slides of the lectures - Coxeter Groups meet Convex Geometry ...

Coxeter groups meets
convex geometry
Lecture I
Finite reflection
groups and
Permutahedra
Christophe Hohlweg
LaCIM, UQAM
jeudi 16 août 2012

jeudi 16 août 2012
A dodecahedron

a
a
Permutahedra
of dim 3
a
jeudi 16 août 2012

s(a)
H a (e 2 −e 1 )
=span{e 1 }
e(a)
s(H a (e 1 ))
=span{e 1 +e 2 }
H a (e 1 )
=span{e 2 −e 1 }
e 2
e 1 + e 2
st(a)
e 2 − e 1
t(a)
st(H a (e 2 −e 1 ))
=span{e 2 }
e 1
t(H a (e 2 −e 1 ))
=span{e 2 }
sts(a)
ts(a)
sts(H a (e 1 ))
=span{e 2 −e 1 }
ts(H a (e 1 ))
=span{e 1 +e 2 }
stst(a)=tsts(a)
tst(a)
tst(H a (e 2 −e 1 ))
=span{e 1 }
jeudi 16 août 2012

s = s e1 ; s(a)
{e,s} = W s
e ; e(a)
s{e,t} = sW t
{e,t} = W t
st
t = s e2 −e 1
; t(a)
e 2 − e 1
st{e,s} = stW s
e 1
t{e,s} = tW s
sts
ts
sts{e,t} = stsW t
tst{e,s} = tstW s
ts{e,t} = tsW t
stst = tsts = w o
tst
jeudi 16 août 2012

s 1
e
s 2
s 1 s 3
s 2 s 3
s 3
s 1 s 2
s 2 s 1
s 1 s 3 s 2
s 2 s 3 s 1
s 1 s 2 s 3
s 3 s 2
s 1 s 2 s 1
s 1 s 2 s 3 s 1
s 3 s 2 s 1
s 2 s 3 s 2
s 2 s 3 s 2 s 1 s1 s s s s
s 1 s 3 s 2 s 1
s 1 s 2 s 3 s 2
2 3 1 2
s 2 s 3 s 1 s 2
s 1 s 2 s 3 s 2 s 1
w ◦
s 2 s 3 s 1 s 2 s 1
jeudi 16 août 2012

Type A
Type B
τ 1
e
τ 2
w o
τ 1 τ 3
τ 2 τ 3
τ 3
τ 1 τ 2
τ 2 τ 1
τ 1 τ 3 τ 2
τ 2 τ 3 τ 1
w o
τ 1 τ 2 τ 3 τ 2
τ 2 τ 3 τ 1 τ 2
τ 1 τ 2 τ 3
τ 3 τ 2
τ 1 τ 2 τ 1
e
τ 1 τ 3 τ 2 τ 1
τ 2 τ 3 τ 2 τ 1 τ1 τ 2 τ 3 τ 1 τ 2
τ 1 τ 2 τ 3 τ 1
τ 3 τ 2 τ 1
τ 2 τ 3 τ 2
e
τ 1 τ 2 τ 3 τ 2 τ 1
w ◦
τ 2 τ 3 τ 1 τ 2 τ 1
τ 1
e
w o = τ 0 τ 1 τ 2 τ 0 τ 1 τ 2 τ 0 τ 1 τ 2
τ 1 τ 3
τ 2 τ 3
τ 1 τ 2
τ 1 τ 2 τ 1
τ 3
τ 1 τ 2 τ 3 τ 2
τ 1 τ 2 τ 3
τ 1 τ 2 τ 3 τ 1
τ 2
τ 1 τ 2 τ 3 τ 1 τ 2
w ◦
e
τ 2 τ 3 τ 2
jeudi 16 août 2012

jeudi 16 août 2012
Type H

Coxeter groups meets
convex geometry
Lecture II
Geometry of infinite
root systems
Christophe Hohlweg
LaCIM, UQAM
jeudi 16 août 2012

The other way around ?
V finite dim. vector space ; B symmetric bilinear form
The isotropic cone
of B: Q = {v ∈ V | B(v, v) =0}
Root of a B-reflection on V: for α/∈ Q and
s α (v) =v − 2B(v, α)α
B(α, α) =1
with .
v ∈ V
B-reflections generate B-reflection group in O B (V )
Examples
Finite (Euclidean) reflection
groups (iff B is a scalar product)
infinite dihedral group
Q
ρ ρ
4
4
ρ 3
ρ 3
Affine reflection
groups (iff B is positive
semidefinite)
ρ ρ
2
2
β = ρ α = ρ 1
1
(a) B(α, β) =−1
jeudi 16 août 2012

The other way around ?
More examples
Affine reflection
groups (iff B is positive
semidefinite)
infinite dihedral group
Q
Non-affine
reflection groups
(e.g. hyperbolic, etc)
infinite dihedral group
ρ ρ
4
4
ρ 4
ρ 4
Q − α = ρ 1
ρ 3
ρ 3
ρ 3
ρ 3
ρ 2
ρ 2
ρ 2
ρ 2
β = ρ 1
β = ρ 1
α = ρ 1
(a) B(α, β) =−1
(b) B(α, β) =−1.01 < −1
Q − = {v ∈ V | B(v, v) ≤ 0}
jeudi 16 août 2012

What is an infinite root system ?
ρ n = nα +(n + 1)β
Q
ρ n =(n + 1)α + nβ
ρ 4
ρ 4
ρ 3
ρ 3
= s α s β (α)
= 3α +2β
s β (α) =
α +2β =
ρ 2
ρ 2
= s α (β)
= β +2α
β = ρ 1
α = ρ 1
(a) B(α, β) =−1
s α (v) =v − 2B(v, α)α.
jeudi 16 août 2012

A simple system
• ∆ is a basis of V ;
What is an infinite root system ?
∆, i.e.,
• B(α, α) = 1 for all α ∈ ∆;
(in fact Positively independent)
• B(α, β) ∈ ] −∞, −1] ∪ {− cos π
k
,k ≥ 2}, for all α, β ∈ ∆
A B-reflection group
W
generated
W infinite dihedral group
ρ ρ
4
4
by S := {s α | α ∈ ∆} .
Q − α = ρ 1
ρ 3
ρ 3
Root system: Φ=W (∆)
ρ 2
ρ 2
β = ρ 1
(b) B(α, β) =−1.01 < −1
jeudi 16 août 2012

What is an infinite root system ?
A simple system
∆, i.e.,
A B-reflection group
W
generated
• ∆ is a basis of V ;
by S := {s α | α ∈ ∆} .
• B(α, α) = 1 for all α ∈ ∆;
• B(α, β) ∈ ] −∞, −1] ∪ {− cos π
k
,k ≥ 2}, for all α, β ∈ ∆
Root system: Φ=W (∆)
W infinite dihedral group
ρ 4
ρ 4
Proposition (see Krammer)
(a) (W, S) is a Coxeter system;
(b) the order of s α s β is k (or ∞ ) if
B(α, β) =− cos( π (or )
k ) B(α, β) ≤−1
(c) Φ + := cone(∆) ∩ Φ is a positive
root system: Φ=Φ + −Φ + .
ρ 3
ρ 2
β = ρ 1
Q − ρ 3
ρ 2
α = ρ 1
(b) B(α, β) =−1.01 < −1
jeudi 16 août 2012

What is an infinite root system ?
A simple system ∆, i.e., A B-reflection group W generated
• ∆ isA a basis useful of V characterization ; by S := {s α | of α ∈the ∆} . length
and biclosed sets
• B(α, α) = 1 for all α ∈ ∆;
• B(α, β) ∈ ] −∞, −1] ∪ {− cos π
k
,k ≥ 2}, for all α, β ∈ ∆
Root system: Φ=W (∆)
W infinite dihedral group
ρ 4
ρ 4
Proposition (see Krammer)
(a) (W, S) is a Coxeter system;
(b) the order of s α s β is k (or ∞ ) if
B(α, β) =− cos( π (or )
k ) B(α, β) ≤−1
(c) Φ + := cone(∆) ∩ Φ is a positive
root system: Φ=Φ + −Φ + .
ρ 3
ρ 2
β = ρ 1
Q − ρ 3
ρ 2
α = ρ 1
(b) B(α, β) =−1.01 < −1
jeudi 16 août 2012

Example: infinite affine dihedral group
ρ n = nα +(n + 1)β
Q
ρ n =(n + 1)α + nβ
ρ 4
ρ 4
ρ 3
ρ 3
= s α s β (α)
= 3α +2β
s β (α) =
α +2β =
ρ 2
ρ 2
= s α (β)
= β +2α
β = ρ 1
α = ρ 1
(a) B(α, β) =−1
s α (v) =v − 2B(v, α)α.
jeudi 16 août 2012

A problem
τ 1
e
τ 2
τ 1 τ 3
τ 2 τ 3
τ 3
τ 1 τ 2
τ 2 τ 1
τ 1 τ 3 τ 2
τ 2 τ 3 τ 1
τ 1 τ 2 τ 3 τ 2 τ 1 τ 2 τ 3
τ 1 τ 2 τ 1
τ 3 τ 2
τ 2 τ 3 τ 1 τ 2
τ 1 τ 3 τ 2 τ 1
τ 1 τ 2 τ 3 τ 1
τ 3 τ 2 τ 1
τ 2 τ 3 τ 2
τ 2 τ 3 τ 2 τ 1 τ1 τ 2 τ 3 τ 1 τ 2
τ 1 τ 2 τ 3 τ 2 τ 1
w ◦
τ 2 τ 3 τ 1 τ 2 τ 1
τ 1
e
What about such a
τ 1 τ 3
τ 2 τ 3
picture for infinite
Coxeter groups?
τ 1 τ 2 τ 3 τ 2
τ 3
τ 1 τ 2
τ 1 τ 2 τ 1
τ 1 τ 2 τ 3
τ 1 τ 2 τ 3 τ 1
τ 2
τ 1 τ 2 τ 3 τ 1 τ 2
w ◦
τ 2 τ 3 τ 2
Type A 3
jeudi 16 août 2012

A problem
τ 1
e
τ 2
τ 1 τ 3
τ 2 τ 3
τ 3
τ 1 τ 2
τ 2 τ 1
τ 1 τ 3 τ 2
τ 2 τ 3 τ 1
τ 1 τ 2 τ 3 τ 2 τ 1 τ 2 τ 3
τ 1 τ 2 τ 1
τ 3 τ 2
τ 2 τ 3 τ 1 τ 2
τ 1 τ 3 τ 2 τ 1
τ 1 τ 2 τ 3 τ 1
τ 3 τ 2 τ 1
τ 2 τ 3 τ 2
τ 2 τ 3 τ 2 τ 1 τ1 τ 2 τ 3 τ 1 τ 2
To obtain, at least, the
exchange graph of a
τ 1 τ 2 τ 3 τ 2 τ 1
w ◦
τ 2 τ 3 τ 1 τ 2 τ 1
τ 1
e
τ 1 τ 3
τ 2 τ 3
τ 1 τ 2
τ 1 τ 2 τ 1
cluster algebra as the
τ 3
graph of a sub(quotient)
lattice of the graph of
the weak order ...
τ 1 τ 2 τ 3 τ 2
τ 1 τ 2 τ 3
τ 1 τ 2 τ 3 τ 1
τ 2
τ 1 τ 2 τ 3 τ 1 τ 2
w ◦
τ 2 τ 3 τ 2
Type A 3
jeudi 16 août 2012

τ1τ3
τ1
τ3
e
τ1τ2
τ1τ3τ2
τ1τ2τ3τ2
τ3τ2
τ1τ2τ3
τ1τ2τ1
τ2
A problem
τ1τ3τ2τ1
τ1τ2τ3τ2τ1
τ2τ1
τ1τ2τ3τ1
τ3τ2τ1
τ2τ3
τ2τ3τ2
τ2τ3τ1
τ2τ3τ2τ1
τ1τ2τ3τ1τ2
w◦
τ2τ3τ1τ2
τ2τ3τ1τ2τ1
τ1
e
τ1τ3
τ1τ2
τ3
τ1τ2τ3
τ1τ2τ1
τ1τ2τ3τ2
τ1τ2τ3τ1
τ2
τ1τ2τ3τ1τ2
w◦
τ2τ3
τ2τ3τ2
To obtain, at least, the
exchange graph of a
cluster algebra as the
graph of a sub(quotient)
lattice of the graph of
the weak order ...
Finite case : N.
Reading, Reading
Speyer
Infinite case : the
group is not
enough, neither are
c-sortable elements
jeudi 16 août 2012

τ1τ3
τ1
τ3
e
τ1τ2
τ1τ3τ2
τ1τ2τ3τ2
τ3τ2
τ1τ2τ3
τ1τ2τ1
τ2
A problem
τ1τ3τ2τ1
τ1τ2τ3τ2τ1
τ2τ1
τ1τ2τ3τ1
τ3τ2τ1
τ2τ3
τ2τ3τ2
τ2τ3τ1
τ2τ3τ2τ1
τ1τ2τ3τ1τ2
w◦
τ2τ3τ1τ2
τ2τ3τ1τ2τ1
τ1
e
τ1τ3
τ1τ2
τ3
τ1τ2τ3
τ1τ2τ1
τ1τ2τ3τ2
τ1τ2τ3τ1
τ2
τ1τ2τ3τ1τ2
w◦
τ2τ3
τ2τ3τ2
What to do
(in my point of view) ?
First, solve:
Conjecture (M. Dyer, 2011)
The set of biclosed sets
with the inclusion is a
lattice
And understand better
infinite root systems!!!!
jeudi 16 août 2012

Other examples of infinite root systems?
ρ n = nα +(n + 1)β
Q
ρ n =(n + 1)α + nβ
ρ 4
ρ 4
ρ 3
ρ 3
= s α s β (α)
= 3α +2β
`Cut’
Φ +
by
s β (α) =
α +2β =
ρ 2
ρ 2
= s α (β)
= β +2α
an affine
hyperplane
β = ρ 1
α = ρ 1
V 1 = {v ∈ V | α∈∆
v α =1}
(a) B(α, β) =−1
jeudi 16 août 2012

Other examples of
infinite root systems?
ρ 4
Q
ρ 4
Affine hyperplane
V 1 = {v ∈ V | α∈∆
v α =1}
ρ 4
ρ 4
ρ 3
ρ 2
β = ρ 1
ρ 3
ρ 2
(a) B(α, β) =−1
= s α s β (α)
= 3α +2β
= s α (β)
= β +2α
α = ρ 1
Normalized isotropic
cone: Q := Q ∩ V1
Normalized roots
ρ := ρ/ ρ α
α∈∆
β = ρ 1
Q − α = ρ 1
ρ 3
ρ 3
ρ 2
ρ 2
(b) B(α, β) =−1.01 < −1
s β (α) =
α +2β =
jeudi 16 août 2012

Other examples of infinite root systems ...
s β (α) =
α +2β =
dim 2
dim 3
jeudi 16 août 2012

Other examples of infinite root systems ...
dim 3
dim 2
dim 3
jeudi 16 août 2012

Other examples of
infinite root systems ...
dim 3 dim 3
dim 2
dim 4
conv(∆)
jeudi 16 août 2012

Other examples of
infinite root systems ...
dim 3 dim 3
dim 2
dim 4
conv(∆)
The `size’ of a generalized root
(in red in this last picture) is
decreasing as the depth of the
root is increasing.
dp(ρ) =1+min{k | ρ = s α1 s α2 ...s αk (α k+1 ),
α 1 ,...,α k ,α k+1 ∈ ∆}.
jeudi 16 août 2012

Joint works with :
Labbé & Ripoll;
Dyer & Ripoll
W infinite dihedral group
ρ 4
ρ 3
ρ 4
ρ 3
Limits of roots
Root system: Φ=W (∆)
Depth of a root:
dp(ρ) =1+min{k | ρ = s α1 s α2 ...s αk (α k+1 ),
α 1 ,...,α k ,α k+1 ∈ ∆}.
ρ 2
Q − α = ρ 1
ρ 2
Euclidean norm for
∆
orthonormal basis
β = ρ 1
(b) B(α, β) =−1.01 < −1
Lemma
∃λ >0, ∀ρ ∈ Φ + , ||ρ|| 2 ≥ 1+λ(dp(ρ) − 1).
Theorem 1
Consider an injective sequence of positive roots (ρ n ) n∈N .
Then the norm ||ρ n || tends to infinity (for any norm on ).
V
jeudi 16 août 2012

Limits of normalized roots
Theorem 1
Consider an injective sequence of positive roots (ρ n ) n∈N .
Then the norm ||ρ n || tends to infinity (for any norm on ).
Normalized set of positive roots: Φ:={ρ | ρ ∈ Φ + }⊆V 1
V
Fact:
Φ ⊆ conv(∆)
a compact.
W infinite dihedral group
ρ 4
ρ 4
ρ 3
ρ 3
Cut by the affine hyperplane
V 1 = {v ∈ V | α∈∆
v α =1}
Q − α = ρ 1
ρ 2
ρ 2
β = ρ
1
jeudi 16 août 2012
(b) B(α, β) =−1.01 < −1

Limits of normalized roots
Theorem 1
Consider an injective sequence of positive roots (ρ n ) n∈N .
Then the norm ||ρ n || tends to infinity (for any norm on ).
Normalized set of positive roots: Φ:={ρ | ρ ∈ Φ + }⊆V 1
V
Fact:
Φ ⊆ conv(∆)
a compact.
Corollary
If
(ρ n ) n∈N
converges to a limit , then
∈ Q ∩ conv(∆).
jeudi 16 août 2012

Limits of normalized roots
Theorem 1
Consider an injective sequence of positive roots (ρ n ) n∈N .
Then the norm ||ρ n || tends to infinity (for any norm on ).
Normalized set of positive roots: Φ:={ρ | ρ ∈ Φ + }⊆V 1
V
Fact:
Φ ⊆ conv(∆)
a compact.
Corollary
If
(ρ n ) n∈N
converges to a limit , then
∈ Q ∩ conv(∆).
Remark: directions of roots converging in Q.
(i) Root systems of Lie algebras (Kac 1990).
(ii) Imaginary cone for Coxeter groups (Dyer, 2012)
jeudi 16 août 2012

Limits of normalized roots
Corollary
If
(ρ n ) n∈N
converges to a limit , then
∈ Q ∩ conv(∆).
Problem: understand the set of accumulation points
E(Φ) = Acc( Φ) ⊆ Q ∩ conv(∆)
dim 3
dim 4
conv(∆)
jeudi 16 août 2012

The set
E(Φ) = Acc( Φ)
Some natural questions:
A `fractal phenomenon’?
Restriction to parabolic subgroups?
How W acts on E(Φ) ?
Link with hyperbolic geometry (hyperbolic reflection groups)?
Link with Apollonian gasket (Kleinian groups)?
dim 3
dim 4
conv(∆)
jeudi 16 août 2012

The set
E(Φ) = Acc( Φ)
Some natural questions:
A `fractal phenomenon’?
Restriction to parabolic subgroups?
How W acts on E(Φ) ?
Link with hyperbolic geometry (hyperbolic reflection groups)?
Link with Apollonian gasket (Kleinian groups)?
dim 4
conv(∆)
jeudi 16 août 2012

A geometric action on
E(Φ) = Acc( Φ)
Remark: V 1 is not stable under W.
New action:
D :=
w · v = w(v)
where
on the set
w∈W
w(V \ V 0 ) ∩ V 1 V 0 := {v ∈ V | α∈∆
Proposition (CH, Labbé, Ripoll)
(a) Φ ⊆ D is stable under W ;
(b) E(Φ) ⊆ D is stable under W ;
(c) Q ∩ L(α, x) ={x, s α · x} .
v α =0}
x
Theorem (Dyer, CH, Ripoll)
Action faithful if irreducible not
affine or finite.
s α · x
L(α, x)
jeudi 16 août 2012

A geometric action on
E(Φ) = Acc( Φ)
Remark: V 1 is not stable under W.
New action:
D :=
w · v = w(v)
where
on the set
w∈W
w(V \ V 0 ) ∩ V 1 V 0 := {v ∈ V | α∈∆
Proposition (CH, Labbé, Ripoll)
(a) Φ ⊆ D is stable under W ;
(b) E(Φ) ⊆ D is stable under W ;
(c) Q ∩ L(α, x) ={x, s α · x} .
v α =0}
x
Theorem (Dyer, CH, Ripoll)
Action faithful if irreducible not
affine or finite.
s α · x
L(α, x)
jeudi 16 août 2012

A geometric action on
E(Φ) = Acc( Φ)
Remark: V 1 is not stable under W.
New action:
D :=
w · v = w(v)
where
on the set
w∈W
w(V \ V 0 ) ∩ V 1 V 0 := {v ∈ V | α∈∆
Proposition (CH, Labbé, Ripoll)
(a) Φ ⊆ D is stable under W ;
(b) E(Φ) ⊆ D is stable under W ;
(c) Q ∩ L(α, x) ={x, s α · x} .
v α =0}
x
Theorem (Dyer, CH, Ripoll)
Action faithful if irreducible not
affine or finite.
s α · x
L(α, x)
jeudi 16 août 2012

Remarkable dense subsets
of
E(Φ) = Acc( Φ)
Dihedral reflection subgroups: ,
Associated root system:
Observation:
W = s ρ ,s γ ρ, γ ∈ Φ +
Φ = W ({ρ, γ})
Limits of roots of dihedral reflection subgroups:
E 2 :=
E(Φ )= Q ∩ L(ρ, γ)
ρ 1 ,ρ 2 ∈Φ + L( ρ 1 , ρ 2 ) ∩ Q
Theorem 2 (CH, Labbé, Ripoll 2011)
The set is dense in .
E 2
E(Φ)
jeudi 16 août 2012

Remarkable dense subsets
of E(Φ) = Acc( Φ)
Theorem 2 (CH, Labbé, Ripoll 2011)
The set is dense in .
E 2
Proof (sketch).
E(Φ)
E 2 = W · E where
E2 ◦ :=
2
◦ L(α, ρ) ∩ Q
α∈∆
ρ∈Φ +
Proposition (CH, Labbé, Ripoll)
E2
◦
The set is dense in E(Φ) .
Two cases: l/∈ V ⊥ or l ∈ V ⊥ (which
is dealt with by Perron-Frobenius)
jeudi 16 août 2012

Parabolic subgroups
and
E(Φ) = Acc( Φ)
I ⊆ ∆
V I = span(I) ∩ V 1
W I := s α | α ∈ I
Φ I := W I (I)
Consider and .
Standard parabolic subgroup: ;
Associated root system: .
Remark: in general (e.g. rank 5).
E(Φ I ) = E(Φ) ∩ V I
α β
Theorem 4 (Dyer, CH, Ripoll 2012)
For I ⊆ ∆ , we have
E 2 (Φ) ∩ V I = E 2 (Φ I ) ;
s α
s γ
4 -1.5
∞
s β
γ
jeudi 16 août 2012

Singleton are generating
E(Φ) = Acc( Φ)
Theorem 3 (Dyer, CH, Ripoll 2012)
The closure of W · x is dense in E(Φ) for E(Φ)
x
s α · x
s α · y
L(α, x)
y
jeudi 16 août 2012

Singleton are generating
E(Φ) = Acc( Φ)
Theorem 3 (Dyer, CH, Ripoll 2012)
The closure of W · x is dense in E(Φ) for E(Φ)
Proof based on a strong
property of the
imaginary cone (Matthew
Dyer)
x
s α · x
s α · y
L(α, x)
y
jeudi 16 août 2012

Singleton are generating
E(Φ) = Acc( Φ)
Theorem 3 (Dyer, CH, Ripoll 2012)
The closure of W · x is dense in E(Φ) for E(Φ)
Corollary (Dyer, CH, Ripoll)
A first fractal Phenomenon
x
s α · x
s α · y
L(α, x)
y
jeudi 16 août 2012

Further
works
Study the action of W on E(Φ) .
the `fractal phenomenon 2’?
Study Dyer’s imaginary cone for
Coxeter groups.
Applications to the study of `biclosed’
sets of roots?
Understand the action on limits relatively to the imaginary
cone
Link with Apollonian gasket (Kleinian groups)?
...
Fin/The end
jeudi 16 août 2012