The descent algebras of Coxeter groups
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A descent algebra is a subalgebra of the group algebra of a Coxeter group. They were first defined over a field of characteristic zero. In this thesis, the main areas of research to be addressed are; 1. The formulation of a rule for multiplying two elements of descent algebra of the Coxeter groups of type D. 2. The identification of properties exhibited by descent algebras over a field of prime characteristic. In addressing the first, a framework which exploits the specific properties of Coxeter groups is set up. With this framework, a new justification is given for existing rules for multiplying together two elements in the descent algebras of the Coxeter groups of type A and B. This framework is then used to derive a new multiplication rule for the descent algebra of the Coxeter groups of type D. To address the second, a descent algebra over a field of prime characteristic, p, is defined. A homomorphism into the algebra of generalised p-modular characters is then described. This homomorphism is then used to obtain the radical, and allows the irreducible modules of the descent algebra to be determined. Results from the two areas addressed are then exploited to give an explicit description of the radical of the descent algebra of the symmetric groups, over a finite field. In this instance, the nilpotency index of the radical and the irreducible representations are also described. Similarly, the descent algebra of the hyper-octahedral groups, over a finite field, has its radical, nilpotency index, and irreducible representations explicitly determined.
Thesis, PhD Doctor of Philosophy
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