35 research outputs found
Barrett-Johnson inequalities for totally nonnegative matrices
Given a matrix , let denote the submatrix of determined by
rows and columns . Fischer's Inequalities state that for each Hermitian positive semidefinite matrix , and each subset of
and its complement , we have . Barrett and Johnson (Linear Multilinear
Algebra 34, 1993) extended these to state inequalities for sums of products of
principal minors whose orders are given by nonincreasing integer sequences
, summing to .
Specifically, if for all
, then where sums are over sequences
of disjoint subsets of satisfying , . We show that these inequalities hold for totally nonnegative matrices
as well
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