Semigroup automorphisms of total positivity
Projesh Nath Choudhury, Shaun Fallat, Chi-Kwong Li
TL;DR
This work classifies semigroup automorphisms of the totally positive and totally nonnegative matrix semigroups, showing they coincide and are precisely given by a determinant-based scaling combined with diagonal or anti-diagonal conjugation: $T(A)=\mu(\det A)(\det A)^{-1/n}\,RAR^{-1}$, with $\mu:(0,\infty)\to(0,\infty)$ bijective multiplicative and $R$ either diagonal or anti-diagonal with positive entries. The authors establish a reduction from TP$(n)$ to ITN$(n)$ via a standard automorphism extension, and then perform a refined analysis of centralizers and block-structure under automorphisms, culminating in an induction on $n$. They prove that automorphisms preserve the bidiagonal generators and the Whitney bidiagonal factorization, effectively forcing any automorphism to act uniformly across the entire semigroup. The results bridge total positivity, semigroup automorphisms, and conjugation symmetries, providing a canonical description of symmetry for these fundamental matrix classes and their intermediate multiplicative 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.