# What is the centralizer of a Coxeter element?

Let $$(W,S)$$ be a Coxeter system (of finite rank) and $$c \in W$$ a Coxeter element.

If $$W$$ is finite, then the centralizer $$C_W(c)$$ is the cyclic group generated by $$c$$ (e.g. see the book "Reflection Groups and Invariant Theory" of Kane).

If $$(W,S)$$ is irreducible and not finite or affine, then the cyclic group $$\langle c \rangle$$ is of finite index in $$C_W(c)$$ (see Theorem 2.6 in the paper "Irreducible Coxeter Groups" by Luis Paris).

Are there more results in this direction? For affine Coxeter groups? Is it conjectured that $$C_W(c) = \langle c \rangle$$ holds in general? Or is there a counterexample?

Update: There is a paper called "On the centralizer of a Coxeter element" by Blokhina (seems like that he/ she was a student of Vinberg), originally in russian, but there is an english translation in Moscow Univ. Math. Bull. You find an entry here on mathscinet.

I just managed to get a copy of that paper (but because of the copyright I'm not sure if I'm allowed to make it available here). The following statement is proven in that paper (including all infinite affine families David is mentioning in his comment):

Let $$(W,S)$$ be a Coxeter system of finite rank and $$c \in W$$ a Coxeter element. If $$(W,S)$$ is of type $$\widetilde{A}_n$$ or if the Coxeter graph of $$(W,S)$$ is a tree and all of its labels are 3 (that is, it is simply-laced), then $$C_W(c)= \langle c \rangle$$.

(The arguments in the proof are somewhat similar to the proof of the statement for finite Coxeter groups given by Carter. Jim mentioned this reference implicitely in his comment. I add it here: It is Proposition 30 in Carter's paper "Conjugacy Classes in the Weyl group".)

• Have you tried to exhibit Coxeter elements in explicit affine Coxeter groups, such as type $\widetilde{A_n}$?
– YCor
Feb 14 '19 at 17:33
• Do you mean "Irreducible Coxeter Groups" by Luis Paris ? That has a relevant Theorem 2.6. Feb 14 '19 at 19:37
• Yes, thanks! The author is L. Paris (I edited it in my question). Feb 14 '19 at 22:52
• Other references for the case of a finite Coxeter group are given in my unpublished notes people.math.umass.edu/~jeh/pub/count.pdf, but I'm unsure about infinite Coxeter groups. Feb 14 '19 at 23:19
• I have a case by case check that $C_W(c) = \langle c \rangle$ in types $\tilde{A}$, $\tilde{B}$, $\tilde{C}$ and $\tilde{D}$, which I'll post if nobody posts something better. (Sorry for deleting then reposting this comment; I thought I found an error in my type $\tilde{D}$ work, but it was right the first time.) Feb 15 '19 at 2:00