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On representation theory of cyclotomic Hecke-Clifford algebras

Lei Shi, Jinkui Wan

TL;DR

The work develops a complete framework for the representation theory of cyclotomic Hecke-Clifford algebras and their degenerate counterparts by introducing separate parameters and multipartition indexing. It provides explicit constructions of irreducible modules $\mathbb{D}(\underline{\lambda})$ and $D(\underline{\lambda})$ via residue data and intertwining elements, and proves semisimplicity in the separate-parameter regime through dimension-count arguments and Wedderburn theory. A key innovation is the separate-parameter polynomial $P_n^{(\bullet)}(q^2,\underline{Q})$ (and its degenerate analogue) whose nonvanishing encodes semisimplicity, and the resulting complete, nonisomorphic irreducible module sets parameterized by multipartitions. These results imply that generic non-degenerate cyclotomic Hecke-Clifford algebras (and their degenerate analogues) are semisimple, with a precise center description and explicit type classification (M vs Q) tied to combinatorial data. The work thus extends Ariki–Koike-type techniques to a superalgebraic setting and supplies concrete, tableau-based realizations of simples useful for further structural and categorification investigations.

Abstract

In this article, we give an explicit construction of the simple modules for both non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras over an algebraically closed field of characteristic not equal to $2$ under certain condition in terms of parameters in defining these algebras. As an application, we obtain a sufficient condition on the semi-simplicity of these cyclotomic Hecke-Clifford superalgebras via a dimension comparison. As a byproduct, both generic non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras are shown to be semisimple.

On representation theory of cyclotomic Hecke-Clifford algebras

TL;DR

The work develops a complete framework for the representation theory of cyclotomic Hecke-Clifford algebras and their degenerate counterparts by introducing separate parameters and multipartition indexing. It provides explicit constructions of irreducible modules and via residue data and intertwining elements, and proves semisimplicity in the separate-parameter regime through dimension-count arguments and Wedderburn theory. A key innovation is the separate-parameter polynomial (and its degenerate analogue) whose nonvanishing encodes semisimplicity, and the resulting complete, nonisomorphic irreducible module sets parameterized by multipartitions. These results imply that generic non-degenerate cyclotomic Hecke-Clifford algebras (and their degenerate analogues) are semisimple, with a precise center description and explicit type classification (M vs Q) tied to combinatorial data. The work thus extends Ariki–Koike-type techniques to a superalgebraic setting and supplies concrete, tableau-based realizations of simples useful for further structural and categorification investigations.

Abstract

In this article, we give an explicit construction of the simple modules for both non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras over an algebraically closed field of characteristic not equal to under certain condition in terms of parameters in defining these algebras. As an application, we obtain a sufficient condition on the semi-simplicity of these cyclotomic Hecke-Clifford superalgebras via a dimension comparison. As a byproduct, both generic non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras are shown to be semisimple.
Paper Structure (18 sections, 45 theorems, 124 equations)

This paper contains 18 sections, 45 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Some basics about superalgebras
  4. Some combinatorics
  5. Cyclotomic Hecke-Clifford superalgebras and separate parameters
  6. Affine Hecke-Clifford algebra $\mathcal{H}_{\Delta}(n)$
  7. Intertwining elements for $\mathcal{H}_{\Delta}(n)$
  8. Cyclotomic Hecke-Clifford algebra $\mathcal{H}^f_{\Delta}(n)$
  9. Separate parameters.
  10. Semi-simplicity on non-degenerate cyclotomic Hecke-Clifford superalgebras
  11. Construction of Simple modules
  12. Comparing Dimension
  13. Degenerate Case
  14. Affine Sergeev algebra
  15. Intertwining elements for $\mathfrak{H}_{\Delta}(n)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $q\neq \pm 1\in\mathbb{K}^*$ and $\underline{Q}=(Q_1,Q_2,\ldots,Q_m)\in(\mathbb{K}^*)^m$. Assume $f=f^{(\bullet)}_{\underline{Q}}$ and $P^{(\bullet)}_{n}(q^2,\underline{Q})\neq 0$, with $\bullet\in\{\mathtt{0},\mathtt{s},\mathtt{ss}\}$. Then $\mathcal{H}^f_{\Delta}(n)$ is a (split) semisimple al forms a complete set of pairwise non-isomorphic irreducible $\mathcal{H}^f_{\Delta}(n)$-module. Mor

Theorems & Definitions (81)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 71 more