The equivariant Kazhdan–Lusztig polynomial of a matroid

Abstract

Abstract. We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

Kazhdan-Lusztig Polynomials of Thagomizer Matroids

“…Remark 4.4. Unlike the analogous statements for uniform matroids, Conjecture 4.1 is less enlightening than Theorem 1.1(1) (see [GPY17], Theorem 3.1 and Remark 3.4).…”

“…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.…”

The Kazhdan-Lusztig polynomials of uniform matroids

“…Let us first follow Gedeon, Proudfoot, and Young [8] to recall some related definitions and notations on equivariant matroid Kazhdan-Lusztig polynomials. Let M be a matroid on the ground set I, and let W be a finite group acting on I and preserving M. Gedeon, Proudfoot, and Young referred to this collection of data as an equivariant matroid W M. Let VRep(W ) denote the ring of virtual representations of W over Z with coefficients in Z, and let VRep(W ) [t] denote the polynomial ring in t over VRep(W ).…”

“…As remarked by Gedeon, Proudfoot, and Young [8], their computation of the equivariant Kazhdan-Lusztig polynomials for uniform matroids relies on a guess on the generating function of these polynomials. Our proof given here is more direct, and uses the concept of the equivariant inverse Kazhdan-Lusztig polynomials of matroids.…”

The $Z$-Polynomial of a Matroid

“…In[GPY17], we always denoted our group by W . Here we use the letter Γ to avoid conflict with our notation for Whitney numbers.…”