A Hopf algebra of subword complexes (Extended abstract)

International audienceWe introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras

A Hopf algebra of subword complexes (Extended abstract)

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras

K-orbit closures and Barbasch-Evens-Magyar varieties

We define the Barbasch-Evens-Magyar variety. We show it is isomorphic to the smooth variety defined in [D. Barbasch-S. Evens '94] that maps finite-to-one to a symmetric orbit closure, thereby giving a resolution of singularities in certain cases. Our definition parallels [P. Magyar '98]'s construction of the Bott-Samelson variety [H. C. Hansen '73, M. Demazure '74]. From this alternative viewpoint, one deduces a graphical description in type A, stratification into closed subvarieties of the same kind, and determination of the torus-fixed points. Moreover, we explain how these manifolds inherit a natural symplectic structure with Hamiltonian torus action. We then prove that the moment polytope is expressed in terms of the moment polytope of a Bott-Samelson variety.Comment: 26 pages, 4 figure

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