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Page 1: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

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Polyhedral Combinatorics of Coxeter GroupsDissertation’s Defense

Jean-Philippe Labbe

July 8th 2013

Page 2: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

A few motivations

Open Problem (Dyer (1993))

Is there, for each infinite Coxeter group, a complete ortholatticethat contains the weak order?

; Introduction of the limit roots of an infinite Coxeter group.

Open Problem (Jonsson (2003))

Is there a polytopal realization of the multi-associahedron?

; Introduction of the multi-cluster complex of a finite Coxetergroup.

Page 3: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

A few motivations

Open Problem (Dyer (1993))

Is there, for each infinite Coxeter group, a complete ortholatticethat contains the weak order?

; Introduction of the limit roots of an infinite Coxeter group.

Open Problem (Jonsson (2003))

Is there a polytopal realization of the multi-associahedron?

; Introduction of the multi-cluster complex of a finite Coxetergroup.

Page 4: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

PART I: Ortholattice for infinite Coxeter groups?Fixed-gear drivetrain

Geometry

AlgebraCombinatorics

Page 5: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

PART I: Ortholattice for infinite Coxeter groups?Fixed-gear drivetrain

Geometry

AlgebraCombinatorics

Page 6: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

PART I: Ortholattice for infinite Coxeter groups?Fixed-gear drivetrain

Geometry

Algebra

Combinatorics

Page 7: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

PART I: Ortholattice for infinite Coxeter groups?Fixed-gear drivetrain

Geometry

AlgebraCombinatorics

Page 8: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

The weak order on Coxeter groups

(W ,S) – infinite Coxeter group(〈s, t ∈ S |e = s2 = (st)ms,t 〉

)

Definition (Weak order)

Let u, v ∈W . Then u ≤ v ⇐⇒ u is a prefix of v .

Example

tus and tustut.

(W ,≤) is a meet-semilattice. There is no maximal element.

QuestionHow to tell when the join of two elements exists?

Page 9: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

The weak order on Coxeter groups

(W ,S) – infinite Coxeter group(〈s, t ∈ S |e = s2 = (st)ms,t 〉

)

Definition (Weak order)

Let u, v ∈W . Then u ≤ v ⇐⇒ u is a prefix of v .

Example

tus and tustut.

(W ,≤) is a meet-semilattice. There is no maximal element.

QuestionHow to tell when the join of two elements exists?

Page 10: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Asymptotical behaviour of roots

Study the directions of the roots.

ρ′1ρ1

ρ′2ρ2

ρ′3ρ3

ρ′4ρ4

Q

V1

V1Q

α = ρ′1β = ρ1 ρ′2ρ2 · · ·

The infinite dihedral group I2(∞).

Page 11: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Limit roots of Coxeter groups

sα sβ

sα sβ∞

sδ∞ ∞

sγ∞ ∞

Theorem (Hohlweg-L.-Ripoll, Canad. J. Math. (2013))

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

Page 12: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Limit roots of Coxeter groups

sα sβ

sα sβ∞

sδ∞ ∞

sγ∞ ∞

Theorem (Hohlweg-L.-Ripoll, Canad. J. Math. (2013))

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

Page 13: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

A first step using biconvexity

Theorem (L. 2012)

Let (W , S) be a Coxeter group of rank n ≤ 3, there is a completeortholattice containing the weak order.

; The join of two elements can be computed using convex hulls.

Fact (L. 2012)

For Coxeter groups of rank n ≥ 4 the join can not be computedusing convex hulls.

The notion of convexity is to restrictive.; biclosedness seems to be the right geometry to look at.

Page 14: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

A first step using biconvexity

Theorem (L. 2012)

Let (W , S) be a Coxeter group of rank n ≤ 3, there is a completeortholattice containing the weak order.

; The join of two elements can be computed using convex hulls.

Fact (L. 2012)

For Coxeter groups of rank n ≥ 4 the join can not be computedusing convex hulls.

The notion of convexity is to restrictive.; biclosedness seems to be the right geometry to look at.

Page 15: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

PART II: Subword Complexes in Discrete Geometry

Geometry

AlgebraCombinatorics

Page 16: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complexes

triangulations

multi-triangulations

gen. triangulationsl

clusters

gen. multi-triangulationsl

multi-clusters

Integer k > 1(Geometric)

Finite Coxeter Groups (Algebraic)

Page 17: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complexes

triangulations

multi-triangulations

gen. triangulationsl

clusters

gen. multi-triangulationsl

multi-clusters

Integer k > 1(Geometric)

Finite Coxeter Groups (Algebraic)

Page 18: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complexes

triangulations

multi-triangulations

gen. triangulationsl

clusters

gen. multi-triangulationsl

multi-clusters

Integer k > 1(Geometric)

Finite Coxeter Groups (Algebraic)

Page 19: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complexes

triangulations

multi-triangulations

gen. triangulationsl

clusters

gen. multi-triangulationsl

multi-clusters

Integer k > 1(Geometric)

Finite Coxeter Groups (Algebraic)

Page 20: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Simplicial complex of multi-triangulations

Fix a convex m-gon.

Multi-triangulation: Maximal set of diagonals not containing k + 1pairwise crossing diagonals.

∆m,k : the simplicial complex with

faces ←→ sets of (relevant) diagonals not containingk + 1 pairwise crossing diagonals

Conjecture (Jonsson, 2003)

∆m,k is isomorphic to the boundary complex of a simplicialpolytope.

Page 21: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Simplicial complex of multi-triangulations

Fix a convex m-gon.

Multi-triangulation: Maximal set of diagonals not containing k + 1pairwise crossing diagonals.

∆m,k : the simplicial complex with

faces ←→ sets of (relevant) diagonals not containingk + 1 pairwise crossing diagonals

Conjecture (Jonsson, 2003)

∆m,k is isomorphic to the boundary complex of a simplicialpolytope.

Page 22: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Simplicial complex of multi-triangulations

Fix a convex m-gon.

Multi-triangulation: Maximal set of diagonals not containing k + 1pairwise crossing diagonals.

∆m,k : the simplicial complex with

faces ←→ sets of (relevant) diagonals not containingk + 1 pairwise crossing diagonals

Conjecture (Jonsson, 2003)

∆m,k is isomorphic to the boundary complex of a simplicialpolytope.

Page 23: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Simplicial complex ∆m,k - Example

Let m = 6 and k = 2

When m = 2k + 2, ∆m,k is a k-simplex.

Page 24: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Simplicial complex ∆m,k - Example

Let m = 6 and k = 2

When m = 2k + 2, ∆m,k is a k-simplex.

Page 25: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Subword complexes

(W ,S) finite Coxeter system of rank n

Definition (Knutson-Miller, 2004)

Let Q = (q1, . . . , qr ) be a word in S and π ∈W .

simplicial complexsubword complex ∆(Q, π) := facets ←→ complements (in Q)

of reduced expressions of π

Theorem (Knutson-Miller, 2004)

Subword complexes are topological spheres or balls.

Page 26: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Subword complexes

(W ,S) finite Coxeter system of rank n

Definition (Knutson-Miller, 2004)

Let Q = (q1, . . . , qr ) be a word in S and π ∈W .

simplicial complexsubword complex ∆(Q, π) := facets ←→ complements (in Q)

of reduced expressions of π

Theorem (Knutson-Miller, 2004)

Subword complexes are topological spheres or balls.

Page 27: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complex - Definition

Definition (Ceballos-L.-Stump, J. of Alg. Comb. 2013)

The multi-cluster complex ∆kc (W ) is the subword complex

∆(ckw◦(c),w◦) of type W .

Theorem (CLS, 2013)

The subword complex ∆(cw◦(c),w◦) is isomorphic to the c-clustercomplex of type W .

Page 28: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complex - Definition

Definition (Ceballos-L.-Stump, J. of Alg. Comb. 2013)

The multi-cluster complex ∆kc (W ) is the subword complex

∆(ckw◦(c),w◦) of type W .

Theorem (CLS, 2013)

The subword complex ∆(cw◦(c),w◦) is isomorphic to the c-clustercomplex of type W .

Page 29: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complexes of type A and B

Theorem (Pilaud-Pocchiola 2012, Stump 2011)

The multi-cluster simplicial complexcomplex ∆k

c (An) ∼= of k-triangulations of aconvex m-gon

where m = n + 2k + 1.

Theorem (CLS, 2013)

The multi-cluster simplicial complex of centrallycomplex ∆k

c (Bm−k) ∼= symmetric k-triangulationsof a regular convex 2m-gon

Corollary

∆symm,k is a vertex-decomposable simplicial sphere.

Page 30: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Multi-cluster complexes of type A and B

Theorem (Pilaud-Pocchiola 2012, Stump 2011)

The multi-cluster simplicial complexcomplex ∆k

c (An) ∼= of k-triangulations of aconvex m-gon

where m = n + 2k + 1.

Theorem (CLS, 2013)

The multi-cluster simplicial complex of centrallycomplex ∆k

c (Bm−k) ∼= symmetric k-triangulationsof a regular convex 2m-gon

Corollary

∆symm,k is a vertex-decomposable simplicial sphere.

Page 31: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Universality and polytopality of ∆kc (W )

Question (Knutson-Miller, 2004)

Charaterize all simplicial spheres that can be realized as a subwordcomplex.

Theorem (CLS, 2013)

A simplicial sphere it is the linkis realized as ⇐⇒ of a face of a multi-cluster

a subword complex complex ∆kc (W ).

Corollary

The following two statements are equivalent.

(i) Every spherical subword complex is polytopal.

(ii) Every multi-cluster complex is polytopal.

Page 32: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Universality and polytopality of ∆kc (W )

Question (Knutson-Miller, 2004)

Charaterize all simplicial spheres that can be realized as a subwordcomplex.

Theorem (CLS, 2013)

A simplicial sphere it is the linkis realized as ⇐⇒ of a face of a multi-cluster

a subword complex complex ∆kc (W ).

Corollary

The following two statements are equivalent.

(i) Every spherical subword complex is polytopal.

(ii) Every multi-cluster complex is polytopal.

Page 33: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Universality and polytopality of ∆kc (W )

Question (Knutson-Miller, 2004)

Charaterize all simplicial spheres that can be realized as a subwordcomplex.

Theorem (CLS, 2013)

A simplicial sphere it is the linkis realized as ⇐⇒ of a face of a multi-cluster

a subword complex complex ∆kc (W ).

Corollary

The following two statements are equivalent.

(i) Every spherical subword complex is polytopal.

(ii) Every multi-cluster complex is polytopal.

Page 34: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Polytopality of multi-cluster complexes

Conjecture (⇒ Knutson-Miller’04, Jonsson’05, Soll-Welker’09)

The multi-cluster complex is the boundary complex of a simplicialpolytope.

I True for k = 1: Chapoton-Fomin-Zelevinsky,Hohlweg-Lange-Thomas, Pilaud-Stump, Stella

I True for I2(m), k ≥ 1: cyclic polytope, Ceballos-L.-Stump

I True for A3, k = 2: Bokowski-Pilaud, Ceballos-L.

Recent work of Bergeron-Ceballos-L.:

I New combinatorial construction giving 60000+ realizations ofA3, k = 2

I Fan realizations of all subword complexes of type A3

Page 35: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Polytopality of multi-cluster complexes

Conjecture (⇒ Knutson-Miller’04, Jonsson’05, Soll-Welker’09)

The multi-cluster complex is the boundary complex of a simplicialpolytope.

I True for k = 1: Chapoton-Fomin-Zelevinsky,Hohlweg-Lange-Thomas, Pilaud-Stump, Stella

I True for I2(m), k ≥ 1: cyclic polytope, Ceballos-L.-Stump

I True for A3, k = 2: Bokowski-Pilaud, Ceballos-L.

Recent work of Bergeron-Ceballos-L.:

I New combinatorial construction giving 60000+ realizations ofA3, k = 2

I Fan realizations of all subword complexes of type A3

Page 36: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Polytopality of multi-cluster complexes

Conjecture (⇒ Knutson-Miller’04, Jonsson’05, Soll-Welker’09)

The multi-cluster complex is the boundary complex of a simplicialpolytope.

I True for k = 1: Chapoton-Fomin-Zelevinsky,Hohlweg-Lange-Thomas, Pilaud-Stump, Stella

I True for I2(m), k ≥ 1: cyclic polytope, Ceballos-L.-Stump

I True for A3, k = 2: Bokowski-Pilaud, Ceballos-L.

Recent work of Bergeron-Ceballos-L.:

I New combinatorial construction giving 60000+ realizations ofA3, k = 2

I Fan realizations of all subword complexes of type A3

Page 37: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Thanks!

Merci! Thank you! Grazie! Danke! Gracias!

e

s1s2s3s1s2s3s1s2s3s1s2s3s1s2s3

e

s1s2s3s1s2s3s1s2s3s1s2s3s1s2s3

1

Page 38: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Recent developments on limit roots

; Relation with the Tits cone [Dyer-Hohlweg-Ripoll, (arxiv:2013)]

; Limit set of Kleinian groups [Hohlweg-Preaux-Ripoll,(arxiv:2013)]

; Sphere packings and geometric invariants for Coxeter groups[Chen-L., (in preparation)]

Page 39: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Recent developments on multi-cluster complexes

e

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1

; Generalized brick polytope, spanning trees [Pilaud-Stump,arxiv:2011-12]

; Denominator vectors of cluster algebras of finite types[Ceballos-Pilaud, (arxiv:2013)]

; Common vertices of permutahedra and generalized associahedra[L., 2013 and L.-Lange (in preparation)]

Page 40: Polyhedral Combinatorics of Coxeter Groups - …page.mi.fu-berlin.de/labbe/pdfs/Defense.pdfPolyhedral Combinatorics of Coxeter Groups Dissertation’s Defense Jean-Philippe Labb e

Geometric computation of the join

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αu

s(αu)

u(αs) u(αt)

su(αt)

ut(αs)ut(αu)

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u

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