Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian

TL;DR

This paper demonstrates that eigenfunctions of the difference Toda Hamiltonian can be represented through fermionic formulas, utilizing Whittaker vectors, the Drinfeld Casimir element, and quantum group representation theory.

Contribution

It introduces a novel fermionic formula representation for these eigenfunctions and proves identities for their coefficients using quantum group techniques.

Findings

Eigenfunctions expressed via fermionic formulas.

Identities for eigenfunction coefficients established.

Representation theory links to combinatorial structures.

Abstract

We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.