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# Pure and Applied Mathematics Quarterly

## Volume 10 (2014)

### Number 3

### Bases of $q$-Schur Module $\mathcal{A}^{\lambda}$

Pages: 439 – 458

DOI: http://dx.doi.org/10.4310/PAMQ.2014.v10.n3.a2

#### Authors

#### Abstract

In this paper, we construct the so-called $q$-Schur modules as left principal ideals of cyclotomic $q$-Schur algebras, and prove that they are isomorphic to those cell modules defined in [3] and [9] in any level $r$. After that, mainly, we prove that these $q$-Schur modules are free and construct their bases. This result gives new versions of some known results such as standard basis and the branching theorem. With the help of this realization and the new basis, we give a new proof of the Branch rule of Weyl modules which was first discovered by Wada in [13].

#### Keywords

$q$-Schur module, cyclotomic $q$-Schur algebra, branching theorem

#### 2010 Mathematics Subject Classification

20G43