Total positivity, grassmannian and modified bessel functions

V. M. Buchstaber, A. A. Glutsyuk

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

2 Citations (Scopus)

Abstract

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 Gl,m of l-dimensional subspaces in Rm is called strictly totally positive (according to A. E. Postnikov) if one can normalize its Plücker 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 Gl,m, l < m.

Original languageEnglish
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages97-107
Number of pages11
Volume733
DOIs
Publication statusPublished - 2019
Externally publishedYes

Publication series

NameContemporary Mathematics
Volume733
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

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