The \(Z\)-polynomial of a matroid. (English) Zbl 1380.05022

Proudfoot, Nicholas; Xu, Yuan; Young, Ben

Electron. J. Comb. 25, No. 1, Research Paper P1.26, 21 p. (2018).

Summary: 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 Poincaré duality.

MSC:

05B35	Combinatorial aspects of matroids and geometric lattices
52B40	Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
55N33	Intersection homology and cohomology in algebraic topology
11B83	Special sequences and polynomials
12D10	Polynomials in real and complex fields: location of zeros (algebraic theorems)

Keywords:

matroids; Kazhdan-Lusztig polynomials