Degree-One Rational Cherednik Algebras for the Symmetric Group

TL;DR

This paper introduces new degree-one deformations of rational Cherednik algebras for symmetric groups, expanding the algebraic structures and characterizing their commutator relations in the context of symmetric group actions.

Contribution

It constructs and characterizes degree-one Drinfeld orbifold algebras for symmetric groups, generalizing rational Cherednik algebras and exploring their commutator relations.

Findings

Degree-one versions of $rak{gl}_n$-type rational Cherednik algebras are produced.

No degree-one Lie orbifold algebra maps exist for the standard irreducible reflection representation.

A three-parameter family of Drinfeld orbifold algebras arises from maps supported only off the identity.

Abstract

Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized families of Drinfeld orbifold algebras for symmetric groups acting on doubled representations that generalize rational Cherednik algebras by deforming in degree one. We characterize rich families of maps recording commutator relations with their linear parts supported only on and only off the identity when the symmetric group acts on the natural permutation representation plus its dual. This produces degree-one versions of $\mathfrak{gl}_n$-type rational Cherednik algebras. When the symmetric group acts on the standard irreducible reflection representation plus its dual there are no degree-one Lie orbifold algebra maps, but there is a three-parameter family of Drinfeld orbifold algebras arising from maps supported only off the identity. These provide degree-one generalizations of the $\mathfrak{sl}_n$-type rational Cherednik algebras $H_{0,c}$.