## 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.