ON CYCLE DECOMPOSITIONS IN COXETER GROUPS

arXiv:1611.03442v1 [] 10 Nov 2016

THOMAS GOBET
Abstract. The aim of this note is to show that the cycle decomposition
of elements of the symmetric group admits a quite natural formulation
in the framework of dual Coxeter theory, allowing a generalization of
it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasiCoxeter element). We show that such an element admits an analogue
of the cycle decomposition. Elements which are not in this family still
admit a generalized cycle decomposition, but it is not unique in general.

1. Introduction
The cycle decomposition in the symmetric group is a powerful combinatorial tool to study properties of permutations. On the other hand, the
symmetric groups can be realized as Coxeter groups. It is easy for example
to determine the order of an element from its cycle decomposition, hence
even if we prefer to view the symmetric groups as Coxeter groups it is sometimes useful to represent their elements as permutations and make use of
their unique cycle decomposition, rather than using Coxeter theoretic representations of the elements as words in the simple generating set.
It therefore appears as natural to wonder whether the cycle decomposition
admits a natural generalization to Coxeter groups. However, when trying
to define cycle decompositions in the symmetric group purely in terms of
the classical Coxeter theoretic data, one rapidly sees an obstruction towards
such a generalization: considering a Coxeter system (W, S) of type An (with
W identified with Sn+1 and S with the set of simple transpositions) and
an element w ∈ W with cycle decomposition w = c1 c2 · · · ck , the Coxeter
length ℓS (w) of w is not equal in general to the sums of the lengths of the
various ci .
However, replacing the generating simple set S by the set T of all transpositions and the classical length by the length function ℓT on W with respect
P
to T , one has that ℓT (w) = ki=1 ℓT (ci ). The set of transpositions forms a
single conjugacy class. From a Coxeter theoretic point of view, it is the set
of reflections of W , i.e., the set of W -conjugates of the elements of S. In
particular the reflection length function ℓT can be defined for an abritrary
Coxeter group. There are deep motivations for the study of a (finite) reflection group as a group generated by the set T of all its reflections instead of
just the set S of reflections through the walls of a chamber. This approach,
nowadays called the dual approach, has been a very active field of research
in the last fifteen years (see for instance [4], [8], [1], [12], [13]).
The author is funded by the ANR Geolie ANR-15-CE40-0012.
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The above basic observation on the reflection length of a permutation indicates that the cycle decomposition has something which is dual in essence.
Each cycle can be thought of as a Coxeter element in an irreducible parabolic
subgroup of W . From the type An picture, it therefore appears as natural
to generalize a cycle decomposition as a decomposition of an element w of
Coxeter system (W, S) into a product of Coxeter elements in irreducible reflection subgroups of W , which pairwise commute, and such that the sum of
their reflection lengths equals the reflection length of w. However, even for
finite W there are in general elements failing to admit such a decomposition.
In order to make it work, one has to relax the definition of Coxeter element
to that of a quasi-Coxeter element (in type An both are equivalent). Namely,
given w ∈ W and denoting by RedT (w) the set of T -reduced expressions of
w, that is, minimal length expressions of w as product of reflections, we say
that w ∈ W is a parabolic quasi-Coxeter element if it satisfies the following
condition:
Condition 1.1. There exists (t1 , t2 , . . . , tk ) ∈ RedT (w) such that W ′ :=
ht1 , t2 , . . . , tk i is a parabolic subgroup of W .
If the parabolic subgroup is the whole group W then we just call w a quasiCoxeter element. In type Dn for instance, there are quasi-Coxeter elements
which fail to be Coxeter elements (that is, with no T -reduced expression
yielding a simple system for W ; see [3]).
The parabolic subgroup in the above Condition is unique in the sense
that if another reduced expression (q1 , q2 , . . . , qk ) ∈ RedT (w) generates a
parabolic subgroup W ′′ of W , then W ′ = W ′′ (see Lemma 2.6). We therefore
denote W ′ by P (w). In this situation it is easy to derive
Proposition 1.2 (Generalized cycle decomposition). Let (W, S) be a Coxeter system. Let w ∈ W satisfying Condition 1.1. There exists a (unique
up to the order of the factors) decomposition w = x1 x2 · · · xm , xi ∈ W such
that
(1) xi xj = xj xi for all i, j = 1, . . . , m,
(2) ℓT (w) = ℓT (x1 ) + ℓT (x2 ) + · · · + ℓT (xm ),
(3) Each xi admits a T -reduced expression generating an irreducible parabolic subgroup Wi of W and
P (w) = W1 × W2 × · · · × Wm .
This statement is not entirely satisfying in the sense that we would like to
state the maximality condition given in point (3) only in terms of the factors
xi and not in terms of the parabolic subgroups. More precisely, we expect
the xi ’s to be indecomposable, that is, to admit no nontrivial decomposition
of the form ui vi with ui vi = vi ui and ℓT (xi ) = ℓT (ui ) + ℓT (vi ). For finite
groups at least this can be achieved (see Proposition 3.5) yielding
Theorem 1.3 (Generalized cycle decomposition in finite Coxeter groups).
Let (W, S) be a finite Coxeter system. Let w ∈ W satisfying Condition 1.1.
There exists a (unique up to the order of the factors) decomposition w =
x1 x2 · · · xm , xi ∈ W such that
(1) xi xj = xj xi for all i, j = 1, . . . , m,
(2) ℓT (w) = ℓT (x1 ) + ℓT (x2 ) + · · · + ℓT (xm ),

ON CYCLE DECOMPOSITIONS IN COXETER GROUPS

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(3) Each xi is indecomposable.
Theorem 1.3 is the analogue of the cycle decomposition for parabolic
quasi-Coxeter elements. In general there are elements in W failing to be
parabolic quasi-Coxeter elements, but the advantage of this definition is that
such an element is always a quasi-Coxeter element in a reflection subgroup
(but this subgroup is never unique when W is finite: in that case by Corollary
3.11 below these reflection subgroups are in bijection with the number of
Hurwitz orbits on RedT (w); by [3, Theorem 1.1] this number is one precisely
when w satisfies Condition 1.1).
2. Coxeter groups and their parabolic subgroups
Let (W, S) be a (not necessarily finite) Coxeter system of rank n = |S|. We
assume the reader to be familiar with the general theory
groups
S of Coxeter
−1
and refer to [6] or [11] for basics on the topic. Let T = w∈W wSw be the
set of reflections of W . Let ℓT : W → Z≥0 be the reflection length, that is,
for w ∈ W the integer ℓT (w) is the smallest possible length of an expression
of w as product of reflections. We write ≤T for the absolute order on W ,
that is, for u, v ∈ W we set
u ≤T v ⇔ ℓT (u) + ℓT (u−1 v) = ℓT (v).
Given w ∈ W , we denote by RedT (w) the set of T -reduced expressions
of w, that is, the set of minimal length expressions for w as products of
reflections.
Definition 2.1. A subgroup W ′ ⊆ W is parabolic if it exists a subset
S ′ = {r1 , r2 , . . . , rn } ⊆ T and m ≤ n such that (W, S ′ ) is a Coxeter system
and W ′ = hr1 , r2 , . . . , rm i.
The above definition, which is borrowed from [2], is more general than the
usual definition of parabolic subgroups as conjugates of subgroups generated
by subsets of S. In [3, 4.4 and 4.6] it is shown that the above definition is
equivalent to the classical one for finite and irreducible 2-spherical Coxeter
groups. The example below shows that the two definitions are not equivalent
in general:
Example 2.2. Let W be a universal Coxeter group on three generators
S = {s, t, u}, that is, with no relation between distinct generators. Then
S ′ := {s, t, tut} ⊆ T is a simple system for W , hence X := hs, tuti is
parabolic. However, using the fact that elements of W have a unique Sreduced expression (because W is universal) it is easy to check that X is not
conjugate to any of the three rank 2 standard parabolic subgroups of W .
Parabolic subgroups provide a family of reflection subgroups of W , that
is, subgroups generated by reflections. Any reflection subgroup W ′ ⊆ W
comes equipped with a canonical structure of Coxeter group (see [10]), in
particular it has a canonical set S ′ of Coxeter generators. Moreover by [10,
Corollary 3.11 (ii)] the set Ref(W ′ ) of W ′ -conjugates of S ′ (the reflections
of W ′ ) coincide with W ′ ∩ T . The rank rank(W ′ ) of W ′ is defined to be |S ′ |.
The following result will be useful:

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Theorem 2.3 ([2, Theorem 1.4]). Let W ′ ⊆ W be a parabolic subgroup.
Let w ∈ W ′ . Then RedT ′ (w) = RedT (w), where T ′ = W ′ ∩ T is the set of
reflections of W ′ .
In this context it seems natural to us to conjecture the following:
Conjecture 2.4. Let w ∈ W . Assume that there is (t1 , t2 , . . . , tk ) ∈ RedT (w)
such that W ′ := ht1 , t2 , . . . , tk i is parabolic. Then for any (q1 , q2 , . . . , qk ) ∈
RedT (w) we have W ′ = hq1 , q2 , . . . , qk i.
Note that
Theorem 2.5 ([3], [2]). Conjecture 2.4 holds in the following cases:
(1) When W is finite,
(2) When w is a parabolic Coxeter element in W , that is, if it exists
S ′ = {r1 , r2 , . . . , rn } ⊆ T and m ≤ n such that w = r1 r2 · · · rm and
S ′ is a simple system for W .
Proof. Conjecture 2.4 for finite W is an immediate consequence of [3, Theorem 1.1]: there it is shown that an element satisfies Condition 1.1 if and
only if the Hurwitz action (see Section 3.3 for the definition) is transitive on
RedT (w); but this action leaves the subgroup generated by the reflections
in a T -reduced expression invariant. It also holds for parabolic Coxeter elements since in that case the Hurwitz action is also transitive on RedT (w)
by [2, Theorem 1.3].

Lemma 2.6. Let w ∈ W . Assume that (t1 , t2 , · · · , tk ), (q1 , q2 , . . . , qk ) ∈
RedT (w) are such that both W ′ := ht1 , t2 , . . . , tk i and W ′′ := hq1 , q2 , . . . , qk i
are parabolic. Then W ′ = W ′′ .
Proof. Since W ′ is parabolic, by Theorem 2.3 we have qi ∈ W ′ for all i,
hence W ′′ ⊆ W ′ . Reversing the roles of W ′ and W ′′ we get W ′ ⊆ W ′′ . 
The lemma above allows the following definition
Definition 2.7. Let w ∈ W satisfying Condition 1.1. We denote by P (w)
the parabolic subgroup of W generated by any T -reduced decomposition of
w generating a parabolic subgroup. This is well-defined by Lemma 2.6.
It follows immediately from Theorem 2.3 that any parabolic subgroup P
containing w must contain P (w). We call P (w) the parabolic closure of w.
Lemma 2.8. Let W ′ ⊆ W be a finitely generated reflection subgroup. Then
there is a unique (up to the order of the factors) decomposition W ′ = W1 ×
W2 × · · · × Wk where W1 , W2 , . . . , Wk are irreducible reflection subgroups of
Ṡk
W ′ and Ref(W ) = i=1 Ref(Wi ).
Proof. Let S ′ be the canonical set of Coxeter
of W ′ (see [10]).
S generators
′
−1
′
= W ′ ∩ T . If
By [10, Corollary 3.11], S is finite and w∈W ′ wS w
Γ1 , Γ2 , . . . , Γk are the irreducible components of the Coxeter graph of (W ′ , S ′ ),
then W ′ = W1 × W2 × · · · × Wk (where Wi is generated by the nodes of Γi )
and each Wi is an irreducible reflection subgroup. Now if there is another
decomposition W = W1′ × W2′ × · · · × Wℓ′ , then all the reflections in Wi′ must
be included in Wj for some j (otherwise irreducibility is not satisfied) and
vice-versa, implying uniqueness of the decomposition.


ON CYCLE DECOMPOSITIONS IN COXETER GROUPS

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3. Generalized cycle decompositions
3.1. Proof of Proposition 1.2.
Proof of Proposition 1.2. Let (t1 , t2 , . . . , tk ) ∈ RedT (w) such that P := ht1 , t2 , . . . , tk i
is parabolic. By Lemma 2.6 we have P = P (w). It follows from the definition of a parabolic subgroup that there is a (unique up to the order of the
factors) factorization
P = W1 × W2 × · · · × Wm
where the Wi ’s are irreducible parabolic subgroups. Moreover we have k =
P
Ṡm
rank(P ) = m
i=1 Ref(Wi ). It follows that for
i=1 rank(Wi ) and Ref(P ) =
each j = 1, . . . , k, there exists j ′ ∈ {1, . . . , m} such that tj ∈ Wj ′ . This
implies that we can transform the T -reduced expression (t1 , t2 , . . . , tk ) by a
sequence of commutations of adjacent letters into a T -reduced expression
(q1 , . . . , qℓ1 , qℓ1 +1 , . . . , qℓ2 , . . . , qℓm−1 , . . . , qk ) of w where
{q1 , . . . , qℓ1 −1 } ⊆ W1 , . . . , {qℓm−1 , . . . , qk } ⊆ Wm .
Note that since the set {t1 , . . . , tk } generates P we must have
hq1 , . . . , qℓ1 −1 i = W1 , . . . , hqℓm−1 , . . . , qk i = Wm .
Setting ℓ0 = 1 and ℓm = k + 1 we define xi := qℓi−1 qℓi−1 +1 · · · qℓi −1 for
all i = 1, . . . , m. Note that since the Wi ’s are irreducible and parabolic
we get (3). As (q1 , . . . qk ) ∈ RedT (w) is given by concatenating T -reduced
expressions of the xi ’s we have ℓT (w) = ℓT (x1 )+ ℓT (x2 )+ · · · + ℓT (xm ) which
shows (2). Since xi ∈ Wi for all i and P = W1 × W2 × · · · × Wm we have
xi xj = xj xi for all i, j = 1, . . . , m, which shows (1).
It remains to show that the decomposition is unique up to the order of
the factors. Hence assume that w = y1 y2 · · · ym′ is another decomposition
of w satisfying the three conditions of Proposition 1.2. By the third condition each of the yi ’s has a reduced expression generating an irreducible
parabolic subgroup Wi′ = P (yi ) of W and concatenating them yields a reduced expression generating P (w). By uniqueness of the decomposition
P (w) = W1 × · · · × Wm we must have m = m′ and there must exist a permutation π ∈ Sm such that Wi′ = Wπ(i) for all i. Up to reordering, we can
therefore assume that Wi′ = Wi for all i. Since x = x1 x2 · · · xm = y1 y2 · · · ym
it follows by uniqueness of the decomposition of w as element of the direct
product W1 × W2 × · · · × Wm that xi = yi for all i = 1, . . . , m.

Remark 3.1. In type An we recover the classical cycle decomposition. In
that case each element w satisfies Condition 1.1. The second condition in
Proposition 1.2 follows from [7, Lemma 2.2].
3.2. Finite Coxeter groups and their parabolic subgroups. This section is devoted to recalling well-known facts on parabolic subgroups of finite
Coxeter groups and their connexion with finite root systems. Most of what
we present here is covered in [6], though often in different notations. From
now on we always assume (W, S) to be finite.
Let (W, S) be finite, of rank n. Let Φ be a root system for (W, S) in an
n-dimensional Euclidean space V with inner product (·, ·). Let Φ+ ⊆ Φ be

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a positive system. Recall that there is a one-to-one correspondence between
T and Φ+ which we denote by t 7→ αt . Let w ∈ W and let
V w := {v ∈ V | w(v) = v}
be the subspace of V consisting of the fixed points under the action of w.
The following well-known result is due to Carter [9, Lemma 3]
Lemma 3.2 (Carter’s Lemma). Let αt1 , αt2 , . . . , αtk ∈ Φ+ . Then {αt1 , αt2 , . . . , αtk }
is linearly independent if and only if ℓT (t1 t2 · · · tk ) = k. In that case one
has dim V t1 t2 ···tk = n − k and W ′ := ht1 , t2 , . . . , tk i is a reflection subgroup
of rank k of W .
The following will be useful (see [4, Lemma 1.2.1 (i)])
Lemma 3.3. Let x ∈ W , t ∈ T . Then t ≤T x ⇔ V x ⊆ V t .
Given w ∈ W , there is an orthogonal decomposition V = V w ⊕ Mov(w)
with respect to (·, ·), where Mov(w) := im(w−1) (see for instance [1, Section
2.4]).
Recall that for finite Coxeter groups, Definition 2.1 is equivalent to the
classical definition, that is, W ′ ⊆ W is parabolic if and only if W ′ is a
conjugate of a standard parabolic subgroup WI of W , where for I ⊆ S
we write WI := hs | s ∈ Ii. There is the following result, characterizing
parabolic subgroups of finite Coxeter groups as centralizers of subspaces of
V (which is a Corollary of [6, 3.3, Proposition 1]).
Proposition 3.4. Let P ⊆ W be a parabolic subgroup. Then
P = Fix(E) := {x ∈ W | x(v) = v, ∀v ∈ E}
for some subspace E ⊆ V . Conversely, given any subspace E ⊆ V , the
subgroup Fix(E) is a parabolic subgroup of W .
In fact, the subspaces E in Proposition 3.4 can be chosen to be intersections of reflection hyperplanes. Given linearly independent reflection hyperplanes Vt1 , . . . , Vtk where ti ∈ T, i = 1, . . . , k and setting w := t1 t2 · · · tk ,
T
it follows from Carter’s Lemma (Lemma 3.2) that V w = ki=1 Vti , and
dim(V w ) = n − ℓT (w) (see also [4, Lemma 1.2.1 (ii)]). It follows from
the discussion above that P (w) = Fix(V w ) and rank(P (w)) = ℓT (w). Every
parabolic subgroup is the parabolic closure of some (in general not uniquely
determined) element.
3.3. Hurwitz action and proof of Theorem 1.3. We now give a few
properties of elements of finite Coxeter groups satisfying Condition 1.1 before
proving Theorem 1.3. These elements were introduced in [3] and called
quasi-Coxeter elements.
Recall that for each w ∈ W with ℓT (w) = k, there is an action of the kstrand Artin braid group Bk on RedT (w) called the Hurwitz action, defined
as follows. The Artin generator σi ∈ Bk acts by
σi · (t1 , . . . , ti−1 , ti , ti+1 , ti+2 , . . . , tk ) = (t1 , . . . , ti−1 , ti ti+1 ti , ti , ti+2 , . . . , tk ).
In [3, Theorem 1.1], it is shown that this action is transitive if and only
if w satisfies Condition 1.1. Since the reflection subgroup generated by the
reflections from a reduced expression is invariant under a Hurwitz move

ON CYCLE DECOMPOSITIONS IN COXETER GROUPS

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as above, this means that either every T -reduced expression generates a
parabolic subgroup (which by Lemma 2.6 is nothing but P (w)) or no reduced
expression does.
With the following Proposition (which requires the above mentioned result from [3] for which we only have a case-by-case proof) it will be easy to
derive a proof of Theorem 1.3
Proposition 3.5. Let (W, S) be a finite irreducible Coxeter system. Let x
be a quasi-Coxeter element in W . Then there is no nontrivial decomposition
x = uv = vu such that u, v ∈ W and
ℓT (x) = ℓT (u) + ℓT (v).
Proof. Assume that there is such a decomposition x = uv. For y ∈ {u, v}
define
Wy := ht ∈ T | t ≤T yi.
We claim that in that case we have W = Wu × Wv and
˙
Ref(W ) = Ref(Wu )∪Ref(W
v ),
contradicting the irreducibility of W .
Firstly we show that Mov(v) ⊆ V u . We have V u ∩ V v ⊆ V uv , hence
Mov(uv) ⊆ Mov(u) + Mov(v). Since moreover ℓT (x) = ℓT (uv) = ℓT (u) +
ℓT (v) by Carter’s Lemma we have Mov(u) ∩ Mov(v) = 0. Let a ∈ Mov(v).
Then since uv = vu we have u(a) ∈ Mov(v), hence u(a) − a ∈ Mov(v) ∩
Mov(u) = 0. Hence a ∈ V u , which shows the claimed inclusion.
Now if t ∈ T is such that t ≤T u, then by Lemma 3.3 we have that
Mov(v) ⊆ V u ⊆ V t which implies that αt ∈ V v . Using 3.3 again, we deduce
that t commutes with any reflection t′ ∈ T such that t′ ≤T v.
Concatenating a T -reduced expression t1 t2 · · · tk of u with a T -reduced
expression tk+1 tk+2 · · · tn of v we get a T -reduced expression t1 t2 · · · tn of
x. By the discussion above we have tt′ = t′ t for all t ∈ {t1 , t2 , . . . , tk },
t′ ∈ {tk+1 , tk+2 , . . . , tn } and any reflection occurring in a T -reduced expression in the orbit Bn · (t1 , t2 , . . . , tn ) lies either in Wu or in Wv . Since
by [3, Theorem 1.1] there is a unique Hurwitz orbit on RedT (x), the set
{t1 , t2 , . . . , tn } generates W . Let t ∈ T . Since a quasi-Coxeter element x
has no nontrivial fixed point in V we have that t ≤T x, hence t occurs in a
reduced expression of RedT (x). But there is only one orbit Bn ·(t1 , t2 , . . . , tk )
of Bn on RedT (w). It follows that t ∈ Wu or in t ∈ Wv . The claimed direct
product decomposition follows.

Proof of Theorem 1.3. The existence of the claimed decomposition w =
x1 x2 · · · xm is given by Propositions 1.2 and 3.5. It remains to show uniqueness. Assume that w = y1 y2 · · · yℓ is another such decomposition. For each
i, choose a reduced expression ti1 · · · tini of yi . Thanks to (2), concatenating
these reduced expressions yields a reduced expression of w. For a fixed i, all
the tji are in P (w) by Theorem 2.3. It follows from Lemma 2.6 that tji lies
in one of the parabolic factor W (i, j) of P (w), and indecomposability of yi
forces W (i, j) to depend only on i. Therefore we set W (i) := W (i, j) and
note that yi ∈ W (i). But since the irreducible parabolic factors of P (w) are

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precisely the P (xj )’s, for each W (i) there exists j such that W (i) = P (xj ).
It follows from the decomposition
(3.6)

P (w) = P (x1 ) × P (x2 ) × . . . P (xm )

and the uniqueness of the decomposition of w in the direct product (3.6)
that each of the xj can be written as a product of yi ’s: more precisely, xj is
the product of all those yi ’s such that W (i) = P (xj ) (and for each j, there
must be at least one yi such that W (i) = P (xj ) otherwise w would have two
distinct decompositions in (3.6)). But by indecomposability of xj , there can
be at most one such yi , which concludes.

3.4. Consequences. We conclude with some facts about elements for which
Condition 1.1 fails. For such an element w ∈ W , by [3, Theorem 1.1] the
Hurwitz operation on RedT (w) is not transitive. Denote by H(w) the set
of orbits. For O ∈ H(w), denote by hOi the reflection subgroup generated
by any T -reduced expression in O. This is well-defined since the Hurwitz
action leaves the subgroup generated by a T -reduced expression unchanged.
Hence we get the following:
Proposition 3.7. Let (W, S) be finite. Let w ∈ W . For each O ∈ H(w),
the element w has a unique generalized cycle decomposition in the sense of
Theorem 1.3 in the Coxeter group hOi.
Proof. The reflection subgroup hOi is a Coxeter group. Denote by S ′ its set
of canonical Coxeter generators. We have (see [10, Corollary 3.11 (ii)]) that
[
T ′ := hOi ∩ T =
uS ′ u−1 .
u∈hOi

Hence w is a quasi-Coxeter element in (hOi, S ′ ). Applying Theorem 1.3 to
w viewed as element of hOi we get the claim.

Lemma 3.8. Let w ∈ W . If O1 , O2 ∈ H(w) with O1 6= O2 , then hO1 i =
6
hO2 i.
Proof. Let (t1 , t2 , . . . , tk ) ∈ O1 . Then ht1 , t2 , . . . , tk i = hO1 i. Let S ′ be the
set of canonical Coxeter generators of the reflection subgroup hO1 i. The
Hurwitz operation is transitive on RedT1 (w) where T1 = T ∩ hO1 i (see the
proof of Proposition 3.7). If we have hO1 i = hO2 i, then in particular for
(q1 , q2 , . . . , qk ) ∈ O2 we have qi ∈ T1 for all i since qi ∈ T ∩ hO2 i = T1 . This
implies that (t1 , t2 , . . . , tk ) and (q1 , q2 , . . . , qk ) lie in the same Hurwitz orbit
since the Hurwitz operation is transitive on RedT1 (w), a contradiction.

Remark 3.9. Proposition 3.7 tells us that any w ∈ W has a unique cycle
decomposition in any reflection subgroup generated by one of its T -reduced
expressions. However distinct such reflection subgroups, equivalently (by
Lemma 3.8) distinct Hurwitz orbits in H(w) can yield the same decomposition of w: more precisely if x1 , x2 , . . . , xm is a cycle decomposition in hO1 i

ON CYCLE DECOMPOSITIONS IN COXETER GROUPS

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and y1 , y2 , . . . , yℓ is a cycle decomposition in hO2 i where O1 and O2 are
distinct elements in H(w), then it is possible that ℓ = m and
{x1 , x2 , . . . , xm } = {y1 , y2 , . . . , ym }.
As an example consider W of type G2 = I2 (6) with S = {s, t}. Then w =
stst is a Coxeter element in both irreducible reflection subgroups hs, tsti and
ht, stsi of type A2 hence its cycle decomposition is the trivial decomposition
x1 = stst = y1 in both subgroups.
However we always have
{P (x1 )1 , P (x2 )1 , . . . , P (xm )1 } =
6 {P (y1 )2 , P (y2 )2 , . . . , P (ym )2 },
where P (xi )1 (resp. P (yi )2 ) is the parabolic closure of xi (resp. yi ) in hO1 i
(resp. hO2 i). Indeed, the reflections in these subgroups have to generate
the parent group hO1 i (resp. hO2 i) in which the cycle decomposition is
considered and they are distinct by Lemma 3.8. In the above example with
W of type G2 we have P (x1 )1 = hs, tsti 6= P (y1 )2 = ht, stsi.
Also note the following
Lemma 3.10. Let w ∈ W with ℓT (w) = k. Let W ′ ⊆ W be a reflection
subgroup of W containing w with set of reflections T ′ = W ′ ∩ T . Then
ℓT (w) = ℓT ′ (w).
Proof. Since T ′ ⊆ T we must have ℓT (w) ≤ ℓT ′ (w). Assume that ℓT ′ (w) =
k′ > k. Let (t1 , t2 , . . . , tk′ ) ∈ RedT ′ (w). Let i be minimal such that
ℓT (t1 t2 · · · ti ) 6= ℓT ′ (t1 t2 · · · ti ). Then ℓT (t1 t2 · · · ti ) = i−2, ℓT ′ (t1 t2 · · · ti ) = i.
Let u = t1 t2 · · · ti−1 . By minimality of i we have ℓT (u) = ℓT ′ (u) = i − 1.
In particular, we have ti ≤T u. The parabolic closure P (u) of u therefore
has rank i − 1 and contains t1 , . . . , ti−1 but also ti . But since (t1 , . . . , tk′ )
is T ′ -reduced, the reflection subgroup W ′′ = ht1 , t2 , . . . , ti i has rank i (as a
reflection subgroup of the Coxeter group W ′ , for instance by Lemma 3.2).
But the reflections of W ′′ as a reflection subgroup of W ′ or W are the same,
hence W ′′ also has rank i as a reflection subgroup of W . Therefore W ′′
cannot be included in P (u) which has smaller rank, a contradiction.

Hence together with Lemma 3.8 we get
Corollary 3.11. For all w ∈ W , there is a one-to-one correspondence between H(w) and reflection subgroups of W in which w is a quasi-Coxeter
element, given by O 7→ hOi.
Example 3.12. Let W be of type D4 with S = {s0 , s1 , s2 , s3 } where s2
commutes with no other simple reflection. Then the element
w = s1 (s2 s1 s2 )(s2 s0 s2 )s3
is a quasi-Coxeter element in W (see [3, Example 2.4]), but it is not a Coxeter
element (in the sense that it has no T -reduced expression yielding a simple
system for W ). It follows from Proposition 3.5 that its cycle decomposition
is the trivial decomposition x1 = w. Now W can be viewed as a reflection
f of type B4 . In that case W is not parabolic
subgroup of a Coxeter group W
f , hence w has no reduced expression generating a parabolic subgroup of
in W
f : indeed, if there was such a decomposition, then the Hurwitz operation
W

10

THOMAS GOBET

f would be transitive by [3,
on the set of T -reduced expressions of w in W
Theorem 1.1], hence each reduced expression would generate a parabolic
f . The element w has a
subgroup, in particular W would be parabolic in W
unique generalized cycle decomposition in W which is the one we gave above
(since W is irreducible), but w can also be realized as a Coxeter element in
a (non irreducible) reflection subgroup W ′ of type B2 × B2 as follows: in the
f of type B4 (see [5, Section
signed permutation model for the Weyl group W
8.1]), we have w = (1, −2, −1, 2)(3, 4, −3, −4), which is a product of two
Coxeter elements of the two type B2 reflection subgroups consisting of those
signed permutations supported on {±1, ±2} and {±3, ±4} respectively. In
W ′ the unique cycle decomposition of w has two factors x1 = (1, −2, −1, 2),
x2 = (3, 4, −3, −4). Note that neither x1 nor x2 lies in W .
Acknowledgments
I thank Barbara Baumeister and Vivien Ripoll for helpful discussions.
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Institut Elie Cartan de Lorraine, Université de Lorraine, B.P. 70239, F54506 Vandoeuvre-lès-Nancy Cedex.

