Monomial Nonnegativity and the Bruhat Order

We show that five nonnegativity properties of polynomials coincide when restricted to polynomials of the form x1, pi(1) ... xn,pi(n) - x1, sigma(1) ... xn, sigma(n), where $\pi and sigma are permutations in Sn. In particular, we show that each of these properties may be used to characterize the Bruhat order on Sn

Equal Entries in Totally Positive Matrices

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ matrix is $\Theta(n^{4/3})$ (resp. $\Theta(n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\textrm{-by-}2$ minors in a $2\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\textrm{-by-}2$ minors and outerplanar graphs.Comment: 15 page

The logarithmic method and the solution to the TP2-completion problem

A matrix is called TP2 if all 1-by-1 and 2-by-2 minors are positive. A partial matrix is one with some of its entries specified, while the remaining, unspecified, entries are free to be chosen. A TP2-completion, of a partial matrix T , is a choice of values for the unspecified entries of T so that the resulting matrix is TP2. The TP2-completion problem asks which partial matrices have a TP2-completion. A complete solution is given here. It is shown that the Bruhat partial order on permutations is the inverse of a certain natural partial order induced by TP2 matrices and that a positive matrix is TP2 if and only if it satisfies certain inequalities induced by the Bruhat order. The Bruhat order on permutations is generalized to a partial order, GBr, on nonnegative matrices, and the concept of majorization is generalized to a partial order, DM, on nonnegative matrices. It is shown that these two partial orders are inverses of each other on the set of nonnegative matrices. Using this relationship and the Hadamard exponential transform on nonnegative matrices, explicit conditions for TP2-completability of a given partial matrix are given. It is shown that an m-by- n partial TP2 matrix T is TP2-completable if and only if tijspecified taijij ≥ 1 for every matrix A = (aij) ∈ Mm,n having (1) aij = 0 if tij is unspecified; (2) each row sum and each column sum of A is zero; and (3) 1≤i≤p1≤j≤ qaij ≥ 0, for all (p, q) ∈ {lcub}1, 2, ..., m{rcub} x {lcub}1, 2, ..., n{rcub}. However, there may be infinitely many such conditions, and some of them may be obtainable from others. In order to find a set of minimal conditions, the theory of cones and generators, and the logarithmic method are used. It is shown that the set of matrices used in the exponents of the inequalities forms a finitely generated cone with integral generators. This gives finitely many polynomial inequalities on the specified entries of a partial matrix of given pattern as conditions for TP2-completability. A computational scheme for explicitly finding the generators is given and the combinatorial structure of TP2-completable pattern is investigated

Two new criteria for comparison in the Bruhat order

We give two new criteria by which pairs of permutations may be compared in defining the Bruhat order (of type A). One criterion uses totally nonnegative polynomials and the other uses Schur functions

Two New Criteria for Comparison in the Bruhat Order

Abstract. We give two new criteria by which pairs of permutations may be compared in defining the Bruhat order (of type A). One criterion utilizes totally nonnegative polynomials and the other utilizes Schur functions. Résumé. Nous donons deux critères nouveaux avec lesquels on peut comparer couples de permutations en definant l’order de Bruhat (de type A). Un critère utilise les polynômes totallement nonnegatifs et l’autre utilise les fonctions symétriques de Schur. 1. Main The Bruhat order on Sn is often defined by comparing permutations π = π(1) · · · π(n) and σ = σ(1) · · · σ(n) according to the following criterion: π ≤ σ if σ is obtainable from π by a sequence of transpositions (i, j) where i < j and i appears to the left of j in π. (See e.g. [7, p. 119].) A second well-known criterion compares permutations in terms of their defining matrices. Let M(π) be the matrix whose (i, j) entry is 1 if j = π(i) and zero otherwise. Defining [i] = {1,..., i}, and denoting the submatrix of M(π) corresponding to rows I and columns J by M(π)I,J, we have the following. Theorem 1.1. Let π and σ be permutations in Sn. Then π is less than or equal to σ in the Bruhat order if and only if for all 1 ≤ i, j ≤ n − 1, the number of ones in M(π) [i],[j] is greater than or equal to th