Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial...

76
γ s α s β s γ 4 4 Experimental Coxeter Group Theory Jean-Philippe Labb´ e Sage Days 88 August 23 rd 2017

Transcript of Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial...

Page 1: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

76 Appendix A. Root systems of rank 3 & 4

α β

γ

sα sβ

sγ4 4

α β

γ

sα sβ

sγ4 5

α β

γ

sα sβ

sγ4

α β

γ

sα sβ

sγ4 4

Figure A.2: The other examples...

Experimental Coxeter Group Theory

Jean-Philippe Labbe

Sage Days 88August 23rd 2017

Page 2: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Plan of the talk

1. Combinatorial basics

2. An Open Problem about Coxeter Groups

3. Geometric Representations of Coxeter Groups

4. Experimenting with limit roots of infinite Coxeter groups

Page 3: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Partially ordered set (poset) :Hasse diagram of a poset :

1 2

1, 3 2, 3

1, 2, 3

a

b c

d e

f

Lattice : Existence of meet and join ∀p, q ∈ P(join = unique least upper bound)(meet = unique greatest lower bound)

Page 4: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Partially ordered set (poset) :Hasse diagram of a poset :

1 2

1, 3 2, 3

1, 2, 3

a

b c

d e

f

Lattice : Existence of meet and join ∀p, q ∈ P(join = unique least upper bound)(meet = unique greatest lower bound)

Page 5: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Partially ordered set (poset) :Hasse diagram of a poset :

1 2

1, 3 2, 3

1, 2, 3

a

b c

d e

f

Lattice : Existence of meet and join ∀p, q ∈ P(join = unique least upper bound)(meet = unique greatest lower bound)

Page 6: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Symmetric group Sn+1 :The group of permutations of 1, . . . , n + 1

generators s1, . . . , sn, si = (i i + 1)

length of w ∈ Sn+1 : smallest r such that w = si1 . . . sir

longest element w : the permutation [n + 1, . . . , 1]

reduced expression of w : expression for w of smallest length

Page 7: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Symmetric group Sn+1 :The group of permutations of 1, . . . , n + 1

generators s1, . . . , sn, si = (i i + 1)

length of w ∈ Sn+1 : smallest r such that w = si1 . . . sir

longest element w : the permutation [n + 1, . . . , 1]

reduced expression of w : expression for w of smallest length

Page 8: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – CombinatoricsCayley graph of a group :

vertices ↔ elements of the groupedges ↔ multiplication by a generator

Examples : S3 : 〈s1, s2 | s21 = s22 = (s1s2)3 = e〉 andI2(∞) : 〈s, t | s2 = t2 = e〉

e

s1s1

s2s2

s1s2s2

s2s1s1

s1s2s1 = s2s1s2

s1 s2

e

s t

st ts

sts tst

Page 9: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – CombinatoricsCayley graph of a group :

vertices ↔ elements of the groupedges ↔ multiplication by a generator

Examples : S3 : 〈s1, s2 | s21 = s22 = (s1s2)3 = e〉 andI2(∞) : 〈s, t | s2 = t2 = e〉

e

s1s1

s2s2

s1s2s2

s2s1s1

s1s2s1 = s2s1s2

s1 s2

e

s t

st ts

sts tst

Page 10: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Weak order of the Symmetric group :Def : Cayley graph of the group =⇒ Hasse diagram of the weakorder

e

s1 s2

s1s2 s2s1

s1s2s1 = s2s1s2

e

s t

st ts

sts tst

Fact (classic) : This is a complete lattice.

Page 11: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Preliminaries – Combinatorics

Weak order of the Symmetric group :Def : Cayley graph of the group =⇒ Hasse diagram of the weakorder

e

s1 s2

s1s2 s2s1

s1s2s1 = s2s1s2

e

s t

st ts

sts tst

Fact (classic) : This is a complete lattice.

Page 12: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Other example

S4 :

e

s1 s2 s3

s1s2 s2s1 s1s3 s2s3 s3s2

s1s2s1 s1s2s3 s1s3s2 s2s3s1 s2s3s2 s3s2s1

s1s2s3s1 s1s2s3s2 s2s3s1s2 s1s3s2s1 s2s3s2s1

s1s2s3s1s2 s1s2s3s2s1 s2s3s2s1s2

w

Page 13: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

2. An Open Problem about Coxeter Groups

Page 14: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Reflection groups

A reflection fixes an hyperplane and flips a complementary vector

R ∈ GL(V ) such that

I 1 is an eigenvalue of geom. mult. n − 1,

I −1 is an eigenvalue of geom. mult. 1.

Page 15: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Reflection groups

A reflection fixes an hyperplane and flips a complementary vector

R ∈ GL(V ) such that

I 1 is an eigenvalue of geom. mult. n − 1,

I −1 is an eigenvalue of geom. mult. 1.

Page 16: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Reflection groups

A reflection fixes an hyperplane and flips a complementary vector

R ∈ GL(V ) such that

I 1 is an eigenvalue of geom. mult. n − 1,

I −1 is an eigenvalue of geom. mult. 1.

Page 17: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Reflection groups

A reflection fixes an hyperplane and flips a complementary vector

R ∈ GL(V ) such that

I 1 is an eigenvalue of geom. mult. n − 1,

I −1 is an eigenvalue of geom. mult. 1.

Page 18: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Reflection groups

A reflection fixes an hyperplane and flips a complementary vector

R ∈ GL(V ) such that

I 1 is an eigenvalue of geom. mult. n − 1,

I −1 is an eigenvalue of geom. mult. 1.

Page 19: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Coxeter groups

Coxeter groups are abstract groups obtained by a presentation withgenerators and relations :

W = 〈S |e = s2 = (st)ms,t ; ∀s, t ∈ S〉

Coxeter matrix : M = (ms,t)s,t∈S

Theorem (Coxeter, 1934)

Finite reflection groups of Euclidean spaces are exactly finiteCoxeter groups.

The classification : An,Bn,Dn,E6,E7,E8,F4,H3,H4, I2(m).

Page 20: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Coxeter groups

Coxeter groups are abstract groups obtained by a presentation withgenerators and relations :

W = 〈S |e = s2 = (st)ms,t ; ∀s, t ∈ S〉

Coxeter matrix : M = (ms,t)s,t∈S

Theorem (Coxeter, 1934)

Finite reflection groups of Euclidean spaces are exactly finiteCoxeter groups.

The classification : An,Bn,Dn,E6,E7,E8,F4,H3,H4, I2(m).

Page 21: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Coxeter groups

Coxeter groups are abstract groups obtained by a presentation withgenerators and relations :

W = 〈S |e = s2 = (st)ms,t ; ∀s, t ∈ S〉

Coxeter matrix : M = (ms,t)s,t∈S

Theorem (Coxeter, 1934)

Finite reflection groups of Euclidean spaces are exactly finiteCoxeter groups.

The classification : An,Bn,Dn,E6,E7,E8,F4,H3,H4, I2(m).

Page 22: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Coxeter groups

Coxeter groups are abstract groups obtained by a presentation withgenerators and relations :

W = 〈S |e = s2 = (st)ms,t ; ∀s, t ∈ S〉

Coxeter matrix : M = (ms,t)s,t∈S

Theorem (Coxeter, 1934)

Finite reflection groups of Euclidean spaces are exactly finiteCoxeter groups.

The classification : An,Bn,Dn,E6,E7,E8,F4,H3,H4, I2(m).

Page 23: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Coxeter groups

Coxeter groups are abstract groups obtained by a presentation withgenerators and relations :

W = 〈S |e = s2 = (st)ms,t ; ∀s, t ∈ S〉

Coxeter matrix : M = (ms,t)s,t∈S

Theorem (Coxeter, 1934)

Finite reflection groups of Euclidean spaces are exactly finiteCoxeter groups.

The classification : An,Bn,Dn,E6,E7,E8,F4,H3,H4, I2(m).

Page 24: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Weak order : infinite case

Infinite Coxeter groups do not have a longest element w, unlikefinite ones.

The Cayley graph (and so the weak order) is more complicated.

Theorem (Bjorner, 1984)

The weak order of a Coxeter group is a meet-semilattice.(Meets exist. Joins not necessarily)

Infinite Coxeter groups act on Euclidean, Lorentz (hyperbolic)spaces and higher rank spaces (as we will see).

; computations and properties in infinite Coxeter groups are morecomplicated

Page 25: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Weak order : infinite case

Infinite Coxeter groups do not have a longest element w, unlikefinite ones.

The Cayley graph (and so the weak order) is more complicated.

Theorem (Bjorner, 1984)

The weak order of a Coxeter group is a meet-semilattice.(Meets exist. Joins not necessarily)

Infinite Coxeter groups act on Euclidean, Lorentz (hyperbolic)spaces and higher rank spaces (as we will see).

; computations and properties in infinite Coxeter groups are morecomplicated

Page 26: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Weak order : infinite case

Infinite Coxeter groups do not have a longest element w, unlikefinite ones.

The Cayley graph (and so the weak order) is more complicated.

Theorem (Bjorner, 1984)

The weak order of a Coxeter group is a meet-semilattice.(Meets exist. Joins not necessarily)

Infinite Coxeter groups act on Euclidean, Lorentz (hyperbolic)spaces and higher rank spaces (as we will see).

; computations and properties in infinite Coxeter groups are morecomplicated

Page 27: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

An Open Problem : Original MotivationProblem : The weak order is only a meet-semilattice.Question : How to determine if two elements have a join ?

Strategy (Dyer) :

I The weak order has a geometric definition

I Add special geometric elements to the weak order

I Define a join in this extended ordering

July 2010 : Cafe Depot discussion with C. Hohlweg presenting theproblem.

Winter 2011 : Experiments in Sage : Sphere packings and fractalsappeared in pictures !

Summer 2011 : Definition of Limit roots (Hohlweg–L.–Ripoll, Dyer)

Fall 2011 : With H. Chen, we investigated the relation ballpackings vs Lorentzian Coxeter groups

Page 28: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

An Open Problem : Original MotivationProblem : The weak order is only a meet-semilattice.Question : How to determine if two elements have a join ?

Strategy (Dyer) :

I The weak order has a geometric definition

I Add special geometric elements to the weak order

I Define a join in this extended ordering

July 2010 : Cafe Depot discussion with C. Hohlweg presenting theproblem.

Winter 2011 : Experiments in Sage : Sphere packings and fractalsappeared in pictures !

Summer 2011 : Definition of Limit roots (Hohlweg–L.–Ripoll, Dyer)

Fall 2011 : With H. Chen, we investigated the relation ballpackings vs Lorentzian Coxeter groups

Page 29: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

An Open Problem : Original MotivationProblem : The weak order is only a meet-semilattice.Question : How to determine if two elements have a join ?

Strategy (Dyer) :

I The weak order has a geometric definition

I Add special geometric elements to the weak order

I Define a join in this extended ordering

July 2010 : Cafe Depot discussion with C. Hohlweg presenting theproblem.

Winter 2011 : Experiments in Sage : Sphere packings and fractalsappeared in pictures !

Summer 2011 : Definition of Limit roots (Hohlweg–L.–Ripoll, Dyer)

Fall 2011 : With H. Chen, we investigated the relation ballpackings vs Lorentzian Coxeter groups

Page 30: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

An Open Problem : Original MotivationProblem : The weak order is only a meet-semilattice.Question : How to determine if two elements have a join ?

Strategy (Dyer) :

I The weak order has a geometric definition

I Add special geometric elements to the weak order

I Define a join in this extended ordering

July 2010 : Cafe Depot discussion with C. Hohlweg presenting theproblem.

Winter 2011 : Experiments in Sage : Sphere packings and fractalsappeared in pictures !

Summer 2011 : Definition of Limit roots (Hohlweg–L.–Ripoll, Dyer)

Fall 2011 : With H. Chen, we investigated the relation ballpackings vs Lorentzian Coxeter groups

Page 31: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

3. Geometric Representations of Coxeter Groups

Page 32: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Recall – Linear algebra

Bilinear forms :Let B be a symmetric bilinear form on a real vector space V ofdim. n,

B : V × V → R

I If the matrix of B is positive definite, then (V ,B) is aEuclidean space

I If the signature of B is (n − 1, 1), then (V ,B) is a Lorentzspace

I isotropic cone, Q := v ∈ V | B(v , v) = 0

Page 33: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Recall – Linear algebra

Bilinear forms :Let B be a symmetric bilinear form on a real vector space V ofdim. n,

B : V × V → R

I If the matrix of B is positive definite, then (V ,B) is aEuclidean space

I If the signature of B is (n − 1, 1), then (V ,B) is a Lorentzspace

I isotropic cone, Q := v ∈ V | B(v , v) = 0

Page 34: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Recall – Linear algebra

Bilinear forms :Let B be a symmetric bilinear form on a real vector space V ofdim. n,

B : V × V → R

I If the matrix of B is positive definite, then (V ,B) is aEuclidean space

I If the signature of B is (n − 1, 1), then (V ,B) is a Lorentzspace

I isotropic cone, Q := v ∈ V | B(v , v) = 0

Page 35: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Recall – Linear algebra

Bilinear forms :Let B be a symmetric bilinear form on a real vector space V ofdim. n,

B : V × V → R

I If the matrix of B is positive definite, then (V ,B) is aEuclidean space

I If the signature of B is (n − 1, 1), then (V ,B) is a Lorentzspace

I isotropic cone, Q := v ∈ V | B(v , v) = 0

Page 36: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Lorentz space

A 3-dim. Lorentz space

image source : wikipedia.org

Page 37: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Sketch of General Strategy

To pass from a Coxeter group to a reflection group :

I Take a real vector space

I Equip with the “right”geometry (bilinear formB)

I Get the reflections σi foreach basis vector αi

I Create a reflection group〈σi 〉 α2α1 α2α1

Root system := Orbit of the basis vectors

Page 38: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Sketch of General Strategy

To pass from a Coxeter group to a reflection group :

I Take a real vector space

I Equip with the “right”geometry (bilinear formB)

I Get the reflections σi foreach basis vector αi

I Create a reflection group〈σi 〉

α2α1

α2α1

Root system := Orbit of the basis vectors

Page 39: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Sketch of General Strategy

To pass from a Coxeter group to a reflection group :

I Take a real vector space

I Equip with the “right”geometry (bilinear formB)

I Get the reflections σi foreach basis vector αi

I Create a reflection group〈σi 〉 α2α1

α2α1

Root system := Orbit of the basis vectors

Page 40: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Sketch of General Strategy

To pass from a Coxeter group to a reflection group :

I Take a real vector space

I Equip with the “right”geometry (bilinear formB)

I Get the reflections σi foreach basis vector αi

I Create a reflection group〈σi 〉 α2α1

α2α1

Root system := Orbit of the basis vectors

Page 41: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Sketch of General Strategy

To pass from a Coxeter group to a reflection group :

I Take a real vector space

I Equip with the “right”geometry (bilinear formB)

I Get the reflections σi foreach basis vector αi

I Create a reflection group〈σi 〉

α2α1

α2α1

Root system := Orbit of the basis vectors

Page 42: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Sketch of General Strategy

To pass from a Coxeter group to a reflection group :

I Take a real vector space

I Equip with the “right”geometry (bilinear formB)

I Get the reflections σi foreach basis vector αi

I Create a reflection group〈σi 〉

α2α1

α2α1

Root system := Orbit of the basis vectors

Page 43: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example

Take the infinite dihedral group

M =

(1 ∞∞ 1

)B =

(1 −1−1 1

)

So ∆ = αs , αt, σs =

(−1 20 1

), σt =

(1 02 −1

)W = 〈σs , σt〉 and Φ = W · (∆) ←− root system

The roots are of the form ±((n + 1)αs + nαt) and±(nαs + (n + 1)αt)

Page 44: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example

Take the infinite dihedral group

M =

(1 ∞∞ 1

)B =

(1 −1−1 1

)

So ∆ = αs , αt, σs =

(−1 20 1

), σt =

(1 02 −1

)W = 〈σs , σt〉 and Φ = W · (∆) ←− root system

The roots are of the form ±((n + 1)αs + nαt) and±(nαs + (n + 1)αt)

Page 45: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example

Take the infinite dihedral group

M =

(1 ∞∞ 1

)B =

(1 −1−1 1

)

So ∆ = αs , αt, σs =

(−1 20 1

), σt =

(1 02 −1

)W = 〈σs , σt〉 and Φ = W · (∆) ←− root system

The roots are of the form ±((n + 1)αs + nαt) and±(nαs + (n + 1)αt)

Page 46: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Root system of rank 2

Where are the roots going ?

αtαs

2αt + αsαt + 2αs

3αt + 2αs2αt + 3αs

4αt + 3αs3αt + 4αs

Q

HHQ

αtαs · · ·

Infinite dihedral group I2(∞).

Page 47: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Root systems of rank 3

76 Appendix A. Root systems of rank 3 & 4

α β

γ

sα sβ

sγ4 4

α β

γ

sα sβ

sγ4 5

α β

γ

sα sβ

sγ4

α β

γ

sα sβ

sγ4 4

Figure A.2: The other examples...

Page 48: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Root systems of rank 4

Appendix A. Root systems of rank 3 & 4 79

4

4

4

4

4

4

4 4

4 4

4

3

3 3

∞ ∞

Figure A.3: The other examples...

Page 49: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

4. Experimenting with Limit roots of infinite Coxeter groups

Page 50: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Limit roots of Coxeter groups

sα sβ

sα sβ∞

sδ∞ ∞

sγ∞ ∞

Theorem (Hohlweg-L.-Ripoll, 2014 (Def. of limit roots))

The set E (Φ) of accumulation points of normalized roots Φ iscontained in the isotropic cone of (V ,B).

Page 51: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Limit roots of Coxeter groups

sα sβ

sα sβ∞

sδ∞ ∞

sγ∞ ∞

Theorem (Hohlweg-L.-Ripoll, 2014 (Def. of limit roots))

The set E (Φ) of accumulation points of normalized roots Φ iscontained in the isotropic cone of (V ,B).

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The Tits Cone and Sphere Packings

Theorem (Chen-L., 2015)

The set E (Ω) of accumulation points of normalized weights Ω iscontained in the isotropic cone of (V ,B). Moreover, if W isLorentzian, then E (Ω) = E (Φ).

Page 53: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

The Tits Cone and Sphere Packings

Theorem (Chen-L., 2015)

The set E (Ω) of accumulation points of normalized weights Ω iscontained in the isotropic cone of (V ,B). Moreover, if W isLorentzian, then E (Ω) = E (Φ).

Page 54: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Back to Dyer’s Strategy : biclosed sets of roots

Φ = Φ+ t −Φ+

w ∈W , inversion set : inv(w) = Φ+ ∩ w−1Φ−

DefinitionA subset A ⊆ Φ+ is closed if

∀α, β ∈ A and aα + bβ ∈ Φ+, a, b ∈ R+ ⇒ aα + bβ ∈ A

A ⊆ Φ+ is biclosed if A and Φ+ \ A are closed.

Lemma (Bourbaki (∼’60), Pilkington (2008), Dyer (2011))

Let A ⊆ Φ+. A = inv(w) for w ∈W ⇔ A is finite and biclosed.

Page 55: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Back to Dyer’s Strategy : biclosed sets of roots

Φ = Φ+ t −Φ+

w ∈W , inversion set : inv(w) = Φ+ ∩ w−1Φ−

DefinitionA subset A ⊆ Φ+ is closed if

∀α, β ∈ A and aα + bβ ∈ Φ+, a, b ∈ R+ ⇒ aα + bβ ∈ A

A ⊆ Φ+ is biclosed if A and Φ+ \ A are closed.

Lemma (Bourbaki (∼’60), Pilkington (2008), Dyer (2011))

Let A ⊆ Φ+. A = inv(w) for w ∈W ⇔ A is finite and biclosed.

Page 56: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Back to Dyer’s Strategy : biclosed sets of roots

Φ = Φ+ t −Φ+

w ∈W , inversion set : inv(w) = Φ+ ∩ w−1Φ−

DefinitionA subset A ⊆ Φ+ is closed if

∀α, β ∈ A and aα + bβ ∈ Φ+, a, b ∈ R+ ⇒ aα + bβ ∈ A

A ⊆ Φ+ is biclosed if A and Φ+ \ A are closed.

Lemma (Bourbaki (∼’60), Pilkington (2008), Dyer (2011))

Let A ⊆ Φ+. A = inv(w) for w ∈W ⇔ A is finite and biclosed.

Page 57: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example – biclosed sets

αtαs

2αt + αsαt + 2αs

3αt + 2αs2αt + 3αs

4αt + 3αs3αt + 4αs

Q

HHQ

αtαs · · ·

The positive roots of the infinite dihedral group I2(∞).

Page 58: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example – biclosed sets

αtαs

2αt + αsαt + 2αs

3αt + 2αs2αt + 3αs

4αt + 3αs3αt + 4αs

Q

HHQ

αtαs · · ·

The positive roots of the infinite dihedral group I2(∞).

Page 59: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example – biclosed sets

αtαs

2αt + αsαt + 2αs

3αt + 2αs2αt + 3αs

4αt + 3αs3αt + 4αs

Q

HHQ

αtαs · · ·

The positive roots of the infinite dihedral group I2(∞).

Page 60: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example – biclosed sets

inv(e) = ∅

inv(s) = αs inv(t) = αt

inv(st) = αs , s(αt) inv(ts) = αt , t(αs)

inv(sts) = αs , s(αt), st(αs) inv(tst) = αt , t(αs), ts(αt)

inv((st)∞) = (n + 1)αs + nαt | n ∈ inv inv((ts)∞) = nαs + (n + 1)αt | n ∈ inv

Φ+ \ inv(ts) Φ+ \ inv(st)

Φ+ \ inv(t) Φ+ \ inv(s)

Φ+

The poset of biclosed set of the infinite dihedral group I2(∞)

Page 61: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example – biclosed sets

inv(e) = ∅

inv(s) = αs inv(t) = αt

inv(st) = αs , s(αt) inv(ts) = αt , t(αs)

inv(sts) = αs , s(αt), st(αs) inv(tst) = αt , t(αs), ts(αt)

inv((st)∞) = (n + 1)αs + nαt | n ∈ inv inv((ts)∞) = nαs + (n + 1)αt | n ∈ inv

Φ+ \ inv(ts) Φ+ \ inv(st)

Φ+ \ inv(t) Φ+ \ inv(s)

Φ+

The poset of biclosed set of the infinite dihedral group I2(∞)

Page 62: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Example – biclosed sets

inv(e) = ∅

inv(s) = αs inv(t) = αt

inv(st) = αs , s(αt) inv(ts) = αt , t(αs)

inv(sts) = αs , s(αt), st(αs) inv(tst) = αt , t(αs), ts(αt)

inv((st)∞) = (n + 1)αs + nαt | n ∈ inv inv((ts)∞) = nαs + (n + 1)αt | n ∈ inv

Φ+ \ inv(ts) Φ+ \ inv(st)

Φ+ \ inv(t) Φ+ \ inv(s)

Φ+

The poset of biclosed set of the infinite dihedral group I2(∞)

Page 63: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Dyer’s conjecture

Theorem (Dyer 2011)

Finite biclosed sets ordered by inclusion is a meet-semilattice.

1. The meet is easy to describe using union and complementationof sets

2. The join is more complicated to describe

Conjecture (Dyer 2011)

Biclosed sets ordered by inclusion is a complete lattice. The topelement is the set Φ+.

Motivation : replace the notion of reflection order in the infinitecase (related to the computation of Kazhdan–Lusztig polynomials)

Page 64: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Dyer’s conjecture

Theorem (Dyer 2011)

Finite biclosed sets ordered by inclusion is a meet-semilattice.

1. The meet is easy to describe using union and complementationof sets

2. The join is more complicated to describe

Conjecture (Dyer 2011)

Biclosed sets ordered by inclusion is a complete lattice. The topelement is the set Φ+.

Motivation : replace the notion of reflection order in the infinitecase (related to the computation of Kazhdan–Lusztig polynomials)

Page 65: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Dyer’s conjecture

Theorem (Dyer 2011)

Finite biclosed sets ordered by inclusion is a meet-semilattice.

1. The meet is easy to describe using union and complementationof sets

2. The join is more complicated to describe

Conjecture (Dyer 2011)

Biclosed sets ordered by inclusion is a complete lattice. The topelement is the set Φ+.

Motivation : replace the notion of reflection order in the infinitecase (related to the computation of Kazhdan–Lusztig polynomials)

Page 66: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

A convexity approach

Theorem (L. 2012)

Let (W ,S) be a Coxeter group of rank n ≤ 3, there exists acomplete lattice containing the weak order. The join is computedusing convex hulls.

Fact (L. 2012)

For groups of rank n ≥ 4, joins cannot be computed this way.

Next best thing : infinite reduced words :

Definition (Cellini–Papi ’98, Ito ’01, Lam–Pylyavskyy ’13, . . .)

An infinite reduced word is

I an infinite path is the Cayley graph

I starting at the identity

I and such that all prefixes are geodesics (reduced)

Infinite reduced word ←→ infinite chains in the weak order.

Page 67: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

A convexity approach

Theorem (L. 2012)

Let (W ,S) be a Coxeter group of rank n ≤ 3, there exists acomplete lattice containing the weak order. The join is computedusing convex hulls.

Fact (L. 2012)

For groups of rank n ≥ 4, joins cannot be computed this way.

Next best thing : infinite reduced words :

Definition (Cellini–Papi ’98, Ito ’01, Lam–Pylyavskyy ’13, . . .)

An infinite reduced word is

I an infinite path is the Cayley graph

I starting at the identity

I and such that all prefixes are geodesics (reduced)

Infinite reduced word ←→ infinite chains in the weak order.

Page 68: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

A convexity approach

Theorem (L. 2012)

Let (W ,S) be a Coxeter group of rank n ≤ 3, there exists acomplete lattice containing the weak order. The join is computedusing convex hulls.

Fact (L. 2012)

For groups of rank n ≥ 4, joins cannot be computed this way.

Next best thing : infinite reduced words :

Definition (Cellini–Papi ’98, Ito ’01, Lam–Pylyavskyy ’13, . . .)

An infinite reduced word is

I an infinite path is the Cayley graph

I starting at the identity

I and such that all prefixes are geodesics (reduced)

Infinite reduced word ←→ infinite chains in the weak order.

Page 69: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Limit roots vs infinite reduced words

Let w be an infinite reduced word.

inv(w) :=⋃p∈P

inv(p),

where P is the set of prefixes of w .

Theorem (Chen-L. 2014-17))

Let (W ,S) be a Lorentzian Coxeter group. The inversion sets ofinfinite reduced words each contain a unique accumulation point(i.e. limit root).

General Question :

infinite reduced words?←→ limit roots

(combinatorics) ←→ (discrete geometry)

Page 70: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Limit roots vs infinite reduced words

Let w be an infinite reduced word.

inv(w) :=⋃p∈P

inv(p),

where P is the set of prefixes of w .

Theorem (Chen-L. 2014-17))

Let (W ,S) be a Lorentzian Coxeter group. The inversion sets ofinfinite reduced words each contain a unique accumulation point(i.e. limit root).

General Question :

infinite reduced words?←→ limit roots

(combinatorics) ←→ (discrete geometry)

Page 71: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Limit roots vs infinite reduced words

Let w be an infinite reduced word.

inv(w) :=⋃p∈P

inv(p),

where P is the set of prefixes of w .

Theorem (Chen-L. 2014-17))

Let (W ,S) be a Lorentzian Coxeter group. The inversion sets ofinfinite reduced words each contain a unique accumulation point(i.e. limit root).

General Question :

infinite reduced words?←→ limit roots

(combinatorics) ←→ (discrete geometry)

Page 72: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Spectral analysis of Geometric Coxeter groupsMathoverflow Question :

; Initiated the study of spectrum of matrices coming from Coxetergroups (with S. Labbe, arXiv :1511.04975)

Page 73: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Spectral analysis of Geometric Coxeter groupsMathoverflow Question :

; Initiated the study of spectrum of matrices coming from Coxetergroups (with S. Labbe, arXiv :1511.04975)

Page 74: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Spectral analysis of Geometric Coxeter groupsMathoverflow Question :

; Initiated the study of spectrum of matrices coming from Coxetergroups (with S. Labbe, arXiv :1511.04975)

Page 75: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Questions and problems

QuestionFor which equiv. relation “∼” do we get

Limit roots ∼= Infinite reduced words/∼ ?

QuestionHow are limit roots related to Kac–Moody algebras ?

QuestionHow to characterize the existence of the join using combinatorics onwords ?

QuestionWhich type of invariant does the Hausdorff dimension of the limitroots give ?

Page 76: Experimental Coxeter Group Theory 0.5cm [width=3cm]lmrk42 · Plan of the talk 1.Combinatorial basics 2.An Open Problem about Coxeter Groups 3.Geometric Representations of Coxeter

Total Eclipse

of the Roots !