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The Regular Representation of the twisted queer $q$-Schur Superalgebra

Zhenhua Li

TL;DR

This work advances the understanding of representations of the quantum queer superalgebra and its twisted q-Schur counterpart by providing explicit realizations and a detailed weight-module theory. It shows that finite-dimensional irreducible weight modules are precisely highest-weight modules and constructs Verma quotients to classify them, while also establishing a comprehensive decomposition of the regular representation of the twisted queer $q$-Schur superalgebra into irreducibles, confirming semisimplicity. The approach hinges on endomorphism realizations linked to a Hecke–Clifford framework, yielding concrete bases and weight structures that facilitate direct analysis of module decompositions. The results connect quantum queer symmetry with Schur–Weyl dualities in a semisimple setting, with potential implications for categorification and combinatorial representation theory.

Abstract

We study the representation theory of the quantum queer superalgebra ${U_{\lcase{v}}(\mathfrak{\lcase{q}}_{n})}$ and obtain some properties of the highest weight modules. Furthermore, based on the realization of ${U_{\lcase{v}}(\mathfrak{\lcase{q}}_{n})}$, we study the representation theory of the twisted queer $q$-Schur superalgebra ${\widetilde{\mathcal{Q}}_{\lcase{v}}(\lcase{n},\lcase{r})}$, and obtain the decomposition of its regular module as a direct sum of irreducible submodules, which also means ${\widetilde{\mathcal{Q}}_{\lcase{v}}(\lcase{n},\lcase{r})}$ is semisimple.

The Regular Representation of the twisted queer $q$-Schur Superalgebra

TL;DR

This work advances the understanding of representations of the quantum queer superalgebra and its twisted q-Schur counterpart by providing explicit realizations and a detailed weight-module theory. It shows that finite-dimensional irreducible weight modules are precisely highest-weight modules and constructs Verma quotients to classify them, while also establishing a comprehensive decomposition of the regular representation of the twisted queer -Schur superalgebra into irreducibles, confirming semisimplicity. The approach hinges on endomorphism realizations linked to a Hecke–Clifford framework, yielding concrete bases and weight structures that facilitate direct analysis of module decompositions. The results connect quantum queer symmetry with Schur–Weyl dualities in a semisimple setting, with potential implications for categorification and combinatorial representation theory.

Abstract

We study the representation theory of the quantum queer superalgebra and obtain some properties of the highest weight modules. Furthermore, based on the realization of , we study the representation theory of the twisted queer -Schur superalgebra , and obtain the decomposition of its regular module as a direct sum of irreducible submodules, which also means is semisimple.
Paper Structure (7 sections, 13 theorems, 68 equations)

This paper contains 7 sections, 13 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The realization of $U_{v}(\mathfrak{q}_{n})$
  3. The queer superalgebra and the quantum queer superalgebra
  4. The twisted queer $q$-Schur superalgebra
  5. The realizations of $U_{v}(\mathfrak{q}_{n})$
  6. The weight modules of $U_{v}(\mathfrak{q}_{n})$
  7. The regular representation of twisted queer $q$-Schur superalgebra ${\widetilde{\mathcal{Q}}}_{v}(n,r)$

Key Result

Theorem 2.3

(See DGLW2025) There is a $\mathbb{Q}(v)$-superalgebra isomorphism $\boldsymbol{\xi}_n: U_{v}(\mathfrak{q}_{n}) \to {\mathfrak{A}[n]}_{v}$, mapping where $1 \le i \le n, 1 \le j \le n-1$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 18 more