Title:

Variations on a theme of Solomon

The main aim of this thesis is to examine the structure of the Hecke algebra HK(G,B) of a finite group G with a split (B,N) pair of rank n and characteristic p, with Weyl group W, over a field K of characteristic p. We then see how this relates to the structure of the KGmodule L induced from the principal KBmodule. Chapters 1, 2 and 3 are introductory. Chapter 1 deals with the properties of finite Coxeter groups, and gives L. Solomon's decomposition of the group algebra of a finite Coxeter group over the field of rational numbers. In Chapter 2 we discuss finite groups with split (B,N) pairs and their irreducible modular representations. Chapter 3 deals with Hecke algebras and the generic ring and some of its specialisations. In Chapter 4, we examine the structure of the OHecke algebra H of type (W,R) over any field, which is defined in Chapter 3; the algebra HK(G,B) is an example of one of these. H has 2n distinct irreducible representations, where n = [R], all of which are onedimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2n indecomposable left (or right) ideals, in a similar manner to the Solomon decomposition of the underlying Weyl group W. In Chapter 5, we also obtain decompositions of the generic ring similar to Solomon's decomposition of the underlying Coxeter group. These decompositions carry over to some specialisations of the generic ring; in particular, we get Solomon's decomposition of the Coxeter group and decompositions of the Hecke algebra of a finite group with (B,N) pair over a field of characteristic zero. Certain homology modules which arise from the Tits building of a finite group with a (B,N) pair, called relative Steinberg modules because of the character of the group they afford, are discussed in Chapter 6. Finally, in Chapter 7, we see that L is a direct sum of 2n indecomposable KGmodules, each of which is a relative Steinberg module. We can deduce what [W] of the composition factors of L are, but there are in general others.
