Comparison Theorem
(Theorem~\ref{properties}), which roughly says that calculations involving Harish-Chandra induction of
unipotent representations of finite general linear groups are essentially equivalent to the analogous calculations
for symmetric groups.
The only additional ingredients in the proof are to check that the Orlik--Solomon algebra
of $U_{n,m}(q)$ is the unipotent $q$-analogue of
the Orlik--Solomon algebra of $U_{n,m}$ (Example~\ref{truncation})
and that the recursive formula for $C_{n,m}^i(q)$ is essentially the same as the recursive formula for $C_{n,m}^i$
(Equations~\eqref{uniform-recursion} and~\eqref{q-niform-recursion}).
\section{Unipotent representations and the Comparison Theorem}\label{sec:comparison}
Given a pair of natural numbers $k\leq n$ and a pair of representations $V$ of $S_k$ and $V'$ of $S_{n-k}$,
we define
$$V*V' := \Ind_{S_k\times S_{n-k}}^{S_n}\Big(V\boxtimes V'\Big).$$
Irreducible representations of the symmetric group $S_n$ are classified by partitions of $n$.
Given a partition $\la$, let $V[\la]$ be the associated representation.
For each cell $(i,j)$ in the Young diagram for $\la$, let $h_\la(i,j)$ be the corresponding hook length;
then the dimension of $V[\la]$ is equal to $$\frac{n!}{\prod h_\la(i,j)}.$$
We now review some analogous statements and constructions in the representation theory of finite general linear groups.
Given a pair of natural numbers $k\leq n$, let $P_{k,n}(q)\subset\Gn$ denote
the parabolic subgroup associated with the Levi $\Gk\times\Gnk$.
Given a pair of representations $V(q)$ of $\Gk$ and $V'(q)$ of $\Gnk$,
we obtain a representation $V(q)\boxtimes V'(q)$ of $\Gk\times\Gnk$, and we may interpret this as a representation of $P_{k,n}(q)$
via the natural surjection $P_{k,n}(q)\to \Gk\times\Gnk$. We then define
$$V(q)*V'(q) := \Ind_{P_{k,n}(q)}^{\Gn}\Big(V(q)\boxtimes V'(q)\Big).$$
This operation is called \emph{Harish-Chandra induction}.
Let $\Bn\subset \Gn$ be the subgroup of upper triangular matrices. An irreducible representation
of $\Gn$ is called \emph{unipotent} if it appears as a direct summand of the representation
$$\Cbb\big[\Gn/\Bn\big] =
\Ind_{\Bn}^{\Gn}\!\big(\triv_{\Gn}\big).$$
(We note that the definition of unipotent representations of finite groups of Lie type outside of type A is more complicated.)
An arbitrary representation is called unipotent if it is isomorphic to a direct sum of irreducible unipotent representations.
\begin{theorem}\label{properties}
Let $q$ be a prime power and $n$ a natural number.
\begin{enumerate}
\item\label{theo2.1_1} Irreducible unipotent representations of $\Gn$ are in canonical bijection with partitions of $n$.
\item\label{theo2.1_2} The irreducible unipotent representation $V(q)[\la]$ associated with the partition $\la$ has dimension
$$q^{\sum (k-1)\la_k}\frac{[n]_q!}{\prod [h_\la(i,j)]_q}.$$
\item\label{theo2.1_3} If $k\leq n$, $V(q)$ is a unipotent representation of $\Gk$, and $V'(q)$ is a unipotent representation of $\Gnk$,
then $V(q)*V'(q)$ is a unipotent representation of $\Gn$.
\item\label{theo2.1_4} Let $\la$, $\mu$, and $\nu$ be partitions of $n$, $k$, and $n-k$, respectively.
The multiplicity of $V(q)[\la]$ in $V(q)[\mu]*V(q)[\nu]$ is equal to the multiplicity of $V[\la]$ in $V[\mu] * V[\nu]$.
\end{enumerate}
\end{theorem}
\begin{proof}
Statements~\ref{theo2.1_1} and~\ref{theo2.1_4} appear in~\cite[Theorem~B]{Curtis75}.
The fact that the dimension of $V(q)[\la]$ is polynomial in $q$ appears in~\cite[Theorem~2.6]{BensonCurtis}.
For an explicit calculation of this polynomial, see~\cite[Equation~(1.1)]{DipperJames}.
Finally, Statement~\ref{theo2.1_3} follows from the fact that $\Cbb\big[\Gk/\Bk\big] * \Cbb\big[\Gnk/\Bnk\big] \cong \Cbb\big[\Gn/\Bn\big]$.
\end{proof}
\begin{rema}
The standard proof of Theorem~\ref{properties}\eqref{theo2.1_1} is very far from constructive.
One proves that the endomorphism algebra of $\Cbb\big[\Gn/\Bn\big]$ is isomorphic to the Hecke algebra
of $S_n$; this implies that the irreducible constituents of $\Cbb\big[\Gn/\Bn\big]$ are in canonical bijection with irreducible
modules over the Hecke algebra, which are in turn in canonical bijection with irreducible representations of $S_n$.
However, a recent paper of Andrews~\cite{Andrews} gives a construction of $V(q)[\la]$ modeled on tableaux, which
is analogous to the usual construction of $V[\la]$.
\end{rema}
\begin{rema}
A generalization of Statement 4 due to Howlett and Lehrer~\cite[Theorem~5.9]{Howlett-Lehrer} is commonly referred to
as the Comparison Theorem. For the purposes of this paper, we will use this terminology to refer to the entirety of Theorem~\ref{properties}.
\end{rema}
\section{Orlik--Solomon algebras}
For any matroid $M$ on the ground set $E$, let $\OS_{\! M}^*$ be the \emph{Orlik--Solomon algebra} of $M$~\cite{OS}, and let
$$\chi_M(t) := \sum_{i=0}^{\rk M} (-1)^i \dim \OS_{\! M}^{i} t^{\rk M - i}$$
be the \emph{characteristic polynomial} of $M$. The Orlik--Solomon algebra is a quotient of the exterior algebra
over the complex numbers with generators $\{x_e\mid e\in E\}$. Let $\bOS_{\! M}^*$ be the \emph{reduced Orlik--Solomon algebra} of $M$,
which is defined as the subalgebra of $\OS_{\! M}^*$ generated by $\{x_e -x_{e'}\mid e,e'\in E\}$.
If $\rk M > 0$, then we have a graded algebra isomorphism
\begin{equation}\label{kunneth}\OS_{\! M}^* \cong \bOS_{\! M}^* \otimes \Cbb[x]/\langle x^2\rangle\end{equation}
and therefore a vector space isomorphism
\begin{equation}\label{kunneth individual}\OS_{\! M}^i \cong \bOS_{\! M}^i \oplus \bOS_{\! M}^{i-1}.\end{equation}
If a finite group $W$ acts on $M$, we obtain induced actions on $\OS^*_{\! M}$ and $\bOS^*_{\! M}$, and the isomorphisms
of Equations~\eqref{kunneth} and~\eqref{kunneth individual} are $W$-equivariant.
\begin{exam}\label{linear}
Suppose that $V$ is a vector space over $\Fq$, and that $\{H_e\mid e\in E\}$ is a collection of hyperplanes
with associated matroid $M$. Fix a prime $\ell$ that does not divide $q$, and fix an embedding of $\Ql$ into $\Cbb$.
Let $$X := V(\Fqb)\smallsetminus \bigcup_{e\in E} H_e(\Fqb)
\andH \Pbb X := \Pbb V(\Fqb)\smallsetminus \bigcup_{e\in E} \Pbb H_e(\Fqb).$$
Then we have canonical isomorphisms
$$\OS_{\! M}^* \cong H^*(X; \Ql)\otimes_{\Ql}\Cbb
\andH
\bOS_{\! M}^* \cong H^*(\Pbb X; \Ql)\otimes_{\Ql}\Cbb,$$
where the cohomology rings are $\ell$-adic \'etale cohomology.
If $\rk M > 0$, then we have an isomorphism $X\cong \Pbb X \times\gm(\Fqb)$,
and Equation~\eqref{kunneth} is simply the Kunneth formula.
If $W$ acts on $V$ by linear automorphisms preserving the collection of hyperplanes, we obtain an induced action on $M$,
and these isomorphisms are $W$-equivariant.
\end{exam}
\begin{exam}
The Boolean matroid $U_{n,0}$ is $S_n$-equivariantly realized by the coordinate hyperplanes in $\Fq^n$.
Its Orlik--Solomon algebra $\OS_{n,0}^*$ is equal to the exterior algebra on $n$ generators, which is isomorphic
to the cohomology of $X_{n,0} \cong \Gm^n(\Fq)$.
As a representation of $S_n$, we have $$\OS_{n,0}^* \cong \Lambda^*\Big( V[n-1,1] \oplus V[n]\Big)
\andH \bOS_{n,0}^* \cong \Lambda^*\Big( V[n-1,1]\Big).$$
In particular, this implies that
\begin{equation}\label{reduced uniform}\bOS_{n,0}^i \cong V[n-i,1^i]\end{equation}
for all $i 0$, then $\deg P_M(t) < \tfrac{1}{2}\rk M$.
\item For every $M$, $\displaystyle t^{\rk M} P_M(t^{-1}) = \sum_{F}\chi_{M_F}(t) P_{M^F}(t).$
\end{enumerate}
If $M$ admits the action of a finite group $W$,
the equivariant Kazhdan--Lusztig polynomial is defined by the three analogous conditions, with the
coefficients of the characteristic polynomial replaced by the graded pieces of the Orlik--Solomon algebra (with corresponding signs),
which are now virtual representations of $W$ rather than integers.
For every flat $F\in L$, let $W_F\subset W$ denote the stabilizer of $F$.
If $C_{M,W}^i$ is the coefficient of $t^i$ in the $W$-equivariant
Kazhdan--Lusztig polynomial of $M$ and $i < \rk M/2$, we have the following explicit recursive formula~\cite[Proposition~2.9]{GPY}:
\begin{equation}\label{ekl-recursion}
C^i_{M,W} = \sum_{\substack{[F]\in L/W\\ 0\leq j\leq \rk F}} (-1)^j\; \Ind_{W_F}^W\!\Big(\OS^{j}_{M_F}
\otimes\; C^{\crk F - i + j}_{M^F,W_F}\Big),\end{equation}
where we take in the sum one flat from each $W$-orbit in $L$.
\begin{exam}
Consider the case of the uniform matroid $U_{n,m}$.
Proper flats are subsets of $[n]$ of cardinality less than $n-m$, and the $S_n$-orbit of a flat is determined
by its cardinality. The stabilizer of a flat of cardinality $k$ is isomorphic to the Young subgroup $S_k\times S_{n-k}\subset S_n$.
Thus Equation~\eqref{ekl-recursion} transforms into the following recursion:
\begin{align}\label{uniform-recursion}
\nonumber C^i_{n,m} &= (-1)^i \OS^{i}_{n-m} + \sum_{k=0}^{n-m-1}\sum_{j=0}^k (-1)^j\; \Ind_{S_k\times S_{n-k}}^{S_n}\!\Big(\OS^{j}_{k,0}
\otimes\; C^{n - m - k - i + j}_{n-k,m}\Big)\\
&= (-1)^i \OS^{i}_{n-m} + \sum_{k=0}^{n-m-1}\sum_{j=0}^k (-1)^j\; \OS^{j}_{k,0} * C^{n - m - k - i + j}_{n-k,m},\end{align}
where the first term corresponds to the maximal flat $F = [n]$.
\end{exam}
\begin{exam}
Consider the case of the $q$-uniform matroid $U_{n,m}(q)$.
Proper flats are collections of linearly independent hyperplanes in $\Fq^n$ of cardinality less than $n-m$,
and the $\Gn$-orbit of a flat is determined
by its cardinality. The stabilizer of a flat of cardinality $k$ is isomorphic to the parabolic subgroup $P_{n,k}(q)\subset \Gn$.
Thus Equation~\eqref{ekl-recursion} transforms into the unipotent $q$-analogue of Equation~\eqref{uniform-recursion}:
\begin{align}\label{q-niform-recursion}
\nonumber C^i_{n,m}(q) &\Mk = \Mk (-1)^i \OS^{i}_{n-m}(q) \Mk +\Mk \sum_{k=0}^{n-m-1}\sum_{j=0}^k (-1)^j \Ind_{P_{n,k}(q)}^{\Gn}\!\left(\OS^{j}_{k,0}(q)
\otimes C^{n - m - k - i + j}_{n-k,m}(q)\right)\\
&\Mk =\Mk (-1)^i \OS^{i}_{n-m}(q) + \sum_{k=0}^{n-m-1}\sum_{j=0}^k (-1)^j\; \OS^{j}_{k,0}(q) * C^{n - m - k - i + j}_{n-k,m}(q).\end{align}
\end{exam}
\begin{proof}[Proof of Theorem~\ref{q-niform}]
By Equations~\eqref{kunneth individual},~\eqref{reduced uniform}, and~\eqref{reduced q-niform}, Equation~\eqref{q-niform-recursion}
is precisely the unipotent $q$-analogue of Equation~\eqref{uniform-recursion}.
Then by Theorem~\ref{properties}, the first part of Theorem~\ref{q-niform} is equivalent to the first part
of Theorem~\ref{uniform}. The second part of Theorem~\ref{q-niform} follows from Theorem~\ref{properties}\eqref{theo2.1_2}.
\end{proof}
\longthanks{
The author is indebted to June Huh for help with formulating the main result and to Olivier Dudas for
help with proving it.
}
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\end{document}