Braids, Shuffles and Symmetrizers

TL;DR

This paper introduces multiplicative shuffle elements in braid group rings, explores their algebraic properties in Hecke and BMW algebras, and analyzes the spectra of associated shuffle operators.

Contribution

It presents new multiplicative analogues of shuffle elements and expresses symmetrizers in terms of these shuffles within Hecke and BMW algebras.

Findings

Explicit formulas for (anti)-symmetrizers in terms of multiplicative shuffles

Spectral analysis of shuffle operators reveals eigenvalues and multiplicities

Introduction of graded associative b-shuffle algebras

Abstract

Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For the Hecke and BMW algebras, the (anti)-symmetrizers have simple expressions in terms of the multiplicative shuffles. The (anti)-symmetrizers can be expressed in terms of the highest multiplicative 1-shuffles (for the Hecke and BMW algebras) and in terms of the highest additive 1-shuffles (for the Hecke algebras). The spectra and multiplicities of eigenvalues of the operators of the multiplication by the multiplicative and additive 1-shuffles are examined.