SIGNED TREE ASSOCIAHEDRA Vincent PILAUD (CNRS & LIX)

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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 {(σ(1), . . . , σ(n + 1)) | σ ∈ Σn+1} \ ≥ 321 = H ∩ H (J) ∅6=J([n+1] 312 231 x1=x3 213 132

x1=x2 123 x2=x3 PERMUTAHEDRON

Permutohedron Perm(n)

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

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 {L(T ) | T triangulation of the (n + 3)-gon} \ ≥ = H ∩ H (δ) δ diagonal of the (n+3)-gon

k δ i r(T, j) `(T, j) j B(δ)

≥ n+1 X |B(δ)| + 1 L(T ) = `(T, j) · r(T, j) H (δ) = x ∈ R xj ≥ j∈[n+1] 2 j∈B(δ)

Loday, Realization of the Stasheﬀ polytope (’04) LODAY’S ASSOCIAHEDRON

Loday’s associahedron = conv {L(T ) | T binary tree on n + 1 nodes} \ ≥ = 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 ) ∈ R>0(ei − ej)

Loday, Realization of the Stasheﬀ polytope (’04) LODAY’S ASSOCIAHEDRON

Loday’s associahedron = conv {L(T ) | T binary tree on n + 1 nodes} \ ≥ = 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 ) ∈ 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 → {triangulations} (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 {HL(T ) | T triangulation of P } = H ∩ H (δ) δ diagonal of P j r(T, j)

`(T, j)

B(δ) i

( `(T, j) · r(T, j) if j down ≥ X |B(δ)| + 1 HL(T )j = H (δ) = x xj ≥ n + 2 − `(T, j) · r(T, j) j 2 if up j∈B(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 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: ( |{π maximal path in S with 2 incoming arcs at j}| if j down a(S)j = n + 2 − |{π maximal path in S with 2 outgoing arcs at j}| 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: ( |{π path in S not using the outgoing arc at j}| if j down a(S)j = n + 2 − |{π path in S not using the incomming arc at j}| 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

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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 ∈ 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 {u} and U r {u}

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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

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 |V| BRAID FAN

V V Braid arrangement on R = collection of hyperplanes x ∈ R xu = xv for u =6 v ∈ V Braid fan BF = complete simplicial fan deﬁned by the braid arrangement on V X |V| + 1 H := x ∈ R xv = 2 v∈V SPINE FAN

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

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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

CORO. For any signed tree T, the signed nested complex N (T) is a simplicial sphere SIGNED TREE ASSOCIAHEDRON

Signed tree associahedron Asso(T) = convex polytope with

(i) a vertex a(S) ∈ RV for each maximal signed spine S ∈ S(T), with coordinates ( − {π ∈ Π(S) | v ∈ π and rv ∈/ π} if v ∈ V a(S)v = + |V| + 1 − {π ∈ Π(S) | v ∈ π and rv ∈/ π} if v ∈ V

− + where rv = unique incoming (resp. outgoing) arc when v ∈ V (resp. when v ∈ V ) Π(S) = set of all (undirected) paths in S, including the trivial paths

(ii) a facet deﬁned by the half-space ≥ V X |B| + 1 H (B) := x ∈ R xv ≥ 2 v∈B for each signed building block B ∈ B(T) EXM: VERTEX DESCRIPTION

1 1 2 1 2 1 2 1 3 1 1 2 1 4 3 3 4 1 1 3 4 2 3 4 3 (4,4,1,1) 2 2 4 2 2 4 2 3 4 2 2 4 3 4 (4,3,2,1) 1 4 4 3 3 (4,3,1,2) 3 4 3 3 (3,4,2,1) (1,7,1,1) (4,2,3,1)(444,2,4,2 4 (3,4,1,2) 2 1 1 1 (4,2,1,3) 3 1 (4,1,4,1)(4( (4,1,2,3)121,21,1,22,3)223) 4 (2,4,3,1) 3 2 (4,1,2,3)(4,1,2,(412,33)3 2 (3,2,4,1)3,2,22,2422,4,4,4 (2,4,1,3) (4,1,1,4) 2 3 4 (2,3,4,1) 4 (3,2,1,4) 1 1 4 (1,4,4,1) (3,1,4,2)3,1,1,4141,4,4 2 2 3 (1,4,3,2) (2,3,1,4) (3,1,2,4) 1 (1,4,2,3) 2 (1,4,1,4) 4 3 1 4 3 (1,3,4,2) 3 3 4 3 4 (1,3,2,4) (2,1,4,3)2,1,1,41,141,4,4,3)4 4 (4,–2,4,4) 1 (2,1,3,4) 2 3 (1,2,4,3) 4 2 2 (1,2,3,4) 2 2 1 4 4 4 3 4 3 1 1 3 4 (1,1,4,4) 4 3 2 1 1 4 2 4 3 1 3 3 3 4 2 2 1 1 2 2 1 2 3 2 1 1 1 EXM: FACET DESCRIPTION

1 1 2 3 1 1 2 3 4 1 2 3 2 3 4 2 3 4 4 1 1 4 1 34 2 3 2 3 2 3 234234 4 4 1 4 4 3 4 2 3 234 3 24 134 4 1 23 1 1 2 3 2 3 14 2 3 124124 2 4 1 4 13 4 1 124124 123 2 3 123 1 1 1 12 4 1 2 3 2 3 2 3 1 4 4 4 1 2 3 1 1 2 3 4 2 3 2 3 4 4 4 MAIN RESULT

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

( − {π ∈ Π(S) | v ∈ π and rv ∈/ π} if v ∈ V a(S)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 ≥ V X |B| + 1 H (B) := x ∈ R xv ≥ 2 v∈B for all signed building blocks B ∈ B(T)

CORO. The signed tree associahedron Asso(T) realizes the signed nested complex N (T) SKETCH OF THE PROOF

STEP 1. We have X |V| + 1 X |sc(r)| + 1 a(S) = and a(S) = v 2 v 2 v∈V v∈sc(r) for any arc r of S. In other words,“each vertex a(S) belongs to the hyperplanes H=(B) it is supposed to”. Proof by double counting SKETCH OF THE PROOF

STEP 2. If S and S0 are two adjacent maximal spines on T, such that S0 is obtained from S by ﬂipping an arc joining node u to node v, then

0 a(S ) − a(S) ∈ R>0 · (eu − ev) U o U o r r' u v u v

i i V V 0 a(S ) − a(S) = (|U| + 1) · (|V | + 1) · (eu − ev) SKETCH OF THE PROOF

STEP 2. If S and S0 are two adjacent maximal spines on T, such that S0 is obtained from S by ﬂipping an arc joining node u to node v, then

0 a(S ) − a(S) ∈ R>0 · (eu − ev)

STEP 3. A general theorem concerning realizations of simplicial fan by polytopes In other words, a characterization of when is a simplicial fan regular

Hohlweg-Lange-Thomas, Permutahedra and generalized associahedra (’11) De Loera-Rambau-Santos, Triangulations: Structures for Algorithms and Applications (’10) FURTHER GEOMETRIC PROPERTIES

PROP. The signed tree associahedron Asso(T) is sandwiched between the permutahedron Perm(V) and the parallelepiped Para(T) X X [eu, ev] = Perm(T) ⊂ Asso(T) ⊂ Para(T)= π(u − v) · [eu, ev] u6=v∈V u−v∈T Common vertices of Asso(T) and Para(T) ≡ orientations of T which are spines on T Common vertices of Asso(T) and Perm(T) ≡ linear orders on V which are spines on T ⇒ no common vertex of the three polytopes except if T is a signed path FURTHER GEOMETRIC PROPERTIES

PROP. Asso(T) and Asso(T0) isometric ⇐⇒ T and T0 isomorphic or anti-isomorphic, up to the sign of their leaves, ie. ∃ bijection θ :V → V0 st. ∀u, v ∈ V • u−v edge in T ⇐⇒ θ(u)−θ(v) edge in T0 • if u is not a leaf of T, the signs of u and θ(u) coincide (resp. are opposite) FURTHER GEOMETRIC PROPERTIES

REM. The vertex barycenter of Asso(T) does not necessarily coincide with that of the permutahedron (but it lies on the linear span of the characteristic vectors of the orbits of V under the automorphism group of T) arXiv:1309.5222 THANK YOU