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Register a new userEquivalence classes for the mu-coefficient of Kazhdan-Lusztig polynomials in S_n
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Authors
Gregory S. Warrington
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We study equivalence classes relating to the Kazhdan-Lusztig mu(x,w) coefficients in order to help explain the scarcity of distinct values. Each class is conjectured to contain a crosshatch pair. We also compute the values attained by mu(x,w) for the permutation groups S_10 and S_11.
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Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan-Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan-Lusztig polynomials, and we determine the equivariant inverse Kazhdan-Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot and Youngs formula for the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr and Radcliffes combinatorial interpretation for the ordinary Kazhdan-Lusztig polynomials of uniform matroids, we further present a new formula for the corresponding equivariant Kazhdan-Lusztig polynomials.
The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [{it Adv. Math. 2016}]. Let $U_{m,d}$ denote the uniform matroid of rank $d$ on a set of $m+d$ elements. Gedeon, Proudfoot, and Young [{it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of $U_{m,d}$ using equivariant Kazhdan-Lusztig polynomials. In this paper we give two alternative explicit formulas, which allow us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of $U_{m,d}$ for $2leq mleq 15$ and all $d$s. The case $m=1$ was previously proved by Gedeon, Proudfoot, and Young [{it S{e}m. Lothar. Combin. 2017}]. We further determine the $Z$-polynomials of all $U_{m,d}$s and prove the real-rootedness of the $Z$-polynomials of $U_{m,d}$ for $2leq mleq 15$ and all $d$s. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Youngs formula for the Kazhdan-Lusztig polynomials of $U_{m,d}$s without using the equivariant Kazhdan-Lusztig polynomials.
The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeons conjecture by the Pieri rule.
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling numbers of the first kind and Stirling numbers of the second kind, as well as a non-recursive formula for the braid matroid Kazhdan-Lusztig polynomials.
We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $textbf{U}^{imath}$-module and a based $textbf{U}$-module is a based $textbf{U}^{imath}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).
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