Total positivity, Grassmannian and modified Bessel functions

Abstract

A rectangular matrix is called {\it totally positive} if all its minors are positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional subspaces in $\mathbb R^m$ is called {\it strictly totally positive} if one can normalize its Pl\ucker coordinates to make all of them positive. Clearly if a $k\times m$-matrix, $k<m$, is totally positive, then each collection of its $l\leq k$ raws generates an $l$-subspace represented by a strictly totally positive point of the Grassmanian manifold $G_{l,m}$. The totally positive matrices and the subsets of strictly totally positive points in Grassmanian manifolds arise in many domains of mathematics, mechanics and physics. F.R.Gantmacher and M.G.Krein considered totally positive matrices in the context of classical mechanics. As it was shown in a joint paper by M.Boiti, F.Pemperini and A.Pogrebkov, each matrix of appropriate dimension with positive minors of higher dimension generates a multisoliton solution of the Kadomtsev-Petviashvili (KP) partial differential equation. There exist several approaches of construction of totally positive matrices due to F.R. Gantmacher, M.G.Krein, A.E.Postnikov and ourselves. In our previous paper we have proved that certain determinants formed by modified Bessel functions of the first type are positive on the positive semi-axis. This yields a one-dimensional family of totally positive points in all the Grassmanian manifolds. In the present paper we provide a construction of multidimensional families of totally positive points in all the Grassmanian manifolds, again using modified Bessel functions of the first type but different from the above-mentioned construction. These families represent images of explicit injective mappings of the convex open subset $\{ x=(x_1,\dots,x_l)\in\mathbb R_+^l | x_1<\dots<x_l\}\subset\mathbb R^l$ to the Grassmanian manifolds $G_{l,m}$, $l<m$.