Last update: 4 January 2014
The classification of the finite real reflection groups involves the study of group presentations in which there are a finite number of generators satisfying relations of the form with for A comprehensive treatement of such groups, known as Coxeter groups after H.S.M. Coxeter, occurs in [Bou1968]. In 1953 Shephard and Todd classified all finite complex reflection groups [STo1954]. Then in 1966 Coxeter gave presentations for all the dimensional finite complex reflection groups generated by reflections [Cox1967]. Most of these presentations can be described as follows. There are a finite number of generators, and a symmetric matrix, in which each entry is a positive integer We denote the diagonal entries of by This matrix defines the relations in our group presentation. For each diagonal entry there is a relation and for each off diagonal entry there is a relation with factors on each side of the equal sign. Notice that if then the relation with symbols on each side of the equal sign is equivalent to the relation Thus we see that the group presentations encountered in the study of real reflection groups are present among those just described.
In [Cox1967] Coxeter introduced a graphical notation for this type of group presentation. If the presentation has generators so the matrix is the graph has vertices. Corresponding to the diagonal entry of the number is written under the vertex except when and then the vertex remains unlabeled. For each off diagonal entry that is greater than an edge is drawn between the and vertices and this edge is labeled with the number except when and then the edge remains unlabeled. If so and commute, no edge is drawn between the and vertices. For example, the graph indicates the group presentation which is given by the matrix Using the symbol to denote such a graph we let denote the corresponding abstract group.
We remark here that if is odd, then the relation can be viewed as forcing to be conjugate to and hence these two generators must have the same order. Looking ahead to Theorem 1 and its Corollary we see that in this case we must require for without this restriction Theorem 1 is false. We thus denote by the collection of all symmetric matrices such that each entry is a positive integer and if an off diagonal entry is odd then the diagonal entries and are the same. For we let denote the graph of the group presentation given by and we let be the collection of all those graphs associated with the matrices in Finally we denote by the subclass of all those graphs in with vertices. We will be interested in studying the groups for
The first result obtained for the groups is stated precisely in Theorem 1 and can be summarized as follows: Let and let be an dimensional complex vector space. Then there is a homomorphism and a Hermitian form on which is invariant under the linear group Further is a reflection of order for all
We begin Section II by showing that for the group admits an antiautomorphism, denoted which fixes the generators Thus defining by we see that is another representation of Now if is a linear graph (see definition on page 15) Coxeter gives a representation of [Cox1967] and we point out that Coxeter's representation is the defined above. If is such that for all (so is a Coxeter group) then the form and the representation of Theorem 1 are precisely the form and representation given in [Bou1968].
For a connected graph we show in Proposition 6 that a necessary condition for the finiteness of is that be positive definite and a large portion of Section II is devoted to the classification of all those connected graphs with positive definite. This result given in Theorem 2. Then a comparison with those graphs listed by Coxeter [Cox1967] yields that the positive definiteness of is also a sufficient condition for to be finite. This is stated in Theorem 4. For the final result of Section II we use the relationship between the representation of Theorem 1 and the representations and to obtain the result that if is finite, then is a faithful representation. We remark that this is false if the hypothesis of the finiteness of is deleted.
In Section III we use a result of Springer [Spr1974] to show that if and is finite then every reflection in is conjugate in to a power of one of the generating reflections For Coxeter groups this is a known result [Bou1968].
In Section IV we begin by verifying that for a connected graph with finite one can obtain a certain set of integers, called the exponents of (see page 47 or [STo1954]) from knowledge of the eigenvalues of the inverse of the group element Then letting denote the order of the element we show that the number of reflections in is This result in combination with an observation of McMullen produces the fact that the number of reflections in plus the number of reflecting hyperplanes in is equal to These results are stated precisely in Theorems 6 and 7 and in Corollaries 7 and 8.
For connected with finite we let denote the smallest exponent of and we let be the set of vertices of For we let be the subgraph of obtained by deleting from all those vertices in and all the edges connected to those vertices. We put We end section IV by verifying that We generalize this arithmetic formula to a conjecture about characters induced from the principal characters and we give some evidence supporting the conjecture.
This is a typed version of David W. Koster's thesis Complex Reflection Groups.
This thesis was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) at the University of Wisconsin - Madison, 1975.