On the quiver of the descent algebra

We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to W.Comment: 31 pages, LaTeX; major revision; to appear in Journal of Algebr

Poset topology and homological invariants of algebras arising in algebraic combinatorics

We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs applications de cette interaction. Par exemple, nous introduisons une nouvelle interpr\'etation du nombre de Leray d'un complexe de clique en termes de la dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159 and an article in preparation. 12 pages, 3 Figure

The down operator and expansions of near rectangular k-Schur functions

We prove that the Lam-Shimozono "down operator" on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of "near rectangles" in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood--Richardson coefficients