Super-Schur polynomials for Affine Super Yangian Y($$ \hat{\mathfrak{gl}} $$1|1)
Dmitry Galakhov, Alexei Morozov, Nikita Tselousov
Abstract
A bstract We explicitly construct cut-and-join operators and their eigenfunctions — the Super-Schur functions — for the case of the affine super-Yangian Y( $$ \hat{\mathfrak{gl}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>gl</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> 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 , θ 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.