Universal graded Specht modules for cyclotomic Hecke algebras

Alexander Kleshchev, Andrew Mathas and Arun Ram

Abstract

The graded Specht module $S^{\lambda }$ for a cyclotomic Hecke algebra comes with a distinguished generating vector $z^{\lambda }\in S^{\lambda }$, which can be thought of as a "highest weight vector of weight $\lambda$". This paper describes the defining relations for the Specht module $S^{\lambda }$ as a graded module generated by $z^{\lambda }$. The first three relations say precisely what it means for $z^{\lambda }$ to be a highest weight vector of weight $\lambda$. The remaining relations are homogeneous analogues of the classical Garnir relations. The homogeneous Garnir relations, which are simpler than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations.

Keywords: Hecke algebras, symmetric groups, Specht modules, grading.