Supersymmetric polynomials and algebro-combinatorial duality

In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynomials form a representation of a super-algebra analog \mathsf{T}(\widehat{\mathfrak{gl}}_{1|1}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mover> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔤</mml:mi> <mml:mi>𝔩</mml:mi> </mml:mstyle> <mml:mo accent="true">̂</mml:mo> </mml:mover> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> of Ding-Iohara-Miki (quantum toroidal) algebra, emerging as a BPS algebra of D-branes on a conifold. A supersymmetric modification for Young tableaux and Kostka numbers are also discussed.