An explicit construction of the Weyl module as a quotient of symmetric tensors by dual Garnir relations

TL;DR

This paper introduces a new explicit construction of Weyl modules as quotients of symmetric tensors, using dual Garnir relations, and extends these ideas to endofunctors on representation categories.

Contribution

It provides a novel explicit construction of Weyl modules via dual Garnir relations, generalizing to endofunctors on representation categories.

Findings

Weyl modules can be constructed as quotients of symmetric tensors.

Dual Garnir relations serve as key relations in the construction.

The approach applies broadly to representations of any group.

Abstract

The Weyl modules are the standard modules for the Schur algebra. Their duals (the costandard modules) have well-known constructions as quotients of exterior powers and as submodules of symmetric powers. This paper presents analogous constructions for the Weyl modules themselves, introducing relations that are a non-trivial dualisation of the Garnir relations. More generally, our constructions describe endofunctors on the category of representations of any group.