Affine quantum Schur algebras and affine Hecke algebras

TL;DR

This paper extends the classification of finite dimensional irreducible modules for affine quantum Schur algebras and affine Hecke algebras, generalizing previous results to the case where the rank parameter n is less than or equal to r.

Contribution

It generalizes the determination of Drinfeld polynomials and classifies irreducible modules for affine quantum Schur algebras in the case n ≤ r, expanding prior work limited to n > r.

Findings

Generalized Drinfeld polynomial results to n ≤ r case

Classified finite dimensional irreducible modules for affine quantum Schur algebras

Extended classical results to the affine case

Abstract

Let ${\mathsf F}$ be the Schur functor from the category of finite dimensional ${\mathcal H}_{\vartriangle}(r)_\mathbb C$-modules to the category of finite dimensional ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$-modules, where ${\mathcal H}_{\vartriangle}(r)_\mathbb C$ is the extended affine Hecke algebra of type $A$ over ${\mathbb C}$ and ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$ is the affine quantum Schur algebras over $\mathbb{C}$. The Drinfeld polynomials associated with ${\mathsf F}(V)$ were determined in \cite[7.6]{CP96} and \cite[4.4.2]{DDF} in the case of $n>r$, where $V$ is an irreducible ${\mathcal H}_{{\vartriangle}}(r)_\mathbb C$-module. We will generalize the result in [loc. cit.] to the case of $n\leq r$. As an application, we will classify finite dimensional irreducible ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$-modules, which has been proved in \cite[4.6.8]{DDF} using a different method. Furthermore we will use it to generalize \cite[(6.5f)]{Gr80} to the affine case.