Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/9912128

Total positivity: tests and parametrizations

Sergey Fomin·Andrei Zelevinsky
View original source ↗Read source document ↗Export metadata ↓

Abstract

An introduction to total positivity (TP), with the emphasis on efficient TP criteria and parametrizations of TP matrices. Intended for general mathematical audience.

Explore related subjects

math.RAmath.COmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sergey Fomin, Andrei Zelevinsky. 1999-12-15. Total positivity: tests and parametrizations. https://arxiv.org/abs/math/9912128

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Galois Rings: Ring-Theoretic Properties and Applications to Coulomb Branches and Affine Hecke Algebras

Galois rings and Galois orders, introduced by Futorny and Ovsienko, are realized as subrings of fixed subrings of skew group (or monoid) rings and have numerous applications in the structure and representation theory of associative algebras. This paper consists of two parts. The first parte investigates ring-theoretic properties that follow from the Galois ring structure alone. In particular, we estabilish natural conditions under which Galois are Ore domains or (semi)prime Goldie rings. We also study several ring-theoretic dimensions and combine the theories of Galois rings and PI-algebras to obtain new structural results. In the second part, we apply these results, together with general techniques from ring theory, to affine Hecke algebras in the sense of Ginzburg, Kapranov, and Vasserot, as well as to spherical Coulomb branch algebras. In particular, we prove that these algebras are Jacobson semiprimitive, satisfy the Nullstellensatz, and determine several of their ring-theoretic dimensions. For affine Hecke algebras, we further prove that they satisfy the maximal Nullstellensatz, are integral over their centers, and determine their T-ideals of polynomial identities, PI-degree, and PI-exponents. For spherical Coulomb branch algebras, we compute the Krull dimension, estabilish that they satisfy the Gelfand-Kirillov conjecture, and prove that they are not PI-algebras

math.RA↗
arXiv2026

Efficient Compression in Semigroups

Straight-line programs are a central tool in several areas of computer science, including data compression, algebraic complexity theory, and the algorithmic solution of algebraic equations. In the algebraic setting, where straight-line programs can be interpreted as circuits over algebraic structures such as semigroups or groups, they have led to deep insights in computational complexity. A key result by Babai and Szemerédi (1984) showed that finite groups afford efficient compression via straight-line programs, enabling the design of a black-box computation model for groups. Building on their result, Fleischer (2019) placed the Cayley table membership problem for certain classes (pseudovarieties) of finite semigroups in NPOLYLOGTIME, and in some cases even in FOLL. He also provided a complete classification of pseudovarieties of finite monoids affording efficient compression. In this work, we complete this classification program initiated by Fleischer, characterizing precisely those pseudovarieties of finite semigroups that afford efficient compression via straight-line programs. Along the way, we also improve several known bounds on the length and width of straight-line programs over semigroups, monoids, and groups. These results lead to new upper bounds for the membership problem in the Cayley table model: for all pseudovarieties that afford efficient compression and do not contain any nonsolvable group, we obtain FOLL algorithms. In particular, we resolve a conjecture of Barrington, Kadau, Lange, and McKenzie (2001), showing that the membership problem for all solvable groups is in FOLL.

math.RA↗
arXiv2026

On the Spectral Region of n-Cycle Stochastic Matrices

For every $n\geq3$, we determine the union of the spectra of row-stochastic matrices supported on a directed $n$-cycle with self-loops and strictly positive cycle edges. These matrices describe progression through cyclic stages with geometric waiting times, and also arise by uniformization of unidirectional continuous-time reaction cycles. For the latter, the classification determines the minimum transition-rate bound required to realize a prescribed nonreal eigenvalue and provides rates attaining that bound. The angles $m=\mathrm{Arg}λ$ and $M=\mathrm{Arg}(λ-1)$ reduce the nonreal eigenvalue problem to determining the range of a strictly convex logarithmic sum over angles with prescribed sum. Jensen's inequality gives its minimum, simplex vertices give its finite maximum, and connectedness makes these bounds sufficient for realization. The resulting branch graphs are strictly decreasing and their horizontal sections are ordered. This ordering gives the complete boundary, which alternates between uniform-cycle segments and one-loop algebraic arcs. We state a direct membership criterion and construct a realizing matrix for every spectral point; every nonreal point admits a realization with at most two distinct self-loop weights. The real spectral set is $[-1,1]$ for even $n$ and $(0,1]$ for odd $n$. Allowing deleted cycle edges adds only $0$ in odd dimension and adds nothing in even dimension. We also apply the classification to a branching network with matched holding probabilities on parallel paths. The proofs are independent of Karpelevich's theorem.

math.RA↗