Tropical totally positive matrices

We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowshi

An Overview of Polynomially Computable Characteristics of Special Interval Matrices

It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some of such classes. In particular, we focus on several special interval matrices and investigate their convenient properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse M-matrices, inverse nonnegative matrices, nonnegative matrices, totally positive matrices and some others. We focus in particular on computing the range of the determinant, eigenvalues, singular values, and selected norms. Whenever possible, we state also formulae for determining the inverse matrix and the hull of the solution set of an interval system of linear equations. We survey not only the known facts, but we present some new views as well

Total positivity in loop groups I: whirls and curls

This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\R[t,t^{-1}]), and for the formal loop group GL_n(\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeroes of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the ``factorization problem'' using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlev\'{e} dynamical systems and Berenstein-Kazhdan's study of geometric crystals.Comment: 49 pages, 7 figure

Some more amplituhedra are contractible

The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current Developments in Mathematics 2014, Int.\ Press). In this paper we show that that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.Comment: 7 pages, to appear in Selecta Mathematic