# Eigenvalues for elements of (infinite) Coxeter groups

My current research requires some knowledge on the eigenvectors of elements (of infinite order) of Coxeter groups view as reflections in their geometric representation.

After some reading, my impression is that many has been done for the spectrum of "Coxeter elements" or "Coxeter transformations", which are (if I understand correctly) the product of a permutation of the generators. However I find few result on the spectrum of other elements (of infinite order).

It's possible that this impression is wrong and I missed something.

Question: Did I miss any reference? If not, why didn't "non-Coxeter elements" interest people? Is the eigenvalues too obvious to study? or is it too complicated?

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The eigenvalues of elements of infinite order are certainly not trivial to study, and as far as I know little has been determined about them. Keep in mind that arbitrary infinite Coxeter groups are quite varied and hard to study systematically beyond the most basic theory.

Maybe I can clarify at least what you've summarized about the special case of Coxeter elements. These were originally studied by Coxeter for finite reflection groups, where the eigenvalues have remarkable properties. But the definition makes sense for any Coxeter group having a finite set $S$ of involutions as standard generators. You just take the product of elements of $S$ is any order to get a "Coxeter element". The problem is that these are not usually all conjugate, unless the Coxeter graph is a forest (union of trees, where there are no cycles).

Most of the affine Weyl groups (irreducible by definition) have trees as graphs, but then it's still tricky to study the eigenvalues of a Coxeter element. The only significant work I'm aware of in cases where the Coxeter elements are all conjugate was done by N. A'Campo: Sur les valeurs propres de la transformation de Coxeter. Invent. Math. 33 (1976), no. 1, 61–67. (This should be available online at http://gdz.sub.uni-goettingen.de/gdz/.)

The main result A'Campo obtains in this note for Coxeter elements outside the affine Weyl group case is that all real parts of square roots of eigenvalues are $>1$.

Probably you haven't missed any important literature (though I'm not certain). The question you raise is definitely nontrivial in any case and might be best approached in narrowly defined cases.

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Thank you very much for your answer (and also for your book of course :P). A'Campo's note is among the documents on my desk. Seems that I didn't miss anything. Then I think I'll start to look into it. For your information, the Coxeter groups that I'm particularly interested in are those of type (n-1,1) ("lorentzian" or "hyperbolic" in literature). We may have observed something interesting about their eigenvectors, but nothing is clear yet. – Hao Chen Jun 21 '13 at 20:07
@Hao: A word of caution. In a "hyperbolic" situation, the standard geometric realization of the Coxeter group might not be the best representation to use. This kind of issue comes up, for instance, in a recent arXiv preprint: front.math.ucdavis.edu/1305.0052 (and it's always a concern when discussing infinite Coxeter groups that come up in geometry). – Jim Humphreys Jun 21 '13 at 21:13
@Humphreys: This happens to be one of the papers that leads us to our current focus. Thanks. – Hao Chen Jun 21 '13 at 21:40
Another good link on the Coxeter element subject is Howlett, "Coxeter groups and M-matrices", ams.org/mathscinet-getitem?mr=647197 . I am also unaware of any good references for elements of the group other than Coxeter elements. – David Speyer Jun 22 '13 at 0:42