SIGNED TREE ASSOCIAHEDRA Vincent PILAUD (CNRS & LIX) 1 2 3 4 4 2 1 3 POLYTOPES FROM COMBINATORICS POLYTOPES & COMBINATORICS polytope = convex hull of a finite set of Rd = bounded intersection of finitely many half-spaces face = intersection with a supporting hyperplane face lattice = all the faces with their inclusion relations Given a set of points, determine the face lattice of its convex hull. Given a lattice, is there a polytope which realizes it? PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g \ ≥ 321 = H \ H (J) ?6=J([n+1] 312 231 x1=x3 213 132 x1=x2 123 x2=x3 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ = \ H≥(J) 2211 4312 H 3421 6=J [n+1] 3412 ? ( 4213 k-faces of Perm(n) 2212 ≡ ordered partitions of [n + 1] 2431 1211 into n + 1 − k parts 2413 ≡ surjections from [n + 1] 2341 3214 to [n + 1 − k] 1221 1432 2314 1423 1212 3124 1342 1222 1112 1324 2134 1243 1234 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ = \ H≥(J) 2211 4312 H 3421 6=J [n+1] 3412 ? ( 4213 k-faces of Perm(n) 2212 ≡ ordered partitions of [n + 1] 2431 1211 into n + 1 − k parts 2413 ≡ surjections from [n + 1] 2341 3214 to [n + 1 − k] 1221 1432 2314 1423 1212 3124 connections to • inversion sets 1342 1222 1112 • weak order 1324 • reduced expressions 2134 1243 • braid moves 1234 • cosets of the symmetric group 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¨oder 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) Lee ('89), Gel'fand-Kapranov-Zelevinski ('94), Billera-Filliman-Sturmfels ('90), . , Ceballos-Santos-Ziegler ('11) Loday ('04), Hohlweg-Lange ('07), Hohlweg-Lange-Thomas ('12), P.-Santos ('12), P.-Stump ('12+), Lange-P. ('13+) LODAY'S ASSOCIAHEDRON Loday's associahedron = conv fL(T ) j T triangulation of the (n + 3)-gong \ ≥ = H \ H (δ) δ diagonal of the (n+3)-gon k δ i r(T; j) `(T; j) j B(δ) ≥ n+1 X jB(δ)j + 1 L(T ) = `(T; j) · r(T; j) H (δ) = x 2 R xj ≥ j2[n+1] 2 j2B(δ) Loday, Realization of the Stasheff polytope ('04) LODAY'S ASSOCIAHEDRON Loday's associahedron = conv fL(T ) j T binary tree on n + 1 nodesg \ ≥ = H \ H (I) I interval of [n+1] k k i∗ ∗ i ∗ i i j j∗ r(T; j) `(T 0; j) r(T 0; j) `(T; j) j j 0 L(T ) − L(T ) 2 R>0(ei − ej) Loday, Realization of the Stasheff polytope ('04) LODAY'S ASSOCIAHEDRON Loday's associahedron = conv fL(T ) j T binary tree on n + 1 nodesg \ ≥ = H \ H (I) I interval of [n+1] i∗ ∗ i j∗ j∗ r(T; j) `(T 0; j) r(T 0; j) `(T; j) 0 L(T ) − L(T ) 2 R>0(ei − ej) Loday, Realization of the Stasheff polytope ('04) ASSOCIAHEDRON AND PERMUTAHEDRON The associahedron is obtained from the permutahedron by removing facets ASSOCIAHEDRON AND PERMUTAHEDRON Relevant connections to combinatorial properties: • the normal fan of Perm(n) refines that of Asso(P ) • it defines a surjection κ : Sn+1 ! ftriangulationsg (connection to linear extensions and insertion in binary search trees) • κ defines a lattice homomorphism from the weak order to the Tamari lattice LODAY'S ASSOCIAHEDRON AND PERMUTAHEDRON 321 312 231 x =x 1 3 141 213 132 123 x1=x2 x2=x3 HOHLWEG & LANGE'S ASSOCIAHEDRA Can also replace Loday's (n + 3)-gon by others. 1 2 3 1 2 1 3 0 + + + 4 0 + + – 4 0 + – + 4 0 4 – – – 3 2 1 2 3 . to obtain different realizations of the associahedron Hohlweg-Lange, Realizations of the associahedron and cyclohedron ('07) HOHLWEG & LANGE'S ASSOCIAHEDRA \ ≥ Asso(P ) = conv fHL(T ) j T triangulation of P g = H \ H (δ) δ diagonal of P j r(T; j) `(T; j) δ B(δ) i k ( `(T; j) · r(T; j) if j down ≥ X jB(δ)j + 1 HL(T )j = H (δ) = x xj ≥ n + 2 − `(T; j) · r(T; j) j 2 if up j2B(δ) Hohlweg-Lange, Realizations of the associahedron and cyclohedron ('07) SPINES Lange-P., Using spines to revisit a construction of the associahedron ('13+) 5 5 3 3 2 7 2 7 5* 5* 2* 2* 7* 7* 3* 0 1* 9 0 1* 9 3* 6* 8* 8* 1 8 1 6* 8 4* 4* 4 6 4 6 Spines = labeled and oriented dual binary trees SPINES Lange-P., Using spines to revisit a construction of the associahedron ('13+) 5* 5* 2* 2* 7* 7* 3* 1* 1* 3* 6* 8* 8* 6* 4* 4* REM. 1. Spines can be defined without their triangulations. <j >j j j <j >j SPINES Lange-P., Using spines to revisit a construction of the associahedron ('13+) 5* 5* 2* 2* 7* 7* 3* 1* 1* 3* 6* 8* 8* 6* 4* 4* REM. 1. Spines can be defined without their triangulations. 2. Alternative vertex description of Hohlweg-Lange's associahedra: ( jfπ maximal path in S with 2 incoming arcs at jgj if j down a(S)j = n + 2 − jfπ maximal path in S with 2 outgoing arcs at jgj if j up SPINES Lange-P., Using spines to revisit a construction of the associahedron ('13+) 5* 5* 2* 2* 7* 7* 3* 1* 1* 3* 6* 8* 8* 6* 4* 4* REM. 1. Spines can be defined without their triangulations. 2. Alternative vertex description of Hohlweg-Lange's associahedra: ( jfπ path in S not using the outgoing arc at jgj if j down a(S)j = n + 2 − jfπ path in S not using the incomming arc at jgj if j up GRAPH ASSOCIAHEDRA NESTED COMPLEX AND GRAPH ASSOCIAHEDRON G graph on ground set V Tube on V = connected induced subgraph of G Compatible tubes = nested, or disjoint and non-adjacent 8 4 9 8 4 9 1 1 2 3 5 2 3 5 0 0 6 7 6 7 Nested complex N (G) = simplicial complex of sets of pairwise compatible tubes = 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 4321 4312 3421 3412 4213 2431 2413 2341 3214 1432 2314 1423 3124 1342 1324 2134 1243 1234 TWO QUESTIONS Qu 1. Which graph associahedra can be realized by removahedra? Lange-P., Which nestohedra are removahedra? ('14+) Qu 2. Can we obtain distinct realizations of graph associahedra? Yes for trees. P., Signed tree associahedra ('13+) SIGNED TREE ASSOCIAHEDRON SIGNED SPINES 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 + 2 8 4 9 2 7 1 35678 368 2 3 5 0 0 49 0 49 5 6 7 1 1 CONTRACTIONS AND SPINE COMPLEX LEM. Contracting an arc in a signed spine on T leads to a new signed spine on T LEM. Let S be a signed spine on T with a node labeled by a set U containing at least two elements. For any u 2 U, there exists a signed spine on T whose nodes are labeled exactly as that of S, except that the label U is partitioned into fug and U r fug 2 7 2 7 6 6 8 3 3 0 8 0 9 5 9 5 1 1 4 4 CONTRACTIONS AND SPINE COMPLEX LEM. Contracting an arc in a signed spine on T leads to a new signed spine on T LEM. Let S be a signed spine on T with a node labeled by a set U containing at least two elements. For any u 2 U, there exists a signed spine on T whose nodes are labeled exactly as that of S, except that the label U is partitioned into fug and U r fug Signed spine complex S(T) = simplicial complex whose inclusion poset is isomorphic to the poset of edge contractions on the signed spines of T CORO. The signed spine complex S(T) is a pure simplicial complex of rank jVj BRAID FAN V V Braid arrangement on R = collection of hyperplanes x 2 R xu = xv for u =6 v 2 V Braid fan BF = complete simplicial fan defined by the braid arrangement on V X jVj + 1 H := x 2 R xv = 2 v2V SPINE FAN For S spine on T, define C(S) := fx 2 H j xu ≤ xv; for all arcs u ! v in Sg 1 2 1 4 2 3 2 2 3 4 1 3 2 1 3 4 4 4 3 4 3 4 2 2 2 3 1 3 1 4 1 1 THEO.