A CONJECTURE FOR THE COMPUTATION OF THE DECOMPOSITION MATRICES OF THE HECKE ALGEBRAS OF TYPE-A AT ROOTS OF UNITY

Abstract : We exhibit a connection behaving rite combinatorics of the Hecke algebras as of type A at roots of unity, and the combinatorics of the crystal basis of the basic representation of the affine Lie algebras of type A((1)). In particular, we conjecture that the decomposition matrices of the algebras H-m ((n) root 1) are easily obtained from the global lower crystal basis of the basic representation ($) over cap sl(n).