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Seminormal bases of cyclotomic Hecke-Clifford algebras

Shuo Li, Lei Shi

TL;DR

This work develops explicit seminormal bases and complete primitive idempotent decompositions for semisimple cyclotomic Hecke-Clifford algebras and their degenerate (Sergeev) analogues. Central techniques include constructing explicit simple modules $ ext{D}(oldsymbol{bla})$ and $ ext{L}(oldsymbol{bla})$ from Clifford-factor data, introducing intertwining elements $ ext{Phi}_{oldsymbol{s},oldsymbol{t}}$, and formulating two complementary seminormal bases for each block. Under a separate condition (nondegenerate via $P^{(ullet)}_n(q^2, ext{Q}) eq0$ or degenerate via $P^{(ullet)}_n(1, ext{Q}) eq0$), the authors obtain complete sets of (super) primitive idempotents and central idempotents, describe block equivalences, and provide explicit multiplication rules in the seminormal basis. The results yield practical, computable bases and operator actions, enabling explicit calculations in cyclotomic Hecke-Clifford and Sergeev algebras and laying groundwork toward cellular-like structures in this superalgebra context.

Abstract

In this paper, we describe actions of standard generators on certain bases of simple modules for semisimple cyclotomic Hecke-Clifford superalgebras. As applications, we explicitly construct a complete set of primitive idempotents and seminormal bases for these algebras.

Seminormal bases of cyclotomic Hecke-Clifford algebras

TL;DR

This work develops explicit seminormal bases and complete primitive idempotent decompositions for semisimple cyclotomic Hecke-Clifford algebras and their degenerate (Sergeev) analogues. Central techniques include constructing explicit simple modules and from Clifford-factor data, introducing intertwining elements , and formulating two complementary seminormal bases for each block. Under a separate condition (nondegenerate via or degenerate via ), the authors obtain complete sets of (super) primitive idempotents and central idempotents, describe block equivalences, and provide explicit multiplication rules in the seminormal basis. The results yield practical, computable bases and operator actions, enabling explicit calculations in cyclotomic Hecke-Clifford and Sergeev algebras and laying groundwork toward cellular-like structures in this superalgebra context.

Abstract

In this paper, we describe actions of standard generators on certain bases of simple modules for semisimple cyclotomic Hecke-Clifford superalgebras. As applications, we explicitly construct a complete set of primitive idempotents and seminormal bases for these algebras.

Paper Structure

This paper contains 29 sections, 55 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Some basics about superalgebras
  4. Clifford algebra $\mathcal{C}_n$
  5. Non-degenerate case
  6. Affine Hecke-Clifford algebra $\mathcal{H}_{c}(n)$
  7. Intertwining elements for $\mathcal{H}_{c}(n)$
  8. Cyclotomic Hecke-Clifford algebra $\mathcal{H}^f_{c}(n)$
  9. Combinatorics
  10. Separate Condition
  11. Degenerate case
  12. Affine Sergeev algebra $\mathfrak{H}_{c}(n)$
  13. Intertwining elements for $\mathfrak{H}_{c}(n)$
  14. Cyclotomic Sergeev algebra $\mathfrak{H}^g_{c}(n)$
  15. Separate Condition
  16. ...and 14 more sections

Key Result

Theorem 1.1

The simple module $\mathbb{D}(\underline{\lambda})$ has a $\mathbb{K}$-basis of the form Moreover, the actions of generators of $\mathcal{H}^f_{c}(n)$ on the above basis are given explicitly in X eigenvalues, Caction and Taction Non-dege.

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • ...and 110 more