Plactic relations for $r$-domino tableaux

TL;DR

This paper establishes plactic relations for signed permutations that determine when two permutations share the same insertion $r$-domino tableaux, advancing understanding in Kazhdan-Lusztig theory of type B.

Contribution

It introduces specific plactic relations for signed permutations that characterize equivalence in Garfinkle's $r$-domino tableau insertion algorithm.

Findings

Defines plactic relations for signed permutations.

Characterizes when two permutations have identical $r$-domino tableaux.

Links plactic relations to Kazhdan-Lusztig theory in type B.

Abstract

The work of C. Bonnaf{\'e}, M. Geck, L. Iancu and T. Lam \cite{Geck-Lam} shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm \cite{Garfinkle1}.