Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$

TL;DR

This paper constructs explicit eigenfunctions called Super-Schur functions for the affine super-Yangian $ ext{Y}( ext{gl}_{1|1})$, extending classical symmetric function theory to a superalgebra context with applications in string theory.

Contribution

It introduces Super-Schur functions as eigenfunctions of the affine super-Yangian and generalizes classical symmetric function formulas to the superalgebra setting.

Findings

Explicit construction of Super-Schur functions

Generalized hook and Cauchy formulas for super-Young diagrams

Connection to string theory origins

Abstract

We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables $(p_k,\theta_k)$. The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided.