Super-Hamiltonians for super-Macdonald polynomials

TL;DR

This paper introduces a super-generalization of the Macdonald Hamiltonian by incorporating Grassmann variables, leading to super-Macdonald polynomials as eigenfunctions, expanding the mathematical framework of symmetric functions.

Contribution

It presents a novel super-Hamiltonian incorporating Grassmann variables and constructs eigenfunctions as super-Macdonald polynomials, extending previous Macdonald polynomial theory.

Findings

Super-Hamiltonian includes Grassmann variables

Eigenfunctions are super-Macdonald polynomials

Comparison with canonical bosonic case discussed

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.