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Total positivity, Grassmannian and modified Bessel functions
A rectangular matrix is called totally positive, (according to F. R.
Gantmacher and M. G. Krein) if all its minors are positive. A point of a
real Grassmannian manifold G(l,m) of l-dimensional subspaces in Rm is called
strictly totally positive (according to A. E. Postnikov) if one can normalize
its Pl¨ucker coordinates to make all of them positive. The totally positive
matrices and the strictly totally positive Grassmannians, that is, the subsets
of strictly totally positive points in Grassmannian manifolds arise in many
areas: in classical mechanics (see the book of F. R. Gantmacher and M. G.
Krein); in a wide context of analysis, differential equations and probability
theory (see the book of S. Karlin); in physics, for example, in construction of
solutions of the Kadomtsev-Petviashvili (KP) partial differential equation (see
a paper by T. M. Malanyuk, a paper by M. Boiti, F. Pemperini, A. Pogrebkov,
a paper of Y. Kodama, L. Williams). Different problems of mathematics,
mechanics and physics led to constructions of totally positive matrices by many
mathematicians, including F. R. Gantmacher, M. G. Krein, I. J. Schoenberg,
S. Karlin, A. E. Postnikov and ourselves. One-dimensional families of totally
positive matrices whose entries are modified Bessel functions of the first kind
have arisen in our study (in collaboration with S. I. Tertychnyi) of model of
the overdamped Josephson effect in superconductivity and double confluent
Heun equations related to it.
In the present paper we give a new construction of multidimensional families
of totally positive matrices different from the above-mentioned families.
Their entries are again formed by values of modified Bessel functions of the
first kind, but now with non-negative integer indices. Their columns are numerated
by the indices of the modified Bessel functions, and their rows are
numerated by their arguments. This yields new multidimensional families of
strictly totally positive points in all the Grassmannian manifolds. These families
represent images of explicit injective mappings of the convex open simplex
{x = (x1, . . . , xl) ∈ Rl | 0 < x1 < · · · < xl} ⊂ Rl to the Grassmannian
manifolds G(l,m), l <m.
In book
