Categorifications of the extended affine Hecke algebra and the affine q-Schur algebra S(n,r) for 2 < r < n

TL;DR

This paper develops a categorification of the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, utilizing diagrammatic methods and extending known 2-representations.

Contribution

It introduces new categorifications of these algebras using Elias-Khovanov and Khovanov-Lauda diagrams, and defines their affine analogues of 2-representations.

Findings

Categorification of extended affine Hecke algebra achieved.

Categorification of affine quantum Schur algebra S(n,r) established.

Extension of 2-category of affine (singular) Soergel bimodules provided.

Abstract

We categorify the extended affine Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define the affine analogue of the Elias-Khovanov and the Khovanov-Lauda 2-representations of these categorifications into an extension of the 2-category of affine (singular) Soergel bimodules.