Remarks on Fermionic Formula

TL;DR

This paper explores fermionic formulae linked to quantum affine algebras, discussing their connections to representation theory, combinatorics, and integrable models, and analyzing their mathematical properties and implications.

Contribution

It provides a comprehensive discussion of fermionic formulae for general non-twisted quantum affine algebras, integrating various aspects like crystal bases, one-dimensional sums, and the string hypothesis.

Findings

Connections between fermionic formulae and crystal base theory

Validation of the combinatorial completeness of the string hypothesis

Formulation of spinon character formulae for quantum affine algebras

Abstract

Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra U_q(X^{(1)}_n) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary X_n.