Twisted immanant and matrices with anticommuting entries

TL;DR

This paper introduces the twisted immanant, a new matrix function for self-conjugate partitions, revealing properties like Cauchy-Binet formulas and connections to matrices with anticommuting entries.

Contribution

It defines the twisted immanant as an analogue of the immanant, extending the theory to matrices with anticommuting entries and establishing related identities.

Findings

Satisfies Cauchy-Binet type formulas

Describes invariants under conjugation for anticommuting matrices

Provides an analogue of Cauchy identities for symmetric polynomials

Abstract

This article gives a new matrix function named "twisted immanant," which can be regarded as an analogue of the immanant. This is defined for each self-conjugate partition through a "twisted" analogue of the irreducible character of the symmetric group. This twisted immanant has some interesting properties. For example, it satisfies Cauchy-Binet type formulas. Moreover it is closely related to the following results for matrices whose entries anticommute with each other: (i) the description of the invariants under the conjugations, and (ii) an analogue of the Cauchy identities for symmetric polynomials.