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

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

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)

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)

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)

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

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

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

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

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.

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.

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

2. An Open Problem about Coxeter Groups

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.

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.

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.

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.

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.

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).

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).

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).

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).

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).

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

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

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

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

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

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

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

3. Geometric Representations of Coxeter Groups

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

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

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

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

Lorentz space

A 3-dim. Lorentz space

image source : wikipedia.org

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

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

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

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

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

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

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)

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)

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)

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(∞).

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...

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...

4. Experimenting with Limit roots of infinite Coxeter groups

Limit roots of Coxeter groups

sα sβ

sδ

sγ

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).

Limit roots of Coxeter groups

sα sβ

sδ

sγ

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).

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).

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).

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.

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.

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.

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(∞).

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(∞).

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(∞).

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(∞)

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(∞)

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(∞)

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)

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)

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)

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.

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.

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.

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)

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)

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)

Spectral analysis of Geometric Coxeter groupsMathoverflow Question :

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

Spectral analysis of Geometric Coxeter groupsMathoverflow Question :

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

Spectral analysis of Geometric Coxeter groupsMathoverflow Question :

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

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 ?

Total Eclipse

of the Roots !