  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 $S_3$.
