A Perron theorem for matrices with negative entries and applications to Coxeter groups

Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a matrix $A$ is a simple root of the characteristic polynomial and is strictly greater than the modulus of any other root, then $A$ is conjugate to a matrix $Z$ some power of which is positive. In this article, we provide an explicit conjugate matrix $Z$, and prove that the spectral radius of $A$ is a simple and dominant eigenvalue of $A$ if and only if $Z$ is eventually positive. For $n\times n$ real matrices with each row-sum equal to $1$, this criterion can be declined into checking that each entry of some power is strictly larger than the average of the entries of the same column minus $\frac{1}{n}$. We apply the criterion to elements of irreducible infinite nonaffine Coxeter groups to provide evidences for the dominance of the spectral radius, which is still unknown.Comment: 14 page

On inversion sets and the weak order in Coxeter groups

In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and their position relative to the imaginary cone. Finally, we discuss inversion sets of infinite reduced words and the notions of biconvex and biclosed sets of positive roots.Comment: 22 pages; 10 figures; v2 some references were added; v2: final version, to appear in European Journal of Combinatoric

Convex Geometry of Subword Complexes of Coxeter Groups

This monography presents results related to the convex geometry of a family of simplicial complexes called ``subword complexes''. These simplicial complexes are defined using the Bruhat order of Coxeter groups. Despite a simple combinatorial definition much of their combinatorial properties are still not understood. In contrast, many of their known connections make use of specific geometric realizations of these simplicial complexes. When such realizations are missing, many connections can only be conjectured to exist. This monography lays down a framework using an alliance of algebraic combinatorics and discrete geometry to study further subword complexes. It provides an abstract, though transparent, perspective on subword complexes based on linear algebra and combinatorics on words. The main contribution is the presentation of a universal partial oriented matroid whose realizability over the real numbers implies the realizability of subword complexes as oriented matroids.Diese Monographie präsentiert Ergebnisse im Zusammenhang mit einer Familie von simplizialen Komplexen, die "Subwortkomplexe" genannt werden. Diese Simplizialkomplexe werden mit Hilfe der Bruhat-Ordnung von Coxeter-Gruppen definiert. Trotz einer einfachen kombinatorischen Definition werden viele ihrer kombinatorischen Eigenschaften immer noch nicht verstanden. Spezifische geometrische Realisierungen dieser Simplizialkomplexe machen neue Herangehensweisen an Vermutungen des Gebiets m\"oglich. Wenn solche Verbindungen fehlen, können viele Zusammenhänge nur vermutet werden. Diese Monographie legt einen Rahmen fest, in dem eine Allianz aus algebraischer Kombinatorik und diskreter Geometrie verwendet wird, um weitere Subwortkomplexe zu untersuchen. Es bietet eine abstrakte und transparente Perspektive auf Teilwortkomplexe, die auf linearer Algebra und Kombinatorik von Wörtern basiert. Der Hauptbeitrag ist die Darstellung eines universellen, nur teilweise orientierten Matroids, dessen Realisierbarkeit über den reellen Zahlen die Realisierbarkeit von Teilwortkomplexen als orientierte Matroide impliziert