The image of the Specht module under the inverse Schur functor in arbitrary characteristic

TL;DR

This paper characterizes when the inverse Schur functor maps Specht modules to dual Weyl modules across different characteristics, providing new insights especially in characteristics 2 and 3, and exploring module decompositions.

Contribution

It offers a necessary and sufficient condition for the isomorphism in characteristic 2 and elementary proofs for other characteristics, advancing understanding of Specht and dual Weyl modules.

Findings

Isomorphism holds in all characteristics except 2 and 3.

New examples of indecomposable Specht modules in characteristic 2.

The image may lack a filtration by dual Weyl modules when the isomorphism fails.

Abstract

This paper gives a necessary and sufficient condition for the image of the Specht module under the inverse Schur functor to be isomorphic to the dual Weyl module in characteristic 2, and gives an elementary proof that this isomorphism holds in all cases in all other characteristics. These results are new in characteristics 2 and 3. We deduce some new examples of indecomposable Specht modules in characteristic 2. When the isomorphism does not hold, the dual Weyl module is still a quotient of the image of the Specht module, and we prove some additional results: we demonstrate that the image need not have a filtration by dual Weyl modules, we bound the dimension of the kernel of the quotient map, and we give some explicit descriptions for particular cases. Our method is to view the Specht and dual Weyl modules as quotients of suitable exterior powers by the Garnir relations.