MANY GEOMETRIC REALIZATIONS OF THE ASSOCIAHEDRON POLYTOPES & - - PowerPoint PPT Presentation

CombinatoireS July 2, 2015

• V. PILAUD

(CNRS & LIX)

MANY GEOMETRIC REALIZATIONS OF THE ASSOCIAHEDRON

POLYTOPES & COMBINATORICS

SIMPLICIAL COMPLEX

simplicial complex = collection of subsets of X downward closed exm:

X = [n] ∪ [n] ∆ = {I ⊆ X | ∀i ∈ [n], {i, i} ⊆ 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

123 321 213 132 312 231

Permutohedron Perm(n)

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

• ∅=J[n+1]
• x ∈ Rn+1
• j∈J

xj ≥

• |J| + 1

2

PERMUTAHEDRON

3412 3421 4321 4312 2413 4213 3214 1423 1432 1342 1243 1234 2134 1324 2341 2431 3124 2314

Permutohedron Perm(n)

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

• ∅=J[n+1]
• x ∈ Rn+1
• j∈J

xj ≥

• |J| + 1

2

PERMUTAHEDRON

3412 3421 4321 4312 2413 4213 3214 1423 1432 1342 1243 1234 2134 1324 2341 2431 3124 2314

Permutohedron Perm(n)

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

• ∅=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

3412 3421 4321 4312 2413 4213 3214 1423 1432 1342 1243 1234 2134 1324 2341 2431 3124 2314 1211 1221 1222 1212 2212 1112 2211 2321 1321 2331 1231 1332 1322 1232 1233 1323 1223 1312 2313 1213 2113 3213 2312 3212 2311 3211 3321 1123 2123 3312

Permutohedron Perm(n)

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

• ∅=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

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

to [n + 1 − k]

PERMUTAHEDRON

3|4|1|2 4|3|1|2 4|3|2|1 3|4|2|1 3|1|4|2 3|2|4|1 3|2|1|4 1|3|4|2 1|4|3|2 1|4|2|3 1|2|4|3 1|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|4 134|2 14|23 1|234 13|24 3|124 123|4 34|12 4|13|2 14|3|2 4|1|23 14|2|3 1|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|14 3|14|2 3|24|1 12|3|4 1|2|34 2|13|4 34|2|1 3|4|12 34|1|2 4|3|12

Permutohedron Perm(n)

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

• ∅=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

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

to [n + 1 − k]

≡ ordered partitions of [n + 1]

into n + 1 − k parts

PERMUTAHEDRON

3|4|1|2 4|3|1|2 4|3|2|1 3|4|2|1 3|1|4|2 3|2|4|1 3|2|1|4 1|3|4|2 1|4|3|2 1|4|2|3 1|2|4|3 1|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|4 134|2 14|23 1|234 13|24 3|124 123|4 34|12 4|13|2 14|3|2 4|1|23 14|2|3 1|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|14 3|14|2 3|24|1 12|3|4 1|2|34 2|13|4 34|2|1 3|4|12 34|1|2 4|3|12

Permutohedron Perm(n)

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

• ∅=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

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

to [n + 1 − k]

≡ ordered partitions of [n + 1]

into n + 1 − k parts

≡ collections of n − k nested

subsets of [n + 1]

COXETER ARRANGEMENT

34|12 134|2 3|124 13|24 123|4 1|234 14|23 3|4|1|2 4|3|1|2 4|3|2|1 3|4|2|1 3|1|4|2 3|2|4|1 3|2|1|4 1|3|4|2 1|4|3|2 1|4|2|3 1|2|4|3 1|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]

≡ ordered partitions of [n + 1]

into n + 1 − k parts

≡ 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 ↔ Schr¨

• der 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

• ✂

✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂✂ ❅ ❅ ❅ ❅ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂

• ❅

❅ ❅ ❅ ❇ ❇ ❇ ❇ ❇ ❇ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣α2

α1+α2 α2+α3 α1+α2+α3 α1 α3

(Pictures by CFZ)

Chapoton-Fomin-Zelevinsky (’02) 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)

Hopf algebra Cluster algebras CHAP.-FOM.-ZEL.’S ASSOCIAHEDRON

• ✂

✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂✂ ❅ ❅ ❅ ❅ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂

• ❅

❅ ❅ ❅ ❇ ❇ ❇ ❇ ❇ ❇ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣α2

α1+α2 α2+α3 α1+α2+α3 α1 α3

(Pictures by CFZ)

Chapoton-Fomin-Zelevinsky (’02) Ceballos-Santos-Ziegler (’11)

Cluster algebras

• ✁

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❅ ❅ ❅ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

• ❅

❅ ❅ ❍ ❍ ❍ ❍ ❍ ✁ ✁ ✁ ❇ ❇ ❇ ❇ ❍ ❍ ❍ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❍ ❍ ❍

• ❆

❆ ❆

• ❍❍❍❍

❍ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣α2

α1+α2 α2+α3

2α2 + α3

α1 α3

2α1 + 2α2 + α3

SECONDARY POLYTOPES

TRIANGULATIONS AND SUBDIVISIONS

triangulation of P ⊂ Rd = collection of triangles with corners in P such that

• covering property: their union cover the convex hull of P,
• intersection property: any two triangles intersect in a proper face.

flip =

FLIP GRAPH

REGULAR SUBDIVISIONS

P point set in Rd ω : P → R height function Sub(P, ω) = projection of the lower convex hull of the point set {(p, ω(p)) | p ∈ P}

regular subdivision = subdivision S such that ∃ ω : P → Rd for which S = Sub(P, ω)

REGULAR SUBDIVISIONS

P point set in Rd ω : P → R height function Sub(P, ω) = projection of the lower convex hull of the point set {(p, ω(p)) | p ∈ P}

regular subdivision = subdivision S such that ∃ ω : P → Rd for which S = Sub(P, ω)

NON-REGULAR TRIANGULATIONS

All triangulations of a convex polygon are regular

SECONDARY FAN

secondary cone of a subdivision S of P = C(S) =

ω ∈ RP

S refines S(P, ω) secondary fan of P = {C(S) | S subdivision of P}

SECONDARY POLYTOPE

volume vector of a triangulation T of P = Φ(T) =

p∈∆∈T vol(∆)

• p∈P

∈ RP

secondary polytope of P = convex hull of {Φ(T) | T triangulation of P}

SECONDARY POLYTOPE

• THM. For a point set P ⊂ RP:
• 1. The secondary polytope of P has dimension |P| − d − 1.
• 2. The secondary fan of P is the inner normal fan of the secondary polytope of P.
• 3. The face lattice of the secondary polytope of P is isom. to the refinement poset of

regular subdivisions of P.

Gelfand-Kapranov-Zelevinsky, Discriminants, resultants, and multidimensional determinants (’94)

SECONDARY POLYTOPE

Non-regular triangulations and subdivisions are invisible

SECONDARY POLYTOPE

Secondary polytope of a convex polygon = associahedron

LODAY’S ASSOCIAHEDRON

BINARY TREES

T binary tree

Infix search labeling = labeling with [n] with the following local rule

j <j >j ?

Rule for binary search trees

7 6 5 4 3 2 1

SYLVESTER FAN

cone of a binary tree T = C(T) = {x ∈ Rn | xi ≤ xj for each edge i → j in T} sylvester fan = {C(T) | T binary tree on n nodes}

123 321 213 132 312 231 x1=x2 x2=x3 x1=x3

LODAY’S ASSOCIAHEDRON

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

• 1≤i≤j≤n+1

H≥(i, j) L(T) := ℓ(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)

7 6 5 4 3 2 1 2 1 3 8 1 12 1

LODAY’S ASSOCIAHEDRON

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

• 1≤i≤j≤n+1

H≥(i, j) L(T) := ℓ(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)

123 321 213 312 141

LODAY’S ASSOCIAHEDRON

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

• 1≤i≤j≤n+1

H≥(i, j) L(T) := ℓ(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)

123 321 213 132 312 231 x1=x2 x2=x3 x1=x3 141

CAMBRIAN TREES

Cambrian tree = directed and labeled (with [n]) trees with the following local rule

j <j >j ? j <j >j ?

Rule for Cambrian trees

7 6 5 4 3 2 1

CAMBRIAN FANS

cone of a Cambrian tree T = C(T) = {x ∈ Rn | xi ≤ xj for each edge i → j in T} Cambrian fan = {C(T) | T binary tree on n nodes}

123 321 213 132 312 231 x1=x2 x2=x3 x1=x3

HOHLWEG-LANGE’S ASSOCIAHEDRA

For any signature ε ∈ ±n+1,

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

with HL(T)j :=

• ℓ(T, j) · r(T, j)

if ε(j) = −

n + 2 − ℓ(T, j) · r(T, j)

if ε(j) = +

Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07) Lange-P., Associahedra via spines (’13+)

1 1 2 2 3 12 7 7 6 5 4 3 2 1

HOHLWEG-LANGE’S ASSOCIAHEDRA

For any signature ε ∈ ±n+1,

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

with HL(T)j :=

• ℓ(T, j) · r(T, j)

if ε(j) = −

n + 2 − ℓ(T, j) · r(T, j)

if ε(j) = +

Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07) Lange-P., Associahedra via spines (’13+)

123 321 132 231 303

HOHLWEG-LANGE’S ASSOCIAHEDRA

For any signature ε ∈ ±n+1,

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

with HL(T)j :=

• ℓ(T, j) · r(T, j)

if ε(j) = −

n + 2 − ℓ(T, j) · r(T, j)

if ε(j) = +

Hohlweg-Lange, Realizations of the associahedron and cyclohedron (’07) Lange-P., Associahedra via spines (’13+)

123 321 213 132 312 231 x1=x2 x2=x3 x1=x3 303

COMPATIBILITY FANS

COMPATIBILITY FANS

T◦ an initial triangulation δ, δ′ two internal diagonals

compatibility degree between δ and δ′

(δ δ′) =

  

−1

if δ = δ′ if δ and δ′ do not cross

1

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

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.

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

WHAT SHOULD I TAKE HOME FROM THIS TALK?

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

• ✂

✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂✂ ❅ ❅ ❅ ❅ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂

• ❅

❅ ❅ ❅ ❇ ❇ ❇ ❇ ❇ ❇ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣α2

α1+α2 α2+α3 α1+α2+α3 α1 α3

(Pictures by CFZ)

Chapoton-Fomin-Zelevinsky (’02) Ceballos-Santos-Ziegler (’11)

TAKE HOME YOUR ASSOCIAHEDRA!

SECONDARY POLYTOPE

A B C C D E E F F G D G B H K L L M K M N N O O H A

LODAY’S ASSOCIAHEDRON

D A E M K A B B C C D F E J G G H H I I J M F L 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 L L M M B F G

TAKE HOME YOUR ASSOCIAHEDRA!

SECONDARY POLYTOPE

A B C C D E E F F G D G B H K L L M K M N N O O H A

LODAY’S ASSOCIAHEDRON

D A E M K A B B C C D F E J G G H H I I J M F L 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 L L M M B F G

THANK YOU

A B C C D E E F F G D G B H K L L M K M N N O O H A

SECONDARY POLYTOPE

Gelfand-Kapranov-Zelevinsky (’94) Billera-Filliman-Sturmfels (’90)

D A E M K A B B C C D F E J G G H H I I J M F L L K

LODAY’S ASSOCIAHEDRON

Loday (’04) Hohlweg-Lange (’07) Hohlweg-Lange-Thomas (’12)

A E B C C D D E A F J G H H I I J K K L L M M B F G

CHAPOTON-FOMIN-ZELEVINSKY’S ASSOCIAHEDRON

Chapoton-Fomin-Zelevinsky (’02) Ceballos-Santos-Ziegler (’11)