Super-Hamiltonians for super-Macdonald polynomials

Abstract

The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables ${\theta }_k$ are introduced, and the Hamiltonian is represented as a differential operator acting on a space of functions of both types of variables $p_k$ and ${\theta }_k$. Eigenfunctions for this Hamiltonian are a suitable generalization of Macdonald polynomials to super-Macdonald polynomials discussed earlier in the literature. Peculiarities of the construction in comparison to the canonical bosonic case are discussed.

References in corpus (19)

• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• W-symmetry, topological vertex and affine Yangian
• Quantum toroidal ${\mathfrak{gl}}_1$ algebra : plane partitions
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• The supersymmetric Ruijsenaars-Schneider model
• Macdonald polynomials in superspace: conjectural definition and positivity conjectures
• Hunt for 3-Schur polynomials
• Kerov functions revisited
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Kerov functions for composite representations and Macdonald ideal
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• On Hamiltonians for Kerov functions
• Hook variables: cut-and-join operators and $\tau$-functions
• Macdonald polynomials for super-partitions
• Algorithms for representations of quiver Yangian algebras
• Supersymmetric polynomials and algebro-combinatorial duality