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: 123 123 123 123 123 123 123 123 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

FANS polyhedral cone = positive span of a finite set of R d = 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 R d = 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 Permutohedron Perm ( n ) = conv { ( σ (1) , . . . , σ ( n + 1)) | σ ∈ Σ n +1 } 321 � � � �� | J | + 1 � � x ∈ R n +1 � = H ∩ x j ≥ � 2 � j ∈ J ∅ � = J � [ n +1] 312 231 213 132 123

PERMUTAHEDRON Permutohedron Perm ( n ) = conv { ( σ (1) , . . . , σ ( n + 1)) | σ ∈ Σ n +1 } 4321 � � � �� | J | + 1 � � x ∈ R n +1 � = H ∩ x j ≥ 4312 � 2 � 3421 j ∈ J ∅ � = J � [ n +1] 3412 4213 2431 2413 2341 3214 2314 1432 1423 3124 1342 1324 2134 1243 1234

PERMUTAHEDRON Permutohedron Perm ( n ) = conv { ( σ (1) , . . . , σ ( n + 1)) | σ ∈ Σ n +1 } 4321 � � � �� | J | + 1 � � x ∈ R n +1 � = H ∩ x j ≥ 4312 � 2 � 3421 j ∈ J ∅ � = J � [ n +1] 3412 connections to 4213 • weak order 2431 • reduced expressions • braid moves 2413 2341 • cosets of the symmetric group 3214 2314 1432 1423 3124 1342 1324 2134 1243 1234

PERMUTAHEDRON Permutohedron Perm ( n ) = conv { ( σ (1) , . . . , σ ( n + 1)) | σ ∈ Σ n +1 } 4321 � � � �� | J | + 1 3211 � � x ∈ R n +1 � = H ∩ x j ≥ 4312 3321 � 2 2211 � 3312 3421 j ∈ J ∅ � = J � [ n +1] 2311 3212 3412 connections to 2321 4213 • weak order 2312 2212 2431 • reduced expressions 1211 3213 2331 • braid moves 2413 1321 2341 • cosets of the symmetric group 3214 2313 1312 1221 2113 k -faces of Perm ( n ) 2314 1432 1231 1322 1212 1423 3124 ≡ surjections from [ n + 1] 1332 1213 to [ n + 1 − k ] 1323 2123 1342 1222 1112 1324 1232 2134 1223 1243 1123 1233 1234

PERMUTAHEDRON Permutohedron Perm ( n ) = conv { ( σ (1) , . . . , σ ( n + 1)) | σ ∈ Σ n +1 } � � � �� 4 | 3 | 2 | 1 | J | + 1 � � 34 | 2 | 1 x ∈ R n +1 � = H ∩ x j ≥ 3 | 4 | 2 | 1 4 | 3 | 12 � 2 34 | 12 � 3 | 4 | 12 4 | 3 | 1 | 2 j ∈ J ∅ � = J � [ n +1] 34 | 1 | 2 3 | 24 | 1 3 | 4 | 1 | 2 connections to 4 | 13 | 2 3 | 2 | 4 | 1 • weak order 3 | 14 | 2 3 | 124 4 | 1 | 3 | 2 • reduced expressions 134 | 2 3 | 2 | 14 4 | 1 | 23 • braid moves 3 | 1 | 4 | 2 14 | 3 | 2 • cosets of the symmetric group 4 | 1 | 2 | 3 3 | 2 | 1 | 4 3 | 1 | 24 13 | 4 | 2 14 | 23 23 | 1 | 4 k -faces of Perm ( n ) 3 | 1 | 2 | 4 1 | 4 | 3 | 2 14 | 2 | 3 13 | 24 1 | 3 | 4 | 2 ≡ surjections from [ n + 1] 2 | 3 | 1 | 4 1 | 4 | 23 13 | 2 | 4 to [ n + 1 − k ] 13 | 2 | 4 2 | 13 | 4 1 | 4 | 2 | 3 ≡ ordered partitions of [ n + 1] 1 | 234 123 | 4 1 | 3 | 2 | 4 1 | 24 | 3 into n + 1 − k parts 2 | 1 | 3 | 4 1 | 23 | 4 1 | 2 | 4 | 3 12 | 3 | 4 1 | 2 | 34 1 | 2 | 3 | 4

PERMUTAHEDRON Permutohedron Perm ( n ) = conv { ( σ (1) , . . . , σ ( n + 1)) | σ ∈ Σ n +1 } � � � �� 4 | 3 | 2 | 1 | J | + 1 � � 34 | 2 | 1 x ∈ R n +1 � = H ∩ x j ≥ 3 | 4 | 2 | 1 4 | 3 | 12 � 2 34 | 12 � 3 | 4 | 12 4 | 3 | 1 | 2 j ∈ J ∅ � = J � [ n +1] 34 | 1 | 2 3 | 24 | 1 3 | 4 | 1 | 2 connections to 4 | 13 | 2 3 | 2 | 4 | 1 • weak order 3 | 14 | 2 3 | 124 4 | 1 | 3 | 2 • reduced expressions 134 | 2 3 | 2 | 14 4 | 1 | 23 • braid moves 3 | 1 | 4 | 2 14 | 3 | 2 • cosets of the symmetric group 4 | 1 | 2 | 3 3 | 2 | 1 | 4 3 | 1 | 24 13 | 4 | 2 14 | 23 23 | 1 | 4 k -faces of Perm ( n ) 3 | 1 | 2 | 4 1 | 4 | 3 | 2 14 | 2 | 3 13 | 24 1 | 3 | 4 | 2 ≡ surjections from [ n + 1] 2 | 3 | 1 | 4 1 | 4 | 23 13 | 2 | 4 to [ n + 1 − k ] 13 | 2 | 4 2 | 13 | 4 1 | 4 | 2 | 3 ≡ ordered partitions of [ n + 1] 1 | 234 123 | 4 1 | 3 | 2 | 4 1 | 24 | 3 into n + 1 − k parts 2 | 1 | 3 | 4 1 | 23 | 4 ≡ collections of n − k nested 1 | 2 | 4 | 3 12 | 3 | 4 1 | 2 | 34 1 | 2 | 3 | 4 subsets of [ n + 1]

COXETER ARRANGEMENT Coxeter fan = fan defined by the hyperplane arrangement 4 | 3 | 2 | 1 3 | 4 | 2 | 1 34 | 12 � x i = x j x ∈ R n +1 � � � 1 ≤ i<j ≤ n +1 4 | 3 | 1 | 2 = collection of all cones 3 | 4 | 1 | 2 � x i < x j if π ( i ) < π ( j ) � x ∈ R n +1 � � 3 | 2 | 4 | 1 4 | 1 | 2 | 3 3 | 124 for all surjections π : [ n + 1] → [ n + 1 − k ] 4 | 1 | 3 | 2 134 | 2 3 | 1 | 4 | 2 3 | 2 | 1 | 4 14 | 23 3 | 1 | 2 | 4 ( n − k ) -dimensional cones 1 | 4 | 3 | 2 1 | 3 | 4 | 2 13 | 24 ≡ surjections from [ n + 1] 1 | 4 | 2 | 3 2 | 3 | 1 | 4 to [ n + 1 − k ] 123 | 4 1 | 3 | 2 | 4 ≡ ordered partitions of [ n + 1] 1 | 234 into n + 1 − k parts 1 | 2 | 4 | 3 2 | 1 | 3 | 4 ≡ collections of n − k nested 1 | 2 | 3 | 4 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 vertices ↔ binary trees edges ↔ flips edges ↔ rotations faces ↔ dissections faces ↔ Schr ¨ o 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 LODAY’S CHAP.-FOM.-ZEL.’S POLYTOPE ASSOCIAHEDRON ASSOCIAHEDRON � � ❅ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ α 2 ❅ (Pictures by CFZ) � � ❅ ✂✂ ❇ ❇ ❅ ❇ ❅ ✂ ❇ ❅ ✂ α 2 + α 3 ❇ ❅ ✂ ❇ ❅ α 1 + α 2 ✂ ✂ ❅ � ✂ ✂ ❅ � ✂ ✂ ❅ � ✂ ✂ ❅ � ✂ ✂ ✂ α 1 + α 2 + α 3 ✂ ✂ ✂ ✂ α 3 ❅ ❅ ✂ ❅ ✂ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ � � ❅ ✂ α 1 � � � � Gelfand-Kapranov-Zelevinsky (’94) Loday (’04) Chapoton-Fomin-Zelevinsky (’02) Billera-Filliman-Sturmfels (’90) Hohlweg-Lange (’07) Ceballos-Santos-Ziegler (’11) Hohlweg-Lange-Thomas (’12)

THREE FAMILIES OF REALIZATIONS SECONDARY LODAY’S CHAP.-FOM.-ZEL.’S POLYTOPE ASSOCIAHEDRON ASSOCIAHEDRON � � ❅ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ α 2 ❅ (Pictures by CFZ) � � ❅ ✂✂ ❇ ❇ ❅ ❇ ❅ ✂ ❇ ❅ ✂ α 2 + α 3 ❇ ❅ ✂ ❇ ❅ α 1 + α 2 ✂ ✂ ❅ � ✂ ✂ ❅ � ✂ ✂ ❅ � ✂ ✂ ❅ � ✂ ✂ ✂ α 1 + α 2 + α 3 ✂ ✂ ✂ ✂ α 3 ❅ ❅ ✂ ❅ ✂ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ � � ❅ ✂ α 1 � � � � Gelfand-Kapranov-Zelevinsky (’94) Loday (’04) Chapoton-Fomin-Zelevinsky (’02) Billera-Filliman-Sturmfels (’90) Hohlweg-Lange (’07) Ceballos-Santos-Ziegler (’11) Hohlweg-Lange-Thomas (’12) �❍❍❍❍ � � � ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ α 2 ✁ ❆ ❍ Hopf ❆ 2 α 2 + α 3 ✁ � ❅ ❆ � Cluster ❍ ❅ ❍ ✁ ❍ � ❅ algebra ✁ ❅ ✁ algebras α 1 + α 2 ❅ ✁ ❅ ✁ ❍ ❍ α 2 + α 3 ✁ ❍ ❅ Cluster ✁ ✁ ❇ � ✁ ✁ ❇ � 2 α 1 + 2 α 2 + α 3 ✁ ✁ ❇ � algebras ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ � ✁ ✁ ❇ � ❍ � ❍ ✁ ❅ ❍ ✁ ❅ � α 3 � ❍ ❍ ✁ ❅ � � α 1 ❅ ✁ ❅ ✁ � ❅ ✁ � �

SECONDARY POLYTOPES