Knuth Relations for the Hyperoctahedral Groups

TL;DR

This paper explores the structure of one-sided cells in type B Hecke algebras with unequal parameters, using domino tableaux and operator constructions to support a conjectured combinatorial description.

Contribution

It adapts Garfinkle's methods to construct operators that verify the conjectured description of cells in type B Hecke algebras with unequal parameters.

Findings

Constructed operators generating the conjectured cell description

Extended Garfinkle's methods to the unequal parameter case

Supported the conjecture with combinatorial and algebraic evidence

Abstract

C. Bonnaf{\'e}, M. Geck, L. Iancu, and T. Lam have conjectured a description of one-sided cells in unequal parameter Hecke algebras of type $B$ which is based on domino tableaux of arbitrary rank. In the integer case, this generalizes the work of D. Garfinkle whose methods we adapt to construct a family of operators which generate the conjectured combinatorial description.