Totally positive matrices and dilogarithm identities

TL;DR

This paper explores involutions on totally positive matrices, linking them to the tetrahedron equation and symmetric group actions, leading to new dilogarithm identities involving matrix minors that are symmetric under S3.

Contribution

It introduces a novel connection between involutions on totally positive matrices, the tetrahedron equation, and symmetric group actions, resulting in new symmetric dilogarithm identities.

Findings

Involutions relate to the tetrahedron equation and S3 actions.

Derived a family of dilogarithm identities involving minors.

Identities are invariant under S3 symmetry.

Abstract

We show that two involutions on the variety $N_n^+$ of upper triangular totally positive matrices are related, on the one hand, to the tetrahedron equation and, on the other hand, to the action of the symmetric group $S_3$ on some subvariety of $N_n^+$ and on the set of certain functions on $N_n^+$. Using these involutions, we obtain a family of dilogarithm identities involving minors of totally positive matrices. These identities admit a form manifestly invariant under the action of the symmetric group $S_3$.