On the boundary of totally positive upper triangular matrices (Q1906761)

The real \(n \times n\) upper triangular matrices with (strictly) positive diagonal entries form a multiplicative group \(B_+\). A matrix \(M \in B_+\) is called totally positive if every determinantal minor that has no row or column lying completely below the main diagonal of \(M\) is positive: the strictly positive members of \(B_+\) form a subsemigroup \(B_p\). The authors prove that the boundary of \(B_p\) in \(B_+\) lies on the hypersurface \(H\) given by \(\Delta_1 \cdots \Delta_{n - 1} = 0\), where \(\Delta_i\) denote the minor of \(M \in B_+\) formed by its first \(i\) rows and last \(i\) columns. Further, the \(2^{n - 1}\) regions obtained from \(B_p\) by conjugating with diagonal matrices with entries \(\pm 1\) are connected components of \(B_+ \backslash H\). Several related problems are raised.