Implementation of the Kazhdan–Lusztig algorithm
ثبت نشده
چکیده
We denote D the set of parameters (of course the existence of the Cayley transform and cross action tables imply that the set D has already been enumerated, and thus identified with a set [0,M [ of integers.) We let M be the free A-module generated by D, where A = Z[v, v−1], and we write q = v2. We replace the canonical basis (Tδ)δ∈D of M by tδ = v Tδ, where l : D → N is the length function (also assumed to be tabulated), and similarly denote tw the corresponding basis of the Hecke algebra H of the complex Weyl group. We denote i the canonical involution on M: the unique A-antilinear involution such that
منابع مشابه
Embedded Factor Patterns for Deodhar Elements in Kazhdan-Lusztig Theory
The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar [16] has given a framework for computing the Kazhdan-Lusztig polynomials which generally invo...
متن کاملRobinson-Schensted algorithm and Vogan equivalence
We provide a combinatorial proof for the coincidence of Knuth equivalence classes, Kazhdan–Lusztig left cells and Vogan classes for the symmetric group, involving only Robinson-Schensted algorithm and the combinatorial part of the Kazhdan–Lusztig cell theory. The determination of Kazhdan–Lusztig cells for the symmetric group is given in the proof of [4, Thm1.4]. The argument is largely combinat...
متن کاملLeading Coefficients of Kazhdan–lusztig Polynomials for Deodhar Elements
We show that the leading coefficient of the Kazhdan–Lusztig polynomial Px,w(q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in [BW01] and [BJ07]. In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm [Deo90], we provide...
متن کاملDeodhar Elements in Kazhdan-Lusztig Theory
The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusz...
متن کاملSpecial matchings and Kazhdan-Lusztig polynomials
In 1979 Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements ofW , which have become known as the Kazhdan-Lusztig polynomials of W , and which have proven to be of importance in several areas of mathematics. In this paper we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these pol...
متن کاملLinear Algebra Construction of Formal Kazhdan-lusztig Bases
General facts of linear algebra are used to give proofs for the (wellknown) existence of analogs of Kazhdan-Lusztig polynomials corresponding to formal analogs of the Kazhdan-Lusztig involution, and of explicit formulae (some new, some known) for their coefficients in terms of coefficients of other natural families of polynomials (such as the corresponding formal analogs of the Kazhdan-Lusztig ...
متن کامل