A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications

Abstract

Let $T$ be a tree with a vertex set $\{1,2,\dots ,N\}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with $T$.

16 pages

References in corpus (2)

• Polynomials with the half-plane property and matroid theory
• The $m-$dissimilarity map and representation theory of $S{L}_m$