Control sets and total positivity (Q1882673)

The goal of this paper is to connect two different mathematical subjects, namely totally positive matrices and control sets. The set \(T\) of totally positive matrices in \(Sl(d,\mathbb{R})\) (subgroup of determinant one matrices of size \(d \times d\)) is also a semigroup with non-empty interior in \(Sl(d,\mathbb{R})\). Taking account of this, the main aim of this work is to study properties of totally positive matrices by exploiting its semigroup structure. The authors study the action of \(T\) on the flag manifolds and describe its control sets. They give geometric interpretations and new proofs of some classical results. In addition, they prove that the control sets of \(T\) coincide with the control sets of the semigroup \(S\) of sign regular matrices in \(Sl(d,\mathbb{R})\), and in fact they work out the properties of \(T\) through the action of \(S\). Finally, the authors also consider, for each multi-index \(r=\{r_1,\ldots,r_k\}\), the semigroups \(T_r\) and \(S_r\) of matrices having positive (respectively fixed sign) minors of orders \(r_1,\ldots,r_k\). Their control sets are described and it is proved that \(S_r\) is a maximal semigroup in the particular case where \(r=\{r_1\}\) is a singleton whereas, for arbitrary \(r\), the semigroup \(S_r\) is maximal of type \(r\).