On the combinatorics of row and corner transfer matrices of the $A_{n-1}^{(1)}$ restricted face models

TL;DR

This paper establishes a bijection linking spectral data of transfer matrices in face models to path energies, deriving fermionic forms of branching functions and a bosonic-fermionic identity.

Contribution

It introduces a weight-preserving bijection connecting spectral data of transfer matrices to path energies, leading to new fermionic forms of branching functions.

Findings

Derived fermionic forms of polynomial branching functions.

Established a bosonic-fermionic polynomial identity.

Linked spectral data to path energies in face models.

Abstract

We establish a weight-preserving bijection between the index sets of the spectral data of row-to-row and corner transfer matrices for $U_q\widehat{sl(n)}$ restricted interaction round a face (IRF) models. The evaluation of momenta by adding Takahashi integers in the spin chain language is shown to directly correspond to the computation of the energy of a path on the weight lattice in the two-dimensional model. As a consequence we derive fermionic forms of polynomial analogues of branching functions for the cosets ${(A^{(1)}_{n-1})_{\ell -1}\otimes (A^{(1)}_{n-1})_{1}} \over (A^{(1)}_{n-1})_{\ell}$, and establish a bosonic-fermionic polynomial identity.