Totally positive matrices and dilogarithm identities

We show that two involutions on the variety Nn+ 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 S3 on some subvariety of Nn+ and on the set of certain functions on Nn+. 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 S3.