## 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.

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 Monday, February 28, 2011