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Super-immanants and Littlewood correspondences

Naihuan Jing, Yinlong Liu, Jian Zhang

Abstract

In this paper, we introduce the notion of super-immanants for supermatrices over a supercommutative algebra. Using the super Schur-Weyl duality we show that the super immanants play a significant role in covariant tensor representations of the general linear Lie superalgebra. Among various things, we obtain a supertrace formula for super-immanants, which generalizes Kostant's trace formula to the super setting. Furthermore, we show that the Littlewood correspondences between super-immanants and supersymmetric polynomials establish an isomorphism between their corresponding algebras.

Super-immanants and Littlewood correspondences

Abstract

In this paper, we introduce the notion of super-immanants for supermatrices over a supercommutative algebra. Using the super Schur-Weyl duality we show that the super immanants play a significant role in covariant tensor representations of the general linear Lie superalgebra. Among various things, we obtain a supertrace formula for super-immanants, which generalizes Kostant's trace formula to the super setting. Furthermore, we show that the Littlewood correspondences between super-immanants and supersymmetric polynomials establish an isomorphism between their corresponding algebras.
Paper Structure (10 sections, 15 theorems, 127 equations)

This paper contains 10 sections, 15 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Super-immanants and super Schur-Weyl duality
  3. Super-immanants
  4. Super Schur-Weyl duality
  5. Gelfand-Tsetlin type basis for covariant representations
  6. Weight space interpretation of super-immanants
  7. Super Littlewood correspondences
  8. Super Littlewood correspondences I and II
  9. MacMahon Master Theorem
  10. Super Goulden-Jackson identities and super Littlewood correspondence III

Key Result

Proposition 2.1

Let $I=(i_1, \ldots , i_r)$ be an ordered multiset of $[m+n]$ and $X$ be the generator supermatrix of $A(\mathrm{Mat}_{m|n})$. Then for any $\lambda\vdash r$, where $\mathcal{E}_{\mathcal{T}}^{\lambda}$ be the primitive idempotent of $\mathfrak S_r$ associated with a standard Young tableau ${\mathcal{T}}$ of shape $\lambda$. Moreover when $\lambda \notin H(m,n;r)$, $\mathrm{Imm}_{\chi^{\lambda}}(

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of theorem \ref{['super-Kostant-thm']}
  • Theorem 3.1
  • Theorem 3.2
  • ...and 17 more