THE ASSOCIAHEDRON AND ITS FRIENDS V. PILAUD (CNRS & LIX) S´eminaire Lotharingien de Combinatoire April 4 { 6, 2016 PROGRAM I. THREE CONSTRUCTIONS OF THE ASSOCIAHEDRON • Compatibility fan and cluster algebras • Loday's associahedron and Hopf algebras • Secondary polytope II. GRAPH ASSOCIAHEDRA • Graphical nested complexes • Compatibility fans for graphical nested complexes • Signed tree associahedra III. BRICK POLYTOPES AND THE TWIST ALGEBRA • [Multi][pseudo]triangulations and sorting networks • The brick polytope • The twist algebra I. THREE CONSTRUCTIONS OF THE ASSOCIAHEDRON α2 @@ p @ ¢ p B ¢ p B @ p B @ ¢ p ¢ p B @ p B α2+α3 @ ¢ p ¢ p B @ α1+p α2 ¢@ ¢ p ¢ @ p ¢ p ¢ @ ¢ p ¢ @ ¢ p ¢ p α1+α2+α¢¢ 3 ¢ p ¢ p @ ¢ α3 p @ ¢ p p @ ¢ p p p p p p p p p p p p p@ p p¢ p p p p p p p p p p p p p p pp p p p p α1 p p p p p p p p p p p p p p p p FANS & POLYTOPES Ziegler, Lectures on polytopes ('95) Matouˇsek, Lectures on Discrete Geometry ('02) SIMPLICIAL COMPLEX simplicial complex = collection of subsets of X downward closed exm: X =[ n] [ [n] 123 123 123 123 123 123 123 123 ∆ = fI ⊆ X j 8i 2 [n]; fi;i g 6⊆ Ig 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 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 = f normal cone of F j F face of P g simple polytope =) simplicial fan =) simplicial complex EXAMPLE: PERMUTAHEDRON Hohlweg, Permutahedra and associahedra ('12) PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ n+1 X jJj + 1 = H \ x 2 R xj ≥ 4312 2 3421 ?6=J([n+1] j2J 3412 4213 2431 2413 2341 3214 1432 2314 1423 3124 1342 1324 2134 1243 1234 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ n+1 X jJj + 1 = H \ x 2 R xj ≥ 4312 2 3421 ?6=J([n+1] j2J 3412 connections to 4213 • weak order 2431 • reduced expressions 2413 • braid moves • cosets of the symmetric group 2341 3214 1432 2314 1423 3124 1342 1324 2134 1243 1234 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ n+1 X jJj + 1 3211 = H \ x 2 R xj ≥ 3321 4312 2 2211 2 3421 3312 ?6=J([n+1] j J 2311 3412 3212 connections to 2321 4213 • weak order 2312 2212 2431 • reduced expressions 1211 3213 2331 2413 • braid moves 1321 • cosets of the symmetric group 2341 2313 3214 1221 1312 2113 k-faces of Perm(n) 1231 1432 2314 1322 1423 1212 3124 ≡ surjections from [n + 1] 1332 1213 to [n + 1 − k] 1342 1323 2123 1222 1112 1232 1324 1223 2134 1243 1233 1123 1234 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4|3|2|1 \ n+1 X jJj + 1 34|2|1 = H \ x 2 R xj ≥ 4|3|12 3|4|2|1 2 34|12 2 4|3|1|2 3|4|12 ?6=J([n+1] j J 34|1|2 3|4|1|2 3|24|1 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 3|1|4|2 • braid moves | | | 14|3|2 • cosets of the symmetric group 4 1 2 3 3|1|24 3|2|1|4 14|23 13|4|2 | | k-faces of Perm(n) | | 1|4|3|2 3|1|2|4 23 1 4 14 2 3 | | | 13|24 | | | ≡ surjections from [n + 1] | | 1 3 4 2 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|24|3 1|3|2|4 into n + 1 − k parts 1|23|4 2|1|3|4 1|2|4|3 12|3|4 1|2|34 1|2|3|4 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4|3|2|1 \ n+1 X jJj + 1 34|2|1 = H \ x 2 R xj ≥ 4|3|12 3|4|2|1 2 34|12 2 4|3|1|2 3|4|12 ?6=J([n+1] j J 34|1|2 3|4|1|2 3|24|1 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 3|1|4|2 • braid moves | | | 14|3|2 • cosets of the symmetric group 4 1 2 3 3|1|24 3|2|1|4 14|23 13|4|2 | | k-faces of Perm(n) | | 1|4|3|2 3|1|2|4 23 1 4 14 2 3 | | | 13|24 | | | ≡ surjections from [n + 1] | | 1 3 4 2 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|24|3 1|3|2|4 into n + 1 − k parts 1|23|4 2|1|3|4 1|2|4|3 12|3|4 ≡ collections of n − k nested 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 = 34|12 3|4|2|1 2 n+1 x R xi = xj 1≤i<j≤n+1 | | | = collection of all cones 4 3 1 2 | | | 3 4 1 2 +1 x 2 n x < x if π(i) < π(j) 4|1|2|3 3|2|4|1 R i j 3|124 for all surjections π :[n + 1] ! [n + 1 − k] 4|1|3|2 134|2 3|1|4|2 14|23 3|2|1|4 3|1|2|4 1|4|3|2 1|3|4|2 13|24 (n − k)-dimensional cones ≡ surjections from [n + 1] 1|4|2|3 2|3|1|4 to [n + 1 − k] 1|3|2|4 123|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 Ceballos-Santos-Ziegler, Many non-equivalent realizations of the associahedron ('11) 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 Tamari ('51) | Stasheff ('63) | Haimann ('84) | Lee ('89) | (Pictures by Ceballos-Santos-Ziegler) . | 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 α2 @@ (Pictures by CFZ) p @ ¢ p B ¢ p B @ p B @ ¢ p ¢ p B @ p B α2+α3 @ ¢ p ¢ α +pα B @ 1 p 2 ¢@ ¢ p ¢ @ p ¢ p ¢ @ ¢ p ¢ @ ¢ p ¢ p α1+α2+α¢¢3 ¢ p ¢ p @ ¢ α3 p @ ¢ p @ ¢ p p p p p p p p p p p p p p p@ p p¢ p p p p p p p p p p p p p p pp p pα p p 1 p p p p p p p p p p p p Gelfand-Kapranov-Zelevinsky ('94) Loday ('04) p p Chapoton-Fomin-Zelevinskyp ('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 α2 @@ (Pictures by CFZ) p @ ¢ p B ¢ p B @ p B @ ¢ p ¢ p B @ p B α2+α3 @ ¢ p ¢ α +pα B @ 1 p 2 ¢@ ¢ p ¢ @ p ¢ p ¢ @ ¢ p ¢ @ ¢ p ¢ p α1+α2+α¢¢3 ¢ p ¢ p @ ¢ α3 p @ ¢ p @ ¢ p p p p p p p p p p p p p p p@ p p¢ p p p p p p p p p p p p p p pp p pα p p 1 p p p p p p p p p p p p Gelfand-Kapranov-Zelevinsky ('94) Loday ('04) p p Chapoton-Fomin-Zelevinskyp ('02) Billera-Filliman-Sturmfels ('90) Hohlweg-Lange ('07) Ceballos-Santos-Ziegler ('11) Hohlweg-Lange-Thomas ('12) H α2 H H ¡¡ A H p A2α2 + α3H Hopf ¡ p A @ p HH @ Cluster ¡ p H ¡ p @ algebra ¡ p @ ¡α1+αp2 @ algebras p @ ¡ p HH ¡ p ¡¡ H α2+α3 @ Cluster p ¡ p ¡ BB 2αp 1 + 2α2 + α3 ¡ p ¡ B ¡ ¡p B algebras pHH p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp p p H ¡@@ α p p H 3 p p H¡ @ p p α1 @@ ¡ p p p p @¡ p p p p p p p p COMPATIBILITY FAN Chapoton-Fomin-Zelevinsky, Polytopal realizations of generalized assoc.