Hélène Barcelo

Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W. We use this correspondence to find all W for which L is supersolvable. In particular, we show that the only infinite Coxeter group for which L is supersolvable is the infinite dihedral group. Also, we show how this isomorphism gives an embedding of L into the partition lattice whenever W is of type An, Bn or Dn. In addition, we give several results concerning non-broken circuit bases (NBC bases) when W is finite. We show that L is supersolvable if and only if all NBC bases are obtainable by a certain specific combinatorial procedure, and we use the lattice of parabolic subgroups to identify a natural subcollection of the collection of all NBC bases.

Perspectives on A-homotopy theory and its applications

This paper contains a survey of the A-theory of simplicial complexes and graphs, a combinatorial homotopy theory developed recently. The initial motivation arises from the use of simplicial complexes as models for a variety of complex systems and their dynamics. This theory diverges from classical homotopy theory in several crucial aspects. It is related to prior work in matroid theory, graph theory, and work on subspace arrangements.

Non-broken circuits of reflection groups and factorization inD n

AbstractThe set of non-broken circuits of a reflection group W, denoted NBC(W), appears as a basis of the Orlik-Solomon algebra ofW. The factorization of their enumerating polynomial

Young Straightening in a Quotient S n -Module

\sum S \in NBC(W)^{t|S|} \prod {_{i = 1}^k (1 + (d_i - 1)t)}

Combinatorial Representation Theory

with respect to their cardinality involves the exponentsdi-1 ofW. A simple explanation of this factorization is known only for the symmetric groupsSn (Whitney [13]) and for the hyperoctahedral groupsBn (Lehrer [7]). In this paper, we present an elementary proof of the fact that the set NBC(W) of any refection groupW is in bijection with the group elements ofW. We give a simple characterization of the non-broken circuits of the Weyl groups of typeDn and we use this characterization to prove the factorization of their enumerating polynomial.

Homotopy theory of graphs

Abstract Let W be a real reflection group, and let L W denote the lattice consisting of all possible intersections of reflecting hyperplanes of reflections in W . Let p W ( t ) be the characteristic polynomial of L W . To every element X of L W there corresponds a parabolic subgroup of W denoted by Gal( X ). If W is irreducible, we show that an element X of L W is modular if and only if p Gal( X ) ( t ) divides p W ( t ). This characterization is not true if W is not irreducible. Also, we show that if W is neither A n nor B n , then the only modular elements are 0 , 1 and the atoms of L W .

On some submodules of the action of the symmetric group on the free Lie algebra

AbstractWe describe a straightening algorithm for the action of Sn on a certain graded ring