Drinfeld Hecke algebras for symmetric groups in positive characteristic

TL;DR

This paper studies new algebraic deformations called Drinfeld Hecke algebras for symmetric groups acting on polynomial rings over fields of positive characteristic, extending known structures from characteristic zero.

Contribution

It classifies deformations of skew group algebras for symmetric groups in prime characteristic, revealing new algebraic structures not present over the complex numbers.

Findings

Identifies new deformations in prime characteristic that alter group action and polynomial relations.

Classifies all such deformations for the symmetric group's natural reflection representation.

Extends the theory of Hecke algebras to positive characteristic fields.

Abstract

We investigate deformations of skew group algebras arising from the action of the symmetric group on polynomial rings over fields of arbitrary characteristic. Over the real or complex numbers, Lusztig's graded affine Hecke algebra and analogs are all isomorphic to Drinfeld Hecke algebras, which include the symplectic reflection algebras and rational Cherednik algebras. Over fields of prime characteristic, new deformations arise that capture both a disruption of the group action and also a disruption of the commutativity relations defining the polynomial ring. We classify deformations for the symmetric group acting in its natural (reducible) reflection representation.