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 !
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