Universal Specht modules for cyclotomic Hecke algebras

TL;DR

This paper provides a new presentation of graded Specht modules for cyclotomic Hecke algebras, defining them via relations that include homogeneous Garnir relations and braid operators, simplifying their structure.

Contribution

It introduces a set of defining relations for Specht modules, including homogeneous Garnir relations and braid operators, offering a clearer algebraic framework.

Findings

Explicit defining relations for Specht modules are established.

Homogeneous Garnir relations are shown to be simpler than classical ones.

Operators satisfying braid relations are constructed on Specht modules.

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 {\em 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 {\em Garnir relations}. The homogeneous Garnir relations, which are {\em simpler} than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations.