GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

arXiv:1707.03137v1 [] 11 Jul 2017

M. J. DYER AND G. I. LEHRER
Abstract. We determine a fundamental domain for the diagonal action of a
finite Coxeter group W on V ⊕n , where V is the reflection representation. This is
used to give a stratification of V ⊕n , which is respected by the group action, and we
study the geometry, topology and combinatorics of this stratification. These ideas
are used to obtain results on the classification of root subsystems up to conjugacy,
as well as a character formula for W .

Introduction
Let Φ be a finite root system in the Euclidean space V = Rℓ , whose inner product
we shall denote h −, − i. Let W = W (Φ) be the corresponding Weyl group. This is
a finite reflection group on V , and the choice of a simple subsystem of Φ defines a
fundamental region, known as the fundamental chamber, for the action of W on V .
The group W acts diagonally on V n := V ⊕n = V ⊕ · · · ⊕ V , and our first main
(n)
result, Theorem 2.2 in §2 below, is the determination of a fundamental region CW
for this W -action. This turns out to be a locally closed, convex subset of V n . In §3
we show how our fundamental region may be used to obtain a stratification of V n
by locally closed subsets which are transformed into each other under the W -action,
and are such that the closure of any one is a union of such sets. This leads to a
combinatorial structure which significantly generalises the Coxeter complex of W .
We study the topology and combinatorics of this stratification in §4.
As applications of these results, we give in §3.18 a character formula for W , which
generalises the usual Solomon formula for the alternating character of W .
Then, in §5, we show how our fundamental region may be used to study W -orbits
of finite subsets of V , both ordered and unordered. In §6, we apply the results of
§5 to show how the conjugacy classes of type A-subsystems of an arbitrary root
system Φ may be determined by inspection of the Dynkin diagram of Φ. Finally, in
§7, we indicate, partly without proof, how the results of §5–6 may be used to study
conjugacy classes of root subsystems of Φ and thus conjugacy classes of reflection
subgroups of W . Related results on conjugacy of subsystems of root systems may
be found in [14], [12], [5] and [16].
1. Preliminaries
1.1. Let V be a real Euclidean space, i.e. a finite dimensional real vector space
equipped with a symmetric, positive definite bilinear form h −, − i : V × V → R.
For non-zero α ∈ V , let sα : V → V denote the orthogonal reflection in α; it is the
2010 Mathematics Subject Classification. Primary: 20F55: Secondary: 17B22.
1

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M. J. DYER AND G. I. LEHRER

2
α. In this paper,
R-linear map defined by sα (v) = v − h v, α∨ iα where α∨ := h α,α
i
by a root system Φ in V , we shall mean a subset Φ of V satisfying the following
conditions (i)–(iii):
(i) Φ is a finite subset of V \ {0}.
(ii) If α, β ∈ Φ, then sa (β) ∈ Φ.
(iii) If α, cα ∈ Φ with c ∈ R, then c ∈ {±1}.
The subgroup W of End(V ) generated by { sα | α ∈ Φ } is a finite (real) reflection
group i.e. a finite Coxeter group. A simple system Π of Φ is a linearly independent
subset Π ⊆ Φ such that Φ = Φ+ ⊔Φ− where Φ+ := Φ∩R≥0 Π and Φ− = −Φ+ (we use
the symbol ⊔ to indicate a disjoint union). Fix a simple system Π (it is well known
that such simple systems exist). Then Φ+ is the corresponding positive system of Φ
and S := { sα | α ∈ Π } ⊆ W is called the set of simple reflections of W . It is well
known that (W, S) is a Coxeter system. The subset T := { sα | α ∈ Φ } = { wsw −1 |
w ∈ W, s ∈ S } of W is called the set of reflections of W .

1.2. Dual root system. If Φ is a root system in V , then Φ∨ := { α∨ | α ∈ Φ } is
also a root system, called the dual root system of Φ; it has a system of simple roots
Π∨ := { α∨ | α ∈ Φ } with corresponding positive roots Φ∨+ := { α∨ | α ∈ Φ+ } and
associated finite Coxeter system (W, S).
1.3. Weyl groups. The root system Φ is said to be crystallographic if for all α, β ∈
Φ, one has h α, β ∨ i ∈ Z. In that case, W is a finite Weyl group and one defines
the root lattice Q(Φ) := ZΠ and weight lattice P (Φ) := { λ ∈ V | h λ, Φ∨ i ⊆ Z }.
The corresponding lattices Q(Φ∨ ) and P (Φ∨ ) for Φ∨ are called the coroot lattice and
coweight lattice of Φ respectively.
1.4. A subgroup W ′ of V generated by a subset of T is called a reflection subgroup.
It has a root system ΦW ′ = { α ∈ Φ | sα ∈ W ′ }. We call ΦW ′ a (root) subsystem of
Φ. A simple system (resp., positive system) of a root subsystem of Φ will be called
a simple (resp., positive) subsystem of Φ. It is well known that ΦW ′ has a unique
simple system ΠW ′ contained in the set of positive roots Φ+ of W ; the corresponding
positive system is ΦW ′ ,+ := Φ+ ∩ ΦW ′ .
The reflection subgroups WI := h I i generated by subsets I of S are called standard parabolic subgroups and their conjugates are called parabolic subgroups. If
W ′ = WI , then ΠW ′ = { α ∈ Π | sa ∈ I } and ΦW ′ ,+ = Φ ∩ R≥0 ΠW ′ .
1.5. Fundamental chamber for the W -action on V . The subset C = CW :=
{ v ∈ V | h v, Π i ⊆ R≥0 } of V is called the fundamental chamber of W . In the
following Lemma, we collect several standard facts concerning this situation, which
may be found in [2, 15].
1.6. Lemma.
(a) Every W orbit on V contains a unique point of C .
(b) For v ∈ C , the stabiliser StabW (v) := { w ∈ W | w(v) = v } is equal to the
standard parabolic subgroup WI where I := { s ∈ S | s(v) = v } = { sα | α ∈
Π, h α, v i = 0 }.
(c) The set of parabolic subgroups is equal to the set of the stabilisers of points
of V . It is also equal to the set of pointwise stabilisers of subsets of V .

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

3

(d) The intersection of any set of parabolic subgroups of W is a parabolic subgroup
of W . In particular, we have for I, J ⊆ S, WI ∩ WJ = WI∩J .
It follows that if α ∈ Φ is any root, then the W -orbit W α contains a unique
element of CW . A root α in Φ ∩ CW is said to be a dominant root. If Φ is irreducible,
there are at most two such roots (cf. [6, Lemma 2]). We note that if V were a
complex vector space, (c) above would not be true.
1.7. Determination of W -orbit representatives. An efficient algorithm for computing the unique element in W v ∩ C for an arbitrary element v ∈ V is as follows.
For v ∈ V , let Φv := { β ∈ Φ+ | h β, v i < 0 } and nv := |Φv |. If Φv ∩ Π = ∅,
then Φv = ∅ and v ∈ C . Otherwise, there exists some β ∈ Φv ∩ Π; one then
has Φsβ (v) = sβ (Φv \ {β}) and nsβ (v) = nv − 1. Continuing thus, one obtains
sβ1 . . . sβr v ∈ C , where r = nv .
1.8. Dynkin diagrams of simple systems and np subsets. Suppose Φ is crystallographic. Define a np subset Γ of Φ to be a subset such that for distinct elements
α, β ∈ Γ, one has h α, β ∨ i ≤ 0. Then a subset Γ of Φ is a simple subsystem if and
only if it is a linearly independent np subset (see [6]).
Define the diagram of a np subset Γ of Φ to be the Dynkin diagram (as in [10,
§4.7]) of the generalised Cartan matrix (mα,β )α,β∈Γ where mα,β := h α∨, β i. This is
a graph with vertex set Γ such that if α, β ∈ Γ are distinct with |mα,β | ≥ |mβ,α |,
the vertices α, β are connected by |mα,β | lines and these lines are equipped with an
arrow pointing towards α if |mα,β | > 1. Note that the arrow points towards the
shorter root if α, β are of different length, and that if α = −β, then α, β are joined
by two lines equipped with a pair of arrows in opposite directions. It is well known
that the connected components of the diagram of Γ are certain Dynkin diagrams of
finite or affine type. Further, the np subset Γ of Φ is a simple subsystem of Φ if and
only if the irreducible components of its diagram are all Dynkin diagrams of finite
type.
By an ordered simple subsystem (resp., ordered np subset) of Φ, we mean a tuple
b = (β1 , . . . , βn ) of pairwise distinct roots whose underlying set [b] := {β1 , . . . , βn }
is a simple subsystem (resp., np subset) of Φ.
2. Orbits under the diagonal action
2.1. Fundamental domain for the diagonal W -action. For each n ∈ N, let
V n := V ⊕. . .⊕V (n factors) with the diagonal W -action defined by w(v1 , . . . , vn ) :=
(wv1 , . . . , wvn ). (For n = 0, V n := {0} with trivial W -action, by convention). We
identify V n × V m with V n+m in the natural way. For v := (v1 , . . . , vn ) ∈ V n and
any m ≤ n in N, define the truncation τm (v) := (v1 , . . . , vm ) and
(2.1.1)

Wv,m := StabW (τm (v)) = StabW (v1 ) ∩ . . . ∩ StabW (vm ).

Thus, Wv,0 = W and each Wv,m is a parabolic subgroup of Wv,m−1 . In particular,
Wv,m is a reflection group acting on V , and since its root system has a natural

4

M. J. DYER AND G. I. LEHRER

positive subsystem given by its intersection with Φ+ , it has a well defined fundamental chamber CWv,m ⊆ V . Note that for each m, Wv,m ⊆ Wv,m−1 , whence
CWv,m ⊇ CWv,m−1 .
Let
(2.1.2)

(n)

CW := { v = (v1 , . . . , vn ) ∈ V n | for m = 1, . . . , n, vm ∈ CWv,m−1 }.

Let v = (v1 , . . . , vn ) ∈ V n and 1 ≤ m ≤ n − 1. Set W ′ := Wv,m , v′ := (v1 , . . . , vm )
and v′′ := (vm+1 , . . . , vn ). The definitions immediately imply that
(n)

(m)

v ∈ CW if and only if v′ ∈ CW

(2.1.3)

(n−m)

and v′′ ∈ CW ′

.

The next result will be crucial for the investigation of the action of W on subsets
of V . It identifies a fundamental region for the action of W on V n .
(n)

2.2. Theorem.
(a) If v = (v1 , . . . vn ) ∈ CW , then StabW (v) = Wv,n is the
standard parabolic subgroup WIv of W , where
Iv = { sα | α ∈ Π, h α, vi i = 0 for i = 1, . . . , n }.
(n)

(b) Every W -orbit on V n contains a unique point of CW .
(n)
(c) CW is a convex (and hence connected) subset of V n .
Proof. If n = 0, (a) is trivial. Let n > 0. Assume by way of induction that Wv,n−1
is the standard parabolic subgroup WIτn−1 (v) of W . Clearly StabW (v) ⊇ WIv , and
so it suffices to show that any element of StabW (v) lies in WIv . Let w ∈ StabW (v);
then evidently w ∈ StabW (τn−1 v) = WIτn−1 v . Moreover vn ∈ CWIτn−1 v implies
that StabWIτ (v) (vn ) is the standard parabolic subgroup of WIτn−1 (v) generated by
n−1

{ sα ∈ ΠIτn−1 (v) | h α, vn i = 0 }, which is precisely the set Iv . This proves (a).
To prove (b), assume by induction that every W -orbit on V n−1 contains a unique
(n−1)
point of CW . By induction, there is an element w ′ ∈ W such that w ′ (τn−1 (v)) ∈
(n−1)
(n−1)
′
CW . Let v′ := w ′ (v) = (v1′ , . . . , vn′ ). Then τn−1 (v′ ) ∈ CW . Let Wn−1
:=
′
StabW (τn−1 v ); by (a), this is the standard parabolic subgroup WI ′ of W , where
I ′ = { sα | α ∈ Π, h α, vi′ i = 0 for i = 1, . . . , n − 1 }. Now there is an element
′
′
w ′′ ∈ Wn−1
such that w ′′ (vn′ ) ∈ CWn−1
. Let w := w ′′ w ′ ∈ W . Then since w ′′
(n)

′
stabilises (v1′ , . . . , vn−1
), it is clear that v′′ = w(v) = w ′′ (v′ ) ∈ CW . This shows that
(n)
every element of V n may be moved into CW by W . We show that the intersection
(n)
of a W -orbit on V n with CW is a single point, that is, that no two distinct points
(n)
of CW are in the same W -orbit.
This will be done by induction on n. The case n = 1 is clear. Suppose that
(n)
(n)
(n−1)
v ∈ CW and that w ∈ W is such that w(v) ∈ CW . Then w(τn−1 (v)) ∈ CW . By
induction, w ∈ StabW (τn−1 v), which by (a) is equal to WI , where I = { sα | α ∈
Π, h α, vi i = 0 for i = 1, . . . , n − 1 }. Since w(vn ) ∈ CWI and vn ∈ CWI , it follows
that wvn = vn and (b) follows.
(n)
To prove (c), let u = (u1 , . . . , un ) and v = (v1 , . . . , vn ) be points in CW , and for
(n)
t ∈ R, 0 ≤ t ≤ 1, write c(t) = tu + (1 − t)v. We wish to show that c(t) ∈ CW . By

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

5

induction, we may assume that tum +(1−t)vm ∈ CWct ,m−1 for m < n, and we require
the corresponding statement for m = n. We shall prove first that for t 6= 0, 1, we
have, for all non-negative integers m,
(2.2.1)

Wc(t),m = Wu,m ∩ Wv,m .

The inclusion of the right hand side in the left is clear. We prove the reverse
inclusion. For m = 0, this is trivial. In general, suppose by way of induction (using
(a)) that Wc(t),m−1 = WI with m ≥ 1, where I = I1 ∩ I2 , and I1 = { sα | α ∈
Π, h α, ui i = 0 for i = 1, . . . , m − 1 } and I2 = { sα | α ∈ Π, h α, vi i = 0 for i =
1, . . . , m − 1 }. Then um ∈ CWI1 , whence h um, α i ≥ 0 for α ∈ ΠI1 , and vm ∈ CWI2 .
Now if α ∈ ΠI , then h α, um i ≥ 0 and h α, vm i ≥ 0. If h α, um i =
6 0 or h α, vm i =
6 0,
then h α, tum + (1 − t)vm i > 0, and sα 6∈ Wc(t),m . This proves the assertion (2.2.1).
It follows that for t 6= 0, 1, CWc(t),n−1 = { v ∈ V | h v, α i ≥ 0 for all α ∈ I },
where I = I1 ∩ I2 is as in the previous paragraph with m = n. But un ∈ CWI1 and
vn ∈ CWI2 , whence for α ∈ ΠI1 ∩ ΠI2 , we have h α, tun + (1 − t)vn i ≥ 0, and (c) is
proved.

2.3. Total orderings on V . A vector space total order ≤ of V is a total order of
V such that the set { v ∈ V | v > 0 } of positive elements is closed under addition
and under multiplication by positive real scalars. One way such orderings arise is as
follows. Take any (ordered) basisP{α1 , . . . , αp } of V , and define the total order ≤ by
declaring that u < v if v − u = pi=1 ci αi and there is some j with ci = 0 for i < j
and cj > 0. It is well known that in fact, all total orders arise in this way (since V
is finite dimensional).
When V contains a root system Φ, fix a vector space total order ≤ of V such that
every positive root of W is positive in that order i.e. Φ+ ⊆ { v ∈ V | v > 0 }. One
such order is the one described above, taking the simple system Π = {α1 , . . . , αl } to
be the initial segment of a basis of V as above.
Given such an ordering of V , we endow V n with the lexicographic total order 
induced by ≤ i.e. (v1 , . . . , vn ) ≺ (u1 , . . . , un ) if there is some j such that vi = ui for
i < j and vj < uj .
(n)

2.4. Proposition. Let v ∈ CW . Then v is the maximum element of the orbit W v
in the total order induced by .
Proof. Let w ∈ W and u := wv ∈ V n . If w ∈ Wv,n , then u = v. Otherwise, there
is some j with 0 ≤ j ≤ n − 1 such that w ∈ Wv,j \ Wv,j+1 . Write v = (v1 , . . . , vn )
and u = (u1 , . . . , un ). Then vi = ui for i ≤ j, vj+1 ∈ CWv,j , and uj+1 = w(vj+1).
Since w ∈ Wv,j \ Wv,j+1 = Wv,j \ StabW (vj+1 ), it follows that 0 6= vj+1 − uj+1 =
vj+1 − w(vj+1) ∈ R≥0 ΠWv,j . Moreover ΠWv,j ⊆ Φ+ = { v ∈ Φ | v > 0 }, and so
vj+1 > uj+1. Since vi = ui for i < j + 1, one has u  v as required.

2.5. Corollary. Given any total order on V such that 0 ≤ Φ+ , let  be the induced
(n)
lexicographic total order on V n . Then the region CW is precisely the set of maximal
elements in the W -orbits on V n .
Proof. This is immediate from Theorem 2.2 and Proposition 2.4.



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M. J. DYER AND G. I. LEHRER

2.6. Ordered and unordered sets. For a group H acting on the left on a set U,
we denote the set of H-orbits on U by U/H. We use frequently below the simple fact
that if H = H1 × H2 is a product of two groups, then H1 (resp., H2 ) acts naturally
on U/H2 (resp., U/H1 )) and there are canonical bijections
(2.6.1)
U/H ∼
= (U/H1 )/H2 ∼
= (U/H2 )/H1 .
We record the following elementary observation.
2.7. Lemma. Let U be a totally ordered set, and let H be a finite group acting on
U. For any finite subset A of U, denote by max(A) the maximum element of A.
Then
(a) The map Hu 7→ max(Hu) (u ∈ U) defines a bijection from the set U/H to
a subset of U, which we denote by MH = MH (U).
(b) If H = H1 × H2 is a direct product of groups H1 , H2 , then for u ∈ U,
max(Hu) = maxh∈H1 (max(hH2 u)).
(c) The set { H2u | u ∈ MH1 ×H2 } is a set of distinct representatives for the
H1 -orbits on U/H2 .
2.8. Let n ∈ N. We regard the symmetric group Symn as the group of all permutations of {1, . . . , n} (acting on the left) and often write its elements in cycle notation. Take U = V n , and let G be a subgroup of the symmetric group
Symn . Then G has a natural left action on V n by place permutations, defined by
σ(v1 , . . . , vn ) = (vσ−1 (1) , . . . , vσ−1 (n) ), which commutes with the diagonal W -action
and induces a W × G-action on V n . Assume chosen, a total order ≤ on V , with
corresponding total order  on U = V n , as in Corollary 2.5. Write H = W × G, and
recall that for any H-action on the ordered space U, MH (U) is the set of elements
of U which are maximal in their H-orbit.
2.9. Corollary.
(a) For v ∈ V n , max(Hv) = max{ max(W σ(v)) | σ ∈ G }.
(b) { Gv | v ∈ MW ×G (V n ) } is a set of orbit representatives for W acting on
V n /G.
(n)
(c) We have MW (V n ) = CW ⊇ MW ×G (V n ).
Proof. Parts (a) and (b) are immediate from Lemma 2.7. Part (c) follows from parts
(a) and (b) and Corollary 2.5.

cbe a subset of V n which is stable under the action of the
2.10. Proposition. Let S
group H = W × G as above.
(n)
c/W .
(a) The set Rb := Sc∩ CW is in canonical bijection with S
c/G is in canonical bijection with the set S
c/H of
(b) The set of W -orbits on S
c and also with the set of G-orbits on S
c/W .
H-orbits on S
(c) Define a left “dot” G-action (g, b) 7→ g · b on Rb by transferring the Gc/W via the bijection of (a). This action is determined by either
action on S
condition {g · b} = W gb ∩ Rb or g · b = max(W gb) for b ∈ Rb and g ∈ G.
(d) The G-orbits in the dot action on Rb are the equivalence classes for the
b that
equivalence relation ≃ on Rb defined by stipulating (for b, b′ ∈ R)

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

7

c/G are in canonical
b ≃ b′ ⇐⇒ b′ ∈ Hb. Hence the W -orbits on S
b
bijection with R/≃.
c−→ S
c/G, then the number of W -orbits on S
c/G
(e) If η is the natural map S
is at most |η(Rb )|.

Proof. Part (a) follows from Theorem 2.2. Part (b) was already observed more
c/W be the canonical bijection of
generally in 2.6. We prove (c). Let φ : Rb → S
(a). By definition, φ(b) = W b for b ∈ Rb and {φ−1 (Γ)} = Γ ∩ Rb for any W -orbit
c/W . Hence the dot action is determined by {g · b} = W gb ∩ Rb for g ∈ G
Γ∈S
b But by Corollary 2.5, W gb ∩ Rb = {max(W gb)} and (c) follows.
and b ∈ R.
Now we prove (d). Let ≃ be the equivalence relation on Rb for which the equivalence classes are the G-orbits in the dot action. Then by (c), for b, b′ ∈ Rb one
has b ≃ b′ ⇐⇒ b′ ∈ { max(W gb) | g ∈ G }. Certainly b ≃ b′ implies b′ ∈ Hb.
But if b, b′ ∈ Rb with b′ ∈ Hb, one has b′ ∈ W gb for some g ∈ G and then
b This proves the first assertion of
b′ = max(W gb) by Corollary 2.5 since b ∈ R.
(d), and the second then follows from (a).
To see (e), we need only note that the fibres of the restriction of η to Rb are subsets
of the ≃-equivalence classes, whence the number of equivalence classes on Rb is at
most the number of such fibres.


2.11. Taking Sc = V n and G = Symn in Proposition 2.10(c) defines a dot action
(n)
(g, b) 7→ g · b = g ·n b of Symn on CW for each n (of which all possible dot actions
as in Proposition 2.10(c) are various restrictions). We record the following trivial
but useful relations between these ·n actions for varying n
(n)

2.12. Proposition. Let notation be as above. Let b = (β1 , . . . , βn ) ∈ CW , σ ∈ Symn
(n)
and m ∈ N with 0 ≤ m ≤ n. Set b′ := τm (b) = (β1 , . . . , βm ) ∈ CW , W ′ := Wb′
(n−m)
(n−m)
and b′′ := (βm+1 , . . . , βn ) ∈ CW ′ . Denote the · action of Symn−m on CW ′
by
·′n−m .
(n)

(a) If σb ∈ CW , then σ · b = σb.
(b) Suppose {1, . . . , m} is σ-stable. Let σ ′ be the restriction of σ to a permutation
of {1, . . . , m}. Then τm (σ ·n b) = σ ′ ·m (τm b).
(c) Suppose that σ fixes i for i = 1, . . . , m. Let σ ′ ∈ Symn−m be defined by
σ ′ (i) := σ(i + m) − m for i = 1, . . . , n − m. Then σ ·n b = σ ·n (b′ , b′′ ) =
′
′
(b′ , σ ′ ·′n−m b′′ ) i.e. σ·n b = (β1 , . . . , βm , βm+1
, . . . , βn′ ) where (βm+1
, . . . , βn′ ) :=
′′
′ ′
σ ·n−m b .
Proof. This is a straightforward consequence of the definitions. Details are left to
the reader.

2.13. Automorphisms. Denote the group of all (linear) isometries of (V, h −, − i)
which restrict to a permutation of the simple roots Π by D. Then D acts diag(n)
onally on V n and it is easily seen that CW is D-stable. It is well known that if
span(Φ) = V and the W -invariant inner product h −, − i is chosen suitably (by

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M. J. DYER AND G. I. LEHRER

rescaling if necessary its restrictions to the linear spans of the components of Φ),
then D identifies with the group D ′ of all diagram automorphisms of Π (i.e. the
group of automorphisms of the Coxeter system (W, S)). It follows that in general,
(n)
CW ∩ Φn is invariant under the diagonal action of D ′ on Φn (even if span(Φ) 6= V
or h −, − i is not chosen in this way).
3. Stratification of V n
3.1. Cones. Recall that a subset of a topological space V is locally closed if it is
open in its closure, or equivalently, if it is the intersection of an open subset and a
closed subset of V . A subset of V is said to be constructible if it is a finite union
of locally closed subsets of V . By a cone in a real vector space, we mean a subset
which is closed under addition and closed under multiplication by positive scalars.
Note that cones may or may not be constructible.
3.2. Facets. The fundamental chamber C = CW and certain notions below depend
not only on W and Φ, but also on the simple system Π; this dependence will be
made explicit in notation to be introduced presently.
For J ⊆ I ⊆ S define
(3.2.1)

CI,J := { v ∈ V | h α, v i = 0 for α ∈ ΠJ and h α, v i > 0 for α ∈ ΠI \ J }

This is a non-zero locally closed cone in V . From [2],
G
G
(3.2.2)
CWI =
CI,J ,
CI,J =
J ⊆I

CI,K

K:J ⊆ K ⊆ I

For J ⊆ I ⊆ S, F
let W J := { w ∈ W | w(ΠJ ) ⊆ Φ+ } and WIJ := WI ∩ W J . From [2],
one has WI = w∈W J wWJ and each element w ∈ WIJ satisfies l(wx) = l(w) + l(x)
I
for all x ∈ WJ .
Fix I ⊆ S. The sets w(CI,J ) for w ∈ WI and J ⊆ I are called the facets (of WI
acting on V ). They are locally closed cones in V , and the closure of any facet is a
union of facets. It is well known that any two facets either coincide or are disjoint.
The setwise stabiliser in WI of CI,J coincides with the stabiliser in WI of any point
of CI,J and is equal to WJ . It follows that for I ⊆ S, one has the decomposition
G
G
(3.2.3)
V =
w(CI,J )
J : I ⊆ J w∈W J
I

of V as a union of pairwise disjoint facets.
3.3. Strata. The family of subsets w(CS,J ) for J ( S and w ∈ W , or rather the
complex of spherical simplices cut out by their intersections with the unit sphere in
V , is known as the Coxeter complex of W on V (see [3], [9], [11]); there are also
other closely related meanings of the term Coxeter complex in the literature, for
instance (cf. [4]) where the Coxeter complex is an (abstract) simplicial complex or
chamber complex with vertices which are cosets of proper parabolic subgroups. We
shall now define a similar “complex” for W acting on V n , where n ∈ N≥1 , and show
that it has significant similarities to, and differences from, the Coxeter complex.

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

9

Let
(3.3.1)

I = I (n) := { (I0 , . . . , In ) | S = I0 ⊇ I1 ⊇ . . . ⊇ In }

denote the set of all weakly decreasing sequences of n + 1 subsets of S with S as
first term. For I = (I0 , . . . , In ) ∈ I , define
(3.3.2)

XI := CI0 ,I1 × CI1 ,I2 × . . . × CIn−1 ,In ⊆ V n .

The sets w(XI ) for I ∈ I (n) and w ∈ W are non-empty locally closed cones in V n
which will be called the strata of V n (for (W, S) or W ). Their basic properties are
listed in the Theorem below.
3.4. Theorem.
(a) If I ∈ I (n) then XI ⊆ C (n) .
(n)
(b) If I ∈ I , w ∈ W and v ∈ w(XI ), then StabW (v) = wWIn w −1 .
(c) Let v, w ∈ W and I, J ∈ I (n) . Then the following conditions are equivalent
(i) v(XI ) ∩ w(XJ ) 6= ∅.
(ii) v(XI ) = w(XJ ).
(iii) I = J and v −1 w ∈ WIn .
(d) If I ∈ I (n) and w ∈ W , then StabW (w(XI )) = wWIn w −1 .
(e) The sets C (n) and V n are the disjoint unions of the strata they contain.
(f) The topological closure of any stratum of V n is a union of strata.
(n)

3.5. Remarks. (1) The fundamental region CW is constructible (in fact, it is a finite
union of locally closed cones).
(n)
(2) If n = 1, the fundamental region CW = C is closed in V , but in general,
(n)
CW is a constructible subset of V n which need not be locally closed; moreover the
(n)
(n)
closure of a stratum in CW may contain strata outside CW .
(3) If n = 1, then the facets (here called strata) of W on V are the maximal
connected subsets of V , all points of which have the same stabiliser. For n > 1,
the stratum containing v is precisely the connected component containing v of the
space of all u ∈ V n such that StabW (τi (u)) = StabW (τi (v)) for i = 1, . . . , n. Recall
that here τi is the truncation map V n → V i given by τi (u1, . . . , un ) = (u1, . . . , ui ).
3.6. An example. Before giving its proof, we illustrate Theorem 3.4 and its following remarks in the simplest non-trivial situation.
Let W = {1, s}, and S = {s}, regarded as Coxeter group of type A1 acting as
reflection group on V := R with s acting by multiplication by −1, with root system
Φ = {±1} and unique positive root α := 1.
Then CW = R≥0 , CW∅ = R, CS,S = {0}, CS,∅ = R>0 and C∅,∅ = R. If n ∈ N≥1 ,
(n)
then CW is the set of all (λ1 , . . . , λn ) ∈ Rn such that if λj 6= 0 and λi = 0 for
all i < j, then λj > 0. In other words, it consists of zero and all non-zero vectors
in Rn with their first non-zero coordinate positive. Note that this is the cone of
non-negative vectors of a vector space total ordering of V .
(n)
A typical stratum in CW is of the form Xi := XS,...,S,∅,...,∅ for some 0 ≤ i ≤ n.
where there are n + 1 subscripts on X , of which the first n − i + 1 are equal to S
and the last i are equal to ∅. One readily checks that X0 = s(X0 ) = {(0, . . . , 0)}

10

M. J. DYER AND G. I. LEHRER

and that for i > 0,
Xi = {0} × . . . × {0} × R>0 × R × . . . × R = {0}n−i × R>0 × Ri−1 .
Thus, there are 2n + 1 distinct strata of W on V n , namely X0 and Xi , s(Xi ) for
i = 1, . . . , n. One readily checks that the closure of a stratum is given by its union
with the strata below it in the following Hasse diagram:
Xn ❖❖ s(Xn )
❖❖
♦♦♦
Xn−1 s(Xn−1 )
X2 ❖❖ s(X2 )
❖♦
❖
♦♦♦
X1 ❄
s(X1 )
❄

X0

⑧⑧

.

3.7. Proof of Theorem 3.4(a)–(e). Let I ∈ I (n) and v ∈ XI . Then vi ∈ CIi−1 ,Ii
for i = 1, . . . , n, so StabWIi−1 (vi ) = WIi . Since Wv,i = StabWv,i−1 (vi ), it follows by
induction that Wv,i = WIi . By definition, v ∈ C (n) and StabW (v) = WIn . This
proves (a) and (b). In (c), (iii) implies (ii) by (b) and it is trivial that (ii) implies
(i). We show that (i) implies (iii). Suppose that (i) holds: i.e. v(XI ) ∩ w(XJ ) 6= ∅.
That is, for i = 1, . . . , n, CIi−1 ,Ii ∩ v −1 w(CJi−1 ,Ji ) 6= ∅. We have I0 = J0 = S and
v −1 w ∈ WI0 = W . The properties of facets in §3.2 imply by induction on i that
for i = 0, . . . , n, Ii = Ji and v −1 w ∈ WIi . This shows that (i) implies (iii), and
completes the proof of (c). Part (d) follows immediately from (c) and (b).
For (e), we claim that
[
[ [
w(XI ).
(3.7.1)
C (n) =
XI ,
Vn =
I∈I (n)

I∈I (n) w∈W In

To prove the assertion about C (n) , note first that the right hand side is included
in the left. To prove the converse, let v ∈ C (n) . Using Theorem 2.2, write Wv,i =
WIi where Ii ⊆ S, for i = 0, . . . , n. Then clearly I := (I0 , . . . , In ) ∈ I (n) . Since
vi ∈ CWv,i−1 and StabWv,i−1 (vi ) = Wv,i , it follows by induction on i that vi ∈ CIi−1 ,Ii .
Hence v ∈ XI , proving the above formula for C (n) . Since V n = ∪w∈W w(C (n) ), the
above assertion concerning V n follows using (c), which also implies that the facets
in the unions 3.7.1 are pairwise distinct and disjoint.
3.8. Distinguished coset representatives. The proof of Theorem 3.4(f) (which
is given in 3.14) involves relationships between closures of facets with respect to
different parabolic subgroups of W . These results actually apply to arbitrary reflection subgroups, and we prove them in that generality (there is no simplification in
the formulations or proofs of the results for parabolic subgroups alone). In order to
formulate the results, we shall need additional background on reflection subgroups
and more detailed notation which indicates dependence of notions such as facets,
coset representatives etc on the chosen simple systems for the reflection subgroups

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

11

involved. The results needed are simple extensions (given in [8]) of facts from [2]
which are well known in the case of standard parabolic subgroups.
Recall that a simple subsystem Γ of Φ is defined to be a simple system Γ of some
root subsystem of Φ. For such a simple subsystem Γ, let SΓ := { sα | α ∈ Γ } and
WΓ = h SΓ i. Then (WΓ , SΓ ) is a Coxeter system, the length function of which we
denote as lΓ . Denote the set of roots of WΓ as ΦΓ = { α ∈ Φ | sα ∈ WΓ } = WΓ Γ
and the set of positive roots of ΦΓ with respect to its simple system Γ as ΦΓ,+
Let Γ, Γ′ be simple subsystems of Φ such that WΓ′ ⊆ WΓ . Let
′

WΓΓ := { w ∈ WΓ | w(Γ′ ) ⊆ ΦWΓ ,+ }.

(3.8.1)
Evidently one has

u(Γ′ )

(3.8.2)

WΓ

′

= WΓΓ u−1 for all u ∈ WΓ .
′

It is known from [8] that under the additional assumption that ΦΓ′ ,+ ⊆ ΦΓ,+ , WΓΓ
′
is a set of coset representatives for WΓ /WΓ′ and that each element w ∈ WΓΓ is the
unique element of minimal length in the coset wWΓ′ of WΓ with respect to the length
function lΓ . Moreover,
(3.8.3)

′

WΓΓ = { w ∈ WΓ | lΓ (wsα ) > lΓ (w) for all α ∈ Γ′ },

if ΦΓ′ ,+ ⊆ ΦΓ,+

Now in general if WΓ′ ⊆ WΓ , there is a unique simple system Γ′′ for ΦΓ′ such that
Γ ⊆ Φ+ and a unique u ∈ WΓ′ such that Γ′′ = u(Γ′ ). It follows from (3.8.2) and the
′
preceding comments that in this more general situation, it is still true that WΓΓ is
a set of coset representatives for WΓ /WΓ′ .
Similarly, define
′′

(3.8.4)

Γ′

′

WΓ := (WΓΓ )−1 = { w ∈ WΓ | w −1 (Γ′ ) ⊆ ΦWΓ ,+ }.

This is a set of coset representatives in WΓ′ \WΓ , each of minimal length in its coset
if ΦΓ′ ,+ ⊆ ΦΓ,+ . Further,
(3.8.5)

u(Γ′ )

′

WΓ = u Γ WΓ for u ∈ WΓ

and
(3.8.6) if ΦΓ′ ,+ ⊆ ΦΓ,+ , then

Γ′

WΓ = { w ∈ WΓ | lΓ (sα w) > lΓ (w) for all α ∈ Γ′ }.

3.9. Further notation. We now expand the notation of §3.2 to include the possibility of non-parabolic reflection subgroups. For any simple subsystem Γ of Φ,
let
(3.9.1)

CΓ := { v ∈ V | h v, Γ i ⊆ R≥0 } = { v ∈ V | h v, ΦΓ,+ i ⊆ R≥0 }.

denote the corresponding closed fundamental chamber for (WΓ , SΓ ) acting on V .
For any ∆ ⊆ Γ, let
(3.9.2)

CΓ,∆ := { v ∈ V | h v, Γ \ ∆ i ⊆ R>0 , h v, ∆ i = 0 }

denote the (unique) facet of WΓ on V which is open in CΓ ∩ ∆⊥ . One easily checks
that for w ∈ W ,
(3.9.3)

Cw(Γ) = w(CΓ ),

Cw(Γ),w(∆) = w(CΓ,∆ ).

12

M. J. DYER AND G. I. LEHRER

The setwise stabiliser of CΓ,∆ in WΓ coincides with the stabiliser in WΓ of any point
of CΓ,∆ , which is W∆ . Moreover,
G
G
(3.9.4)
CΓ =
CΓ,∆ ,
CΓ,∆ =
CΓ,∆′ .
∆′ : ∆ ⊆ ∆′ ⊆ Γ

∆⊆Γ

3.10. Lemma. Let Γ, Γ′ be simple subsystems of Φ with ΦΓ′ ,+ ⊆ ΦΓ,+ . Then
(a) CΓ ⊆ CΓ′
(n)
(n)
(n)
(n)
(b) CWΓ ⊆ CWΓ′ where CWΓ and CWΓ′ are the fundamental domains we have defined for (WΓ , SΓ ) and (WΓ′ , SΓ′ ) respectively acting on V n .

Proof. For (a), observe that
CΓ = { v ∈ V | h v, ΦΓ,+ i ⊆ R≥0 } ⊆ { v ∈ V | h v, ΦΓ′ ,+ i ⊆ R≥0 } = CΓ′ .
To prove (b), let v ∈ V n . Let W ′ = WΓ and W ′′ := WΓ′ . Recall that Wv,i =
′′
′
=
StabW (τi (v)). Similarly define Wv,i
= StabW ′ (τi (v)) = W ′ ∩ Wv,i and Wv,i
′′
′
′
StabW ′′ (τi (v)) = W ∩ Wv,i ⊆ Wv,i ; they are parabolic subgroups of (WΓ , SΓ ) and of
(WΓ′ , SΓ′ ) respectively with standard positive systems
′′ ,+ = ΦW ′′ ∩ ΦΓ′ ,+ ⊆ ΦW ′
′ ,+
ΦWv,i
∩ ΦΓ,+ = ΦWv,i
v,i
v,i
′′
′
for all i. If v ∈
⊆ CWv,i
by the assumption that ΦΓ′ ,+ ⊆ ΦΓ,+ . Hence by (a), CWv,i

(n)

(n)

′′
′
and so v ∈ CW ′′ by definition.
⊆ CWv,i−1
CW ′ , then for all i = 1, . . . , n, vi ∈ CWv,i−1


3.11. The main lemma. We now prove the main technical lemma required for the
proof of Theorem 3.4(f).
3.12. Lemma. Let Γ, Γ′ be simple subsystems of Φ with WΓ′ ⊆ WΓ .
S
(a) CΓ′ = w∈ Γ′ W w(CΓ ).
Γ
(b) If ∆′ ⊆ Γ′ , then
[
[
w(CΓ,∆ ).
CΓ′ ,∆′ =
′

w∈ Γ WΓ

∆⊆Γ
Ww(∆) ⊇ W∆′

Proof. Suppose that (a) holds for Γ and Γ′ . Then it also holds for Γ and u(Γ′ ) for
any u ∈ WΓ . For by (3.9.3) and (3.8.5), one would have

 [
[
[
uw(CΓ ) =
w(CΓ) =
w ′ (CΓ ).
Cu(Γ′ ) = u(CΓ′ ) = u
′

w∈ Γ WΓ

′

w∈ Γ WΓ

w′ ∈

u(Γ′ )

WΓ

A similar argument shows that if (b) is true for Γ, ∆′ and Γ′ , it is true for Γ, u(∆′ )
and u(Γ′ ) for any u ∈ WΓ . Since there is u ∈ WΓ with u(Γ′ ) ⊆ ΦΓ,+ , we may and do
assume for the proofs of (a)–(b) that ΦΓ′ ,+ ⊆ ΦΓ,+ .
′
To prove (a), note that if w ∈ WΓΓ , then (3.9.3) and Lemma 3.10(a) imply that
w(CΓ′ ) = Cw(Γ′ ) ⊇ CΓ . Hence ∪w∈ Γ′ W w(CΓ ) = ∪w∈W Γ′ w −1 (CΓ ) ⊆ CΓ′ . To prove the
Γ
Γ
reverse inclusion, let v ∈ CΓ′ . Write v = w(v ′) where v ′ ∈ CΓ and w ∈ WΓ is of
minimal length lΓ (w). By (3.8.3)–(3.8.4), it will suffice to show that lΓ (sα w) ≥ lΓ (w)

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

13

for all α ∈ Γ′ . Suppose first that h α, v i = 0. Then v = sa (v) = (sα w)(v ′) with
sα w ∈ WΓ . By choice of w, lΓ (sα w) ≥ lΓ (w). On the other hand, suppose h α, v i =
6 0.
Since v ∈ CΓ′ , this forces 0 < h v, α i = h w(v ′), α i = h v ′, w −1(α) i. Since v ′ ∈ CΓ
and w −1 (α) ∈ ΦWΓ , it follows that w −1(α) ∈ ΦWΓ ,+ and so lΓ (sα w) > lΓ (w) as
required.
′
Now we prove (b). Let w ∈ Γ WΓ , ∆ ⊆ Γ with Ww(∆) ⊇ W∆′ . Let v ∈ CΓ,∆ . Then
v ∈ CΓ , so by (a), w(v) ∈ CΓ′ . Since h v, ∆ i = 0, it follows that h w(v), w(∆) i = 0
and therefore h w(v), ∆′ i = 0 since Ww(∆) ⊇ W∆′ . Hence w(v) ∈ CΓ′ ,∆′ by (3.9.4).
Thus the right hand side of (b) is included in the left hand side. For the reverse
′
inclusion, let v ∈ CΓ′ ,∆′ ⊆ CΓ′ . By (a), there exists w ∈ Γ WΓ with v ′ := w −1 (v) ∈ CΓ .
Thus, v ′ ∈ CΓ,∆ for some ∆ ⊆ Γ. It remains to prove that Ww−1 (∆′ ) ⊆ W∆ . Let
′
α ∈ ∆′ ⊆ Γ′ . Since w ∈ Γ WΓ , it follows that w −1 (α) ∈ ΦWΓ ,+ . Note that
0 = h α, v i = h w −1(α), v ′ i.
Since v ′ ∈ CΓ,∆ , one has sw−1 (α) ∈ StabWΓ (v ′ ) = W∆ . Therefore Ww−1 (∆′ ) ⊆ W∆
since the elements sw−1 (α) for α ∈ ∆′ generate the left hand side. This completes
the proof of (b).

′

3.13. Remarks. One may show that in the union in (b), Γ WΓ may be replaced by
′
Γ′
WΓ∆ := Γ WΓ ∩ WΓ∆ , which is a set of (WΓ′ , W∆ ) double coset representatives in
WΓ′ \WΓ /W∆ , and is the set of all double coset representatives which are of minimal
(and minimum) length in their double coset if ΦΓ′ ,+ ⊆ ΦΓ,+ . (This uses the fact that
standard facts on double coset representatives with respect to standard parabolic
subgroups on both sides generalise to double coset representatives with respect to
an arbitrary reflection subgroup on one side and a standard parabolic subgroup on
the other side.) After this replacement, the union in (b) is one of pairwise disjoint
facets. This leads to a similar refinement in (3.15.1).
3.14. Proof of Theorem 3.4(f). For a simple subsystem Γ of Φ, and n ∈ N, let
(3.14.1)

(n)

IΓ = IΓ

:= { Γ = (Γ0 , . . . , Γn ) | Γ = Γ0 ⊇ Γ1 ⊇ . . . ⊇ Γn }

denote the set of all weakly decreasing sequences Γ of n + 1 subsets of Γ with Γ as
first term. For Γ = (Γ0 , . . . , Γn ) ∈ IΓ , define
(3.14.2)

XΓ := CΓ0 ,Γ1 × CΓ1 ,Γ2 × . . . × CΓn−1 ,Γn ⊆ V n .
(n)

The sets w(XΓ ) for Γ ∈ IΓ and w ∈ WΓ are the (WΓ , SΓ )-strata of V n . If
(n)
I = (I0 , . . . , In ) ∈ I (n) , then Γ := (ΠI0 , . . . , ΠIn ) ∈ IΠ and XI = XΓ . It is easy
to see that the collection of strata of V n with respect to (WΓ , SΓ ) depends only on
the reflection subgroup WΓ and not on the chosen simple system Γ.
(n)
There is a left action of W on ∪Γ IΓ , where the union is over simple subsystems
(n)
Γ of Φ, defined as follows: for w ∈ W and Γ = (Γ0 , . . . , Γn ) ∈ IΓ , one has
(n)
w(Γ) := (w(Γ0 ), . . . , w(Γn )) ∈ Iw(Γ) . By (3.9.3), this action satisfies
(3.14.3)

w(XΓ ) = Xw(Γ) , for all w ∈ W .

The setwise stabiliser of XΓ in WΓ is equal to the stabiliser in WΓ of any point of
XΓ , which is WΓn .

14

M. J. DYER AND G. I. LEHRER

Theorem 3.4(f) follows from the special case Γ = Λ = Π and W = WΓ = WΛ
of the following (superficially stronger but actually equivalent) result, which has a
simpler inductive proof because of the greater generality of its hypotheses.
3.15. Theorem. Let Γ and Λ be simple subsystems of Φ with WΓ ⊆ WΛ . Then for all
n ∈ N≥1 , the closure of any (WΓ , SΓ )-stratum F ′ of V n is a union of (WΛ , SΛ )-strata
F of V n .
Proof. A typical stratum F ′ of V n for WΓ is, by the definitions and (3.14.3), of the
(n)
form F ′ = u(XΓ ) = Xu(Γ) for some u ∈ WΛ and Γ ∈ IΓ . Replacing Γ by u(Γ),
we may assume without loss of generality that u = 1. It will therefore suffice to
(n)
establish the following formula: for Γ = (Γ = Γ0 , . . . , Γn ) ∈ IΓ :
[
(3.15.1)
wn · · · w1 (XΛ ).
XΓ =
w∈W n
(n)
Λ∈IΛ
P (w,Λ)

The union in (3.15.1) is taken over certain sequences w = (w1 , . . . , wn ) ∈ W n and
(n)
Λ = (Λ = Λ0 , . . . , Λn ) ∈ IΛ satisfying the conditions P (w, Λ)(i)–(ii) below
Γ

(i) For i = 1, . . . , n, wi ∈ i−1Wwi−1 ···w1 (Λi−1 ) .
(ii) For i = 1, . . . , n, Wwi ···w1 (Λi ) ⊇ WΓi .
For fixed P (w, Λ), we denote these conditions as (i)–(ii). Note that the condition (ii)
with i = n implies that for F ′ := XΓ and F := wn · · · w1 (XΛ ), one has StabWΛ (F ) =
Wwn ···w1 (Λn ) ⊇ WΓn = StabWΓ (F ′ ).
We shall prove (3.15.1) (and that the conditions (i)–(ii) make sense) by induction
on n. If n = 1, then (3.15.1) reduces to Lemma 3.12(b). Now assume by way of
induction that (3.15.1) holds and consider Γ′ = (Γ, Γn+1 ) = (Γ0 , . . . , Γn , Γn+1 ) ∈
(n+1)
IΓ
. Then
[
wn · · · w1 (XΛ ) × CΓn ,Γn+1
CΓ′ = XΓ × CΓn ,Γn+1 =
w∈W n
(n)
Λ∈IΛ
P (w,Λ)

(n)

Fix w ∈ W n and Λ ∈ IΛ satisfying P (w, Λ) and write w := wn · · · w1 . Then since
Ww(Λn ) ⊇ WΓn , Lemma 3.12(b) gives
[
wn · · · w1 (XΛ ) × CΓn ,Γn+1 = Xw(Λ) × CΓn ,Γn+1 = Xw(Λ) ×
w ′ (Cw(Λn ),Σ )
w ′ ,Σ

where the union is over all w ′ ∈ ΓnWw(Λn ) and Σ ⊆ w(Λn ) with Ww′ (Σ) ⊇ WΓn+1 .
Writing w ′ = wn+1 and Σ = w(Λn+1) gives
[
[
Xw(Λ) × wn+1 (Cw(Λn ),w(Λn+1 ) )
w ′(Cw(Λn ),Σ ) =
Xw(Λ) ×
w ′ ,Σ

wn+1 ,Λn+1

where the union on the right is taken over all wn+1 ∈ ΓnWw(Λn ) and Λn+1 ⊆ Λn
with Wwn+1 w(Λn+1 ) ⊇ WΓn+1 . Since wn+1 ∈ Ww(Λn ) = StabWΛ (Xw(Λ) ), it follows using

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

15

(3.14.3) and (3.9.3) that
Xw(Λ) × wn+1 (Cw(Λn ),w(Λn+1 ) ) = wn+1 (Xw(Λ) ) × wn+1 (Cw(Λn ),w(Λn+1 ) )
= wn+1 w(XΛ × CΛn ,Λn+1 ) = wn+1 w(XΛ′ )
(n+1)

where Λ′ := (Λ0 , . . . , Λn+1 ) ∈ IΛ
. Observe that the conditions imposed on
wn+1 , Λn+1 in the last union are precisely those which ensure that wn+1 , Λn+1 satisfy
the conditions on wi , Λi in (i), (ii) with i = n + 1, and that Λ′ := (Λ0 , . . . , Λn+1 ) ∈
(n+1)
IΛ
. Combining the last four displayed equations with this observation establishes the validity of (3.15.1) with n replaced by n + 1. This completes the inductive
step, and the proof of Theorem 3.15.

The proof of Theorem 3.4 is now complete.
3.16. Geometry. Define the dimension dim(C) of any non-empty cone C in a finitedimensional real vector space by dim(C) = dimR (RC), where RC is the subspace
spanned by C. It is well-known that the dimension of any cone contained in C \ C
is strictly smaller than the dimension of C.
3.17. Corollary. Maintain the hypotheses of Theorem 3.15, so that in particular
WΓ ⊆ WΛ .
(a) The closure of any (WΓ , SΓ )-facet F of V n is the disjoint union of F and of
(WΓ , SΓ )-strata of V n whose dimension is strictly less than dim(F ).
(b) Any (WΓ , SΓ )-stratum F ′ of V n is a union of certain (WΛ , SΛ )-strata F of
V n.
Proof. Write d := dim V . Let Γ be a simple subsystem and ∆ ⊆ Γ. Then one has
dim(CΓ,∆ ) := dim(RCΓ,∆ ) = d − |∆|. In fact, there is an isomorphism of vector
∼
∼
=
=
|Γ \ ∆|
→ Rd−|Γ| × R>0
and
→ Rd which induces a homeomorphism CΓ,∆ −
spaces V −
∼
=
|Γ \ ∆|
d−|Γ|
0
0
m
m−i
→R
× R≥0 where R≥0 = R := {0} and R is identified with R
× Ri
CΓ,∆ −
for 0 ≤ i ≤ m.
(n)
It follows from the above and the definitions that for Γ ∈ IΓ , one has dim(XΓ ) =
Pn
Pn
∼
n =
−
→ Rnd ini=1 |Γi |. Also, there is an isomorphism V
i=1 dim CΓi−1 ,Γi = nd −
ducing homeomorphisms
(3.17.1)

XΓ ∼
= Rnd−

Pn

i=1

|Γi−1 |

|Γ −Γn |

× R>00

,

XΓ ∼
= Rnd−

Pn

i=1

|Γi−1 |

|Γ −Γn |

× R≥00

Note that any cone in XΓ \ XΓ has dimension strictly smaller than that of XΓ
(either by the general fact mentioned above or by a direct check in this case). Part
(a) follows readily from this fact and the special case of Theorem 3.15 in which
Λ = Γ. Given (a), (b) follows from Theorem 3.15 by induction on dim(F ′ ) as
follows: (b) holds vacuously for strata F ′ of negative dimension (since there are
none). In general, F ′ is a union of (WΛ , SΛ )-strata by Theorem 3.15, F ′ \ F ′ is
a union of (WΓ , SΓ )-strata of dimension less than dim(F ′ ) by (a) and hence is a
union of (WΛ , SΛ )-strata by induction, and therefore F ′ = F ′ \ (F ′ \ F ′) is a union
of (WΛ , SΛ )-strata as asserted.


16

M. J. DYER AND G. I. LEHRER

3.18. Character formulae. We finish this section with a character-theoretic application of Theorem 3.4. Assume for simplicity that RΠ = V . The Coxeter complex
{w(CS,J ) for J ( S, w ∈ W } provides a subdivision of the unit sphere S(V ) in
V ∼
= Rℓ into spherical simplices. Applying the Hopf trace formula to the resulting
chain complex, and recalling that dim(w(CS,J ) ∩ S(V )) = ℓ − 1 − |J|, one obtains
the familiar character formula (due to Solomon)
(3.18.1)

det V (w) =

X

(−1)|J| IndW
WJ (1)(w) for w ∈ W.

J⊆S

It follows from Theorem 3.4 and (3.17.1) that the intersections of the strata
w(CXI ), where w ∈ W and I 6= (S, S, . . . , S), with the unit sphere S(V n ) give
a subdivision of S(V n ) into spherical cells (homeomorphic to open balls), and one
may again apply the Hopf trace formula to the resulting chain complex. A straightforward computation, using the fact that for any u ∈ W , dim(uXI ∩ S(V n )) =
nℓ − (|I1 | + |I2 | + · · · + |In |) − 1, then shows that, given the formula (3.18.1), we have
the following formula for w ∈ W .
(3.18.2)
det V n (w) = det V (w)n =

X

(−1)(|I1 |+|I2 |+···+|In |) IndW
WIn (1)(w).

I=(I0 ,I1 ,...,In )∈I (n)

3.19. Remark. It is an easy exercise to show that for fixed In ⊆ S, we have

n|S|

if In = S
(−1)
X
(|I1 |+|I2 |+···+|In |)
(−1)
= 0 if In ( S and n is even

(−1)|In | if I ( S and n is odd.
(n)
I=(I0 ,I1 ,...,In )∈I
n

It follows that when n is even, (3.18.2) is amounts to the statement that detn =
1W , while if n is odd, the right side of (3.18.2) reduces to the right side of (3.18.1),
and therefore amounts to the statement that for n odd, detn = det.
4. Topological and combinatorial properties of the stratification.
Maintain the assumptions of §3.3. Write d := dim(V ). Let F = F (n) :=
(n)
{ w(XI ) | I ∈ IW , w ∈ W } denote the set of all W -strata of V n , partially ordered by inclusion of closures of strata; i.e. for F, G ∈ F (n) , we say that F ≤ G
if F ⊆ G. The fact that this defines a partial order follows from Corollary 3.17(a).
Note that W acts naturally on F (n) as a group of order preserving automorphisms.
(n)
(n)
(n)
Let F
:= { (w, I) | I ∈ IW , w ∈ W In }. The map (w, I) 7→ w(XI ) : F
→
(n)
F is a bijection, by Theorem 3.4(b). We use this bijection to transfer the partial
(n)
order and the W -action on F (n) to a partial order and the W -action on F .
(n)
Using (3.15.1), one sees that this partial order and W -action F
have a purely
combinatorial description in terms of the Coxeter system (W, S); this is in analogy
with the description of the Coxeter complex in terms of cosets of standard parabolic

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

17

(n)

would be unchanged (as poset with W -action) if it
subgroups. In particular, F
had been defined using the diagonal W -action on (RΦ)n instead of that on V n .
(n)

(n)

4.1. Lemma. Let I− := (S, S, . . . , S) ∈ IW and I+ := (S, ∅, . . . , ∅) ∈ IW .
(a) The poset F (n) has a minimum element 0̂ := XI− .
(b) The elements w(XI+ ) for w ∈ W are the distinct maximal elements of F (n) .
Proof. Note that by Theorem 3.4, 0̂ is fixed by the W -action. To show that 0̂ is
(n)
the minimum element of F , it therefore suffices to show that if I ∈ IW , one has
0̂ ⊆ XI . This is clear since by (3.3.2) and (3.2.2)
XI = CI0 ,I1 × . . . × CIn−1 ,In ⊇ CI0 ,I0 × . . . × CIn−1 ,In−1 ⊇ CS,S × . . . × CS,S = XI−
since CI,I is the set of all points of V fixed by WI . This proves (a).
(n)
A similar argument using C∅,∅ = V shows that XI+ ⊇ XI for all I ∈ IW . Since by
(n)
definition F = { w(XI) | w ∈ W, I ∈ IW }, this implies that any maximal stratum
in F is of the form w(XI+ ) for some w ∈ W . But W acts simply transitively on
the set of these strata and there is at least one maximal element, so (b) follows. 
4.2. Topology. In this subsection, we discuss basic topological facts about the
stratification of V n . For m ∈ N, let Bm denote the standard m-ball in Rm and Sm−1
its boundary, the standard m-sphere (with S−1 := ∅).
Let U be a finite-dimensional real vector space. A ray in U is a subset of U of the
form R>0 v for some non-zero v ∈ U. Let R = RU := { R>0 v | v ∈ U, v 6= 0 } denote
the set of all rays in U. Topologise R as follows. Let K be a convex body (i.e. a
compact convex set with non-empty interior) in U with 0 in its interior, so that K
contains a small ball with centre 0. Let ∂(K) denote the boundary of K i.e. the set
of all non-interior points of K. The map v 7→ R>0 v : ∂(K) → R is a bijection and
we topologise R so this map is a homeomorphism. A compactness argument shows
the resulting topology is independent of choice of K. Taking K as the unit sphere
in U (with respect to some Euclidean space structure on U) gives R ∼
= Sdim(U )−1 .
There is a map C 7→ [C] := { R>0 v | v ∈ C \ {0} } taking convex cones C in U
to subsets of R. Clearly, [C] = [C ′ ] if and only if C ′ ∪ {0} = C ∪ {0}. This map
satisfies [C] = [C] and [Int(C)] = Int([C]) where X and Int(X) denote respectively
the closure and interior of a subspace X of the ambient topological space (U or R).
We apply the above considerations with U = V n . Recall that here dim(V ) = d.
4.3. Lemma.
(a) [0̂] = [XI− ] ∼
= Sn(d−|S|)−1 .
Pn
(n)
(b) If F = w(XI ) ∈ F \ {0̂}, then [F ] ∼
= BN where N = nd − 1 − i=1 |Ii |
and [F ] ∼
= BN \ ∂(BN ) := BN \ SN −1 .
(n)

Proof. Note that for I ∈ IW , one has XI = XI− if and only if I0 = In . Using
(3.17.1) and the independence of the topology on RV n from the choice of compact
body K in its definition, it suffices to verify that for m ≤ n ≤ M in N with n ≥ 1,

18

M. J. DYER AND G. I. LEHRER

the following equations hold in RM = Rm × Rn−m × RM −n :
(
Sn−1 , if m = n
n−m
(Rm × R≥0
) ∩ SM −1 ∼
=
Bn−1 , if m < n
n−m
and (Rm × R>0
) ∩ SM −1 ∼
= ∂(Bn−1 ) if m < n. Details are omitted.



4.4. Regular cell decompositions. We shall use below notions of regular cell
complexes and their face posets, and shellability of posets and complexes. A convenient reference for the definitions and facts required is [1, 4.7].
4.5. Proposition.
(a) Suppose that V = RΠ. Then { [F ] | F ∈ F \ {0̂} } is
(the set of open cells of ) a regular cell decomposition of RV n ∼
= Snd−1 where
d := dim V = |S|.
(n)
(b) The poset F \ {0̂} is the face poset of a regular cell decomposition of
Sn|S|−1.
Proof. First we prove (a). Let Ω := F \ {0̂}, regarded as poset. For F ∈ Ω, call
[F ] ⊆ R an open cell and its closure [F ] = [F ] a closed cell. By Corollary 3.17 and
Lemma 4.3 (and the discussion preceding its statement), each closed cell [F ] = [F ]
is a ball in R with [F ] as its interior and with boundary
[F ] \ [F ] = ∪G∈Ω,G<F [G] = ∪G∈Ω,G<F [G]
equal to a union of closed cells. Since R is Hausdorff, (a) follows by the definition
(n)
in [1]. By the discussion immediately preceding Lemma 4.1, F \ {0̂} is the face
poset of the regular cell complex in (a), and (b) follows.

4.6. Proposition 4.5 has a number of combinatorial and topological consequences
c(n) := F (n) ∪ {1̂} obtained by
listed in [1, 4.7]. In particular, the finite poset F
(n)
formally adjoining a maximum element 1̂ to F is graded (i.e. has a minimum element and a maximum element, and all its maximal chains (totally ordered subsets)
c(n) is called the extended face poset of the
have the same cardinality). Note that F
regular cell complex in Proposition 4.5(a).
We conclude with the remark that significant parts of the above results (though
not the regular cell subdivisions of spheres in Proposition 4.5, for example) extend
mutatis mutandis to the diagonal action of infinite Coxeter groups on powers U n of
their Tits cone U.
5. Applications to conjugacy of sets of roots and vectors
5.1. Let Rm×n denote the set of real m × n matrices and A 7→ At denote the matrix
transpose. Identify Rn = Rn×1 .
We shall be addressing the classification of certain tuples (v1 , . . . , vn ) ∈ V n up to
order, under the action of W . Evidently the W -action leaves invariant the set of
inner products hvi , vj i, 1 ≤ i, j ≤ n. Arranging these inner products in a matrix
motivates the following definition.

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

19

5.2. Definition. A genus of rank n is a n × n matrix σ = (ai,j )ni,j=1 of real numbers.
The symmetric group Symn acts on the set G (n) = Rn×n of genera of rank n in
such a way that, for ρ ∈ Symn and σ ∈ G (n) , one has ρσ := σ ′ = (a′i,j )ni,j=1 where
a′i,j = aρ−1 (i),ρ−1 (j) . The automorphism group Gσ of the genus σ is the stabiliser of
σ in this action. Two genera of the same rank n are said to be of the same type if
they are in the same Symn orbit on G (n) . More formally, we define a type of genera
of rank n to be a Symn orbit on G (n) and write I(σ) := Symn σ (the Symn -orbit of
σ) for the type of the genus σ.
For example, the Cartan matrices of fixed rank n can be regarded as genera, and
two of the them have the same type (in the above sense) if and only if they have the
nA
nG
same type (of form A1 1 × · · · × G2 2 ) in the usual sense. Similar remarks apply
to generalised Cartan matrices. Thus, we may regard types, in the usual sense, of
(generalised) Cartan matrices as (special) types of genera in the above sense
5.3. For a natural number n, an ordered system of rank n in V is by definition, an
n-tuple b = (β1 , . . . , βn ) ∈ V n . The Gram genus of b = (β1 , . . . , βn ) is defined to be
the Gram matrix C ′ (b) := (ci,j )i,j=1,...,n ∈ Rn×n where ci,j := h βi , βj i. This defines
a Symn -equivariant map C ′ : V n → G (n) .
Note that σ := C ′ (b) is a positive semidefinite matrix, and is positive definite
(equivalently, it is invertible) if and only if P
[b] is linearly independent. More generally, the space {a = (a1 , . . . , an )t ∈ Rn | ni=1 ai βi = 0} of linear relations on b
identifies with the radical of the quadratic form a 7→ at C ′ (b)a : Rn → R with matrix
C ′ (b) with respect to the standard basis of Rn . It readily follows that the non-empty
fibres of the map C ′ are the orbits of the orthogonal group OV := O(V, h −, − i) acting diagonally on V n (and the set of matrices over which the fibres are non-empty
is precisely the set of positive semidefinite, symmetric matrices in Rn×n ). Also, for
1 ≤ i < j ≤ n, one has βi = βj if and only if the i-th and j-th columns (or rows)
of C ′ (b) are equal. In particular, the columns of σ are pairwise distinct if and only
if β1 , . . . , βn are pairwise distinct. In that case, Gσ is isomorphic to the group of
isometries of the metric space [b] (with metric induced by the norm from h −, − i).
5.4. Suppose that Φ is crystallographic. If b ∈ Φn is an ordered system of
rank n consisting of roots in Φ, its Cartan genus is the matrix of inner products
C ′′ (b) := (h βi∨, βj i)ni,j=1. This defines a Symn -equivariant map C ′′ : Φn → G (n) .
Again, one has βi = βj if and only if the i-th and j-th columns of C ′′ (b) are equal.
Note that b is an ordered simple system if and only if its Cartan genus C ′′ (b) is a
Cartan matrix. Similarly, b is an ordered np subset of rank n, if and only if C ′′ (b)
is a generalised Cartan matrix (which then necessarily has only finite and affine
components). Clearly, the Cartan genus C ′′ (b) of b ∈ Φn is completely determined
by the Gram genus C ′ (b).
5.5. Remark. If b is an ordered simple system in Φ, the automorphism group (see
Definition 5.2) of C ′′ (b) is known as the group of diagram automorphisms of Φ.
5.6. In 5.6–5.10, fix a subgroup W ′ of OV and a W ′ -stable subset Ψ of V . The
main situation of interest is that in which (W ′ , Ψ) = (W, Φ), but other cases such

20

M. J. DYER AND G. I. LEHRER

as when (W ′ , Ψ) is (W, V ) or (OV , V ) are also of interest. Fix a natural number n.
Let
 
Ψ
:= { Γ ⊆ Ψ | |Γ| = n } ⊆ P(Ψ)
(5.6.1)
n

be the “configuration space” of n distinct unordered points of Ψ. This has a natural
W ′ action given by (w, Γ) 7→ w(Γ) := { wγ | γ ∈ Γ } for w ∈ W ′ and Γ ∈ Ψn .

Our main interest will be the study of W ′ -orbits on Ψn . With this in mind, define
the configuration space
Ψ(n) := { b = (β1 , . . . , βn ) ∈ Ψn | βi 6= βj if i 6= j }

of n ordered distinct points of Ψ. Then Ψ(n) admits the diagonal W ′ -action and a
commuting Symn -action by place permutations, hence a natural W ′ × Symn -action.

Moreover, there is a natural W ′ -equivariant surjection π̂ : b 7→ [b] : Ψ(n) → Ψn .
The fibres of π̂ are the Symn -orbits on Ψ(n) , and Symn acts simply transitively on
each fibre. As in (2.6.1), we may canonically identify the set of W ′ -orbits on Ψn as

Ψ
(5.6.2)
/W ′ ∼
= (Ψ(n) / Symn )/W ′ ∼
= Ψ(n) /(W ′ × Symn ) ∼
= (Ψ(n) /W ′ )/ Symn .
n

5.7. There is a natural Symn -equivariant map b 7→ C ′ (b) : Ψn → G (n) which assigns to b ∈ Ψn its Gram genus. As a variant in the case (W ′ , Ψ) = (W, Φ), one
can consider instead the map b 7→ C ′′ (b) : Ψn → G (n) which assigns to b its Cartan
genus; C ′′ is also Symn -equivariant. Let C : Ψn → G (n) be one of the maps C ′ or C ′′ .
(n)
Let G0 be the subset of G (n) consisting of matrices with pairwise distinct columns.
As remarked already, for b = (β1 , . . . , βn ) ∈ Ψn , one has b ∈ Ψ(n) (i.e. βi 6= βj for
(n)
i 6= j) if and only if one has C(b) ∈ G0 .
For a genus σ of rank n and type τ := I(σ), we write S (σ) := C −1 (σ) for the
fibre of C over σ and
[
ρ(S (σ))
T (τ ) := { b ∈ Ψ(n) | C(b) ∈ τ } =
ρ∈Symn

for the union of the fibres of C over all genera of the same type as σ. Let

U (τ ) := { [b] | b ∈ T (τ ) } = { [b] | b ∈ S (σ) } ⊆ P(Ψ).

(n)
(n)
One has U (τ ) ⊆ Ψn if and only if σ ∈ G0 (or equivalently, τ ⊆ G0 ). If it is
necessary to indicate dependence on C, we write SC (τ ) etc.
For example, if (W ′ , Ψ, C) = (W, Φ, C) and τ is a type of Cartan matrices, then
T (τ ) (resp., U (τ )) is the set of all ordered (resp., unordered) simple subsystems of
Φ of that type. Similarly if τ is a type of generalised Cartan matrices, then T (τ )
(resp., U (τ )) is the set of ordered (resp., unordered) np subsets of that type (this
set being empty unless all components of τ are of finite or affine type).
In general, each set S (σ), T (τ ) and U (τ ) is W ′ -stable. In particular, the classification of W ′ -orbits on Ψn is reduced to the classification of W ′ -orbits on U (τ )
(n)
for each type of genus τ ⊆ G0 (with U (τ ) 6= ∅). We describe an approach to this
classification along lines similar to (5.6.2), by study of S (σ) and T (σ)

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

21

5.8. Let σ ∈ G (n) and τ := I(σ). One has a commutative diagram
S (σ) 

/

T (τ )

πσ

π




U (τ )

U (τ )

of W ′ -sets in which the top horizontal map is an inclusion, and π and πσ are the
indicated restrictions of the map π̂ : b 7→ [b]. Since C is Symn -equivariant and
τ = I(σ) is the Symn -orbit of σ, Symn acts naturally on T (τ ), commuting with
its W ′ -action. In this way, T (τ ) acquires a natural structure of W ′ × Symn -set.
By restriction, S (σ) has a natural structure of W ′ × Gσ -set. This W ′ × Gσ -set
S (σ) depends only on the type of σ up to natural identifications. More precisely,
let σ ′ = (a′i,j )i,j=1,...,n be another genus of type τ , say σ ′ = ρσ where ρ ∈ Symn .
Then Gσ′ = ρGσ ρ−1 and S (σ ′ ) = ρS (σ); in fact, the map p : S (σ) → S (σ ′ )
given by b 7→ ρb is a W ′ -equivariant bijection and satisfies p(νb) = (ρνρ−1 )p(b) for
b ∈ S (b) and ν ∈ Gσ .
(n)

(n)

5.9. Assume henceforth that σ ∈ G0 , so τ ⊆ G0 . Then the Symn -orbits on
T (τ ) (resp, Gσ -orbits on S (σ)) are precisely the fibres of π (resp., πσ ) and Symn
(resp., Gσ ) acts simply transitively on each fibre of π (resp., πσ ). (There is even a
natural isomorphism of W′ × Symn -sets Symn ×Gσ S (σ) ∼
= T (τ ).) Hence we may
∼
∼
naturally identify U (τ ) = T (τ )/ Symn = S (σ)/Gσ as W′ -sets. From (2.6.1), one
gets canonical identifications
(5.9.1)

U (τ )/W ′ ∼
= (T (τ )/W ′ )/ Symn ∼
= (S (σ)/W ′)/Gσ .
(n)

In the case (W ′ , Ψ, C) = (OV , V, C ′ ), each non-empty set S (σ) forσ ∈ G0 is a
single W ′ -orbit on Ψ(n) (see 5.3) and thus U (τ ) is a W ′ -orbit on Ψn . The above
equations and the discussion of 5.3 therefore give rise to the following closely related
parameterisations of the set of OV -orbits of unordered sets of n (distinct) points in
V : the set of such orbits corresponds bijectively to the set of symmetric, positive
semidefinite real n × n matrices with distinct columns, modulo simultaneous row
and column permutation (resp., to a set of representatives of such matrices under
such permutations).
5.10. Remarks. For any groups G, H there is a natural analogue in the category of Gsets of the standard notion of principal H-bundle. Transferring standard terminology
from the theory of principal bundles to this setting, the above shows that (assuming
(n)
σ ∈ G0 ), π
b and π are principal Symn -bundles and πσ is a principal Gσ -bundle
which affords a reduction of the structure group of the bundle π from Symn to
Gσ . Moreover, the bundle πσ depends (up to natural identifications of the various
possible structure groups Gσ induced by inner automorphisms of Symn ) only on the
type of σ and not on σ itself. Simple examples show that in general, these bundles
are not trivial bundles.

22

M. J. DYER AND G. I. LEHRER

5.11. From now on, we restrict to the case W ′ = W , which is of primary interest
here. Recall the definition of the dot action of Symn on the fundamental region
(n)
(n)
(n)
CW : for ρ ∈ Symn and b ∈ CW , one has {ρ · b} = W ρb ∩ CW . Recall also the
maps C ′ , C ′′ which associate to an n-tuple of roots its associated Gram genus and
Cartan genus respectively.
(n)

5.12. Proposition. Let σ ∈ G0 and τ := I(σ). The set U (τ )/W of W -orbits
on U (τ ) may be canonically identified with S (σ)/Gσ where RC (σ) = R(σ) :=
(n)
(n)
S (σ) ∩ CW (resp., with (T (τ ) ∩ CW )/ Symn ) and the Gσ (resp., Symn -action)
used to form the coset space is a restriction of the dot action.
Proof. This follows immediately from (5.9.1) (with W ′ = W ) on identifying T (τ )/W
(n)
(n)
with CW ∩ T (τ ) and S (σ)/W with CW ∩ S (σ) (the corresponding actions by
Symn and Gσ identify with restrictions of the dot action).

5.13. We now record some consequences of the bijection U (τ )/W ∼
= R(σ)/Gσ in
the case (W ′ , Ψ) = (W, Φ), for the classification of W -orbits of simple subsystems
of Φ; similar results hold for W -orbits of arbitrary subsets of roots (or indeed, of
vectors in V ). For any b ∈ Φ(n) , we call I(C ′ (b)) (resp., I(C ′′ (b))) the Gram type
(resp., Cartan type) of b and of [b] ∈ Φn . Note that specifying the Gram type of a
(unordered or ordered) simple system amounts to specifying its Cartan type (i.e. its
Cartan matrix up to reindexing) together with the function taking each irreducible
component to the maximal root length in that component.
By a Cartan (resp., Gram) genus of simple systems we mean the Cartan (resp.,
Gram) genus of some ordered simple system of some crystallographic (resp., arbitrary) finite root system. Thus, a Cartan genus of ordered simple systems is just a
Cartan matrix (ai,j )ni,j=1 (with a specified indexing by 1, . . . , n).
5.14. Proposition. Let σ be a Cartan (resp., Gram) genus of ordered simple systems
and C := C ′′ (resp., C := C ′ ). Let τ := I(σ).
(a) There is a natural bijection between W -orbits of simple systems of Cartan
(resp., Gram) type τ in Φ and Gσ -orbits for the dot action on RC (σ).
(b) The conjugacy classes of simple subsystems of Cartan (resp., Gram) type τ
are exactly the conjugacy classes of simple systems [b] for b ∈ RC (σ).
(c) The number of W -orbits of simple systems of Cartan (resp., Gram) type τ
in Φ is at most the number |{ [b] | b ∈ RC (σ) }| of simple systems [b] which
underlie some ordered simple system b in RC (σ).
(d) If |RC (σ)| = 1, or, more generally, RC (σ) 6= ∅ and all ordered simple systems
b in RC (σ) have the same underlying simple system [b], then there is a single
W -conjugacy class of simple systems in Φ of Cartan (resp., Gram) type τ .
Proof. Since SC (σ) is the set of ordered simple systems of Φ with Cartan (resp.,
Gram) matrix σ, part (a) follows from Proposition 5.12. We prove (b). Note that
if ∆ is a simple subsystem of Cartan (resp. Gram) type τ , then ∆ = [b] for some
b ∈ S (σ). We have wb ∈ R(σ) for some w ∈ W . Thus, w∆ = [wb]. So the W orbit of any simple subsystem ∆ of Cartan (resp., Gram) type τ has a representative

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

23

w∆ in RC (σ). On the other hand, for any b ∈ R(σ), one has [b] ∈ T (τ ). Part (b)
follows. Part (c) follows directly from (b), and part (d) follows from (c).

5.15. Remarks. For fixed Φ and any Cartan genus σ of ordered simple systems of type
τ , there are finitely many Gram genera σ1 , . . . , σn of types τ1 , . . . , τn respectively,
such that SC ′′ (σ) = ⊔ni=1 SC ′ (σi ), RC ′′ (σ) = ⊔ni=1 RC ′ (σi ) TC ′′ (τ ) = ∪ni=1 TC ′ (τi ) and
UC ′′ (τ ) = ∪ni=1 UC ′ (τi ). Gram genera are convenient in that there are typically fewer
simple systems of a specified Gram genus than a corresponding Cartan genus; on
the other hand, it may be inconvenient to specify the several Gram genera necessary to describe all (ordered) simple systems of a specified Cartan genus (e.g. an
isomorphism type of simple subsystems).
As an application, we shall show that if L is the length of some root of Φ, there
is a unique conjugacy class of simple subsystems of Gram type I(L · Idn ) with n
maximal.
5.16. Proposition. Let L ∈ R>0 . Consider simple subsystems ∆ of Φ satisfying the
three conditions below:
(i) ∆ is of type An1 for some n.
(ii) All roots in ∆ have the same length L.
(iii) ∆ is inclusion maximal subject to (i) and (ii).
Then any two subsystems of Φ satisfying (i)–(iii) are conjugate under W .
Proof. Following Proposition 2.10, we begin by describing all the ordered simple
(n)
subsystems b = { β1 , . . . , βn } in CW satisfying (i)–(ii). First, β1 is a dominant
root of length L of some component of Φ. The possibilities for β2 are identified by
deleting from the affine Dynkin diagram of Π ⊔ {−β1 } the root −β1 and all roots
attached to −β1 . One then obtains a simple system for a new root (sub)system and
repeats the above process. At each stage, the only choice is in which component
(containing roots of length L) of the relevant root system one takes the dominant
root of length L. The condition (iii) will be satisfied if and only if none of the
components at that stage contain roots of length L. By the maximality assumption
(iii), the integer n and the set [b] = { β1 , . . . , βn } is clearly independent of the order
in which these operations are done.
This implies that all ordered simple systems which satisfy (i)–(iii) and which
lie in the fundamental region are (certain) permutations of each other. It follows
by Proposition 5.14(c), there is just one conjugacy class of unordered subsystems
satisfying (i)–(iii).

5.17. Action of the longest element. For any simple subsystem ∆ of Φ, we
denote by ω∆ the longest element of the Coxeter system (W∆ , S∆ ) where S∆ =
{ sα | α ∈ ∆ } and W∆ := h S∆ i. It is well known that there is some permutation
ρ∆ of ∆ (which may and often will be regarded as a diagram automorphism of the
Dynkin diagram of ∆) such that ω∆ (α) = −ρ∆ (α) for all α ∈ ∆. In case ∆ = Π,
we write ωΠ = ω and ρΠ = ρ. Extending the permutation action on Π linearly,
we sometimes regard ρΠ as an automorphism of Φ which fixes the set Π. If b is an

24

M. J. DYER AND G. I. LEHRER

ordered simple subsystem of Φ, then [b] is a simple subsystem of Φ and so ωb := ω[b]
and ρb := ρ[b] are defined as above.
Recall from Proposition 5.12 that the set RC (σ) of ordered np sets of genus σ
in the fundamental region has a Symn dot action. In general, this action may not
be easy to describe explicitly. However, the following useful special result gives
a simple description of the effect of applying diagram automorphisms induced by
longest elements, to simple subsystems in the fundamental region for W .
(n)

5.18. Proposition. Let b := (β1 , . . . , βn ) ∈ Φn ∩ CW be an ordered simple system
(for some root subsystem Ψ of Φ) which lies in the fundamental region for W on V n .
(n)
Then W (ρb (β1 ), . . . , ρb (βn )) ∩ CW = {(ρ(β1 ), . . . , ρ(βn ))} i.e. (ρ(β1 ), . . . , ρ(βn )) is
the representative, in the fundamental region for W on V n , of the ordered simple
subsystem (ρb (β1 ), . . . , ρb (βn )) of Ψ.
Proof. One has ωb (β1 , . . . , βn ) = −(ρb (β1 ), . . . , ρb (βn )) and therefore (β1 , . . . , βn ) =
(n)
−ωb (ρb (β1 ), . . . , ρb (βn )) ∈ CW since ωb is either an involution or the identity. By
(n)
2.13, CW ∩ Φn is −ω-invariant. Hence
(n)

−ω(β1 , . . . , βn ) = (ωωb)(ρb (β1 ), . . . , ρb (βn )) ∈ CW ∩ Φn .
Since ωωb ∈ W and −ω(βi ) = ρ(βi ), the proposition follows from Theorem 2.2. 
6. Simple subsystems of type A
In this section, Φ denotes a crystallographic root system. We shall define a standard genus σn of type An and determine all ordered simple subsystems of Φ of
Cartan genus σn in the fundamental region for W . The following result of Oshima
plays an important role in the proof of this and related results in arbitrary type.
6.1. Lemma. [14] Suppose that W is an irreducible finite Weyl group with crystallographic root system Φ. Fix ∆ ⊆ Π, and scalars cβ ∈ R for β ∈ ∆, not all zero,
and l ∈ R. Let γ[α] denote the coefficient of α ∈ Π in γ, expressed as a linear
combination of Π, and write
X := { γ ∈ Φ | h γ, γ i1/2 = l and γ[β] = cβ for all β ∈ ∆ }.
If X is non-empty, then it is a single WΠ \ ∆ -orbit of roots. Equivalently, |X ∩
CΠ \ ∆ | ≤ 1.
This is proved in [14, Lemma 4.3] using facts in [13] from the representation theory
of semisimple complex Lie algebras. We shall give a direct proof of a more general
version of Lemma 6.1 in a future work [7], but in this paper make use of it only as
stated.
6.2. Definition. The standard genus σn of Cartan type An is the Cartan matrix of
type An in its standard ordering as in [2] i.e. σn = (ai,j )ni,j=1 where for 1 ≤ i ≤ j ≤ n,
one has aij = 2 if i = j, ai,j = aj,i = −1 if j − i = 1 and ai,j = aji = 0 if j − i > 1.
Thus, for b := (β1 , . . . , βn ) ∈ Φn , one has b ∈ SC ′′ (σn ) if and only if one has
(h βi∨ , βj i)i,j=1,...,n = σn . Clearly, the automorphism group Gσn of σn is trivial if

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

25

n = 1 and is cyclic of order 2 if n > 1, with non-trivial element the longest element
(n)
of Symn in the latter case. Recall that RC ′′ (σn ) := SC ′′ (σn ) ∩ CW .
6.3. Proposition. Let b = (β1 , . . . , βn ) ∈ RC ′′ (σn ) where n ≥ 1. Then β1 is a
dominant root of Φ and β2 , . . . , βn ∈ −Π.
Proof. The proof is by induction on n. Since h βi∨, βi+1 i = −1 for i = 1, . . . , n − 1,
β2 , . . . , βn lie in the same component of Φ as β1 . Without loss of generality, we may
replace Φ by the component containing β1 and assume that Φ is irreducible. For
n = 1, one has β1 ∈ Φ ∩ CW and the result is clear. Now suppose that n > 1. Since
b′ := τn−1 (b) = (β1 , . . . , βn−1 ) ∈ RC ′′ (σn−1 ), induction gives β2 , . . . , βn−1 ∈ −Π. Let
Γ := Π ∩ {β1 , . . . , βn−1 }⊥ and ∆ := Π ∩ {β1 , . . . , βn−2 }⊥ ⊇ Γ. Then WΓ = Wb,n−1
and W∆ = Wb,n−2 by Theorem 2.2. One has sβn ∈ Wb,n−2 = W∆ since h βn , βi i = 0
for i < n − 1, so Σ := supp(βn ) ⊆ ∆. Define
P

Λ := { γ ∈ Σ | h γ, βn−1 i =
6 0 }.

Since 0 6= h βn , βn−1 i = γ∈Σ βn [γ]h γ, βn−1 i, it is clear that |Λ| ≥ 1.
We claim that |Λ| = 1. Suppose to the contrary that |Λ| ≥ 2. We have βn−1 6∈ ∆.
If n ≥ 3, then the full subgraph of the Dynkin diagram on vertex set Σ∪{−βn−1 } ⊆ Π
contains a cycle, contrary to finiteness of W . Hence n = 2, βn−1 ∈ Φ ∩ CW , ∆ = Π
and |{γ ∈ Π | h βn−1 , γ i > 0}| ≥ |Λ| ≥ 2. The classification implies that Φ is of type
Am with m > 1. Regard Π ∪ {−β1 } as the vertex set of the affine Dynkin diagram
corresponding to Φ. Then the full subgraph on {−β1 } ∪ Σ contains a cycle, so its
vertex set is equal to Π ∪ {−β1 }. This implies that Σ = Π and β2 = βn = ±β1
contrary to h βn−1, βn∨ i = −1. This proves the claim.
Write Λ = {α} where α ∈ Π and abbreviate c := βn [α] 6= 0. We have
X
h βn , βn−1 i =
βn [γ]h γ, βn−1 i = βn [α]h α, βn−1 i = ch α, βn−1 i.
γ∈Σ

∨
Writing h βn , βn i = h βn−1 , βn−1 i = dh α, α i, it follows that −1 = h βn , βn−1
i =
∨
∨
∨
ch α, βn−1 i. Since c, h α, βn−1 i ∈ Z, this implies c = −h α, βn−1 i ∈ {±1}. We have
∨
h α, βn−1
i ≥ 0 since α ∈ Π and βn−1 is a dominant root if n = 2, and −βn−1 6= α
∨
are both in Π if n > 2. Hence c = h α, −βn−1
i = −1. Also, −1 = h βn−1, βn∨ i =
c
h βn−1, α∨ i so h βn−1 , α∨ i = −dc−1 = d ∈ Z≥1 . From
d
X
X h γ, γ i
2
2βn
βn [γ]γ =
=
βn [γ]γ ∨
βn∨ =
h βn , βn i
h βn , βn i γ∈Π
h βn , βn i
γ∈Π

i
βn [α] = dc = −d−1 so d = 1.
one sees that Z ∋ βn∨ [α∨ ] = h hβα,α
n ,βn i
We have now established that h α, α i = h βn−1 , βn−1 i = h βn , βn i. This implies
∨
h βn−1 , α∨ i = h α, βn−1
i = 1. To complete the proof, it will suffice to show that
βn = −α. Taking Σ = supp(βn ) as above, let Ψ := WΣ Σ; this is an irreducible
crystallographic root system. From above, one has βn , −α ∈ Ψ, βn [α] = −1 =
(−α)[α] and h βn , βn i = h α, α i. By Lemma 6.1, it follows that βn and −α are in the
⊥
⊥
⊆ ∆∩βn−1
=
same WΣ′ -orbit on ΦΣ where Σ′ := Σ \ {α}. But Σ′ = Σ \ Λ = Σ∩βn−1
Γ. Hence βn ∈ CWΓ ⊆ CWΣ′ . We also have −α ∈ CWΣ′ . From Lemma 1.6, it follows
that βn = −α as required.


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M. J. DYER AND G. I. LEHRER

The following Corollary 6.4 and Theorem 6.7 are closely related to the type An
case of the result [14, Theorem 3.5(i)] of Oshima, while Remark 6.5 is related to
that result in some other classical types.
6.4. Corollary. Let Pn denote the set of all pairs (β, Γ) such that β is a dominant
root of Φ, Γ is the vertex set of a type An subdiagram of the Dynkin diagram of
{β} ∪ −Π and β is a terminal vertex of Γ. Then there is a bijection RC ′′ (σn ) → Pn
defined by b = (β1 , . . . , βn ) 7→ (β1 , [b]) for all b ∈ RC ′′ (σn ).
Proof. Let b ∈ RC ′′ (σn ). Then Proposition 6.3 implies that [b] is the vertex set of a
subdiagram of the Dynkin diagram of {β1 } ∪ −Π. By definition of σn , this diagram
is of type An with β1 terminal and βi joined to βi+1 for i = 1, . . . , n − 1. Conversely,
suppose (β, Γ) ∈ Pn . Since Γ is of type An with β terminal, we may uniquely write
Γ = [b] for some b = (β1 , . . . , βn ) ∈ S (σn ) with β1 = β. We have to show that
(n)
b ∈ CW i.e. for i = 1, . . . , n, βi ∈ CWΠi . where Πi := Π ∩ {β1 , . . . , βi−1 }⊥ . One has
β1 ∈ C by the assumption β1 is dominant. To show βi ∈ WΠi for 2 ≤ i ≤ n, we
have to show that h βi , γ i ≥ 0 for γ ∈ Πi . But γ 6= βi since h βi , βi−1 i < 0. Hence
h γ, βi i ≥ 0, since γ ∈ Π and βi ∈ −Π.

6.5. Remark. An argument similar to that in the last part of the proof shows the
following: Let β1 ∈ Φ be a dominant root and β2 , . . . , βn ∈ −Π be pairwise distinct
and such that for each i = 2, . . . , n, there is some j < i with h βi , βj i =
6 0. Then
(n)
(β1 , . . . , βn ) is an ordered np subset of Φ which lies in CW .
6.6. Conjugacy of simple subsystems of type A. Let ρ denote the automorphism ρ = ρΠ = −ωΠ of Φ induced by the action of the negative of the longest
element ωΠ of W (see 5.17). One has ρ2 = IdΦ and ρ fixes each dominant root.
Hence H := h ρ i = {1, ρ} acts naturally on Pn , where Pn is defined in Corollary 6.4.
6.7. Theorem. The set of W -conjugacy classes of simple systems of type An in Φ
is in bijection with Pn /H.
Proof. Let σ = σn and G = gσ . By Proposition 5.14, conjugacy classes of simple
systems of type An in Φ correspond bijectively to G-orbits in the dot action of G on
RC ′′ (σ). Now G = {1, θ} ⊆ Symn where θ(i) = n − i + 1 for i = 1, . . . , n. For b =
(β1 , . . . , βn ) ∈ RC ′′ (σn ), one has θ(β1 , . . . , βn ) = (βn , . . . , β1 ) = (ρb (β1 ), . . . , ρb (βn ))
for the diagram automorphism ρb defined as in 5.17. For b ∈ R(σ), Proposition
5.18 implies that θ · b = ρ(b) and so G · b = Hb. This shows that conjugacy
classes of simple systems of type An in Φ correspond bijectively to H-orbits on
∼
=
→ Pn
RC ′′ (σ). Finally, the theorem follows on observing that the bijection RC ′′ (σ) −
of Proposition 6.4 is H-equivariant.

7. Simple subsystems of arbitrary types
In this section, we indicate, without proofs, how Propositions 5.14 and 6.3 provide an approach to studying conjugacy of simple subsystems of arbitrary types
in finite crystallographic root systems. We confine the discussion to Cartan genera, though similar results apply to Gram genera and may be more convenient in

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

27

practice. Throughout this section, Φ is a (possibly reducible) crystallographic root
system.
7.1. Standard genera of irreducible type. Choose for each type Xn of finite irreducible crystallographic root system a standard genus σXn = (ai,j )i,j=1n as follows.
Let l be the maximal rank of a type Al standard parabolic subsystem of a simple
system of type Xl (one has l = n − 1 unless Xn = An , when l = n, or Xn = F4 ,
when l = n − 2 = 2). We require first that σXn be a Cartan matrix of type Xn , that
(ai,j )i,j=1l = σl and that for each i = l + 1, . . . , n, there is some (necessarily unique)
pi < i such that api ,i 6= 0. Second, we require that σXn be chosen so (pl+1 , . . . , pn ) is
lexicographically greatest for all possible Cartan matrices satisfying these requirements (i.e. the index of any branch vertex is taken as large as possible subject to
the previous conditions). Finally, if σXn is not uniquely determined by the previous
conditions (that is, if it is B2 = C2 , F4 or G2 ), we require that al,l+1 = 1 (i.e. a root
corresponding to al is longer than that corresponding to al+1 ). In types An with
n ≥ 1, Bn with n ≥ 2, Cn with n ≥ 3, Dn with n ≥ 4 and F4 , σXn is the Cartan
matrix of type Xn indexed exactly as in [2]; in other types the indexing differs.
7.2. The above conditions are chosen on heuristic grounds to make determination of RC ′′ (σXn ) as computationally simple as possible. They assure that if b =
(β1 , . . . , βn ) ∈ RC ′′ (σXn ) then at most one of −β2 , . . . , −βn is not a simple root
(except for Xn = F4 which requires a separate check, this follows from Proposition
6.3 since τl b = (β1 , . . . , βl ) ∈ RC ′′ (σAl ) and l ≥ n − 1).
If such a non-simple root amongst −β2 , . . . , −βn does not exist, then b must arise
as in Remark 6.5 and the possible such b may be determined by inspection of the
Dynkin diagram. It turns out that if there is a non-simple root in −β2 , . . . , −βn ,
then, with two types of exceptions, it is a dominant root for an irreducible standard
parabolic subsystem of the component of Φ containing β1 (and thus containing [b]).
(The exceptions occur when Xn = G2 and that component is of type G2 , or when
Xn = E6 and that component is of type E8 .) In fact, general results (depending on
Lemma 6.1) can be given which reduce the determination of RC ′′ (Xn ) in all cases to
inspection of Dynkin diagrams (alternatively, the results could be listed as tables).
Moreover, the action of Gσ , where σ = σXn can be explicitly determined (in most
cases it is either trivial or given by Proposition 5.18).
7.3. We give some examples. Suppose that Φ is of type Bn with n ≥ 3. Enumerate
the simple roots α1 , . . . , αn exactly as in [2], so α1 and αn are terminal in the Coxeter
graph and αn is short. Denote the highest long (resp., short) root of Φ as α0l (resp.,
s
l
α0s ). We write, for example, αm,...,n
(resp., αm,...,n
) for the highest short (resp., long)
root of the subsystem with simple roots {αm , . . . , αn }. Then for 2 ≤ m ≤ n one has
s
)}
RC ′′ (σBm ) = {(α0l , −α2 , . . . , −αm−1 , −αm,...,n

Proposition 5.14 implies there is a single conjugacy class of subsystems of type Bm .
For 4 < m ≤ n, one has
l
)}.
RC ′′ (σDm ) = {bm := (α0l , −α2 , . . . , −αm−1 , −αm−1,...,n

28

M. J. DYER AND G. I. LEHRER

For m = 4 < n, one has RC ′′ (σD4 ) = {b4 } ∪ {di := (α0l , −α2 , −αi , −α4−i ) | i = 1, 3}.
Observe that [b4 ], [d1 ] = [d3 ] are simple systems for the same subsystem of Φ (of
type D4 ). Hence they are conjugate. We conclude there is a unique conjugacy class
of simple systems of type Dm in Φ if 4 ≤ m ≤ n.
These results may also be seen by (or used in) calculating the dot actions. For example, consider the dot action of G := GσD4 = h (1, 3), (3, 4) i ∼
= Sym3 on RC ′′ (σD4 )
where 4 < n. One obviously has (3, 4) · d1 = d3 (noting (3, 4)d1 = d3 and using
Proposition 2.12(a)) and hence (3, 4) · b4 = b4 . A more tedious calculation shows
(1, 3) · d3 = b4 and that (1, 3) · d1 = d1 . This can also be seen more indirectly as
follows. By Propositions 2.12 and 5.18, (1, 3) · τ3 (u) = τ3 (u) for all u ∈ RC ′′ (σD4 ).
This implies {d3 , b4 } must be stable under the dot action of (1, 3). If the restricted
dot action of (1, 3) on this set is trivial, then (1, 3) would fix RC ′′ (σD4 ) pointwise.
But from above, G acts transitively on RC ′′ (σD4 ), so (1, 3) can’t act trivially.
Note that the dot action provides finer information than just the the description
of conjugacy classes. For example, one may determine from above which diagram
automorphism of, say, [d1 ] = [d3 ] are induced by the action of an element of W .
Although the above results on conjugacy in classical type are well known, the same
techniques can be applied in general to determine even by hand the conjugacy classes
of irreducible subsystems for Φ of exceptional type. We conclude by indicating how
the methods can be extended to the study of possibly reducible subsystems.
7.4. Standard genera of reducible type. The next Lemma permits some reductions in the study of conjugacy of simple subsystems Γ of Φ to the case when both
Φ and Γ are irreducible.
7.5. Lemma. Let a := (α1 , . . . , αn ) ∈ Φn where n ≥ 1.
(a) Let Ψ be a union of components of Φ such that [a] ⊆ Ψ and let W ′ := WΨ .
(n)
(n)
Then a ∈ CW if and only if a ∈ CW ′ .
(b) Suppose that 1 ≤ m < n and that h αi , αj i = 0 for all 1 ≤ i ≤ m and
m+1 ≤ j ≤ n. Suppose also that for each 1 ≤ j ≤ m, there is some i < j with
h βi , βj i =
6 0. Let W ′ := Wα,m , a′ := (α1 , . . . , αm ) and a′′ := (αm+1 , . . . , αn ).
(n)
Let Ψ be the component of Φ containing α1 . Then a ∈ CW if and only if
(m)
(n−m)
a′ ∈ CWΨ and a′′ ∈ CW ′ .
(c) In (b), [a] is a simple subsystem of Φ if and only if [a′ ] is a simple subsystem
of Ψ and [a′′ ] is a simple subsystem of ΦW ′′ .
′
Proof. Note that Wa,i
= W ′ ∩ Wa,i for i = 1, . . . , n. Since Ψ ⊥ (Φ \ Ψ), it follows
′ ∩ RΨ = CW
∩ RΨ. By the definitions, this implies (a). Part (b) follows
that CWa,i
a,i
from (2.1.3) and (a), and (c) holds since [a] = [a′ ] ⊔ [a′′ ] with [a′ ] ⊥ [a′′ ].


7.6. We refer to the genera σAn = σn with n ≥ 1, σBn with n ≥ 3, σCn with n ≥ 2,
σDn with n ≥ 4, σEn with n = 6, 7 or 8, σF4 or σG2 as the standard irreducible genera.
If σ is one of these standard irreducible genera, of rank n, say, and b = (β1 , . . . , βn ) ∈
RC ′′ (σ) then [b] is an irreducible simple system and hence it is entirely contained
in one component Ψ of Φ. In fact, b ∈ SC ′′ (σ) ∩ Ψn . The stabiliser of b in W is
the standard parabolic subgroup generated by Π ∩ [b]⊥ = (Π ∩ Ψ ∩ [b]⊥ ) ⊔ (Π \ Ψ),

GEOMETRY OF CERTAIN FINITE COXETER GROUP ACTIONS

29

which may be readily determined by inspection of Dynkin diagrams since (with two
types of exceptions which may be easily dealt with), each root β ∈ [b] is, up to sign,
a dominant root for an irreducible subsystem of Φ.
7.7. Given genera ρi of ranks ni for i = 1, . . . , k, define the genus ρ = (ρ1 , . . . , ρk )
P
of rank n = ki=1 ni as follows. Let Ni := n1 + . . . + ni for i = 0, . . . , k, Say that
b := (β1 , . . . , βn ) ∈ Φn is of genus ρ if
(i) For i = 1, . . . , k, (βNi−1 +1 , . . . , βNi ) is of genus ρi .
(ii) For 1 ≤ i < j ≤ k and p, q ∈ N with Ni−1 + 1 ≤ p ≤ Ni and Nj−1 + 1 ≤ q ≤
Nj , one has h βp , βq i = 0.
7.8. Fix an arbitrary total order ≤ of the standard genera (e.g. order them by
decreasing rank, with genera of equal rank ordered in the order in which they were
listed at the start of 7.6). We say that a genus ρ is standard if it is of the form
ρ = (ρ1 , . . . , ρk ) where each ρi is a standard irreducible genus and ρ1 ≤ ρ2 ≤ . . . ≤ ρk .
Then for every isomorphism type X of crystallographic root system of rank n, there
is a unique standard genus σX of that type. Using Lemma 7.5, for any root system
Φ, the elements b of RC ′′ (σX ) can be readily determined. Every unordered simple
subsystem of Φ of type X is conjugate to one or more of the subsystems [b] for
b ∈ RC ′′ (σX ). The determination of the conjugacy classes of simple systems of
type X then reduces to the description of the action of GσX on b ∈ RC ′′ (σX ); this
is trivial in many cases (e.g. when |RC ′′ (σX )| = 1 or |GσX | = 1) but can require
significant computational effort (if calculated by hand) in other situations.
7.9. Conclusion. The process outlined above provides an alternative procedure to
the well known one (see e.g. [6]) which is based on the results of Borel and de
Siebenthal for determining the isomorphism classes of unordered simple subsystems
of Φ. However, the results here include additional information such as explicit representatives of the simple subsystems, which can be used in the study of more refined
questions such as conjugacy of subsystems and whether an isomorphism between two
subsystems can be realised as a composite of a specified (diagram) automorphism
of Φ with the action of an element of W . The classification of conjugacy classes
of simple subsystems has been completed computationally; for instance, it can be
found in [5] (see also [14]). In classical types, even more refined information is readily accessible (e.g. from the explicit list in [8] of all simple subsystems contained in
the positive roots together with standard descriptions of the actions of the groups
involving (signed) permutations). However, the techniques discussed in this paper
provide a uniform conceptual approach, applicable to all types. Moreover their geometric underpinning, our results on fundamental domains and stratification, have
more general applicability.
References
[1] Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler.
Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge
University Press, Cambridge, 1999.

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[2] N. Bourbaki. Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre
IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions.
Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann,
Paris, 1968.
[3] Roger W. Carter. Simple groups of Lie type. Wiley Classics Library. John Wiley & Sons Inc.,
New York, 1989.
[4] C. W. Curtis and G. I. Lehrer. A new proof of a theorem of Solomon-Tits. Proc. Amer. Math.
Soc., 85(2):154–156, 1982.
[5] J. Matthew Douglass, Götz Pfeiffer, and Gerhard Röhrle. On reflection subgroups of finite
Coxeter groups. Comm. Algebra, 41(7):2574–2592, 2013.
[6] M. J. Dyer and G. I. Lehrer. Reflection subgroups of finite and affine Weyl groups. Trans.
Amer. Math. Soc., 363(11):5971–6005, 2011.
[7] M. J. Dyer and G. I. Lehrer. On Oshima’s lemma on orbits of parabolic subgroups of finite
root systems. In preparation, 2017.
[8] Matthew Dyer. Reflection subgroups of Coxeter systems. J. Algebra, 135(1):57–73, 1990.
[9] James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Studies
in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.
[10] Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge,
1990.
[11] G. I. Lehrer. The spherical building and regular semisimple elements. Bull. Austral. Math.
Soc., 27(3):361–379, 1983.
[12] Gustav I. Lehrer and Donald E. Taylor. Unitary reflection groups, volume 20 of Australian
Mathematical Society Lecture Series. Cambridge University Press, Cambridge, 2009.
[13] Hiroshi Oda and Toshio Oshima. Minimal polynomials and annihilators of generalized Verma
modules of the scalar type. J. Lie Theory, 16(1):155–219, 2006.
[14] Toshio Oshima. A classification of subsystems of a root system. arXiv:math/0611904
[math.RT], 2006.
[15] Robert Steinberg. Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968.
Notes prepared by John Faulkner and Robert Wilson.
[16] D. E. Taylor. On reflection subgroups of unitary reflection groups. Unpublished Manuscript,
2011.
Department of Mathematics, 255 Hurley Building, University of Notre Dame,
Notre Dame, Indiana 46556, U.S.A.
E-mail address: dyer.1@nd.edu
Department of Mathematics, University of Sydney, Sydney. NSW. 2006
E-mail address: gustav.lehrer@sydney.edu.au

