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.