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On Fundamental Theorems of Invariant Theory for the Special Linear Supergroup

Junaid Razzaq, Rita Fioresi, Maria A. Lledo

TL;DR

This work extends classical invariant theory to the complex special linear supergroup SL(r|s) by developing the super Plücker framework. It establishes the First Fundamental Theorem in the super setting, proves a super Jacobi identity as a pivotal tool, and derives the SL(1|1) Plücker relations with a complete Second Fundamental Theorem. For general SL(r|s), it proposes a generalized set of super Plücker relations and conjectures a full Second Fundamental Theorem, connecting to Voronov’s super Grassmannian embeddings. The results lay groundwork for a broader supergeometric invariant theory and its representation-theoretic implications.

Abstract

The purpose of this paper is to prove the First and Second Fundamental Theorems of invariant theory for the complex special linear supergroup and discuss the superalgebra of invariants, via the super Plucker relations.

On Fundamental Theorems of Invariant Theory for the Special Linear Supergroup

TL;DR

This work extends classical invariant theory to the complex special linear supergroup SL(r|s) by developing the super Plücker framework. It establishes the First Fundamental Theorem in the super setting, proves a super Jacobi identity as a pivotal tool, and derives the SL(1|1) Plücker relations with a complete Second Fundamental Theorem. For general SL(r|s), it proposes a generalized set of super Plücker relations and conjectures a full Second Fundamental Theorem, connecting to Voronov’s super Grassmannian embeddings. The results lay groundwork for a broader supergeometric invariant theory and its representation-theoretic implications.

Abstract

The purpose of this paper is to prove the First and Second Fundamental Theorems of invariant theory for the complex special linear supergroup and discuss the superalgebra of invariants, via the super Plucker relations.

Paper Structure

This paper contains 10 sections, 11 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First Fundamental Theorem of invariant theory for $\mathrm{SL(r|s)}$
  4. The super Jacobi identity
  5. The Super Plücker Relations for $\mathrm{SL}(1|1)$
  6. Second Fundamental Theorem of super invariant theory for $\mathrm{SL}(1|1)$
  7. Super Plücker relations for $\mathrm{SL(r|s)}$
  8. Supergeometry
  9. The super Cramer rule
  10. The Jacobi identity and the Plücker relations

Key Result

Theorem 1.1

Let the notation be as above.

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Example 2.6
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • ...and 19 more