Super-immanants and Littlewood correspondences

TL;DR

This paper introduces super-immanants for supermatrices, explores their role in tensor representations of Lie superalgebras, and establishes a super-analogue of Kostant's trace formula along with algebraic isomorphisms.

Contribution

It defines super-immanants, connects them with super Schur-Weyl duality, and establishes a super Littlewood correspondence, extending classical results to the super setting.

Findings

Super-immanants are introduced for supermatrices.

A supertrace formula generalizes Kostant's trace formula.

Super Littlewood correspondences establish algebra isomorphisms.

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.