Robert B. Howlett

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], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/WJ, where the WJ are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.

On groups of central type

A finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements. (Equivalently, the degree of this character is the square root of the index of the center.) In 1964, Iwahori and Matsumoto [15] conjectured that a group of central type must be solvable. The paper of Liebler and Yellen [16] aims to prove this, but there is a gap in their proof (as explained below). Nevertheless, they do correctly obtain substantial information about a minimal counterexample to the conjecture. The aim of this paper is to fill the gap in the Liebler-Yellen proof (using as they do, the classification of finite simple groups) and to provide further information about which solvable groups can have central type. We show, for instance, that every normal subgroup of the central factor group of a (solvable) group of central type has the property that its system normalizers have square index. It is interesting to contrast this restrictive condition with the result of Gagola that every solvable group is embeddable in the central factor group of a group of central type. (See [6, Theorem 1.2]). The arguments in [16] show that a minimal counterexample G to the Iwahori -Matsumoto conjecture must have the following structure: If K/Z is any minimal normal subgroup of G/Z (where Z = Z ( G ) , the center), then K is abelian, K/Z is a 2-group, S = CG(K ) is the maximal solvable normal subgroup of G and any minimal normal subgroup HIS of G/S is the direct product of a number l of copies of GL(3, 2). Indeed, the Liebler-Yellen methods can be used to show that the Sylow 2-subgroup of K, considered as an (H/S)-module is a product o f / copies of a GL(3,2)-module of dimension 4, this module having a unique minimal submodule, which is trivial, and an irreducible factor module of dimension 3. The minimal submodules are amalgamated in the product.

On outer automorphism groups of coxeter groups

SummaryIt is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ1 andΠ2 any two bases of the root system ofW, thenΠ2 = ±ωΠ1 for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett.

Writing representations over minimal fields

The chief aim of this paper is to describe a procedure which, given a

On the degrees of Steinberg characters of Chevalley groups

d

Extraspecial towers and weil representations

-dimensional absolutely irreducible matrix representation of a finite group over a finite field

On reflection length in reflection groups

\mathbb{E}

Embeddings of Hecke algebras in group algebras

, produces an equivalent representation such that all matrix entries lie in a subfield

On the first cohomology of a group with coefficients in a simple module

\mathbb{F}

On the laws of certain varieties of groups

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