R\'eflexions dans un cristal

TL;DR

This paper computes the descending string parameters of elements in crystals associated with symmetrizable Kac-Moody algebras, relating them to Lusztig parameters, thus advancing understanding of crystal bases in representation theory.

Contribution

It provides a formula linking descending string parameters to Lusztig parameters within the crystal bases of symmetrizable Kac-Moody algebra representations.

Findings

Derived explicit relations between string and Lusztig parameters.

Enhanced understanding of crystal structure in Kac-Moody algebra representations.

Facilitated calculations in crystal basis theory.

Abstract

Let g = n^- + h + n^+ be a symmetrizable Kac-Moody algebra. Let B(\infty) be the Kashiwara crystal of U_q(n^-), let \lambda be a dominant integral weight, let T_\lambda = {t_\lambda} be the crystal with one element of weight \lambda, and let B(\lambda) \subset B(\infty) \otimes T_\lambda be the crystal of the integrable representation of highest weight \lambda. We compute the descending string parameters of an element b \otimes t_\lambda in B(\lambda) in terms of the Lusztig parameters of b.