Generalization of cycle decomposition to Coxeter groups

Question

I'm looking for a generalization of cycle decompositions for permutations to elements of Coxeter groups.

(For the purposes of this question, any conjugate of a parabolic subgroup is also a parabolic subgroup.)

One can think of an $r$-cycle in $S_n$ as a (conjugate of a) Coxeter element for some parabolic subgroup $S_r$. This means one can think of the cycle decomposition as finding a parabolic subgroup - or equivalently a decomposition of the root space as a direct sum of root subspaces - and realizing your permutation as a Coxeter element for that parabolic subgroup.

In the Coxeter group of type B, one can realize elements as signed permutations and take a signed cycle decomposition; the cycles with an odd number of signed elements are Coxeter elements in a parabolic subgroup of type B and the cycles with an even number of signed elements are Coxeter elements in a parabolic subgroup of type A.

As far as I can tell, this doesn't seem to work so well for type D. Is there some other way to recover this idea, ideally for all Coxeter groups, but I'd be happy with finite reflection groups or even just type D?

If it helps, the property of Coxeter elements I'm interested in is that they have maximal reflection length. Uniqueness of decomposition would be nice but might not be necessary.

Answer 1

Following Nathan's advice let me elaborate a bit on my comment and also provide an answer.

1. As pointed out it is not true in general that any element $w$ in a finite Coxeter group is a Coxeter element in some reflection subgroup (see the counterexample above).

2. Actually I have been thinking recently about this question of generalizing the cycle decomposition to arbitrary finite Coxeter groups and here is how I did it. I am stating the result here and briefly explaining. The details and proofs can be found on a preprint which I just put on my webpage http://www.mathematik.uni-kl.de/fileadmin/AGs/agag/gobet/ex/cycle.pdf (it will appear on arxiv soon but there is some additional stuff which I have not written yet which will be added).

Any element $w$ in a finite Coxeter group $(W,S)$ has a generalized cycle decomposition in a suitable sense, but unicity fails in general. It works as follows.

Let $(W,S)$ be finite and let $w\in W$. Denote by $T$ the set of reflections of $W$, that is, the set of all conjugates of the elements of $S$, and by $\ell_T$ the length with repect to the generating set $T$ and by $\leq_T$ the absolute order on $W$ induced by $\ell_T$. Write $\mathrm{Red}_T(w)$ for the set of reduced $T$-decompositions of $w$ (i.e., smallest length factorizations of $w$ as products of reflections). I denote by $P(w)$ the parabolic closure of $w$, that is, the smallest parabolic subgroup containing $w$. I am considering the following condition on $w$:

(Condition A) There exists $(t_1, \dots, t_k)\in\mathrm{Red}_T(w)$ such that the reflection subgroup $\langle t_1, t_2, \dots, t_k\rangle$ is parabolic.

Then I claim the following

Theorem [Generalized cycle decomposition] Let $w\in W$ satisfying Condition A. Then there exist $x_1,x_2,\dots, x_m\in W$ such that

1. $x_i x_j= x_j x_i$ for all $i, j=1,\dots, m$,

2. $w=x_1 x_2\cdots x_m$ and $\ell_T(w)=\ell_T(x_1)+\ell_T(x_2)+\cdots+\ell_T(x_m)$,

3. Each $x_i$ admits a reduced $T$-decomposition generating an irreducible parabolic subgroup $P_i$ (in fact, $P(x_i)$) of $W$ and $$P(w)=P_1\times P_2\times\cdots\times P_m=P(x_1)\times P(x_2)\times\cdots\times P(x_m).$$

Moreover, such a decomposition of $w$ is unique up to the order of the factors.

In the case of the symmetric group of course the $x_i$'s are just the cycles occuring in the cycle decomposition of $w$. Moreover in type $A$ every element satisfies Condition $A$.

Now let me say a few words about elements for which Condition A fails. In fact, elements satisfying Condition A above are the so-called "(parabolic) quasi-Coxeter elements", which we characterized together with Baumeister, Roberts and Wegener as the elements $w$ for which the Hurwitz operation is transitive on $\mathrm{Red}_T(w)$ in this paper http://www.degruyter.com/view/j/jgth.ahead-of-print/jgth-2016-0025/jgth-2016-0025.xml which Christian pointed out. Now if $w$ does not satisfy Condition A, it precisely means that the Hurwitz operation on $\mathrm{Red}_T(w)$ has several orbits. Taking a reduced expression of $w$ in one of the orbits, $w$ is a quasi-Coxeter element in the reflection subgroup generated by the reflections in this reduced expression (which, as any reflection subgroup, is a Coxeter group), hence $w$ has a generalized cycle decomposition in that subgroup. In this situation $w$ will have as many generalized cycle decompositions as the number of Hurwitz orbits on $\mathrm{Red}_T(w)$.

So in fact, somewhat surprinsingly, any (parabolic) quasi-Coxeter element (as the element in $D_4$ above which yielded a counterexample to Nathan's claim) still has a generalized cycle decomposition (I mean: including unicity). The obstruction to unicity comes from elements where no $T$-reduced expression generates a parabolic subgroup, such as for instance the longest element in type $B_2$.

Finally, note that this heavily requires properties which only hold in finite Coxeter groups (such as the characterization of parabolic subgroups as centralizers of subspaces), hence I have no idea about possible generalizations to arbitary Coxeter groups... It would be very interesting to be able to characterize parabolic subgroups as centralizers of subspaces in a suitable faithful representation of $W$ in the infinite case...

EDIT: I realized that Brady and Watt is not needed for unicity - unicity is already clear from the direct product decomposition.

Answer 2

I think the most general thing you can say is the following: Any element $w$ is the Coxeter element of a reflection subgroup (a subgroup generated by reflections). To show this, write a "reduced $T$-word" $t_1t_2\cdots t_k$ for $w$ (a shortest possible word for $w$ as a product of reflections). Note that the usual notion of a reduced word is different, as it uses only simple reflections. One can show (for example, I think this is in Drew Armstrong's AMS Memoir) that $w$ is a Coxeter element in the reflection subgroup generated by the $t_i$.

UPDATE: This answer is wrong. For now, I'm leaving it here so that the comments---which point out why it is wrong---will make sense, and so everyone will know to down-vote it :).

I had gotten concerned when I couldn't actually find this in Armstrong and was looking for a proof. Thomas' counterexample below puts that effort to rest.