Schurification of polynomial quantum wreath products

TL;DR

This paper develops a new algebraic framework for quantum wreath products using Schur algebras, providing uniform duality proofs and explicit bases, with applications to various Hecke algebras.

Contribution

It introduces a Schurification process for quantum wreath products, combining geometric and algebraic methods, and offers explicit bases and duality results.

Findings

Established a uniform proof of Schur duality.

Constructed explicit bases for new Schur algebras.

Applied results to various Hecke algebras.

Abstract

We study the Schur algebra counterpart of a vast class of quantum wreath products. This is achieved by developing a theory of twisted convolution algebras, inspired by geometric intuition. In parallel, we provide an algebraic Schurification via a Kashiwara-Miwa-Stern-type action on a tensor space. We give a uniform proof of Schur duality, and construct explicit bases of the new Schur algebras. This provides new results for, among other examples, Vign\'eras' pro-$p$ Iwahori Hecke algebras of type $A$, degenerate affine Hecke algebras, Kleshchev-Muth's affine zigzag algebras, and Rosso-Savage's affine Frobenius Hecke algebras.