============================== Bar-Ilan Combinatorics Seminar ============================== The next meeting of the seminar will take place, IYH, on (when) Tuesday, 18 Iyar (May 12) 12:00-13:30 (where) Room 201 (Math Dept Seminar Room), Math and CS Building (216), Bar-Ilan University (who) Stuart Margolis (Bar-Ilan University) will talk about (what) "A semigroup approach to descent algebras for Coxeter groups and wreath products: combinatorial aspects" Abstract: The Coxeter complex C of any Coxeter group W has the structure of a left regular band, a semigroup satisfying the laws x^2 = x and xyx = xy, via the projections originally defined by Tits. Furthermore, the usual action of W on C is an action by automorphisms of the semigroup structure of C. In 1997, Bigidare proved that the invariant subalgebra of the semigroup algebra of C relative to this action is anti- isomorphic to the Solomon Descent Algebra of W, D(W). Solomon's original proof that the descent algebra was indeed a subalgebra of the group algebra of W was quite difficult. The present approach is elementary and illuminates the structure of D(W). For example, D(W) is a basic algebra, because this is true of the algebra of any finite idempotent semigroup. Saliola computed the quiver of an arbitrary left regular band and in particular the quiver of a Coxeter complex and other hyperplane arrangements. Hsiao showed how to generalize the above by associating to every finite group G of exponent n and every Coxeter group W , a finite monoid whose idempotents form a left regular band, but satisfies the identity x=x^{n+1}. He showed that the wreath product of W with G acts on this semigroup and that this time the invariant algebra maps onto the "Colored" Descent Algebra, that is the Descent Algebra for the wreath product of G and W studied intensely by Reutenauer, Mantaci and others over the past few years. Indeed this is part of a general attempt to work with colored versions of classical objects associated to Coxeter groups and it seems that these semigroups play an important role. The speaker and Ben Steinberg recently developed the homology theory of regular semigroups in order to generalize Saliola's work and compute the quiver associated to Hsiao's algebra. Left regular bands arise naturally in many contexts on the border of algebra, algebraic combinatorics, group theory, Lie theory and semigroup theory. We begin from scratch with the examples of importance and then develop the theory of left regular bands, their semigroup algebras and their connections with groups and algebraic combinatorics as detailed above. This is a two part talk that will take place in the Combinatorics and the Representation Theory Seminars this week. We will discuss the combinatorial aspects of the subject in the Combinatorics Seminar and the more algebraic aspects in the Representation Theory Seminar. Forthcoming Events: ------------------- * 25 Iyar (May 19) Noa Nitzan (Hebrew University): "A planar 3-convex set is indeed a union of six convex sets" * 3 Sivan (May 26) Dan Romik (Hebrew University): "New enumerstion results for alternating sign matrices" ************************************************************************* You are all invited ! (Graduate students especially welcome) If you want to give a talk at the seminar, or know a prospective speaker, please contact Ron Adin . Seminar's homepage: *************************************************************************