The representation theory of hecke algebras at roots of unity has been studied to some extent with connections to other topics found see geck and jacon 11 but has not been completely . 33 representations of hecke algebras at roots of unity by meinolf geck seminaire bourbaki 50eme annee 1997 98 n 836 novembre 1997 hecke algebras arise naturally in the representation theory of finite or p adic cheval ley groups as endomorphism algebras of certain induced representations see carter 9 lusztig 44 and the references there. For hecke algebras associated with s n lascoux leclerc and thibon 37 have conjectured a solution of the first step by translating the original problem to that of computing 836 02 the canonical basis in the sense of kashiwara and lusztig see 44 of a certain highest weight module for an affine kac moody algebra. Title representations of hecke algebras at roots of unity algebra and applications author i 1 2 i 1 2 andrea faber subject i 1 2 i 1 2 representations of hecke algebras at roots of unity algebra and applications. We present a fast algorithm for computing the global crystal basis of the basic u q widehatmathfraksmathfrakl n modulethis 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