Super-Schur polynomials for Affine Super Yangian Y($$ \hat{\mathfrak{gl}} $$1|1)

Abstract

Abstract We explicitly construct cut-and-join operators and their eigenfunctions — the Super-Schur functions — for the case of the affine super-Yangian Y($$ \hat{\mathfrak{gl}} $$ 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 (pk, θk). 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.