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

Xingyu Dai (Department of Mathematics, Tongji University, Shanghai, China; and Center of Mathematical Sciences, Zhejiang University, Zhejiang, China)

Fang Li (Center of Mathematical Sciences, Zhejiang University, Zhejiang, China)

Kefeng Liu (Center of Mathematical Sciences, Zhejiang University, Zhejiang, China; and Department of Mathematics, University of California at Los Angeles)

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

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