Conjugacy Classes of Involutions and Kazhdan–lusztig Cells
نویسنده
چکیده
According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group Sn are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger’s result is a special case of a general result on “smooth” two-sided cells. Furthermore, we consider Kottwitz’s conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general.
منابع مشابه
M ay 2 00 8 Leading coefficients of the Kazhdan - Lusztig polynomials for an Affine Weyl group of type
In this paper we compute the leading coefficients μ(y,w) of the Kazhdan-Lusztig polynomials Py,w for an affineWeyl group of type B̃2. When a(y) ≤ a(w) or a(y) = 2 and a(w) = 1, we compute all μ(y,w) clearly, where a(y) is the a-function of a Coxeter group defined by Lusztig (see [L1]). With these values μ(y,w), we are able to show that a conjecture of Lusztig on distinguished involutions is true...
متن کاملPositivity Conjectures for Kazhdan-lusztig Theory on Twisted Involutions: the Universal Case
Let (W,S) be a Coxeter system and let w → w∗ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements w ∈ W with w−1 = w∗) naturally generates a module of the Hecke algebra of (W,S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials Pσ y,w w...
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 کاملCalculating Canonical Distinguished Involutions in the Affine Weyl Groups
Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...
متن کاملThe Partial Order on Two-sided Cells of Certain Affine Weyl Groups
In their famous paper [6], Kazhdan and Lusztig introduced the concept of equivalence classes such as left cell, right cell and two-sided cell in a Coxeter group W . We inherit the notations 6 L , 6 R , 6 LR , ∼ L , ∼ R and ∼ LR in [6]. Thus w ∼ LR y (resp. w ∼ L y, resp. w ∼ R y) means that the elements w, y ∈ W are in the same two-sided cell (resp. left cell, resp. right cell) of W , etc. Conc...
متن کامل