Problems of Enumeration and Realizability on Matroids, Simplicial - PowerPoint PPT Presentation

Problems of Enumeration and Realizability on Matroids, Simplicial Complexes, and Graphs Yvonne Kemper August 6, 2014 Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram g s r a h p Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Definition A graph G = ( V , E ) is a set of vertices V = { v 1 , . . . , v n } and a set of edges E = { v i v j : v i , v j ∈ V } . Example Here is a graph! 3 G = ( V , E ) 1 2 ( { 1 , 2 , 3 , 4 } , = { 12 , 13 , 14 , 23 , 24 } ) 4 Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram g s r h a p Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram g s r h a p m a s t d r o i Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Definition A matroid M = ( E ( M ) , I ( M )) consists of a ground set E ( M ) and a family of subsets I ( M ) ⊆ 2 E ( M ) called independent sets such that (1) ∅ ∈ I ; (2) if I ∈ I and J ⊂ I , then J ∈ I ; and (3) if I , J ∈ I , and | J | < | I | , then there exists some e ∈ I \ J such that J ∪ { e } ∈ I . Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Example Here is a graph matroid! 1 2 ( E ( M ) , I ( M )) M = = ( { 1 , 2 , 3 , 4 , 5 } , {∅ , 1 , 2 , 3 , 3 4 , 5 , 12 , 13 , 14 , 15 , 23 , 24 , 25 , 34 , 35 , 45 , 124 , 125 , 134 , 4 5 135 , 145 , 234 , 235 , 245 } ) Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Example Here is a graph matroid! 1 2 ( E ( M ) , I ( M )) M = = ( { 1 , 2 , 3 , 4 , 5 } , {∅ , 1 , 2 , 3 , 3 4 , 5 , 12 , 13 , 14 , 15 , 23 , 24 , 25 , 34 , 35 , 45 , 124 , 125 , 134 , 4 5 135 , 145 , 234 , 235 , 245 } ) Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram g s r h a p m a s t r d o i Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram g s r h a p s e m x a s t r d e o i l p s m i m o p c l i l c i a Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Definition An (abstract) simplicial complex ∆ on a vertex set V is a set of subsets of V . These subsets are called the faces of ∆, and we require that (1) for all v ∈ V , { v } ∈ ∆, and (2) for all F ∈ ∆, if G ⊆ F , then G ∈ ∆. Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Example Here is a graph matroid simplicial complex! 3 ∆ = ( V , F ) 1 2 = ( { 1 , 2 , 3 , 4 } , {∅ , 1 , 2 , 3 , 4 , 12 , 13 , 14 , 23 , 24 } ) 4 Yvonne Kemper PoEaRoMSCaG

Wait, what were those things again? Example Here is a graph matroid simplicial complex! 3 ∆ = ( V , F ) 1 2 = ( { 1 , 2 , 3 , 4 } , {∅ , 1 , 2 , 3 , 4 , 12 , 13 , 14 , 23 , 24 } ) 4 Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram g s r h a p s e m x a s d t r i e o l p s m i m o p c l i l c i a Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram s h p a r g d n a , s e x e l p m o c l a i c g s i r h a p l p s e m m x a s d t r i e i o s l p s m , i s m o p d c l i l c i a i o p r r t o a b m l e m n s o Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram x e s e l p , m a o n c d l a g i r c a i p l p h m s i s , s d i o r t a m g s r h a p n s o e m x a s d n t r i e o l o p s m i i t m o a p c l i l r c i a e p m r o u b n l e e m f s o n o i t a r e m u n e Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram m i s , s p d l i i o c r i a t l a m c o n m o p l y e t x i l e i b s , a z a i n l a d e r g r d g s a r h a p n p s a e h m x s a s d n t r i e o l o p s m i i t m o a p c l i l r c i a e p m r r o e u b n l e a e m f s o l i z a b n i o l i i t t y a r e m u n e Yvonne Kemper PoEaRoMSCaG

In Honor of a Diagram m i s , s p d l i i o c r i a t l a m c o n m o p l y e t x i l e i b s , a z a i n l a d e r g r d g s a r h a p n p s a Lookin’ good! e h m x s a s d n t r i e o l o p s m i i t m o a p c l i l r c i a e p m r r o e u b n l e a e m f s o l i z a b n i o l i i t t y a r e m u n e Yvonne Kemper PoEaRoMSCaG

The Problems Three ◮ h -Vectors of Small Matroids ◮ Flows on Simplicial Complexes ◮ Polytopal Embeddings of Cayley Graphs Yvonne Kemper PoEaRoMSCaG

h -Vectors of Small Matroids Yvonne Kemper PoEaRoMSCaG

Simplicial Complexes: a Few More Definitions Definition The dimension of a face F is | F | − 1, and the dimension of ∆ is d = max {| F | : F ∈ ∆ } − 1. Definition A simplicial complex is pure if all maximal elements of ∆ have the same cardinality. In this case, a facet is a maximal face, a ridge is a face of one dimension lower. Yvonne Kemper PoEaRoMSCaG

Faces: A Natural Quantity to Measure ◮ The f -vector of a simplicial complex ∆, dim ∆ = d − 1, is f (∆) := ( f − 1 (∆) , f 0 (∆) , . . . , f d − 1 (∆)) , where f i (∆) := |{ F ∈ ∆ : dim F = i }| . Yvonne Kemper PoEaRoMSCaG

Faces: A Natural Quantity to Measure ◮ The f -vector of a simplicial complex ∆, dim ∆ = d − 1, is f (∆) := ( f − 1 (∆) , f 0 (∆) , . . . , f d − 1 (∆)) , where f i (∆) := |{ F ∈ ∆ : dim F = i }| . ◮ The h -vector , h (∆) := ( h 0 (∆) , . . . , h d (∆)), is given by: d d � � h j (∆) λ j = f i − 1 (∆) λ i (1 − λ ) d − i . j =0 i =0 Yvonne Kemper PoEaRoMSCaG

Characterizations of f - and h -Vectors Definition Given two integers k , i > 0, write � n i � � n i − 1 � � n j � k = + + · · · + , i i − 1 j where n i > n i − 1 > · · · > n j ≥ j ≥ 1. Define � n i � � n i − 1 � � n j � k ( i ) = + + · · · + . i + 1 i j + 1 Theorem (Sch¨ utzenberger, Kruskal, Katona) A vector (1 , f 0 , f 1 , . . . , f d − 1 ) ∈ Z d +1 is the f -vector of some ( d − 1) -dimensional simplicial complex ∆ if and only if 0 < f i +1 ≤ f ( i +1) 0 ≤ i ≤ d − 2 . , i Yvonne Kemper PoEaRoMSCaG

Other Characterizations? Question Can we characterize subclasses of simplicial complexes? Example ◮ Cohen-Macaulay complexes ◮ Flag complexes ◮ Shifted complexes ◮ Independence complexes of matroids Yvonne Kemper PoEaRoMSCaG

Matroid Complexes: An Example The corresponding complex: Let M be given by: 5 I ( M ) = {∅ , 1 , 2 , 3 , 4 , 5 , 3 12 , 13 , 14 , 15 , 23 , 1 2 24 , 25 , 34 , 35 , 45 , 124 , 125 , 134 , 135 , 145 , 234 , 235 , 245 } 4 Then: f ( M ) = (1 , 5 , 10 , 8) and h ( M ) = (1 , 2 , 3 , 2). Yvonne Kemper PoEaRoMSCaG

O -Sequences ◮ A non-empty set of monomials M is a multicomplex if m ∈ M and n | m ⇒ n ∈ M . ◮ A sequence h = ( h 0 , h 1 , . . . , h d ) of integers is an O -sequence if there exists a multicomplex with precisely h i monomials of degree i . ◮ An O -sequence is pure if all maximal elements have the same degree. Yvonne Kemper PoEaRoMSCaG

O -Sequences ◮ A non-empty set of monomials M is a multicomplex if m ∈ M and n | m ⇒ n ∈ M . ◮ A sequence h = ( h 0 , h 1 , . . . , h d ) of integers is an O -sequence if there exists a multicomplex with precisely h i monomials of degree i . ◮ An O -sequence is pure if all maximal elements have the same degree. Example Let M = { 1 , x 1 , x 2 , x 1 x 2 , x 2 1 , x 2 2 , x 1 x 2 2 , x 2 1 x 2 } . Then, the corresponding (pure) O -sequence is: O ( M ) = (1 , 2 , 3 , 2) . Yvonne Kemper PoEaRoMSCaG

Stanley’s Conjecture Conjecture (Stanley, 1977) The h-vector of a matroid complex is a pure O-sequence. Little progress was made for twenty years, but since 1997, the conjecture has been proved for matroids which are: ◮ of rank 4 (Klee, Samper), ◮ of rank less than or equal to 3 (Stokes, H´ a et al.), ◮ cographic (Biggs, Merino), ◮ lattice-path (Schweig), ◮ cotransversal (Oh), ◮ paving (Merino, et al.). Yvonne Kemper PoEaRoMSCaG

Results Theorem (De Loera, K., Klee) ◮ Let M be a matroid of rank 2 . Then h ( M ) is a pure O-sequence. ◮ Let M be a matroid of corank 2 . Then h ( M ) is a pure O-sequence. ◮ Let M be a matroid of rank d ≥ 4 . Then, the subsequence (1 , h 1 ( M ) , h 2 ( M ) , h 3 ( M )) of h ( M ) is a pure O-sequence. ◮ Let M be a matroid of rank 3 . Then h ( M ) is a pure O-sequence. ◮ Let M be a matroid on at most 9 elements. Then h ( M ) is a pure O-sequence. Yvonne Kemper PoEaRoMSCaG

An Experimental Result: Matroids on at Most Nine Elements ◮ Royle and Mayhew generated list of all matroids on at most nine elements - why not check them all? ◮ Used database to generate all h -vectors for these matroids. ◮ Generated list of all possible O -sequences of multicomplexes (up to maximal degree 9 on at most 9 variables), then checked that every h -vector appeared on this list. Yvonne Kemper PoEaRoMSCaG

Example: A Multicomplex in Two Variables 1. Pick a point ( a , b ) ∈ Z 2 on the hyperplane x + y = r , where r is the rank of the matroid. Yvonne Kemper PoEaRoMSCaG

Example: A Multicomplex in Two Variables 1. Pick a point ( a , b ) ∈ Z 2 on the hyperplane x + y = r , where r is the rank of the matroid. Let’s say r = 3. Yvonne Kemper PoEaRoMSCaG