Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Twists of representations of complex reflection groups and rational Cherednik algebras

Yuri Bazlov, Edward Jones-Healey

TL;DR

This work develops a comprehensive framework for cocycle (Drinfeld) twists in the setting of H-module algebras, their smash products, and induced representations, and applies it to the representation theory of complex reflection groups and rational Cherednik algebras. By systematically twisting algebras and modules, the authors establish explicit correspondences between rational Cherednik algebras and braided (mystic) Cherednik algebras, characterize actions on standard modules, and construct a noncommutative coinvariant algebra underline S_W that mirrors Chevalley’s theorem in a quantum setting. They provide precise descriptions of how twists permute irreducible characters for B_n and D_n via maps like ηφ and J_1, and show that in favorable cases (e.g., m/p even) the twisted and untwisted objects are isomorphic, sometimes yielding automorphisms of group algebras and permutations of characters. The paper also develops the theory of twisted coinvariants and restricted Cherednik algebras, proving trace-invariance under twisting and giving explicit counterexamples to non-isomorphism, thereby highlighting arithmetic phenomena and proposing conjectures for further study. Overall, the work bridges quantum algebra techniques with classical representation theory, delivering new structural insights and tools for understanding twisted representations and their algebraic counterparts.

Abstract

Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to reflection groups and rational Cherednik algebras. In particular, we describe how a twist acts on characters of Coxeter groups of type $B_n$ and $D_n$ and relate them to characters of mystic reflection groups. This is used to characterise twists of standard modules of rational Cherednik algebras as standard modules for certain braided Cherednik algebras. We introduce the coinvariant algebra of a mystic reflection group and use a twist to show that an analogue of Chevalley's theorem holds for these noncommutative algebras. We also discuss several cases where the negative braided Cherednik algebras are, and are not, isomorphic to rational Cherednik algebras.

Twists of representations of complex reflection groups and rational Cherednik algebras

TL;DR

This work develops a comprehensive framework for cocycle (Drinfeld) twists in the setting of H-module algebras, their smash products, and induced representations, and applies it to the representation theory of complex reflection groups and rational Cherednik algebras. By systematically twisting algebras and modules, the authors establish explicit correspondences between rational Cherednik algebras and braided (mystic) Cherednik algebras, characterize actions on standard modules, and construct a noncommutative coinvariant algebra underline S_W that mirrors Chevalley’s theorem in a quantum setting. They provide precise descriptions of how twists permute irreducible characters for B_n and D_n via maps like ηφ and J_1, and show that in favorable cases (e.g., m/p even) the twisted and untwisted objects are isomorphic, sometimes yielding automorphisms of group algebras and permutations of characters. The paper also develops the theory of twisted coinvariants and restricted Cherednik algebras, proving trace-invariance under twisting and giving explicit counterexamples to non-isomorphism, thereby highlighting arithmetic phenomena and proposing conjectures for further study. Overall, the work bridges quantum algebra techniques with classical representation theory, delivering new structural insights and tools for understanding twisted representations and their algebraic counterparts.

Abstract

Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to reflection groups and rational Cherednik algebras. In particular, we describe how a twist acts on characters of Coxeter groups of type and and relate them to characters of mystic reflection groups. This is used to characterise twists of standard modules of rational Cherednik algebras as standard modules for certain braided Cherednik algebras. We introduce the coinvariant algebra of a mystic reflection group and use a twist to show that an analogue of Chevalley's theorem holds for these noncommutative algebras. We also discuss several cases where the negative braided Cherednik algebras are, and are not, isomorphic to rational Cherednik algebras.
Paper Structure (24 sections, 26 theorems, 60 equations)

This paper contains 24 sections, 26 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Twists of algebras and representations
  3. H-module algebras and smash products.
  4. The cocycle twist of an algebra.
  5. Cocycle twists of representations.
  6. Twists via adjoint actions.
  7. Twists of algebra factorisations.
  8. Induced representations.
  9. Twisting irreducible characters of Coxeter groups
  10. Twisting irreducible characters of .
  11. Twisting irreducible characters of .
  12. Rational Cherednik algebras
  13. Application to Cherednik algebras.
  14. Twisting standard modules of rational Cherednik algebras.
  15. Twisted coinvariant algebras
  16. ...and 9 more sections

Key Result

Proposition 2.2

A $2$-cocycle $\mathcal{F}\in H\otimes H$ for a Hopf algebra $H$ gives rise to the functor which takes an object $(A,m)$ to $(A,m_\mathcal{F} = m(\mathcal{F}^{-1}\rhd - ))$, and an arrow $(A,m)\xrightarrow{\phi}(B,m')$ to $(A,m_\mathcal{F})\xrightarrow{\phi_\mathcal{F}}(B,m'_\mathcal{F})$, where $\phi_\mathcal{F}=\phi$ as $H$-module morphisms.

Theorems & Definitions (56)

  • Definition 2.1: majid_1995, Example 2.3.1 and Theorem 2.3.4
  • Proposition 2.2: twistsrcas, Proposition 4.3
  • Definition 2.3: giaquinto1998bialgebra, Definition 1.6
  • Proposition 2.4: etingof2015tensor, Exercise 7.8.32
  • proof : Sketch of proof
  • Proposition 2.5: kulish_2011, Proposition 2.5
  • Corollary 2.6: giaquinto1998bialgebra, Theorem 1.7
  • Definition 2.7: montgomery_1993, Example 7.3.3
  • Proposition 2.8
  • proof
  • ...and 46 more