Hecke algebras at roots of unity and crystal bases of quantum affine algebras

Abstract

We present a fast algorithm for computing the global crystal basis of the basic $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$ -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an th root of 1.