Equivalence classes in the Weyl groups of type B n

Abstract

Abstract. We consider two families of equivalence classes in the Weyl groups of type Bn which are suggested by the study of left cells in unequal parameter Iwahori-Hecke algebras. Both families are indexed by a non-negative integer r. It has been shown that the first family coincides with left cells corresponding to the equal parameter Iwahori-Hecke algebra when r = 0; the equivalence classes in the second family agree with left cells corresponding to a special class of choices of unequal parameters when r is sufficiently large. Our main result shows that the two families of equivalence classes coincide, suggesting the structure of left cells for remaining choices of the Iwahori-Hecke algebra parameters.

A Relation for Domino Robinson-Schensted Algorithms

“…In the more general setting of unequal parameter H , [1] conjectures a parametrization of cells via domino tableaux of rank r, where the specific choice of r depends on the underlying parameters of H . In [12], the results of the present paper are used to reconcile the above conjecture and Garfinkle's original work on primitive ideals. In related work, [6] and [5] provide a geometric interpretation of these combinatorial results in the setting of rational Cherednik algebras.…”

Kazhdan-Lusztig left cells in type $B_n$ for intermediate parameters

“…In prior work, for instance [18] and [16], it has only been necessary to address regular cycles for a rank r tableau, but opposite cycles will play an important role herein.…”

“…Following G. Lusztig's construction in [15] and the combinatorics of standard domino tableaux detailed in [2] and [16], we describe two one-parameter families of partitions of W n , as well as a conjecture relating them.…”

Knuth relations for the hyperoctahedral groups

“…Recent work of M. Taskin finds a family of relations on W n which generates the irreducible combinatorial left cells of rank r. Using the results of [16] which show that reducible combinatorial left cells are common refinements of irreducible ones, it is then possible to describe a set of generators of the reducible combinatorial left cells of rank r based on Taskin's work. This set; however, does not coincide with Λ r+1 , but can be shown to be equivalent.…”

“…The following appears in [3]. Proof The main result of [16] shows that reducible combinatorial left cells of rank r are the least common refinements of irreducible combinatorial left cells of ranks r and r + 1. Hence the corollary follows directly from the above proposition.…”