Extended Nestohedra and their Face Numbers Quang Dao, Christina - PowerPoint PPT Presentation

Extended Nestohedra and their Face Numbers Quang Dao, Christina Meng, Julian Wellman, Zixuan Xu, Calvin Yost-Wolff, Teresa Yu UMN REU July 24, 2019 (UMN REU) Project 7 July 24, 2019 1 / 41

Introduction Nestohedra are a well-understood class of convex polytopes Generalized by Lam–Pylyavskyy ’15 and Devadoss–Heath–Vipismakul ’11 independently LP-algebras Moduli space of a Riemann surface (UMN REU) Project 7 July 24, 2019 2 / 41

What is known so far Non-extended Extended ( � ) When flag Y Link decomposition Y Polytopality Y Gal’s conjecture Y Combinatorial interpretation for γ -vector chordal B Shellings B K n Cluster/LP algebras Y How are they related? Goal: fill in the column! (UMN REU) Project 7 July 24, 2019 3 / 41

Building Sets Definition A (connected) building set B on [ n ] := { 1 , . . . , n } is a collection of subsets of [ n ] such that 1 B contains all singletons { i } and the whole set [ n ] 2 if I , J ∈ B with I ∩ J � = ∅ , then I ∪ J ∈ B . Definition For an undirected graph G , its corresponding graphical building set B G is B G = { I ⊆ V ( G ) | G [ I ] is connected } . (UMN REU) Project 7 July 24, 2019 4 / 41

Examples of Building Sets Complete graph K n all subsets of [ n ] 1 2 B K 4 = { 1 , 2 , 3 , 4 , 12 , 13 , 14 , 23 , 24 , 34 , 123 , 234 , 124 , 134 , 1234 } Path graph P n 4 3 all interval subsets of [ n ] 1 2 3 B P 3 = { 1 , 2 , 3 , 12 , 23 , 123 } 1 Star graph K 1 , n 4 All singletons and all subsets of [ n + 1 ] that contain 3 2 n + 1 B K 1 , 3 = { 1 , 2 , 3 , 4 , 14 , 24 , 34 , 124 , 134 , 234 , 1234 } (UMN REU) Project 7 July 24, 2019 4 / 41

Nested Collections Definition For a building set B , a nested collection N of B is a collection of elements { I 1 , . . . , I m } of B \ { [ n ] } such that 1 for any i � = j , I i and I j are either nested or disjoint 2 for any I i 1 , . . . , I i k pairwise disjoint, their union is not an element of B Consider B = B P 4 = { 1 , 2 , 3 , 4 , 12 , 23 , 34 , 123 , 234 , 1234 } . { 1 , 3 , 34 } is a nested collection { 1 , 2 , 23 } is not a nested collection since { 1 } ∪ { 2 } ∈ B . (UMN REU) Project 7 July 24, 2019 5 / 41

Nested Complexes Definition For a connected building set B on [ n ] , the nested set complex N ( B ) is the simplicial complex with vertices { I | I ∈ B \ [ n ] } faces { I 1 , . . . , I m } that are nested collections of B Definition The nestohedron P ( B ) is the polytope dual to the nested set complex N ( B ) . In the literature, P ( B P n ) is known as the associahedron , and P ( B K n ) is known as the permutohedron . (UMN REU) Project 7 July 24, 2019 6 / 41

Extended Nested Collections Definition For a building set B on [ n ] , an extended nested collection N � of B is a collection of elements { I 1 , . . . , I m , x i 1 , . . . , x i r } such that 1 I k ∈ B for all k , and { I 1 , . . . , I m } form a nested collection of B 2 i j ∈ [ n ] for all j , and i j / ∈ I k for all 1 ≤ k ≤ m B = B P 4 { 1 , 3 , 34 , x 2 } is an extended nested collection { 1 , 3 , 34 , x 4 } is not an extended nested collection (UMN REU) Project 7 July 24, 2019 7 / 41

Extended Nested Complexes and Nestohedra Definition For a building set B on [ n ] , the extended nested set complex N � ( B ) is the simplicial complex with vertices { I | I ∈ B} ∪ { x i | i ∈ [ n ] } faces { I 1 , . . . , I m , x i 1 , . . . , x i r } that are extended nested collections of B 1 x 3 x 2 12 3 123 B = { 1 , 2 , 3 , 12 , 23 , 123 } x 1 23 2 (UMN REU) Project 7 July 24, 2019 8 / 41

Extended Nested Complexes and Nestohedra Definition For a building set B on [ n ] , the extended nested set complex N � ( B ) is the simplicial complex with vertices { I | I ∈ B} ∪ { x i | i ∈ [ n ] } faces { I 1 , . . . , I m , x i 1 , . . . , x i r } that are extended nested collections of B Definition The extended nestohedron P � ( B ) is the polytope dual to the extended nested set complex (UMN REU) Project 7 July 24, 2019 9 / 41

What is known so far Non-extended Extended ( � ) When flag Y Link decomposition Y Polytopality Y Gal’s conjecture Y Combinatorial interpretation for γ -vector chordal B Shellings B K n Cluster/LP algebras Y N � ( B ) ≃ N ( B ′ ) sometimes How are they related? (UMN REU) Project 7 July 24, 2019 10 / 41

When is N � ( B ) ≃ N ( B ′ ) ? Theorem (Manneville – Pilaud ’17) Let G , G ′ be undirected graphs such that N � ( B G ) ≃ N ( B G ′ ) . Then G is a spider and G ′ is the corresponding octopus. spider octopus (UMN REU) Project 7 July 24, 2019 11 / 41

When is N � ( B ) ≃ N ( B ′ ) ? Theorem (Manneville–Pilaud ’17) Let G , G ′ be undirected graphs such that N � ( B G ) ≃ N ( B G ′ ) . Then G is a spider and G ′ is the octopus. spider octopus (UMN REU) Project 7 July 24, 2019 11 / 41

When is N � ( B ) ≃ N ( B ′ ) ? Corollary (Manneville–Pilaud ’17) N � ( B K n ) ≃ N ( B K 1 , n ) is the dual of the stellohedron . N � ( B P n ) ≃ N ( B P n + 1 ) is the dual of the ( n − 2 ) -associahedron . Remark (REU ’19) When G = C 4 , we do not have N � ( B G ) ≃ N ( B ′ ) for any other building set B ′ . Theorem (REU ’19) If B is a building set on [ n ] such that all elements I ∈ B are intervals, then there exists B ′ such that N � ( B ) ≃ N ( B ′ ) . (UMN REU) Project 7 July 24, 2019 12 / 41

What is known so far Non-extended Extended ( � ) When flag Y Y Link decomposition Y Polytopality Y Gal’s conjecture Y Combinatorial interpretation for γ -vector chordal B Shellings B K n Cluster/LP algebras Y N � ( B ) ≃ N ( B ′ ) sometimes How are they related? (UMN REU) Project 7 July 24, 2019 13 / 41

When is N � ( B ) flag? Definition A simplicial complex ∆ is flag if ∆ has no minimal non-faces of degree greater than 2. In other words, ∆ is determined by its 1-skeleton. Proposition (REU ’19) N ( B ) is flag if and only if N � ( B ) is flag. For a graphical building set B = B G , it was shown in (PRW ’08) that N ( B ) is a flag simplicial complex. Corollary (REU ’19) If G is an undirected graph, then N � ( B G ) is flag. (UMN REU) Project 7 July 24, 2019 14 / 41

What is known so far Non-extended Extended ( � ) When flag Y Y Link decomposition Y Y Polytopality Y Gal’s conjecture Y Combinatorial interpretation for γ -vector chordal B Shellings B K n Cluster/LP algebras Y N � ( B ) ≃ N ( B ′ ) sometimes How are they related? (UMN REU) Project 7 July 24, 2019 15 / 41

Link Decompositions of N ( B ) and N � ( B ) Theorem (Zelevinsky ’06) Let B be a building set on S . Then the link of C ∈ B in N ( B ) N ( B ) C ≃ N ( B| C ) ∗ N ( B / C ) . Theorem (REU ’19) For the extended nested complex N � ( B ) , we have: N � ( B ) x i ≃ N � ( B 1 ) ∗ · · · ∗ N � ( B k ) where B 1 , . . . , B k are the connected components of B| [ n ] \{ i } , and N � ( B ) C ≃ N ( B| C ) ∗ N � ( B / C ) for C ∈ B . (UMN REU) Project 7 July 24, 2019 16 / 41

What is known so far Non-extended Extended ( � ) When flag Y Y Link decomposition Y Y Polytopality Y Y Gal’s conjecture Y Combinatorial interpretation for γ -vector chordal B Shellings B K n Cluster/LP algebras Y N � ( B ) ≃ N ( B ′ ) sometimes How are they related? (UMN REU) Project 7 July 24, 2019 17 / 41

Polytopality Theorem (REU ’19) For any building set B , N � ( B ) can be realized as the boundary of a polytope N B . (UMN REU) Project 7 July 24, 2019 18 / 41

Polytopality Consider R n with standard basis 3 vectors e 1 , . . . , e n . Start with cross polytope in R n with vertices e i labeled { i } ∈ B and x 1 vertices − e i labeled x i for all i ∈ [ n ] . x 2 2 1 x 3 B = { 1 , 2 , 3 , 12 , 123 } (UMN REU) Project 7 July 24, 2019 19 / 41

Polytopality Consider R n with standard basis 3 vectors e 1 , . . . , e n . Start with cross polytope in R n with vertices e i labeled { i } ∈ B and x 1 vertices − e i labeled x i for all 123 123 i ∈ [ n ] . x 2 Order the non-singletons of B by decreasing cardinality, then for 2 each I ∈ B a non-singleton, perform stellar subdivision on 1 the face I = {{ i } | i ∈ I } , with the new added vertex labeled I . x 3 B = { 1 , 2 , 3 , 12 , 123 } (UMN REU) Project 7 July 24, 2019 19 / 41

Polytopality Consider R n with standard basis 3 vectors e 1 , . . . , e n . Start with cross polytope in R n with vertices e i labeled { i } ∈ B and x 1 vertices − e i labeled x i for all 123 123 i ∈ [ n ] . x 2 Order the non-singletons of B by decreasing cardinality, then for 2 each I ∈ B a non-singleton, 12 perform stellar subdivision on 1 the face I = {{ i } | i ∈ I } , with the new added vertex labeled I . x 3 The boundary of the resulting polytope N B will be isomorphic B = { 1 , 2 , 3 , 12 , 123 } to N � ( B ) . (UMN REU) Project 7 July 24, 2019 19 / 41