A decomposition of the group algebra of a finite Coxeter group

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14]...

Longest element of a finite Coxeter group

The biHecke monoid of a finite Coxeter group

For any finite Coxeter group W , we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W . The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H0(W ), that is, for the symmetric group, the algebra (o...

On the planarity of a graph related to the join of subgroups of a finite group

‎Let $G$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎We define an undirected simple graph $Delta(G)$ whose‎ ‎vertices are the proper subgroups of $G$, which are not contained in the‎ ‎Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge‎ ‎if and only if $G=langle H‎ , ‎Krangle$‎. ‎In this paper we classify finite groups with planar graph‎. ‎...

The Hecke Group Algebra of a Coxeter Group and Its Representation Theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving op...