The Onsager Algebra Symmetry of $\tau^{(j)}$-matrices in the Superintegrable Chiral Potts Model

TL;DR

This paper shows that the $ au^{(j)}$-matrices in the superintegrable chiral Potts model exhibit Onsager algebra symmetry, linking their eigenvalues to algebraic structures and relating functional relations to the eight-vertex model.

Contribution

It establishes the Onsager algebra symmetry of $ au^{(j)}$-matrices and connects the functional relations of the chiral Potts model with the eight-vertex model through $Q$-operators.

Findings

$ au^{(j)}$-matrices have Onsager algebra symmetry for degenerate eigenvalues

Identifies relations between $Q$-operators and fusion matrices in the eight-vertex model

Links the $T ilde{T}$-relation in the chiral Potts model to functional relations in the eight-vertex model

Abstract

We demonstrate that the $\tau^{(j)}$-matrices in the superintegrable chiral Potts model possess the Onsager algebra symmetry for their degenerate eigenvalues. The Fabricius-McCoy comparison of functional relations of the eight-vertex model for roots of unity and the superintegrable chiral Potts model has been carefully analyzed by identifying equivalent terms in the corresponding equations, by which we extract the conjectured relation of $Q$-operators and all fusion matrices in the eight-vertex model corresponding to the $T\hat{T}$-relation in the chiral Potts model.