# Injective Schur Modules

## TL;DR

This paper characterizes partitions for which certain modules in the representation theory of general linear groups and Hecke algebras are injective or projective, extending classical results to the quantum case.

## Contribution

It provides a complete classification of partitions with injective Schur modules and Young modules in both classical and quantum settings, linking to Carter's condition.

## Key findings

- Classifies partitions with injective Schur modules.
- Determines when Specht modules are Young modules at q ≠ -1.
- Connects results to Carter's condition on regular partitions.

## Abstract

We determine the partitions $\lambda$ for which the corresponding induced module (or Schur module in the language of Buchsbaum et. al., [1]) $\nabla(\lambda)$ is injective in the category of polynomial modules for a general linear group over an infinite field, equivalently which Weyl modules are projective polynomial modules. Since the problem is essentially no more difficult in the quantised case we address it at this level of generality. Expressing our results in terms of the representation theory of Hecke algebras at the parameter $q$ we determine the partitions $\lambda$ for which the corresponding Specht module is a Young module, when $1+q\neq 0$. In the classical case this problem was addressed by D. Hemmer, [12]. The nature of the set of partitions appearing in our solution gives a new formulation of Carter's condition on regular partitions. On the other hand, we note, in Remark 2.22, that the result on irreducible Weyl modules for the quantised Schur algebra $S_q(n,n)$, [17], Theorem 5.39, given in terms of Carter partitions, may be also used to obtain the main result presented here.