Kazhdan-Lusztig polynomials of thagomizer matroids

We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.


Introduction
The main objects of study in this paper are the Kazhdan-Lusztig polynomials of a particular family of matroids. The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield [EPW16]. In the appendix of that paper, the authors (along with Young) explicitly computed the coefficients of these polynomials for some uniform and braid matroids of small rank. Proudfoot, Wakefield and Young studied uniform matroids of rank n − 1 on n elements [PWY16] and gave a combinatorial description for the coefficients of the associated Kazhdan-Lusztig polynomial.
Let M n be the matroid associated with the graph obtained from the bipartite graph K 2,n by adding an edge between the two distinguished vertices. We call M n a thagomizer matroid 1 . The ground set of M n has size 2n + 1 and the rank of M n is n + 1. We give a description of the flats of M n in Section 3.  Let c n,k be the k-th coefficient of P n (t) and note that the degree of P n (t) is n 2 . The following theorem is our main result.
(1) For all n and k, c n,k is the number of Dyck paths of semilength n with k long ascents.
(2) The generating function Φ(t, u) is equal to Remark 1.2. It is known that the number of Dyck paths of semilength n with k long ascents is equal to the quantity 1 n+1 n+1 k n j=2k (see [STT06] and sequence A091156 in [Slo16]).
Remark 1.3. The total number of Dyck paths of semilength n is equal to the nth Catalan number C n = 1 n+1 2n n . Thus Theorem 1.1 implies that P n (1) = C n and Remark 1.2 implies that the leading coefficient of P 2n (t) is C n . Interestingly, C n also appears as the leading coefficient of the Kazhdan-Lusztig polynomial of the uniform matroid of rank 2n − 1 on 2n elements (see [EPW16] Appendix A and [PWY16]).
Remark 1.4. Prior to this paper, uniform matroids were the only infinite family of matroids for which the Kazhdan-Lusztig polynomial has been computed. For example, it is still an open problem to compute the Kazhdan-Lusztig polynomial of the braid matroid; see [EPW16] and [GPY17] for partial results.
We conclude this section with a description of the structure of the paper. In Section 2, we recall the definition of the Kazhdan-Lusztig polynomial of a matroid and review Dyck paths. Section 3 is dedicated to proving Theorem 1.1.
In Section 4, we recall the definition of the equivariant Kazhdan-Lusztig polynomial of a matroid and explore the S n action on M n which allows us to make a conjecture for the S n -equivariant Kazhdan-Lusztig polynomial of M n . This categorification of Kazhdan-Lusztig coefficients was first considered for a uniform matroid of rank n − 1 on n elements by Proudfoot, Wakefield and Young [PWY16] where they were given by an irreducible representation of S n . The equivariant Kazhdan-Lusztig polynomial for a general matroid was subsequently defined by the author, Proudfoot and Young [GPY17] where we further studied uniform matroids in this context and computed the S n -equivariant Kazhdan-Lusztig polynomials of braid matroids of small rank.
Acknowledgements. The author would like to thank Nicholas Proudfoot and Benjamin Young for many helpful discussions and suggestions. The success of this project is due in large part to the On-Line Encyclopedia of Integer Sequences [Slo16] and SageMath [Dev16]. The author was partially supported by NSF grant DMS-1565036.

Preliminaries
2.1. Kazhdan-Lusztig polynomials of a matroid. In this section we follow [EPW16] to define the non-equivariant Kazhdan-Lusztig polynomial of a matroid (which we simply refer to as the Kazhdan-Lusztig polynomial).
Let M be a matroid on the finite ground set E. Denote by L(M ) the lattice of flats of M and χ M (t) the characteristic polynomial of M . The matroid M F is called the contraction of M at F ; it is the matroid on the ground set E \ F whose lattice of flats is L F := {G \ F | G ∈ L(M ) and G ≥ F }. The matroid M F is called the localization of M at F and is the matroid with ground set F whose lattice of flats is is characterized by the following three properties [EPW16, Theorem 2.2].
2.2. Dyck paths. A Dyck path of semilength n is a lattice path in N 2 beginning at (0, 0) and ending at (2n, 0) with up-steps of the form u = (1, 1) and down-steps of the form d = (1, −1). Such a Dyck path may be expressed as a word α ∈ {u, d} 2n . A long ascent of a Dyck path is an ascent of length at least 2. Equivalently, a long ascent of a Dyck path α is a maximal subword consisting of at least two consecutive u's. The Dyck path given in Figure 2 has two long ascents. Let D n be the set of all Dyck paths of semilength n. We denote by a n,k the number of elements in D n with exactly k long ascents. As noted in [STT06], a n,k is also the number of words α ∈ D n with k occurrences of the subword uud. Additional interpretations of a n,k are known; see sequence A091156 in [Slo16].

Main results
We begin this section with a description of the flats F ∈ L(M n ) given by the underlying graph. Let AB be the distinguished edge. For any j ∈ {1, . . . , n}, we call the subgraph with edges Aj and Bj a spike.
If rk F = i, then either (1) F contains exactly one edge from i distinct spikes, or (2) F is the union of i − 1 spikes and AB. For example, when n = 4, a rank 2 flat of the first type is given by {A1, B3} and a rank 2 flat of the second type is given by {AB, A4, B4} (see Figure 1).
In the first case, the localization (M n ) F yields a Boolean matroid of rank i, and the contraction M F n gives a matroid whose lattice of flats is isomorphic to that of M n−i . In the second case, the localization (M n ) F gives a matroid whose lattice of flats is isomorphic to that of M i−1 , and the contraction M F n is a Boolean matroid of rank n − i + 1.
The characteristic polynomial of a rank i Boolean matroid is equal to (t − 1) i . For thagomizer matroids, it is clear that χ M i (t) = (t − 1)(t − 2) i by a simple deletion/contraction argument.
We first turn our attention towards proving the following lemma.
Lemma 3.1. We have the following (equivalent) equations.
(1) For all n, t n+1 P n (t −1 ) ( Proof. There are n i · 2 n−i flats of the first type of rank n − i and n i flats of the second type of rank i + 1. Note that for any Boolean matroid M, P M (t) = 1 [EPW16, Corollary 2.10]. Then we have which is the formula given in Lemma 3.1(1). Now our defining recursion tells us that We let m = n − i which allows us to write the second summand as Recall the identity and set = m and x = 2u(t − 1). This gives This completes the proof of Lemma 3.1.
Finally we are ready to prove Theorem 1.1. Let a n,k be as in Section 2.2, and set F (t, u) := n,k≥0 a n,k t k u n .
It was shown in [STT06, Section 1] that F (t, u) satisfies A priori, this formula should have a ± sign. However, a plus sign would not give F (t, u) as a formal power series. Hence we use a negative sign instead. Let f (t, u) := u · F (t, u). Since we'd like to show that Φ(t, u) = u · F (t, u), we first check that f (t, u) satisfies the functional equation in Lemma 3.1(2).
We have Lastly, we note that both c n,k and a n,k are zero if n > 2k and that f (t, 0) = Φ(t, 0) = 1. Then f (t, u) = Φ(t, u) which equivalently tells us that c n,k = a n,k . This completes the proof of Theorem 1.1.

The S n action
Recall the notation set in Section 2.1. That is, let M be a matroid on a finite ground set E. • If rk M = 0, P W M (t) is equal to the trivial representation in degree 0.
The polynomial P W M (t) is an element of grVRep(W ) and we can recover P M (t) from P W M (t) by taking the graded dimension. Now we turn our attention back to the thagomizer matroid M n . Though the full automorphism group of M n is S n × S 2 (unless n = 1 in which case it is S 3 ), here we only consider the action of the symmetric group S n . Let Let Υ n be all partitions of n of the form [a, n − a − 2i − η, 2 i , η] where η ∈ {0, 1}, i ≥ 0 and 1 < a < n. For any partition λ of n, we let V λ be the irreducible representation of S n indexed by λ.
For any partition λ, we set Conjecture 4.1. For all n > 0, we have Remark 4.2. We have checked this conjecture for thagomizer matroids of rank at most 20 using SageMath [Dev16]. For our calculations, we worked in the symmetric function setting (see Proposition 4.7).
Remark 4.3. We know the coefficients of P n (t) will be honest representations by [GPY17, Corollary 2.12] since M n is S n -equivariantly realizable.
That is, the coefficients of the Kazhdan-Lusztig polynomial of a uniform matroid are more cleanly expressed when given as the dimension of a certain representation of the symmetric group. This is not the case for thagomizer matroids.
The remainder of this section is devoted to understanding the results that allow us to derive the recursive formula and functional equation for the Frobenius characteristic of P n (t). Let W (t) := (t − 1)C and V (t) := (t − 2)C as virtual vector spaces. Then W (t) ⊗r is the equivariant characteristic polynomial of a rank r Boolean matroid and W (t) ⊗ V (t) ⊗r is the equivariant characteristic polynomial of M r . Both W (t) ⊗r and W (t) ⊗ V (t) ⊗r are virtual representations of S r , where S r acts by permuting the factors of the graded tensor product. Note that the equivariant Kazhdan-Lusztig polynomial of a Boolean matroid is the trivial representation in degree zero.
We'd like to categorify the recursive formula given in Lemma 3.1(1). Recall Equation 3.2 The first sum is over flats of rank n − i of the first type mentioned in Section 3. For flats of this type, summing over where S m permutes the vertices that are connected to A by an edge in F , S j permutes the vertices that are connected to B by an edge in F , and S i permutes the vertices that are not adjacent to any edge in F . Similarly, summing over flats of the second type gives where S n−i is acting trivially. As in [GPY17, Section 3.1], we now translate to symmetric functions. We consider the Frobenius characteristic where Λ n is the space of symmetric functions of degree n in infinitely many formal variables {x i | i ∈ N}.
Let s[λ] := ch V λ be the Schur function corresponding to λ and set p n (t) := ch P n (t), w n (t) := ch W (t) ⊗n and v n (t) := ch V (t) ⊗n .
Applying Frobenius characteristic to Equations 4.5 and 4.6, we obtain Note that w(t, u) = s(tu) s(u) by [GPY17, Proposition 3.9]. The results of this section can be summarized in the following proposition.
Proposition 4.7. We have the following (equivalent) equations.
Remark 4.8. In [GPY17], we were able to compute the equivariant Kazhdan-Lusztig polynomial for uniform matroids by showing that our "guess" satisfied a recursion analogous to the one found in Proposition 4.7(2). That case was much simpler; we only had to consider singular applications of the Pieri rule. In this case, w(t, u) 2 requires multiple applications of the Pieri rule while v n (t) = s[n] v 1 (t) involves a plethysm. This makes proving Conjecture 4.1 much more difficult.