Affine quantum Schur algebras and affine Hecke algebras

Qiang Fu

Vol. 270 (2014), No. 2, 351–366

DOI: 10.2140/pjm.2014.270.351

Abstract

Let $F$ be the Schur functor from the category of finite-dimensional ${\mathcal{\mathscr{H}}}_{▵}{\left(r\right)}_{ℂ}$-modules to that of finite-dimensional ${\mathcal{S}}_{▵}{\left(n,r\right)}_{ℂ}$-modules, where ${\mathcal{\mathscr{H}}}_{▵}{\left(r\right)}_{ℂ}$ is the extended affine Hecke algebra of type $A$ over $ℂ$ and ${\mathcal{S}}_{▵}{\left(n,r\right)}_{ℂ}$ is the affine quantum Schur algebras over $ℂ$. The Drinfeld polynomials associated with $F\left(V\right)$, where $V$ is an irreducible ${\mathcal{\mathscr{H}}}_{▵}{\left(r\right)}_{ℂ}$-module, have been previously determined when $n>r$. Here we generalize these results to the case $n\le r$. As an application, we recover the classification of finite-dimensional irreducible ${\mathcal{S}}_{▵}{\left(n,r\right)}_{ℂ}$-modules proved by Deng, Du and Fu using a different method. As another application, we generalize a result of Green to the affine case.

Keywords

affine quantum Schur algebras, affine Hecke algebras, Schur functor

Mathematical Subject Classification 2010

Primary: 17B37, 20C08, 20G43

Milestones

Received: 31 May 2013

Accepted: 25 November 2013

Published: 22 August 2014