Drinfeld Orbifold Algebras for Symmetric Groups

TL;DR

This paper classifies all Drinfeld orbifold algebras for symmetric groups, revealing new infinite families of such algebras beyond graded Hecke algebras, with explicit descriptions of their defining maps.

Contribution

It provides a complete classification of Drinfeld orbifold algebras for symmetric groups, including explicit descriptions and parameter families, expanding known examples in the field.

Findings

Identified one-parameter families supported on the identity.

Discovered three-parameter families supported on 3-cycles and 5-cycles.

Established reduction techniques using group element orbits.

Abstract

Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincar\'{e}-Birkhoff-Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces.