Equivariant Kazhdan-Lusztig polynomials of $q$-niform matroids

We introduce $q$-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.


Introduction
For any matroid M , the Kazhdan-Lusztig polynomial P M (t) ∈ Z[t] was introduced in [EPW16]. In the case where the matroid M admits the action of a finite group W , one can define the equivariant Kazhdan-Lusztig polynomial P W M (t) [GPY17]; 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 [Ged17] and fan, wheel, and whirl matroids [LXY]. Kazhdan-Lusztig polynomials of braid matroids have been studied extensively, both in the equivariant [PY17] and non-equivariant [KW] 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 λ of n, let V [λ] be the associated irreducible representation of S n . The following theorem was proved in [GPY17, Theorem 3.1]; an independent proof of the nonequivariant statement was later given in [GLX + , Theorem 1.2].
Theorem 1.1. Let C i n,m be the coefficient of t i in the S n -equivariant Kazhdan-Lusztig polynomial of U n,m , and let c i n,m := dim C i n,m be the corresponding non-equivariant coefficient.
, and for all i > 0, • c 0 n,m = 1, and for all i > 0, The purpose of this note is to obtain a q-analogue of Theorem 1.1. 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 F n q , which we regard as the q-analogue of the Boolean matroid of rank n. For any natural number m ≤ 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 the q-analogue of the uniform matroid U n,m , and we will therefore refer to it as a q-niform matroid. This matroid was also studied in [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 GL n (q) of invertible n × n matrices with coefficients in F q , which is the q-analogue of S n .
The representation theory of GL n (q) is much more complicated than the representation theory of S n . However, there is a certain subset of irreducible representations of GL n (q), known as irreducible unipotent representations, that correspond bijectively to the irreducible representations of S n . For any partition λ of n, let V (q)[λ] be the associated irreducible unipotent representation of GL n (q). For any positive integer k, we use the standard notation [k] q := 1 + q + · · · + q k−1 and 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 .
Theorem 1.2. Let C i n,m (q) be the coefficient of t i in the GL n (q)-equivariant Kazhdan-Lusztig polynomial of U n,m (q), and let c i n,m (q) := dim C i n,m (q) be the corresponding non-equivariant coefficient.
, and for all i > 0, • c 0 n,m (q) = 1, and for all i > 0, c i n,m (q) is equal to , but no proof exists in the general case. The matroid U n,m is always realizable, but it is not equivariantly realizable unless m ∈ {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 1.1 and 1.2 both provide significant evidence for the equivariant non-negativity conjecture.
Remark 1.4. Theorem 1.1 implies that {C i n,m | n ≥ m} admits the structure of a finitely generated FI-module [CEF15, Theorem 1.13], while Theorem 1.2 implies that {C i n,m (q) | n ≥ m} admits the structure of a finitely generated VI-module [GW18, Theorem 1.6]. In order to define these structures in a natural way, we would need need to be able to define C i n,m and C i n,m (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 i n,1 , and we obtain a canonical FI opmodule structure from [PY17, Theorem 3.3(1)]; 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 FI op -module structure on {C i n,m | n ≥ m} and a canonical VI op -module structure on {C i n,m (q) | n ≥ m}, each with finitely generated duals.
Our proof of Theorem 1.2 relies heavily on Theorem 1.1 along with the Comparison Theorem (Theorem 2.1), 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 3.4) and that the recursive formula for C i n,m (q) is essentially the same as the recursive formula for C i n,m (Equations (7) and (8)).
Acknowledgments: The author is indebted to June Huh for help with formulating the main result and to Olivier Dudas for help with proving it. The author is supported by NSF grant DMS-1565036.

Unipotent representations and the Comparison Theorem
Given a pair of natural numbers k ≤ n and a pair of Irreducible representations of the symmetric group S n are classified by partitions of n. Given a partition λ, let V [λ] be the associated representation. For each cell (i, j) in the Young diagram for λ, let h λ (i, j) be the corresponding hook length; then the dimension of V [λ] is equal to n! h λ (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 ≤ n, let P k,n (q) ⊂ GL n (q) denote the parabolic subgroup associated with the Levi GL k (q) × GL n−k (q). Given a pair of representations V (q) of GL k (q) and V ′ (q) of GL n−k (q), we obtain a representation V (q) ⊠ V ′ (q) of GL k (q) × GL n−k (q), and we may interpret this as a representation of P k,n (q) via the natural surjection P k,n (q) → GL k (q) × GL n−k (q). We then define This operation is called Harish-Chandra induction.
Let B n (q) ⊂ GL n (q) be the subgroup of upper triangular matrices. An irreducible representation of GL n (q) is called unipotent if it appears as a direct summand of the representation An arbitrary representation is called unipotent if it is isomorphic to a direct sum of irreducible unipotent representations.
Theorem 2.1. Let q be a prime power and n a natural number.
1. Irreducible unipotent representations of GL n (q) are in canonical bijection with partitions of n.

The irreducible unipotent representation V (q)[λ] associated with the partition λ has dimension
3. If k ≤ n, V (q) is a unipotent representation of GL k (q), and V ′ (q) is a unipotent representation of GL n−k (q), then V (q) * V ′ (q) is a unipotent representation of GL n (q).
4. Let λ, µ, and ν be partitions of n, k, and n − k, respectively. The multiplicity of Remark 2.2. The standard proof of Theorem 2.1(1) is very far from constructive. One proves that the endomorphism algebra of C GL n (q)/ B n (q) is isomorphic to the Hecke algebra of S n ; this implies that the irreducible constituents of C GL n (q)/ B n (q) 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 [And18] gives a construction of V (q)[λ] modeled on tableaux, which is analogous to the usual construction of V [λ].
Remark 2.3. A generalization of Statement 4 due to Howlett and Lehrer [HL83, Theorem 5.9] 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 2.1. Example 3.1. Suppose that V is a vector space over F q , and that {H e | e ∈ E} is a collection of hyperplanes with associated matroid M . Fix a prime ℓ that does not divide q, and fix an embedding of Q ℓ into C. Let

Orlik-Solomon algebras
and Then we have canonical isomorphisms where the cohomology rings are ℓ-adicétale cohomology. If rk M > 0, then X ∼ = PX × G m (F q ), and Equation (1) 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.
Example 3.2. The Boolean matroid U n,0 is S n -equivariantly realized by the coordinate hyperplanes in F n q . 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 ∼ = G n m (F q ). As a representation of S n , we have and OS * n,0 In particular, this implies that OS i n,0 for all i < n.
Example 3.3. The matroid U n,0 (q) is (by definition) GL n (q)-equivariantly realized by the collection of all hyperplanes in F n q . The variety PX n,0 (q) is an example of a Deligne-Lusztig variety for the group GL n (q). The techniques developed by Lusztig [Lus77] imply that the action of GL n (q) on the cohomology group of PX n,0 (q) is given by the unipotent q-analogue of Equation (3): for all i < n. See [Dud18, Examples 6.1 and 6.4] for a concise and explicit statement of this result.  Example 4.3. Proper flats of U n,m (q) are collections of linearly independent hyperplanes in F n q of cardinality less than n − m. For such an F , U n,m (q) F ∼ = U |F |,0 (q), while U n,m (q) F ∼ = U n−|F |,m (q).
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 ∈ L, let W F ⊂ W denote the stabilizer of F . If C i M,W 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 [GPY17, Proposition 2.9]: where we take in the sum one flat from each W -orbit in L.
Example 4.4. 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 × S n−k ⊂ S n . Thus Equation (6) transforms into the following recursion: where the first term corresponds to the maximal flat F = [n].
Example 4.5. Consider the case of the q-uniform matroid U n,m (q). Proper flats are collections of linearly independent hyperplanes in F n q of cardinality less than n − m, and the GL n (q)-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) ⊂ GL n (q). Thus Equation (6)