Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids

Abstract

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, which provides positive evidence for a conjecture due to Gedeon, Proudfoot and Young. Based on the results on the Kazhdan-Lusztig polynomials, we also determine the Z-polynomials of fan matroids, wheel matroids and whirl matroids, and prove their real-rootedness, which provides further evidence in support of a conjecture of Proudfoot, Xu, and Young.

Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids

“…One of such results is that P M 1 ⊕M 2 = P M 1 • P M 2 , and analogously for Q and Z. On the other hand, explicit formulas exist for some classes of matroids such as the so-called thagomizer matroids [Ged17], fan, wheel and whirl matroids [LXY18], and uniform matroids [GLX + 21, GX21]. Recently, by exploiting the formulas for uniform matroids and the notion of circuit-hyperplane relaxation, in [FV21] explicit formulas for P M , Q M and Z M when M belongs to the large family of all sparse paving matroids are derived.…”

“…Although it is conjectured that these polynomials possess further nice properties, such as P M and Z M being real-rooted [EPW16,PXY18] and Q M having log-concave coefficients [GX21], closed or explicit formulas for their coefficients were known only for a very limited number of matroids: for instance, uniform matroid [GLX + 21], braid matroids [KW19], thagomizer matroids [Ged17,XZ19] and fans, wheels and whirls [LXY18].…”

The Kazhdan-Lusztig polynomials of uniform matroids

“…Since the introduction of the matroid Kazhdan-Lusztig polynomials, due to Elias, Proudfoot and Wakefield [5], these polynomials have attracted much attention, for instance see [9,10,15,6,1,22] and references therein. As noted by Proudfoot [17], the Kazhdan-Lusztig polynomials of matroids can also be considered as a special case of the Kazhdan-Lusztig-Stanley polynomials, which were first introduced by Stanley [20] and further studied by Brenti [3,4].…”

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $ρ$-Removed Uniform Matroids

“…Since their introduction, these polynomials have drawn active research efforts. Mostly, this is due to their (conjecturally) nice properties, such as positivity and real-rootedness (see [2,3,4,8,13]). There has also been much effort put into finding relations between these polynomials or generalizations thereof (see [1,9,12]).…”

“…There has also been much effort put into finding relations between these polynomials or generalizations thereof (see [1,9,12]). However, these polynomials have been explicitly calculated only for very special classes of matroids (for instance, see [7,4,6,8,10]), and yet many of the known formulas have left much room for improvement. In particular, as of now, there is no enlightening interpretation for such coefficients.…”