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On the Grothendieck ring of a quasireductive Lie superalgebra

Maria Gorelik, Vera Serganova, Alexander Sherman

TL;DR

The article develops a framework for studying the Grothendieck ring of finite-dimensional modules over a quasireductive Lie superalgebra $ rak g$ via restriction to a maximal quasitoral subalgebra $ rak h$. It introduces two quotient Grothendieck rings $ extswab{K}_ ext{±}( rak g)$ and shows that the reduced ring $ extswab{K}_ ext{−}( rak g)$ carries nontrivial, rich structure; for $ rak g= rak q_n$ the authors establish a sharp isomorphism between $ ext{sch}_{ rak h}( extswab{K}_ ext{−}( rak q_n))_{Q}$ and the $W$-invariants in the corresponding $ rak h$-category, thereby encoding $ rak g$-representations via $ rak h$-supercharacters. They provide concrete realizations of $ extswab{K}_ ext{−}( extbf{Fin}( rak q_n)_{int})$ as an exterior-algebra quotient, relate DS-functor images to restriction to smaller quasitoral subalgebras, and develop a detailed description of the reduced Grothendieck ring for $ extbf{Fin}( rak q_n)$ through cores and the monoid $oldsymbol ext{Xi}$. The work synthesizes Clifford-algebra techniques, DS-functors, and invariant theory to yield explicit, computable character-theoretic invariants for a central class of Lie superalgebras with broad implications for representation theory and its applications. Overall, the paper advances a robust, algebraically rich picture of how restriction to maximal quasitorial subalgebras governs the Grothendieck-ring structure and supercharacter theory, especially in the queer family $ rak q_n$.

Abstract

Given a Lie superalgebra $\mathfrak{g}$ and a maximal quasitoral subalgebra $\mathfrak{h}$, we consider properties of restrictions of $\mathfrak{g}$-modules to $\mathfrak{h}$. This is a natural generalization of the study of characters in the case when $\mathfrak{h}$ is an even maximal torus. We study the case of $\mathfrak{g}=\mathfrak{q}_n$ with $\mathfrak{h}$ a Cartan subalgebra, and prove several special properties of the restriction in this case, including an explicit realization of the $\mathfrak{h}$-supercharacter ring.

On the Grothendieck ring of a quasireductive Lie superalgebra

TL;DR

The article develops a framework for studying the Grothendieck ring of finite-dimensional modules over a quasireductive Lie superalgebra via restriction to a maximal quasitoral subalgebra . It introduces two quotient Grothendieck rings and shows that the reduced ring carries nontrivial, rich structure; for the authors establish a sharp isomorphism between and the -invariants in the corresponding -category, thereby encoding -representations via -supercharacters. They provide concrete realizations of as an exterior-algebra quotient, relate DS-functor images to restriction to smaller quasitoral subalgebras, and develop a detailed description of the reduced Grothendieck ring for through cores and the monoid . The work synthesizes Clifford-algebra techniques, DS-functors, and invariant theory to yield explicit, computable character-theoretic invariants for a central class of Lie superalgebras with broad implications for representation theory and its applications. Overall, the paper advances a robust, algebraically rich picture of how restriction to maximal quasitorial subalgebras governs the Grothendieck-ring structure and supercharacter theory, especially in the queer family .

Abstract

Given a Lie superalgebra and a maximal quasitoral subalgebra , we consider properties of restrictions of -modules to . This is a natural generalization of the study of characters in the case when is an even maximal torus. We study the case of with a Cartan subalgebra, and prove several special properties of the restriction in this case, including an explicit realization of the -supercharacter ring.
Paper Structure (81 sections, 43 theorems, 165 equations)

This paper contains 81 sections, 43 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Maximal toral subalgebras and restriction
  3. $\mathscr{K}_-(\mathfrak g)$ and $\mathscr{K}_+(\mathfrak g)$
  4. Results for $\mathfrak g=\mathfrak q_n$
  5. Supercharacter isomorphism
  6. Realization of $\mathscr{K}_-(\mathcal{F}(\mathfrak q_n)_{int})$
  7. Relation to the Duflo-Serganova functor
  8. Acknowledgements
  9. Table of contents
  10. List of notation
  11. Preliminaries: maximal quasitoral subalgebras and Clifford algebras
  12. Maximal (quasi)toral subalgebras
  13. Definition
  14. Triangular decomposition
  15. Examples
  16. ...and 66 more sections

Key Result

Theorem 1.1

Equivalently, if $\mu\not\sim\lambda$ then $[L(\lambda):C(\mu)]=[L(\lambda):\Pi C(\mu)]$, where $C(\mu)$ is an irreducible $\mathfrak h$-module of weight $\mu$.

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 67 more