Totally positive matrices and dilogarithm identities. (English) Zbl 1390.15108
Summary: 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 \(S_3\).
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 11C20 | Matrices, determinants in number theory |
| 33B30 | Higher logarithm functions |
| 20B30 | Symmetric groups |
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