Séminaire Lotharingien de Combinatoire, 89B.29 (2023), 12 pp.
Mark Skandera and Daniel Soskin
Barrett-Johnson Inequalities for Totally Nonnegative Matrices
Abstract.
Given a matrix A, let AI,J denote the submatrix of A determined by
rows I and columns J.
The Barrett-Johnson Inequalities
relate sums of products of principal minors of
positive semidefinite (PSD) matrices, when orders of the minors
are given by integer partitions λ = (λ1, ..., λr,
μ = (μ1, ..., μs of n.
Specifically,
we have
for all PSD n × n matrices A,
where sums are over ordered set partitions
of {1,...,n} satisfying |Ik| = λk,
|Jk| = μk,
if and only if λ is majorized by μ.
We show that these inequalities hold for totally nonnegative matrices
as well.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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