Higher level affine Schur and Hecke algebras

TL;DR

This paper introduces higher level affine Schur and Hecke algebras, establishing their isomorphisms with tensor product and quiver Schur algebras, and provides diagrammatic presentations for these structures.

Contribution

It defines higher level affine Schur and Hecke algebras and proves their isomorphisms with existing algebraic structures, extending the theory of affine and cyclotomic algebras.

Findings

Isomorphism between higher level affine Hecke algebra and Webster's tensor product algebra

Isomorphism between higher level affine Schur algebra and quiver Schur algebra

Surjection of higher level affine Schur algebra onto cyclotomic q-Schur algebra

Abstract

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.