The Z-polynomial of a matroid

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomological interpretation of the Z-polynomial in which the symmetry is a manifestation of Poincare duality.


Introduction
The Kazhdan-Lusztig polynomial P M (t) of a matroid M was introduced by Elias, Wakefield, and the first author in [EPW16]. This invariant has shown itself to be surprisingly rich, with many beautiful properties (most of them still conjectural). For example, the coefficients of P M (t) are conjecturally non-negative; in the case where M is realizable, this is proved by interpreting the coefficients as intersection cohomology Betti numbers of the reciprocal plane of the realization [EPW16, Theorem 3.10]. The polynomial P M (t) is conjecturally log concave [EPW16, Conjecture 2.5] and, even stronger, real rooted [GPY,Conjecture 3.2]. Furthermore, if M ′ is obtained from M by contracting a single element, the roots of P M ′ (t) are conjectured to interlace with those of P M (t) [GPY,Remark 3.5].
If the matroid M has a finite symmetry group Γ, then one can study the equivariant Kazhdan-Lusztig polynomial P Γ M (t) [GPY17], whose coefficients are virtual representations of Γ with dimension equal to the coefficients of P M (t). In the case where M is equivariantly realizable over the complex numbers, the same cohomological interpretation allows us to prove that the coefficients are honest representations [GPY17, Corollary 2.12]. The equivariant polynomial P Γ M (t) is conjectured to be equivariantly log concave [GPY17,Conjecture 5.3(2)].
Despite all of the surprising structure that these polynomials are conjectured to have, very few examples are completely understood. Kazhdan-Lusztig polynomials of thagomizer matroids coincide with Dyck path polynomials [Gedb, Theorem 1.1(1)], and Kazhdan-Lusztig polynomials of fan matroids conjecturally coincide with Motzkin polynomials [Geda]. The equivariant Kazhdan-Lusztig coefficients of uniform matroids have been computed explicitly [GPY17, Theorem 3.1] and shown to admit the structure of finitely generated FI-modules. In contrast, the equivariant Kazhdan-Lusztig coefficients of braid matroids admit the structure of finitely generated FS op -modules [PY,Theorem 6.1], and no explicit formula has appeared. Indeed, the problem of computing Kazhdan-Lusztig coefficients of braid matroids was the main motivation for this work.
In this paper we introduce the Z-polynomial Z M (t), which is defined as a weighted sum of the Kazhdan-Lusztig polynomials of all possible contractions of M . The Z-polynomial is palindromic (Proposition 2.3), reflecting the fact that, when M is realizable, the coefficients of Z M (t) may be interpreted as intersection cohomology Betti numbers of a projective variety (Theorem 7.2), for which Poincaré duality holds.
Surprisingly, this symmetry of the Z-polynomial translates into a recursive formula for Kazhdan-Lusztig coefficients that is different from any of the recursive formulas seen before (Corollary 3.2). In particular, it yields a method for computing Kazhdan-Lusztig coefficients of braid matroids that is much faster than any previously available approach. Furthermore, we are able to use this recursion to obtain a formula that expresses each Kazhdan-Lusztig coefficient of M as a finite alternating sum of multi-indexed Whitney numbers (Theorem 3.3). In the case of braid matroids, this becomes a finite alternating sum of products of Stirling numbers of the second kind (Corollary 4.5). We also obtain an equivariant version of our formula (Theorem 6.1), which takes a particularly nice form for uniform matroids (Proposition 6.3).
Our Theorem 3.3 bears a close resemblance to a recent result of Wakefield [Wak,Theorem 5.1], who also obtained a formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. It is likely that our formula is equivalent to Wakefield's, but the combinatorics involved in the two formulas are very different; see Remark 3.6 for further discussion of this point.
Our paper is structured as follows. Section 2 contains the definition of the Z-polynomial, the proof of panlindromicity, and the recursion for Kazhdan-Lusztig coefficients that follows from this symmetry. Section 3 uses this recursion to derive the formula for Kazhdan-Lusztig coefficients in terms of multi-indexed Whitney numbers. Section 4 interprets these results in the case where we have a family of matroids that is closed under contractions, such as braid matroids or uniform matroids. One of the results of this section is that Narayana polynomials are special cases of Zpolynomials (Proposition 4.9). Section 5 contains conjectures about the roots of the Z-polynomial, analogous to the conjectures in [GPY] about the roots of the Kazhdan-Lusztig polynomial. Section 6 explains how to extend our results and conjectures to the equivariant setting.
Finally, Section 7 contains the cohomological interpretation of the Z-polynomial. This section provides the key motivation for the definition of the Z-polynomial, so in some sense it ought to appear at the very beginning of the paper. However, the methods used Section 7 are quite technical, in contrast with the elementary and purely combinatorial methods employed in the rest of the paper, so we relegated it to the end.

Definition and palindromicity
Let M be a matroid on the ground set I, and let L be the lattice of flats of M . Given a flat F ∈ L, let M F be the localization of M at F ; this is the matroid on the ground set F whose lattice of flats is isomorphic to L F := {G ∈ L | G ≤ F }. Dually, let M F be the contraction of M at F ; this is the matroid on the ground set I F whose lattice of flats is isomorphic to For any flat F , we have the rank rk F := rk M F and the corank crk F : be the characteristic polynomial of M , and let P M (t) ∈ Z[t] be the Kazhdan-Lusztig polynomial of M , as defined in [EPW16, Theorem 2.2]. The Kazhdan-Lusztig polynomial is characterized by the following three properties: • If rk M = 0, then P M (t) = 1.
Definition 2.1. For any matroid M , we define the Z-polynomial where µ : L × L → Z is the Möbius function.
Proof. We have Proposition 2.3. For any matroid M , Z M (t) is palindromic of degree rk M . That is, Proof. We have This completes the proof.
Remark 2.4. In Section 7, we will give a geometric interpretation of the Z-polynomial of a realizable matroid, and in this context Proposition 2.3 can be interpreted as Poincaré duality (see Remark 7.3).
Despite the simplicity of the proof, Proposition 2.3 implies a previously unknown recursive formula for Kazhdan-Lusztig coefficients. Let c M (i) and z M (i) denote the coefficients of t i in P M (t) and Z M (t), respectively.
Isolating the first term in the left-hand sum, we obtain the desired equation.
Remark 2.6. Suppose that 2i < rk M , which is a necessary condition for c M (i) to be nonzero provided that rk M > 0. Then the F = ∅ term vanishes from the first sum, and we in fact have Furthermore, if i > 0, then c M F (crk F − i) = 0 unless crk F < 2i, which means that crk F − i < i. This tells us that our recursion expresses c M (i) in terms of other Kazhdan-Lusztig coefficeints c N (j) where j is strictly smaller than i and N has strictly smaller rank than M .

Kazhdan-Lusztig coefficients and Whitney numbers
In this section we will regard c(i) as a function that takes as input a matroid and produces as output an integer. As we observed in Remark 2.6, the function c(i) can be expressed recursively in terms of the functions c(0), . . . , c(i − 1). If we iterate this procedure i times, we obtain an expression for c(i) that does not involve any Kazhdan-Lusztig coefficients except for c(0), which is the constant function with value 1 [EPW16, Proposition 2.11]. This is exactly what we do in this section.
Given a sequence i r , . . . , i 1 of integers and a matroid M with lattice of flats L, we define the r-Whitney number W M (i r , . . . , i 1 ) := (F r , . . . , F 1 ) ∈ L r | F r ≤ · · · ≤ F r and crk F j = i j for all j .
We will usually just write W (i r , . . . , i 1 ), which we regard as a function that takes matroids to numbers. For example, W (i) is the function that counts the number of flats of corank i, while W (i 2 , i 1 ) is the function that counts the number of pairs of comparable flats with coranks i 2 and i 1 .
Remark 3.1. Our conventions differ from the usual ones in that we index our Whitney numbers by corank rather than rank; this will make Theorem 3.3 significantly simpler to state.
Lemma 3.2. Let M be a matroid and i r , . . . , i 1 a sequence of integers. Then Proof. This is immediate from our description of the lattice of flats of M F .
Remark 3.4. If we try to compute c M (i) for a matroid M that does not satisfy the inequality 2i < rk M , then the sum will be empty, because the condition i = a r < a r+1 = rk −i is not satisfied. We will therefore obtain the number zero, which is what we expect. Similarly, we can replace the sum over r from 1 to i with a sum over all r, because the conditions 0 = a 0 < · · · < a r = i can only be satisfied if 1 ≤ r ≤ i.
Remark 3.5. Assuming that we are evaluating this function on a matroid whose rank is greater than 2i, the number of tuples (a 0 , . . . , a r+1 ) satisfying the given conditions is equal to the number of compositions of i into r parts, which is in turn equal to the binomial coefficient i−1 r−1 . Thus the total number of terms in our expression for c(i) is equal to Remark 3.6. Theorem 3.3 bears a strong similarity to [Wak,Theorem 5.1], where c(i) is also expressed as an alternating sum of r-Whitney numbers. It seems likely that there is a bijection between our index set and Wakefield's index set that makes the signed Whitney numbers in our formula match with those in his. However, this bijection is not at all obvious; in particular, it is not even clear to us how to compute the size of Wakefield's index set for general i. Using a computer, Gedeon determined that the index sets do have the same size when i ≤ 4.
Proof of Theorem 3.3: We induct on i. When i = 1, our formula says We have t 1 (∅) = 1 and t 1 ([1]) = 2, so this says c(1) = W (1) − W (rk −1), which was proved in [EPW16, Proposition 2.12]. Now assume that our formula holds for all j < i. Fix a matroid M . By Remark 2.6, we may assume that 2i < rk M , for otherwise c M (i) = 0 and the sum is empty. By Remarks 2.6 and 3.4, we have We can simplify these expressions by first fixing the corank of F to be some number k and then applying Lemma 3.2. This gives us the formula Next, we eliminate k from both sums by observing that k = a r+1 + a r = a t r+1 (S) + a r , and the inequality k < rk M turns into an inequality involving a r . In the first sum, we get the inequality a r < rk M − i, but this is implied by the fact that a r < a r+1 = i < rk M − i. In the second sum, we get the inequality a r < i, which is not implied by the other conditions. Thus we have We now proceed to reindex the two sums. Given a natural number r and a subset S ⊂ [r], let S 0 := S and S 1 := S ∪ {r + 1}, both regarded as subsets of [r + 1]. Then for all j, so we can replace S with S 0 in the first sum. On the other hand, , and the second sum becomes (Note that, by replacing (−1) |S| with (−1) |S 1 | , we have absorbed the external minus sign.) All together, this gives us Finally, we observe that summing over all subsets S ⊂ [r] and then separately considering S 0 and S 1 is the same as summing over all subsets of [r + 1]. If we now re-index the outer sum by letting s = r + 1, we obtain the desired formula for c M (i), and the induction is complete.

Nice families
We define a nice family to be a sequence of matroids Examples of nice families include the following.  Fix a nice family. For ease of notation, we will write and In the four families described above, we have the following.
2. For the matroid associated with the type B d Coxeter arrangement, The first formula appears in [Sut00,Proposition 3]. The second appears in [Slo14, Sequence A028338], using the fact that the exponents of this arrangement are 1, 3, . . . , 2d − 1.
3. For the uniform matroid U m,d ,

For the matroid represented by all vectors in
Corollary 2.5 and Remark 2.6 translate to the following statement.
Corollary 4.2. If 2i < d, then Remark 4.3. Corollary 4.2 has proved to be faster than any previously known formula for computing the Kazhdan-Lusztig coefficients of the braid matroid.
We may also interpret r-Whitney numbers in terms of the numbers W d (k). The following result follows from Lemma 3.2.
Corollary 4.4. If we set i r+1 := d, then we have Combining Corollary 4.4 with Theorem 3.3, we obtain the following result. (−1) |S| Given a nice family, it is natural to use generating functions to collect the Kazhdan-Lusztig polynomials and the Z-polynomials. Let We will also be interested in the exponential generating functions In addition, consider the generating functions along with their exponential analogues Proposition 4.6. We have and alsoP which is equal to P (t, u) by Equation (1). The proofs of the other three statements are identical.
Example 4.7. In type A (the first example), Proposition 4.6 is most elegant in its exponential version. We haveg so Proposition 4.6 says that Example 4.8. In type B (the second example), we havẽ so Proposition 4.6 says that We next consider the third example when m = 1, so that M d is the uniform matroid of rank d on d + 1 elements. In this case, we can use Proposition 4.6 to derive a precise formula for the Z-polynomial.
Proposition 4.9. If M d is the uniform matroid of rank d on d + 1 elements, then the coefficient Proposition 4.6 therefore tells us that In [PWY16, Section 2], we showed that Setting v = u 1−tu , we obtain an explicit algebraic expression for Z(t, u). On the other hand, it is shown in [Pet15, Equation (2.6)] that It is an elementary exercise to check that this formula coincides with our expression for Z(t, u).

Roots of the Z-polynomial
In [GPY, Conjecture 3.2], we conjectured that the polynomial P M (t) is real rooted. Here we make the analogous conjecture for the Z-polynomial. We also gave a conjectural relationship between the roots of P M (t) and the roots of a contraction of P M/e (t), where e ∈ I is a non-loop of M [GPY, Conjecture 3.3], assuming certain nondegeneracy conditions. Here we make a similar conjecture for Z-polynomials, but rather than attempting to formulate the correct nondegeneracy conditions, we focus on the case of a nice family, where the conjecture takes a particularly clean form. If f (t) is a polynomial of degree d with roots α 1 ≤ · · · ≤ α d and g(t) is a polynomial of degree d − 1 with roots β 1 ≤ · · · ≤ β d−1 , we say that f (t) If the inequalities are strict, we say that f (t) strictly interlaces g(t). Proof. We will prove a slightly stronger statement by induction on d. We will prove that, for every d, Z d (t) has roots α 1 , . . . , α d < 0 with α i < qα i+1 for all 0 < i < d, and that Z d (t) strictly interlaces Z d−1 (t). The statement is trivial when d = 1.
As observed in Example 4.10, we have Using the identity In particular, we have This tells us that the numbers Z d (β i ) alternate in sign, and therefore that for all 1 is positive for t sufficiently negative, so there must exist a root α 1 < β 1 . This proves that the roots of Z d (t) lie on the negative real axis and Z d (t) strictly interlaces Z d−1 (t).
To complete the induction, we still need to prove that α i < qα i+1 for all 0 < i < d. For all such i, we have We know that α i Z d−1 (α i ) and α i+1 Z d−1 (α i+1 ) have opposite signs, therefore so do Z d−1 (qα i ) and Z d−1 (qα i+1 ). It follows there there is a root β j i of Z d−1 (t) in between qα i and qα i+1 . Since β j 1 < · · · < β j d−1 , we must have j i = i, and therefore α i < β i < qα i+1 .
Remark 5.6. We have proved Conjectures 5.1 and 5.2 for our third family when m = 1 (Example 5.3) and for our fourth family (Proposition 5.5). For the first two families, and for the third family when 2 ≤ m ≤ 10, we have checked the conjectures on a computer for all d ≤ 30.

Equivariant matroids
An equivariant matroid Γ M consists of a finite group Γ, a matroid M with ground set I, and an action of Γ on I that takes flats of M to flats of M . In [GPY17], we defined the Kazhdan-Lusztig polynomial P Γ M (t) of an equivariant matroid 1 Γ M . This is a polynomial whose coefficients are virtual representations of Γ; equivalently, it is a graded virtual representation. If we forget the action of Γ and take the graded dimension, we recover the ordinary Kazhdan-Lusztig polynomial of M .
All of the material in Sections 2 and 3 generalizes easily to equivariant matroids, starting with the definition of the Z-polynomial. Let L denote the lattice of flats of M . For any flat F ∈ L, let Γ F ⊂ Γ denote the stabilizer of M . We may then define The generalization of Theorem 3.3 comes from interpreting r-Whitney numbers as permutation representations. More precisely, given an equivariant matroid Γ M and a sequence of integers i r , . . . , i 1 , let W Γ M (i r , . . . , i 1 ) be the representation of Γ with basis (F r , . . . , F 1 ) ∈ L r | F r ≤ · · · ≤ F r and crk F j = i j for all j .
We omit the proof of the following result, as it does not differ significantly from the proof of Theorem 3.3.
Theorem 6.1. For all i > 0, we have Theorem 6.1 takes a particularly nice form for uniform matroids. Let ch n be the Frobenius characteristic, which takes representations of the symmetric group S n to symmetric functions of degree n in infinitely many variables. Let s[n] := ch n triv be the complete homogeneous symmetric function of degree n.
Proposition 6.2. We have Proof. The symmetric group S m+d acts transitively on the set (F r , . . . , F 1 ) ∈ L r | F r ≤ · · · ≤ F r and crk F j = i j for all j , with stabilizers conjugate to the Young subgroup G := S d−ir × S ir−i r−1 × · · · × S i 2 −i 1 × S m+i 1 . It follows that W S m+d U m,d (i r , . . . , i 1 ) is isomorphic to Ind S m+d G triv, and the Frobenius characteristic of the induction of the trivial representation from a Young subgroup is equal to the product of the corresponding complete homogeneous symmetric polynomials.
Proof. By Theorem 6.1 and Proposition 6.2, we need to show that is equal to the summand in the statement of the corollary. First, we note that a 0 = 0, so the last factor is equal to s[m + a t 1 (S) ]. Next, we note that d = a r+1 + a r = a t r+1 (S) + a r , so the first r factors of the product may be written uniformly as For each j ∈ [r], we have If V = ⊕V i is a graded virtual representation of a group Γ, we say that V is equivariantly log is isomorphic to an honest representation. We say that V is strongly equivariantly log concave if, for all i ≤ j ≤ k ≤ l with i + l = j + k, V j ⊗ V k − V i ⊗ V l is isomorphic to an honest representation. If Γ is the trivial group, then log concavity and strong log concavity are equivalent, and agree with the usual notion of log concavity for a sequence of integers. For nontrivial Γ, however, strong equivariant log concavity is a strictly stronger condition with the desirable property of being preserved under tensor product [GPY17, Remark 5.8]. The following conjecture is the Z-version of [GPY17, Conjecture 5.3(2)].
Remark 6.6. Polynomials whose roots lie on the negative real axis are log concave in the usual sense, hence if Γ is the trivial group, Conjecture 6.5 is a weaker version of Conjecture 5.1.
Proposition 6.7. Fix a natural number d and a prime power q. Let M be the matroid represented by all vectors in F d q and let Γ = GL n (F q ). Conjecture 6.5 holds for Γ M .
ℓ-adicétale intersection cohomology of X(V ) vanishes in odd degree, and If k = C, the same result holds for topological intersection cohomology.
In this section we prove the analogous result for the Z-polynomial.
Theorem 7.2. If k is a finite field and ℓ is a prime not equal to the characteristic of k, then the ℓ-adicétale intersection cohomology of Y (V ) vanishes in odd degree, and If k = C, the same result holds for topological intersection cohomology. Remark 7.4. Any matroid that can be realized over some field can be realized over a finite field, so Theorems 7.1 and 7.2 apply to all realizable matroids.
A nonempty subset C ⊂ I is called a circuit if and only if, for every flat F , |C ∩ F c | = 1. Conversely, a subset F ⊂ I is a flat if and only if, for every circuit C, |C ∩ F c | = 1. Given a circuit C, there exist elements (C i ) i∈C ⊂ (k × ) C such that i C i v i = 0 for all v ∈ V , and these elements are unique up to scale. The homogeneous coordinate ring of Y (V ) ⊂ (P 1 k ) I has the following description [AB16, Theorem 1.3(a)]: Given a point p ∈ Y (V ), let F p := {i ∈ I | p i = ∞}.
Lemma 7.5. The set F p is a flat.
Proof. If F p is not a flat, then there exists a circuit C and an element i ∈ I such that F c p ∩ C = {i}. For all j ∈ C {i}, y C {j} is a multiple of y i , which vanishes at p. But x i does not vanish at p, nor does y C {i} . This contradicts the fact that f C vanishes at p.
For any flat F , let V F ⊂ A F c k be the intersection of V with A F c k inside of A I k , and let V F ⊂ A F k be the image of V along the projection from A I k . Concretely, V F is cut out of A F k by the linear equations f C (x, 1) for all circuits C ⊂ F . Then we have M ( Proof. The affine coordinate ring of Y (V ) F is obtained from k[Y (V )] by setting x i = 1 and y i = 0 for all i ∈ F c and y j = 1 for all j ∈ F . This ring is isomorphic to As observed above, these are exactly the equations that define V F inside of A F k .
Fix a prime ℓ different from the characteristic of k. The ℓ-adicétale intersection cohomology group of Y (V ) is defined as For any point p ∈ Y (V ), we define to be the cohomology of the stalk of the IC sheaf at p.
Proof. Since the IC sheaf is locally constant along strata, we may assume that p i = 0 for all i, By Lemmas 7.6 and 7.7 and Poincaré duality, We know that IH p−q X(V F ); Q ℓ vanishes unless p − q is even [EPW16,Proposition 3.9]. This implies that the spectral sequence degenerates at the E 1 page, IH m Y (V ); Q ℓ = 0 unless m is even, and IH 2i Y (V ); Q ℓ ∼ = p+q=2i crk F =p IH p−q X(V F ); Q ℓ = F IH 2(crk F −i) X(V F ); Q ℓ .
We now apply Poincaré duality for IH * Y (V ); Q ℓ to see that we can replace i with rk M − i, which has the effect of replacing crk F − i with i − rk F . Thus IH 2i Y (V ); Q ℓ ∼ = F IH 2(i−rk F ) X(V F ); Q ℓ .
The same argument works for topological intersection cohomology when k = C.
Remark 7.8. Theorems 7.1 and 7.2 also hold equivariantly. That is, if Γ acts on I in such a way so that V ⊂ k I is a subrepresentation, then Γ acts on M (V ), X(V ), and Y (V ), and we have as graded representations of Γ. This holds for ℓ-adic intersection cohomology when k is a finite field as well as for topological intersection cohomology when k = C. The first statement for k = C appears in [GPY17, Corollary 2.12]; see also [PY,Theorem 3.1]. The finite field version can be proved similarly; the only technical point is that in the k = C case we argue that the maps in a certain spectral sequence 3 must strictly preserve weights in the mixed Hodge filtration, and in the finite field version we instead use the fact that these maps are equivariant for the action of the Frobenius automorphism.
Once we know the first statement, the proof of Theorem 7.2 extends without modification to the equivariant setting, and the second statement is proved, as well.
Remark 7.9. Consider the category O(V ) of perverse sheaves on Y (V ) that are smooth with respect to the stratification described in this section. This category has some very nice properties; see for example [BGS96,3.3.1] when k = C and [BGS96,4.4.4] when k is a finite field. In particular, which in turn is given by the (backward) graded dimension of the Ext group from the skyscraper sheaf at the point ∞ to the IC sheaf of Y (V ). Other Ext groups from standard objects to simple objects are measured by Kazhdan-Lusztig polynomials of localizations of contractions of M (V ).