many geometric realizations of graph associahedra€¦ · geometric realizations of graph...

Sage Days 64.5

June 3, 2015

MANYGEOMETRIC

REALIZATIONSOF GRAPH

ASSOCIAHEDRAT. MANNEVILLE V. PILAUD

(LIX) (CNRS & LIX)

POLYTOPES & COMBINATORICS

SIMPLICIAL COMPLEX

simplicial complex = collection of subsets of X downward closed

exm:

X = [n] ∪ [n]

∆ = {I ⊆ X | ∀i ∈ [n], {i, i} 6⊆ I}12 13 13 12 23 23 23 23 12 13 13 12

1 2 3 3 2 1

123 123 123 123 123 123 123 123

FANS

polyhedral cone = positive span of a finite set of Rd

= intersection of finitely many linear half-spaces

fan = collection of polyhedral cones closed by faces

and where any two cones intersect along a face

simplicial fan = maximal cones generated by d rays

POLYTOPES

polytope = convex hull of a finite set of Rd

= bounded intersection of finitely many affine half-spaces

face = intersection with a supporting hyperplane

face lattice = all the faces with their inclusion relations

simple polytope = facets in general position = each vertex incident to d facets

SIMPLICIAL COMPLEXES, FANS, AND POLYTOPES

P polytope, F face of P

normal cone of F = positive span of the outer normal vectors of the facets containing F

normal fan of P = { normal cone of F | F face of P }

simple polytope =⇒ simplicial fan =⇒ simplicial complex

PERMUTAHEDRON

34123421

43214312

2413

4213

3214

14231432

1342

12431234

21341324

2341

2431

31242314

Permutohedron Perm(n)

= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}

= H ∩⋂

∅6=J([n+1]

{x ∈ Rn+1

∣∣∣∣ ∑j∈J

xj ≥(|J | + 1

2

)}

PERMUTAHEDRON

34123421

43214312

2413

4213

3214

14231432

1342

12431234

21341324

2341

2431

31242314


= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}

= H ∩⋂

∅6=J([n+1]

{x ∈ Rn+1

∣∣∣∣ ∑j∈J

xj ≥(|J | + 1

2

)}connections to

• weak order

• reduced expressions

• braid moves

• cosets of the symmetric group

PERMUTAHEDRON

34123421

43214312

2413

4213

3214

14231432

1342

12431234

21341324

2341

2431

31242314

1211

1221

1222

1212

2212

1112

2211

2321

13212331

12311332

1322

1232

1233

1323

1223

1312 2313

1213

2113

32132312

32122311

32113321

1123

2123

3312


= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}

= H ∩⋂

∅6=J([n+1]

{x ∈ Rn+1

∣∣∣∣ ∑j∈J

xj ≥(|J | + 1

2

)}connections to

• weak order


• braid moves


k-faces of Perm(n)≡ surjections from [n + 1]

to [n + 1− k]

PERMUTAHEDRON

3|4|1|24|3|1|2

4|3|2|13|4|2|1

3|1|4|2

3|2|4|1

3|2|1|4

1|3|4|21|4|3|2

1|4|2|3

1|2|4|31|2|3|4

2|1|3|41|3|2|4

4|1|2|3

4|1|3|2

2|3|1|43|1|2|4

134|2

14|23

1|234

13|24

3|124

123|4

34|12

4|13|2

14|3|24|1|23

14|2|31|4|23

1|24|3

13|2|4

1|23|4

13|4|2 3|1|24

13|2|4

23|1|4

3|2|143|14|2

3|24|1

12|3|41|2|34

2|13|4

34|2|13|4|1234|1|2

4|3|12


= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}

= H ∩⋂

∅6=J([n+1]

{x ∈ Rn+1

∣∣∣∣ ∑j∈J

xj ≥(|J | + 1

2

)}connections to

• weak order


• braid moves



to [n + 1− k]

≡ ordered partitions of [n + 1]

into n + 1− k parts

PERMUTAHEDRON

3|4|1|24|3|1|2

4|3|2|13|4|2|1

3|1|4|2

3|2|4|1

3|2|1|4

1|3|4|21|4|3|2

1|4|2|3

1|2|4|31|2|3|4

2|1|3|41|3|2|4

4|1|2|3

4|1|3|2

2|3|1|43|1|2|4

134|2

14|23

1|234

13|24

3|124

123|4

34|12

4|13|2

14|3|24|1|23

14|2|31|4|23

1|24|3

13|2|4

1|23|4

13|4|2 3|1|24

13|2|4

23|1|4

3|2|143|14|2

3|24|1

12|3|41|2|34

2|13|4

34|2|13|4|1234|1|2

4|3|12


= conv {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1}

= H ∩⋂

∅6=J([n+1]

{x ∈ Rn+1

∣∣∣∣ ∑j∈J

xj ≥(|J | + 1

2

)}connections to

• weak order


• braid moves



to [n + 1− k]



≡ collections of n− k nested

subsets of [n + 1]

COXETER ARRANGEMENT

34|12

134|23|124

13|24

123|41|234

14|23

3|4|1|24|3|1|2

4|3|2|1 3|4|2|1

3|1|4|2

3|2|4|1

3|2|1|41|3|4|21|4|3|2

1|4|2|3

1|2|4|31|2|3|4

2|1|3|4

1|3|2|4

4|1|2|3

4|1|3|2

2|3|1|4

3|1|2|4

Coxeter fan

= fan defined by the hyperplane arrangement{x ∈ Rn+1

∣∣ xi = xj}1≤i<j≤n+1

= collection of all cones{x ∈ Rn+1

∣∣ xi < xj if π(i) < π(j)}

for all surjections π : [n + 1]→ [n + 1− k]

(n− k)-dimensional cones

≡ surjections from [n + 1]

to [n + 1− k]



≡ collections of n− k nested

subsets of [n + 1]

ASSOCIAHEDRA

ASSOCIAHEDRON

Associahedron = polytope whose face lattice is isomorphic to the lattice of crossing-free

sets of internal diagonals of a convex (n + 3)-gon, ordered by reverse inclusion

vertices ↔ triangulations

edges ↔ flips

faces ↔ dissections

vertices ↔ binary trees

edges ↔ rotations

faces ↔ Schroder trees

VARIOUS ASSOCIAHEDRA

Associahedron = polytope whose face lattice is isomorphic to the lattice of crossing-free

sets of internal diagonals of a convex (n + 3)-gon, ordered by reverse inclusion

(Pictures by Ceballos-Santos-Ziegler)Tamari (’51) — Stasheff (’63) — Haimann (’84) — Lee (’89) —

. . . — Gel’fand-Kapranov-Zelevinski (’94) — . . . — Chapoton-Fomin-Zelevinsky (’02) — . . . — Loday (’04) — . . .

— Ceballos-Santos-Ziegler (’11)

THREE FAMILIES OF REALIZATIONS

SECONDARY

POLYTOPE

Gelfand-Kapranov-Zelevinsky (’94)

Billera-Filliman-Sturmfels (’90)

LODAY’S

ASSOCIAHEDRON

Loday (’04)

Hohlweg-Lange (’07)

Hohlweg-Lange-Thomas (’12)

CHAP.-FOM.-ZEL.’S

ASSOCIAHEDRON

��

��

@@@@

��

��

��

@@@@BBBBBB

@@

@@@

@@@

��

��

p p p p p p p pp p p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pppppppppppppppppppppppppppppppppppppppppα2

α1+α2

α2+α3

α1+α2+α3

α1

α3

(Pictures by CFZ)

Chapoton-Fomin-Zelevinsky (’02)

Ceballos-Santos-Ziegler (’11)


SECONDARY

POLYTOPE



LODAY’S

ASSOCIAHEDRON

Loday (’04)



Hopf

algebra

Cluster

algebras

CHAP.-FOM.-ZEL.’S

ASSOCIAHEDRON

��

��

@@@@

��

��

��

@@@@BBBBBB

@@

@@@

@@@

��

��



α1+α2

α2+α3

α1+α2+α3

α1

α3

(Pictures by CFZ)



Cluster

algebras

��

��

@@@

��

��

��

@@@

HHH

HH

��BBBB

HHH

@@

@@

@@@

HHH

��A

AA��

HHHHH

p p p p p p pp p p p p p p

p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p ppppppppppppppppppppppppppα2

α1+α2α2+α3

2α2 + α3

α1

α3

2α1 + 2α2 + α3

GRAPH ASSOCIAHEDRA

NESTED COMPLEX AND GRAPH ASSOCIAHEDRON

G graph on ground set V

Tube of G = connected induced subgraph of G

Compatible tubes = nested, or disjoint and non-adjacent

Tubing on G = collection of pairwise compatible tubes of G

24 5

3

1 2 3

6

87 9

24 5

3

1 2 3

6

87 9

Nested complex N (G) = simplicial complex of tubings on G

= clique complex of the compatibility relation on tubes

G-associahedron = polytopal realization of the nested complex on G

Carr-Devadoss, Coxeter complexes and graph associahedra (’06)

EXM: NESTED COMPLEX

EXM: GRAPH ASSOCIAHEDRON

SPECIAL GRAPH ASSOCIAHEDRA

Path associahedron Cycle associahedron Complete graph associahedron

= associahedron = cyclohedron = permutahedron

341234214321 4312

2413

4213

3214

14231432

1342

1243 12342134

1324

2341

2431

31242314

LINEAR LAURENT PHENOMENON ALGEBRAS

Laurent Phenomenon Algebra = commut. ring gen. by cluster variables grouped in clusters

seed = pair (x,F) where

• x = {x1, . . . , xn} cluster variables

• F = {F1, . . . , Fn} exchange polynomials

seed mutation = (x,F) 7−→ µi(x,F) = (x′,F′) where

• x′i = Fi/xi while x′j = xj for j 6= i

• F ′j obtained from Fj by eliminating xi

THM. Every cluster variable in a LP algebra is a Laurent polynomial in the cluster

variables of any seed.Lam-Pylyavskyy, Laurent Phenomenon Algebras (’12)

Connection to graph associahedra: Any (directed) graph G defines a linear LP algebra

whose cluster complex contains the nested complex of G

Lam-Pylyavskyy, Linear Laurent Phenomenon Algebras (’12)

COMPATIBILITY FANSFOR GRAPHICAL NESTED COMPLEXES

Thibault Manneville & VP

arXiv:1501.07152

http://arxiv.org/abs/1501.07152

COMPATIBILITY FANS FOR ASSOCIAHEDRA

T◦ an initial triangulation

δ, δ′ two internal diagonals

compatibility degree between δ and δ′

(δ ‖ δ′) =

−1 if δ = δ′

0 if δ and δ′ do not cross1 if δ and δ′ cross

compatibility vector of δ wrt T◦:

d(T◦, δ) =[(δ◦ ‖ δ)

]δ◦∈T◦

compatibility fan wrt T◦

D(T◦) = {R≥0 d(T◦,D) | D dissection}

Fomin-Zelevinsky, Y -Systems and generalized associahedra (’03)

Fomin-Zelevinsky, Cluster algebras II: Finite type classification (’03)

Chapoton-Fomin-Zelevinsky, Polytopal realizations of generalized associahedra (’02)

Ceballos-Santos-Ziegler, Many non-equivalent realizations of the associahedron (’11)

COMPATIBILITY FANS FOR ASSOCIAHEDRA

Different initial triangulations T◦ yield different realizations

THM. For any initial triangulation T◦, the cones {R≥0 d(T◦,D) | D dissection} form a

complete simplicial fan. Moreover, this fan is always polytopal.


COMPATIBILITY FANS FOR GRAPHICAL NESTED COMPLEXES

T◦ an initial maximal tubing on G

t, t′ two tubes of G

compatibility degree between t and t′

(t ‖ t′) =

−1 if t = t′

0 if t, t′ are compatible|{neighbors of t in t′r t}| otherwise

compatibility vector of t wrt T◦:

d(T◦, t) =[(t◦ ‖ t)

]t◦∈T◦

THM. For any initial maximal tubing T◦ on G,

the collection of cones

D(G,T◦) = {R≥0 d(T◦,T) | T tubing on G}

forms a complete simplicial fan, called com-

patibility fan of G.

Manneville-P., Compatibility fans for graphical nested complexes

GRAPH CATALAN MANY SIMPLICIAL FAN REALIZATIONS

THM. When none of the connected components of G is a spider,

# linear isomorphism classes of compatibility fans of G

= # orbits of maximal tubings on G under graph automorphisms of G.

Manneville-P., Compatibility fans for graphical nested complexes (’15)

POLYTOPALITY?

QU. Are all compatibility fans polytopal?

POLYTOPALITY?


Polytopality of a complete simplicial fan ⇐⇒ Feasibility of a linear program

Exm: We check that the compatibility fan on the complete graph K7 is polytopal by

solving a linear program on 126 variables and 17 640 inequalities

POLYTOPALITY?





=⇒ All compatibility fans on complete graphs of ≤ 7 vertices are polytopal...

=⇒ All compatibility fans on graphs of ≤ 4 vertices are polytopal...

POLYTOPALITY?





=⇒ All compatibility fans on complete graphs of ≤ 7 vertices are polytopal...

=⇒ All compatibility fans on graphs of ≤ 4 vertices are polytopal...

To go further, we need to understand better the linear dependences between the compat-

ibility vectors of the tubes involved in a flip

THM. All compatibility fans on the paths and cycles are polytopal


Manneville-P., Compatibility fans for graphical nested complexes (’15)

POLYTOPALITY?


Remarkable realizations of the stellohedra

POLYTOPALITY?


Remarkable realizations of the stellohedra

Convex hull of the orbits under coordinate permutations of the set{∑

i>k i ei∣∣ 0 ≤ k ≤ n

}

SIGNED TREE ASSOCIAHEDRA

arXiv:1309.5222


LODAY’S ASSOCIAHEDRON

Asso(n) := conv {L(T) | T binary tree} = H ∩⋂

1≤i≤j≤n+1

H≥(i, j)

L(T) :=[`(T, i) · r(T, i)

]i∈[n+1]

H≥(i, j) :=

{x ∈ Rn+1

∣∣∣∣ ∑i≤k≤j

xi ≥(j − i + 2

2

)}Loday, Realization of the Stasheff polytope (’04)

7654321

2

1

38

1

12

1123

321

213

312

141


Asso(n) := conv {L(T) | T binary tree} = H ∩⋂

1≤i≤j≤n+1

H≥(i, j)

L(T) :=[`(T, i) · r(T, i)

]i∈[n+1]

H≥(i, j) :=

{x ∈ Rn+1

∣∣∣∣ ∑i≤k≤j

xi ≥(j − i + 2

2

)}Loday, Realization of the Stasheff polytope (’04)

7654321

2

1

38

1

12

1123

321

213 132

312 231

x1=x2 x2=x3

x1=x3 141

• Asso(n) obtained by deleting inequalities in the facet description of the permutahedron

• normal cone of L(T) in Asso(n) = {x ∈ H | xi < xj for all i→ j in T}=⋃σ∈L(T) normal cone of σ in Perm(n)

SPINES

spine of a tubing T = Hasse diagram of the inclusion poset of T

24 5

3

1 2 3

6

87 9

22

7

3

4

5

8

31

9

6

tube t of the tubing T 7−→ node s(t) of the spine S labeled

by t r⋃{t′ | t′ ∈ T, t′ ( t}

tube t(s) :=⋃{s′ | s′ ≤ s in S} ←− [ node s of the spine S

of the tubing T

S spine on G ⇐⇒ for each node s of S with children s1 . . . sk, the tubes t(s1) . . . t(sk)

lie in distinct connected components of G[t(s) r s

]

SPINES

spine of a tubing T = Hasse diagram of the inclusion poset of T

24 5

3

1 2 3

6

87 9

73

5

891

246

tube t of the tubing T 7−→ node s(t) of the spine S labeled

by t r⋃{t′ | t′ ∈ T, t′ ( t}

tube t(s) :=⋃{s′ | s′ ≤ s in S} ←− [ node s of the spine S

of the tubing T

S spine on G ⇐⇒ for each node s of S with children s1 . . . sk, the tubes t(s1) . . . t(sk)

lie in distinct connected components of G[t(s) r s

]

NESTED FANS AND GRAPH ASSOCIAHEDRA

THM. The collection of cones{{x ∈ H | xi < xj for all i→ j in T}

∣∣ T tubing on G}

forms a complete simplicial fan, called the nested fan of G. This fan is always polytopal.

Carr-Devadoss, Coxeter complexes and graph associahedra (’06)

Postnikov, Permutohedra, associahedra, and beyond (’09)

Zelevinsky, Nested complexes and their polyhedral realizations (’06)

4

2

3

1

4

2

1

34

2

31

3

2

4

1 (2,1,4,3)

(3,1,4,2)

(3,2,4,1)(4,1,2,3)

(4,2,3,1)

(4,1,2,3)

(1,2,3,4)(1,2,4,3)

(1,4,1,4)(3,1,2,4)

(2,1,3,4)

(3,2,1,4)

(4,2,1,3)

(4,4,1,1)

(1,7,1,1)

(1,4,4,1)

4

2

3

1 3

2

4

1

4

2

1

3

4

2

31

HOHLWEG-LANGE’S ASSOCIAHEDRA

11

22

312

776

54

3

21

for an arbitrary signature ε ∈ ±n+1,

Asso(ε) := conv {HL(T) | T ε-Cambrian tree}

with HL(T)j :=

{`(T, j) · r(T, j) if ε(j) = −

n + 2− `(T, j) · r(T, j) if ε(j) = +

Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07)

Lange-P., Using spines to revisit a construction of the associahedron (’15)

j

<j >j

?j

<j >j

?Rule for Cambrian trees

123

321

132

231

303

HOHLWEG-LANGE’S ASSOCIAHEDRA

11

22

312

776

54

3

21

for an arbitrary signature ε ∈ ±n+1,

Asso(ε) := conv {HL(T) | T ε-Cambrian tree}

with HL(T)j :=

{`(T, j) · r(T, j) if ε(j) = −

n + 2− `(T, j) · r(T, j) if ε(j) = +

Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07)

Lange-P., Using spines to revisit a construction of the associahedron (’15)

j

<j >j

?j

<j >j

?Rule for Cambrian trees

123

321

213 132

312 231

x1=x2 x2=x3

x1=x3303

• Asso(n) obtained by deleting inequalities in the facet description of the permutahedron

• normal cone of HL(T) in Asso(ε) = {x ∈ H | xi < xj for all i→ j in T}

SIGNED SPINES ON SIGNED TREES

T tree on the signed ground set V = V− t V+ (negative in white, positive in black)

Signed spine on T = directed and labeled tree S st

(i) the labels of the nodes of S form a partition of the signed ground set V

(ii) at a node of S labeled by U = U−tU+, the source label sets of the different incoming

arcs are subsets of distinct connected components of TrU−, while the sink label sets

of the different outgoing arcs are subsets of distinct connected components of TrU+

0

1

49

2 31

0

8 4 9

5

6 7

2

35678

2 7

0

1

49

368

5

SPINE COMPLEX

Signed spine complex S(T) = simplicial complex whose inclusion poset is isomorphic to

the poset of edge contractions on the signed spines of T

2

43

1

3

1

4

2

4

31

2

3

41

2

1

43

2

3

1

4

2

4

2

31

3

2

4

1

4

2

1

3

4

2

3

1

2

3

1

4

4

1

2

3

32

1423

14

132

412

34

42

13342

1

142

3 134

2134

2

132

4

342

1

142

3

12

34

24

13

123

4

124

3

234

1

234

1

123

4

124

3

SPINE FAN

For S spine on T, define C(S) := {x ∈ H | xu ≤ xv, for all arcs u→ v in S}

4

2

3

1

4

2

1

3

1

43

2

3

41

2

4

2

31

3

2

4

1 3

2

1

4

1

2

4

3

THEO. The collection of cones F(T) := {C(S) | S ∈ S(T)} defines a complete simplicial

fan on H, which we call the spine fan


THM. The spine fan F(T) is the normal fan of the signed tree associahedron Asso(T),

defined equivalently as

(i) the convex hull of the points

a(S)v =

{∣∣ {π ∈ Π(S) | v ∈ π and rv /∈ π}∣∣ if v ∈ V−

|V| + 1−∣∣ {π ∈ Π(S) | v ∈ π and rv /∈ π}

∣∣ if v ∈ V+

for all maximal signed spines S ∈ S(T)

(ii) the intersection of the hyperplane H with the half-spaces

H≥(B) :=

{x ∈ RV

∣∣∣∣ ∑v∈B

xv ≥(|B| + 1

2

)}for all signed building blocks B ∈ B(T)


The signed tree associahedron Asso(T) is sandwiched between the permutahedron Perm(V)

and the parallelepiped Para(T)∑u6=v∈V

[eu, ev] = Perm(T) ⊂ Asso(T) ⊂ Para(T) =∑u−v∈T

π(u − v) · [eu, ev]

WHAT SHOULD I TAKE HOMEFROM THIS TALK?


SECONDARY

POLYTOPE



LODAY’S

ASSOCIAHEDRON

Loday (’04)



CHAP.-FOM.-ZEL.’S

ASSOCIAHEDRON

��

��

@@@@

��

��

��

@@@@BBBBBB

@@

@@@

@@@

��

��



α1+α2

α2+α3

α1+α2+α3

α1

α3

(Pictures by CFZ)



TAKE HOME YOUR ASSOCIAHEDRA!

SECONDARY

POLYTOPEA

B

C

C

D

E

E

FF

G

D

G

B

H

K

L

L

M

K

M

N

N

OO

H

A

LODAY’S

ASSOCIAHEDRON

D

A

E

M

K

AB

B

C C

D

F

E

J

G

G

H

H

II

J

M

FL

L

K

CHAP.-FOM.-ZEL.’S

ASSOCIAHEDRON

A

E

B

C

C

D

D

E

A

F

J

G

H

H

I

I

J

K

K

LLMM

B

F

G

Thibault Manneville & VP

Compatibility fans for graphical nested complexes

arXiv:1501.07152

VP

Signed tree associahedra

arXiv:1309.5222

THANK YOU



A

B

C

C

D

E

E F F

GD

G

B

H

K

L

L

M

K

M

N

N

OO

H

A

SECONDARY POLYTOPEGelfand-Kapranov-Zelevinsky (’94)


D

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Loday (’04)



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CHAPOTON-FOMIN-ZELEVINSKY’S ASSOCIAHEDRONChapoton-Fomin-Zelevinsky (’02)


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