Last update: 10 January 2014
Let be an dimensional complex vector space and let denote the ring of polynomial functions on If then acts naturally on and the elements of invariant under form a subring of In [STo1954] it is shown that is a finite group generated by reflections if and only if is generated by algebraically independent homogeneous polynomials Let denote the degree of The integers are uniquely determined by and the numbers are called the exponents of group From [STo1954] we have and that the number of reflections in is In [STo1954], Shephard and Todd further show that if may be generated by reflections then one may choose generating reflections so that the eigenvalues of the product are where and is the order of In their argument there is no general algorithm for choosing such generators. However, if for some connected then there is a natural choice for such generators.
Theorem 6. Let be connected. As usual we let be the generators for and we let Put and let be any permutation of Then
|(i)||The conjugacy class of does not depend on|
|(ii)||The eigenvalues of are where and is the order of|
By Theorem 2 is a tree and thus (i) is given in [Bou1968] (Lemma 1, Chapter V, section 6). For a graph with just one vertex labelled with the integer we see from [STo1954] that and thus (ii) and (iii) are obvious. Now if satisfies so is a Coxeter group, (ii) and (iii) are given by a general argument in [Bou1968] (Proposition 3, Chapter V, section 6). If say is Coxeter observes [Cox1962] that and He also obtains and as the eigenvalues of and thus we have and as the eigenvalues of
We now consult List 1 of Theorem 2 to determine which graphs still remain and we will verify (ii) and (iii) for these groups in a case by case manner.
Suppose is By comparing with we see that Now the matrix for in the basis is where and
The characteristic equation is thus which has roots where
For the graph we find from and that the degrees are We compute that the matrix for in the basis is where and Thus the characteristic equation is and the eigenvalues of are where
If is we find again from and that This time the matrix for in the basis is where and The characteristic equation is thus which has roots where
We finally consider the infinite family of graphs Here, comparing and yields that the degrees are given by Letting we define a new basis by Then Further if interchanges and and fixes every other Hence the matrix for in this basis is The characteristic equation is thus Hence and letting the roots are
This completes the verification.
Let be connected. Put let denote the order of and let We define the integers and by requiring that are the eigenvalues for and are the eigenvalues for Thus, using Theorem 6, In order to give a general argument for Theorem 6 along the same lines as that given in [Bou1968] for the case where is a Coxeter group one needs to know three things:
|(b)||There is an eigenvector corresponding to the eigenvalue of which does not lie in the reflecting hyperplane of any reflection in|
|(c)||The number of reflections in is|
The quantity arises also in another setting. We have On the other hand, Thus, Similarly, by computing in two ways we obtain In case is a Coxeter group, all is conjugate to and both (7) and (8) become These congruences are in fact equalities [Bou1968, p.118] and this leads one to suspect that (7) and (8) are also equalities. Since equality holds in (7) if and only if it holds in (8).
Theorem 7. Let be connected. With and defined as above we have
We may assume the are not all If the statement is obvious. If is From [Cox1962] we have that and Thus we have If we have just computed the numbers and in the verification of Theorem 6. It thus becomes a trivial arithmetic task to verify the desired result for the remaining cases.
Corollary 7. Let be connected. With defined as above we have that the number of reflections in is
Since and is the number of reflections in this follows immediately from Theorem 7.
Coxeter [Cox1974, p.153] credits McMullen with the observation that for a connected graph the number of reflecting hyperplanes in corresponding to reflections in is Now McMullen's observation does not seem to be a consequence of Corollary 7 nor does the corollary appear to follow from the observation. However, taken together we get the marvelous fact
Corollary 8. Let be connected. If is defined as above then the number of reflecting hyperplanes in plus the number of reflections in is
Let and let denote the set of vertices of For we denote by the subgraph of obtained by deleting from all those vertices in and the edges connected to those vertices. Also we put Here we agree that if then Finally we write for
Proposition 10. Let be connected. Put and let denote the vertex set of Put where are the exponents of Then
If is a Coxeter group, and thus (10) occurs in [Sol1966, p.378] where Solomon credits it to Witt.
If has just one vertex labelled with the integer we have and so (10) is trivial.
Suppose next that has two vertices; say is Letting denote the order of we know from Theorem 6 that and thus from Theorem 7 that Using this together with the previously mentioned fact that we compute
Now ignoring the infinite family for a moment and using the table in [Cox1967] in conjunction with that in [STo1954] the verification of (10) for the other three non-Coxeter groups is a trivial arithmetic task.
Finally suppose is the graph Consulting [STo1954] and [Cox1967] we see that and We number the vertices from left to right and denote the vertex by We must show that Let be the set of all subsets of the vertex set Write a disjoint union where we put a subset in if and Let for and put If then where and thus (Here of course
Hence But is the symmetric group on letters, a Coxeter group, so (10) implies Thus
In light of the character formula which is proved in [Sol1966] (Theorem 2, p. 379) one is tempted to view (10) as the result of evaluating an equation involving an alternating sum of induced characters at the identity.
Let be connected and put For we denote by the multiplicity of 1 as an eigenvalue of and we put
Define by where is the character of induced from the principal character of We offer the following
Conjecture: Let Then (Here the smallest exponent of
Note that if is a Coxeter group, The non real eigenvalues of occur in conjugate pairs and the real eigenvalues are Thus and hence, in this case, our conjecture is just Theorem 2 of [Sol1966] with
If the graph has only one vertex (11) is obvious.
If has just two vertices we will give an argument verifying (11) but we need to do some preliminary work first.
Lemma 4. Suppose is a linear group. Let be the subgroup of all those scalar matrices in Put and denote the natural map by for Define Then
|(a)||If satisfies then|
|(b)||If consists of matrices, and satisfies then|
For both (a) and (b) it is clear that the right side is included in the left. So suppose Hence some Taking traces on both sides of this equation we have so implies Thus and yielding (a). In the situation of (b) we take determinants on both sides of the equation to obtain Thus so and
Corollary 9. Suppose is connected; say is As usual we denote the generators for by and let with Then
Suppose Since is irreducible consists of scalar matrices. Put Since is a reflection has order From [STo1954] we know that is the alternating group on four letters, the symmetric group on four letters, or the alternating group on five letters, and any non identity element of order different from two in any of these groups is self centralizing. Hence Now further forces and applying Lemma 4(a) we have Thus
Now suppose Consulting List 1 of Theorem 2 we see must be one of From [STo1954] we have so that Hence is a reflection in It follows immediately from Theorem 5 that is conjugate to for all of the above graphs with the possible exception of where we must rule out the possibility that is conjugate to
In the group has order 24 and is generated by an element of order 8. Hence If is a linear character of and and thus Thus the commutator subgroup of So, and have the same images in but a glance at the proof of Lemma 3 reveals that and have distinct images in Hence, is not conjugate to
So using the notation of Lemma 4 we have In fact, we have In light of Lemma 4(b) we see that to show it suffices to show If is the alternating group on tour or five letters (12) is obvaous. So we assume is the symmetric group on four letters. Consulting [STo1954] we see that is or If is a transposition (12) is again obvious.
So assume is a product of two disjoint transpositions. Thus is an element of the alternating group on four letters. But if is then an element of order three, is also in the alternating group and hence so is a contradiction. If is then as mentioned before has order three and hence is in the alternating group. This again is impossible since has order four and does not lie in the alternating group.
Using Corollary 9 we can now proceed to verify our conjecture (11).
If and is not a reflection, then is not conjugate to any element of either or and so Hence
If is a reflection, then by Theorem 5, is conjugate to an element of or an element of By looking at the matrices one easily sees that if then and thus by Corollary 9, Further from [Cox1974] we know that and that We next separate two cases.
Case (i) is even. Without loss of generality we assume is to an element of By Corollary 6 is not conjugate to an element of Hence Since have distinct non identity eigenvalues the conjugacy classes determined by are all distinct. Hence So
Case (ii) is odd. Thus, say and is conjugate to so is conjugate to an element of and to an element of Hence So again, This completes the verification of our conjecture in case
Let Put and let denote the vertex set of Let and put As usual we let where are the exponents of In [STo1954] Shephard and Todd introduced and verified and later in [Sol1963] Solomon gave a proof for the polynomial identity:
For let be the subspace of spanned by the basis vectors corresponding to the vertices in Also let be the exponents of acting on Further defining on as we defined on and putting one easily sees that if then Finally define a class function by Notice that the left hand side of (13) can be written as
Consider the sum In the last step we have used (13) applied to the groups
Next consider the sum Here, in the next to the last step we have used (13).
So, if our conjecture (11) is true, we have a new polynomial identity: In case is a Coxeter group, and (14) becomes Corollary 2.4 of [Sol1966]. We have verified (14) in a case by case manner for all the non Coxeter groups associated with connected graphs in with the exception of the infinite family This lends another measure of credibility to our conjecture (11).
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.