A generalization of immanants based on partition algebra characters

TL;DR

This paper introduces a new generalization of immanants called the recombinant, based on partition algebra characters, extending the concept beyond symmetric group characters and involving partition diagrams.

Contribution

It defines the recombinant as a new immanant-like function using partition algebra characters, broadening the scope of immanant generalizations.

Findings

Recombinant coincides with classical immanants for specific tableaux shapes.

Uses partition diagrams instead of permutations for summation.

Establishes a connection between partition algebra characters and matrix functions.

Abstract

We introduce a generalization of immanants of matrices, using partition algebra characters in place of symmetric group characters. We prove that our immanant-like function on square matrices, which we refer to as the recombinant, agrees with the usual definition for immanants for the special case whereby the vacillating tableaux associated with the irreducible characters correspond, according to the Bratteli diagram for partition algebra representations, to the integer partition shapes for symmetric group characters. In contrast to previously studied variants and generalizations of immanants, as in Temperley-Lieb immanants and $f$-immanants, the sum that we use to define recombinants is indexed by a full set of partition diagrams, as opposed to permutations.