Schurification of polynomial quantum wreath products
Chun-Ju Lai, Alexandre Minets
TL;DR
The paper extends Schurification to algebras with Bernstein--Lusztig presentations by introducing twisted convolution algebras and a Demazure-type twist, enabling a uniform Schur duality for polynomial-type quantum wreath products ($ ext{PQWP}$) with both coil and laurel Schur algebras. It constructs explicit bases for the coil and laurel Schur algebras, proves Schur duality under invertibility conditions, and develops a Dipper–James–style reformulation that yields a basis $ heta_{A,P}$ for wreath Schur algebras. The framework unifies several known structures (affine Hecke, degenerate affine Hecke, pro-$p$ Iwahori, affine zigzag, Rosso–Savage algebras) under a common convolution-algebra approach, including cases with infinite-dimensional bases. It also provides scalable tools to relate the representation theory of wreath products to centralizer algebras and offers potential applications to imaginary strata of affine KLR algebras and $p$-adic representation theory via new Schur-algebra constructions.
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, Vignéras' 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.