Module structure of cells in unequal parameter Hecke algebras

TL;DR

This paper verifies a conjecture relating Kazhdan-Lusztig left cells in unequal parameter Hecke algebras of type Bn to standard domino tableaux, describing their module structure for the Weyl group.

Contribution

It confirms a conjecture linking cell structures to domino tableaux and details the modules' structure in this context.

Findings

Verification of the conjecture for type Bn Hecke algebras.

Description of each cell's module structure for the Weyl group.

Establishment of a combinatorial parametrization of cells.

Abstract

A conjecture of C. Bonnaf\'e, M. Geck, L. Iancu, and T. Lam parameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by G. Lusztig and the recent work of C. Bonnaf\'e, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.