Parametrizations of k-nonnegative matrices: Cluster algebras and k-positivity tests

Brosowsky, Anna; Chepuri, Sunita; Mason, Alex

doi:10.1016/j.jcta.2020.105217

Cited by 7 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In a sense, assertions (1) ⇐⇒ (2) in Theorems A and C resemble this result in the structure of the second assertions and the simplicity of the statements. Similarly, recall the well-known paper by Fomin and Zelevinsky [10] about tests for totally positive matrices, as well as recent follow-ups such as [5]. Our results may be regarded as being similar in spirit.…”

Section: )supporting

confidence: 80%

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull

Choudhury

Kannan

Khare

2021

Bull. London Math. Soc.

View full text Add to dashboard Cite

A matrix A is totally positive (or non-negative) of order k, denoted T P k (or T N k ), if all minors of size k are positive (or non-negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non-reversal property. We do away with the former, and show that A is T P k if and only if every submatrix formed from at most k consecutive rows and columns has the sign non-reversal property. In fact, this can be strengthened to only consider test vectors in R k with alternating signs. We also show a similar characterization for all T N k matrices -more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing T P matrices by Gantmacher-Krein (Compos. Math. 4 (1937) 445-476) and P -matrices by Gale-Nikaido (Math. Ann. 159 (1965) 81-93).As an application, we study the interval hull I(A, B) of two m × n matrices A = (aij) and B = (bij). This is the collection of C ∈ R m×n such that each cij is between aij and bij. Using the sign non-reversal property, we identify a two-element subset of I(A, B) that detects the T P k property for all of I(A, B) for arbitrary k 1. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non-negativity (of any order) of an interval hull I(A, B).

show abstract

Section: )supporting

confidence: 80%

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull

Choudhury

Kannan

Khare

2021

Bull. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…2021/05/03 19:18 20 Question 6.2 When is Imm v a cluster monomial in O(GL n (C))? When is Imm v a cluster monomial in a k-positivity cluster sub-algebra from [Brosowsky et al, 2017]? Interestingly, the Kazhdan-Lusztig immanants of 123-, 2143-, 1423-, and 3214avoiding permutations do appear in sub-cluster algebras of this kind for k = 2.…”

Section: Final Remarksmentioning

confidence: 99%

1324- and 2143-avoiding Kazhdan–Lusztig immanants and k-positivity

Chepuri

Sherman-Bennett²

2021

Can. J. Math.-J. Can. Math.

Self Cite

View full text Add to dashboard Cite

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan-Lusztig immanants, which are indexed by permutations, involve q = 1 specializations of Type A Kazhdan-Lusztig polynomials, and were defined by Rhoades and Skandera in [Rhoades and Skandera, 2006]. Using results of [Haiman, 1993] and [Stembridge, 1991], Rhoades and Skandera showed that Kazhdan-Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan-Lusztig immanants are positive on k-positive matrices (matrices whose minors of size k × k and smaller are positive). The Kazhdan-Lusztig immanant indexed by v is positive on k-positive matrices if v avoids 1324 and 2143 and for all non-Our main tool is Lewis Carroll's identity.

show abstract

“…In 1937, Gantmacher-Krein [20] gave a fundamental characterization of totally positive matrices of order k by the positivity of the spectra of all submatrices of size at most k. There is a well known article [17] by Fomin-Zelevinsky about tests for TP matrices; there have been numerous subsequent papers along this theme, e.g. [7]. The present paper may be regarded as being similar in spirit.…”

Section: Introduction and Main Resultsmentioning

confidence: 86%

Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution

Choudhury¹

2021

Preprint

View full text Add to dashboard Cite

A matrix A is called totally positive (or totally non-negative) of order k, denoted by T P k (or T N k ), if all minors of size at most k are positive (or non-negative). These matrices have featured in diverse areas in mathematics, including algebra, analysis, combinatorics, differential equations and probability theory. The goal of this article is to provide a novel connection between total positivity and optimization/game theory. Specifically, we draw a relationship between totally positive matrices and the Linear Complementarity Problem (LCP), which generalizes and unifies linear and quadratic programming problems and bimatrix games -this connection is unexplored, to the best of our knowledge. We show that A is T P k if and only if for every submatrix Ar of A formed from r consecutive rows and r consecutive columns (with r ≤ k), LCP(Ar, q) has a unique solution for each vector q < 0. In fact this can be strengthened to check the solution set of LCP at a single vector for each such square submatrix. These novel characterizations are in the spirit of classical results characterizing T P matrices by Gantmacher-Krein [Compos. Math. 1937] and P -matrices by Ingleton [Proc. London Math. Soc. 1966].Our work contains two other contributions, both of which characterize TP using single test vectors whose co-ordinates have alternating signs -i.e., lie in a certain open bi-orthant. First, we improve on one of the main results in recent joint work [Bull. London Math. Soc., in press], which provided a novel characterization of T P k matrices using sign non-reversal phenomena. We further improve on a classical characterization of total positivity by Brown-Johnstone-MacGibbon [J. Amer. Statist. Assoc. 1981] (following Gantmacher-Krein, 1950) involving the variation diminishing property. Finally, we use a Pólya frequency function of Karlin [Trans. Amer. Math. Soc. 1964] to show that our aforementioned characterizations of total positivity, involving (single) test-vectors drawn from the 'alternating' bi-orthant, do not work if these vectors are drawn from any other open orthant.

show abstract

Parametrizations of k-nonnegative matrices: Cluster algebras and k-positivity tests

Cited by 7 publications

References 12 publications

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull

Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull

1324- and 2143-avoiding Kazhdan–Lusztig immanants and k-positivity

Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution

Contact Info

Product

Resources

About