A Basis of the $q$-Schur Module

TL;DR

This paper constructs $q$-Schur modules as ideals in cyclotomic $q$-Schur algebras, proves their isomorphism to existing cell modules, and provides new bases and proofs for branching rules.

Contribution

It introduces a new realization of $q$-Schur modules as ideals, establishes their bases, and offers alternative proofs for known branching rules.

Findings

$q$-Schur modules are free modules with constructed bases.

Established isomorphism between $q$-Schur modules and cell modules.

Re-proved the Branch rule of Weyl modules using new bases.

Abstract

In this paper, we construct the $q$-Schur modules as left principle ideals of the cyclotomic $q$-Schur algebras, and prove that they are isomorphic to those cell modules defined in \cite{8} and \cite{15} at any level $r$. Then we prove that these $q$-Schur modules are free modules and construct their bases. This result gives us new versions of several results about the standard basis and the branching theorem. With the help of such realizations and the new bases, we re-prove the Branch rule of Weyl modules which was first discovered and proved by Wada in \cite{23}.