Compatibility fans realizing graph associahedra - LIX

Compatibility fans realizing graph
associahedra
Thibault Manneville (LIX, Polytechnique)
joint work with
Vincent Pilaud (CNRS, LIX Polytechnique)
Journées du GDR-IM CombAlg
September 22th , 2015
The ip operation
Flip graph on the triangulations of the polygon:
Vertices: triangulations
Edges: ips
(n + 3)-gon ⇒ n diagonals ⇒ the ip graph is n-regular.
Denition
An associahedron is a polytope whose graph is the ip graph
of triangulations of a convex polygon.
Faces ↔ dissections of the polygon
Useful conguration (Loday's)
Gn+3 =
0
n+2
1
n+1
2
n
Graph point of view
{diagonals of Gn+3 } ←→ {strict subpaths of the path [n + 1]}
0
8
1
7
2
6
3
4
5
Non-crossing diagonals
Two ways to be non-crossing in Loday's conguration:
8
0
7
1
6
2
3
4
5
nested subpaths
8
0
7
1
6
2
3
4
5
non-adjacent subpaths
Caution with the second case:
The right condition is indeed non-adjacent, disjoint is not
enough!
8
0
7
1
6
2
3
4
5
Now do it on graphs
G = (V , E ) a (connected) graph.
Denition
Now do it on graphs
G = (V , E ) a (connected) graph.
Denition
A tube of G is a proper subset t ⊆ V inducing a
connected subgraph of G ;
Now do it on graphs
G = (V , E ) a (connected) graph.
Denition
A tube of G is a proper subset t ⊆ V inducing a
connected subgraph of G ;
t and t 0 are compatible if they are nested or
non-adjacent;
Now do it on graphs
G = (V , E ) a (connected) graph.
Denition
A tube of G is a proper subset t ⊆ V inducing a
connected subgraph of G ;
t and t 0 are compatible if they are nested or
non-adjacent;
A tubing on G is a set of pairwise compatible tubes of G .
2
2
4
5
4
1
5
1
3
6
3
6
A tube
A maximal tubing
(generalizes a diagonal)
(generalizes a triangulation)
Graph associahedra
Theorem (Carr-Devadoss '06)
There exists a polytope AssoG , the graph associahedron
of G , realizing the complex of tubings on G .
Faces ↔ tubings of G .
Some classical polytopes...
4321
3421
4312
3412
4213
2431
2413
2341
1432
1342
1243
The associahedron
The cyclohedron
3214
2314
3124
1423
1324
2134
1234
The permutahedron
...can be seen as graph associahedra
4321
3421
4312
3412
4213
2431
2413
2341
1432
1342
1243
The associahedron
The cyclohedron
3214
2314
3124
1423
1324
2134
1234
The permutahedron
Many dierent associahedra
Hohlweg-Lange [HL]: O(2n )
Ceballos-Santos-Ziegler [CSZ] (Santos): O(Cat(n))
[HL]
T
[CSZ] = Chapoton-Fomin-Zelewinsky [CFZ] (type A): 1
Few graph associahedra
Carr-Devadoss [CD]: 1 ⊂ Postnikov [P]: 1
Volodin [Vol]: ???
Probably many, but not explicit.
Many dierent associahedra
Hohlweg-Lange [HL]: O(2n )
Ceballos-Santos-Ziegler [CSZ] (Santos): O(Cat(n))
[HL]
T
[CSZ] = Chapoton-Fomin-Zelewinsky [CFZ] (type A): 1
Fans
Polyhedral Cone: positive span of nitely many vectors.
Fans
Polyhedral Cone: positive span of nitely many vectors.
Simplicial Cone: positive span of independent vectors.
Fans
Polyhedral Cone: positive span of nitely many vectors.
Simplicial Cone: positive span of independent vectors.
Fan = set of polyhedral cones intersecting properly.
Fans
Polyhedral Cone: positive span of nitely many vectors.
Simplicial Cone: positive span of independent vectors.
Fan = set of polyhedral cones intersecting properly.
Simplicial Fan: fan whose cones all are simplicial.
Complete Fan: fan whose cones cover the whole space.
polytope ⇒ complete fan (normal fan).
simple polytope ⇒ complete simplicial fan.
Santos' construction for the fan
→ choose an initial triangulation T0 of the polygon.
d
d1
T0
d4
d2
d3
set udi = −ei
Santos' construction for the fan
→ choose an initial triangulation T0 of the polygon.
d
d1
T0
d2
d3
set udi = −ei
d4
→ for a diagonal d ∈
/ T0 , dene ud = (11d crosses di )di ∈T0 .
→ for a triangulation T , dene C (T ) = cone(ud |d ∈ T ).
→ Dene F = {C (T )|T triangulation}.
Theorem (Ceballos-Santos-Ziegler 13)
F is a complete simplicial fan realizing the associahedron.
Theorem (Ceballos-Santos-Ziegler 13)
F is a complete simplicial fan realizing the associahedron.
T0 =
T0 =
T0 =
Idea of the proof
→ The cone C (T0 ) is the negative orthant.
⇒ full-dimensional and simplicial
Idea of the proof
→ The cone C (T0 ) is the negative orthant.
⇒ full-dimensional and simplicial
→ Local condition on ips T ↔ T 0 = T r {d} ∪ {d 0 }.
ud
C(T )
C(T 0)
< uδ >δ∈T \{d}
ud0
Checking local conditions
→ Formulation: αud + α0 ud 0 +
X
βδ uδ = 0 ⇒ α.α0 > 0.
δ∈T r{d}
Checking local conditions
→ Formulation: αud + α0 ud 0 +
X
βδ uδ = 0 ⇒ α.α0 > 0.
δ∈T r{d}
→ Reduction:
→ Finite number of linear dependences to check explicitly.
For graphs?
2
2
1
1
4
3
3
T0
→ impossible to choose −1, 0, 1 coordinates.
4
The compatibility degree
→ notion of compatibility degree between two tubes (t k t 0 ).
The compatibility degree
→ notion of compatibility degree between two tubes (t k t 0 ).

 −1 if t = t 0 ,
(t k t 0 ) = #(neighbors of t 0 in t r t 0 ) if t 0 6⊆ t,

0 otherwise.
→ Counts compatibility obstructions.
t0
t
(t k t 0 ) = 2
(t 0 k t) = 3
The result!
→ Dene ut = ((t k t1 ), . . . , (t k tn ))
→ For a maximal tubing T , dene C (T ) = cone(ut |t ∈ T ).
→ Dene FG = {C (T )|T triangulation}.
The result!
→ Dene ut = ((t k t1 ), . . . , (t k tn ))
→ For a maximal tubing T , dene C (T ) = cone(ut |t ∈ T ).
→ Dene FG = {C (T )|T triangulation}.
Theorem (M.,Pilaud 15)
FG is a complete simplicial fan realizing AssoG .
The result!
→ Dene ut = ((t k t1 ), . . . , (t k tn ))
→ For a maximal tubing T , dene C (T ) = cone(ut |t ∈ T ).
→ Dene FG = {C (T )|T triangulation}.
Theorem (M.,Pilaud 15)
FG is a complete simplicial fan realizing AssoG .
T0 =
T0 =
T0 =
Link with cluster complexes
→ [CFZ]: compatibility degrees between roots in nite types
to construct generalized associahedra.
T
{Generalized Associahedra} {Graph Associahedra} = A, B, C .
Link with cluster complexes
→ [CFZ]: compatibility degrees between roots in nite types
to construct generalized associahedra.
T
{Generalized Associahedra} {Graph Associahedra} = A, B, C .
type graph
A
path
B
cycle
C
cycle
Link with cluster complexes
→ [CFZ]: compatibility degrees between roots in nite types
to construct generalized associahedra.
T
{Generalized Associahedra} {Graph Associahedra} = A, B, C .
type graph
A
path
B
cycle
C
cycle
roots
tubes
0
(α k α ) (t k t 0 )
(α k α0 ) (t k t 0 )
(α k α0 ) (t 0 k t)
THANK YOU FOR
YOUR AMAZED
ATTENTION!

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