Semigroup automorphisms of total positivity

TL;DR

This paper classifies all automorphisms of the semigroups of invertible totally nonnegative and totally positive matrices, revealing they are identical and preserve the fundamental bidiagonal factorizations.

Contribution

It provides a complete classification of automorphisms of semigroups of ITN and TP matrices, showing they are identical and respect the generators.

Findings

Automorphisms of ITN and TP semigroups are the same.

Automorphisms preserve the bidiagonal factorization generators.

Complete characterization of automorphisms of these matrix semigroups.

Abstract

Totally positive (TP) and totally nonnegative (TN) matrices connect to analysis, mechanics, and to dual canonical bases in reductive groups, by well-known works of Schoenberg, Gantmacher-Krein, Lusztig, and others. TP matrices form a multiplicatively closed semigroup, contained in the larger monoid of invertible totally nonnegative (ITN) matrices. Whitney and Berenstein-Fomin-Zelevinsky found bidiagonal factorizations of all $n\times n$ ITN and TP matrices into multiplicative generators; a natural question now is to classify the multiplicative automorphisms of these semigroups. In this article, we classify all automorphisms of these semigroups of ITN and TP matrices. In particular, we show that the automorphisms are the same, and they respect the multiplicative generators.