Factorial supersymmetric skew Schur functions and ninth variation determinantal identities

TL;DR

This paper extends determinantal identities to a tableau-based ninth variation of supersymmetric skew Schur functions, exploring their properties and independence from entry ordering, with implications for factorial parameters and supersymmetry.

Contribution

It introduces a new tableau-based ninth variation of supersymmetric skew Schur functions and analyzes their determinantal identities and independence from entry ordering.

Findings

Determinantal identities extend to supersymmetric skew Schur functions.

Supersymmetry is lost at the ninth variation but restored at the sixth variation.

Factorial supersymmetric skew Schur functions are independent of entry ordering.

Abstract

The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. These tableaux are built on entries taken from an alphabet of unprimed and primed numbers and that may be ordered in a myriad of different ways, each leading to a determinantal identity. At the level of the ninth variation the corresponding determinantal identities are all distinct but the original notion of supersymmetry is lost. It is shown that this can be remedied at the level of the sixth variation involving a doubly infinite sequence of factorial parameters. Moreover it is shown that the resulting factorial supersymmetric skew Schur functions are independent of the ordering of the unprimed and primed entries in the alphabet.