A combinatoric invariant of simple polytopes 1 Simple polytopes 2 - PowerPoint PPT Presentation

A combina- toric invariant of simple polytopes Bo Chen A combinatoric invariant of simple polytopes § 1 Simple polytopes § 2 graded Boolean Bo Chen ring of simple polytopes School of Mathematics and Statistics, HUST § 3 Jiont-work with Zhi L¨ u and Li Yu n -colorable simple bobchen@hust.edu.cn polytopes § 4 Binary Apr. 2014, SJTU, Shanghai self-dual codes § 5 Motivation Reference § 0 Thanks

convex polytope A combina- toric A convex polytope is defined to be invariant of simple polytopes a convex hull of finite many points in R n . Bo Chen § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

convex polytope A combina- toric A convex polytope is defined to be invariant of simple polytopes a convex hull of finite many points in R n . Bo Chen the intersection of finite number of half-spaces of R n . § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

face A combina- Definition toric invariant of Let H be a half-space of R n . H ∩ P is called a face of a convex simple polytopes polytope P , if H ∩ int ( P ) = ∅ . Bo Chen A 0-face is called a vertex, a 1-face is called an edge and a ( n − 1) -face is called a facet. § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

simple convex polytope A combina- toric invariant of simple polytopes Definition A convex polytope P n is simple Bo Chen if each vertex adjoint to exactly n facets. § 1 Simple polytopes if each vertex adjoint to exactly n edges. § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

simple convex polytope A combina- toric invariant of simple polytopes Definition A convex polytope P n is simple Bo Chen if each vertex adjoint to exactly n facets. § 1 Simple polytopes if each vertex adjoint to exactly n edges. § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

simple convex polytope A combina- toric invariant of simple polytopes Definition A convex polytope P n is simple Bo Chen if each vertex adjoint to exactly n facets. § 1 Simple polytopes if each vertex adjoint to exactly n edges. § 2 graded Boolean if each k -face is the intersection of exactly k facets. ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

A combina- toric invariant of simple polytopes Bo Chen § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

combinatoric of convex polytope A combina- toric invariant of simple polytopes Definition Bo Chen A face lattice of a convex polytope P is a poset ( F , ≤ ) , where § 1 Simple polytopes F is the set of faces of P , and a ≤ b iff a ⊂ b , a, b ∈ F . § 2 graded Boolean ring of simple polytopes § 3 Definition n -colorable simple Two polytopes are called combinatorially isomorphic if their polytopes § 4 Binary face lattices are isomorphic. self-dual codes § 5 Motivation Reference § 0 Thanks

some combinatoric invariants of simple polytopes A combina- toric f-vector,h-vector invariant of simple polytopes Definition Bo Chen Let P n be a simple convex polytope, and suppose f i be the § 1 Simple number of ( n − i − 1) -faces of P n , i = − 1 , 0 , 1 , · · · , n − 1 . polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

some combinatoric invariants of simple polytopes A combina- toric f-vector,h-vector invariant of simple polytopes Definition Bo Chen Let P n be a simple convex polytope, and suppose f i be the § 1 Simple number of ( n − i − 1) -faces of P n , i = − 1 , 0 , 1 , · · · , n − 1 .Let polytopes § 2 graded Boolean k � d − i � ring of � ( − 1) k − i h k = f i − 1 . simple k − i polytopes i =0 § 3 n -colorable f ( P ) = ( f − 1 , f 0 , f 1 , · · · , f n − 1 ) is called the f-vector of P , and simple polytopes h ( P ) = ( h 0 , h 1 , · · · , h n ) is called the h-vector of P . § 4 Binary self-dual codes Actually, n n § 5 f i − 1 ( t − 1) n − i = Motivation � � h k t n − k . Reference i =0 k =0 § 0 Thanks

Stanley-Reisner ring A combina- toric invariant of simple polytopes Stanley-Reisner ring Bo Chen § 1 Simple polytopes § 2 graded Order the facets of P , say { F 1 , · · · , F m } . Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

Stanley-Reisner ring A combina- toric invariant of simple polytopes Stanley-Reisner ring Bo Chen § 1 Simple polytopes § 2 graded Order the facets of P , say { F 1 , · · · , F m } . Define a ideal I ( P ) Boolean of k [ x 1 , · · · , x m ], ring of simple polytopes I ( P ) = � x i 1 · · · x i s | F i 1 ∩ · · · ∩ F i s = ∅� . § 3 n -colorable simple polytopes § 4 Binary A Stanley-Reisner ring k [ P ] of a simple polytope P is defined self-dual codes to be k [ P ] = k [ x 1 , · · · , x m ] /I ( P ) . § 5 Motivation Reference § 0 Thanks

vertex-face incident vectors A combina- toric invariant of simple Definition polytopes For each face f of a polytope P , define a vector of face f , Bo Chen § 1 Simple � if p ∈ f ; 1 , polytopes ζ f : V ( P ) → Z 2 , p �→ § 2 graded 0 , if p / ∈ f . Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 ζ f 0 = (1 , 0 , 1 , 1 , 0) Motivation Reference § 0 Thanks

graded Boolean ring of polytopes A combina- toric invariant of Definition simple polytopes Bo Chen � span { ζ f | f is a codim- k face of P } , if k ≤ n ; B k ( P ) � § 1 Simple V ∗ = Map ( V ( P ) , Z 2 ) ∼ = Z s 2 , if k > n . polytopes § 2 graded Boolean � B k t k . B ( P ) � ring of simple polytopes k ≥ 0 § 3 n -colorable ∀ faces f and g of P , define ( ζ f ◦ ζ g )( v ) = ζ f ( v ) ζ g ( v ) . Then simple polytopes ζ f ∩ g = ζ f ◦ ζ g . § 4 Binary self-dual Suppose P n is simple. Each k -face of a simple convex poly- codes § 5 tope is the intersection of exactly k facets, So B ( P ) becomes Motivation a graded ring. Reference § 0 Thanks

A combina- toric invariant of simple polytopes Proposition Bo Chen dim B 1 ( P ) ≥ h 0 + h 1 . § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes Remark § 3 Actually, choose any vertex v of P and n − 1 facets incident to n -colorable simple v , then face-vectors of remaining facets(of number m − n +1 = polytopes h 0 + h 1 ) are linearly independent in B 1 ( P ) . § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

A combina- toric invariant of simple polytopes Bo Chen § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

n -colorable simple polytopes A combina- toric invariant of simple polytopes Bo Chen Definition § 1 Simple polytopes A simple polytope P n is n -colorable, if there is a map § 2 graded Boolean ring of c : F → [ n ] , simple polytopes § 3 such that c ( F i ) � = c ( F j ) if F i ∩ F j � = ∅ . n -colorable simple polytopes § 4 Binary self-dual codes § 5 Motivation Reference § 0 Thanks

A combina- toric invariant of simple Theorem polytopes Bo Chen The following statements are equivalent. P n is n -colorable. § 1 Simple polytopes § 2 graded Boolean ring of simple polytopes § 3 n -colorable simple Remark polytopes § 4 Binary The proof of equivalence of the first two statements can be self-dual codes found in [Jos]. § 5 Motivation Reference § 0 Thanks

A combina- toric invariant of simple Theorem polytopes Bo Chen The following statements are equivalent. P n is n -colorable. § 1 Simple polytopes § 2 graded each 2 -face of P has even number of vertices. Boolean ring of simple polytopes § 3 n -colorable simple Remark polytopes § 4 Binary The proof of equivalence of the first two statements can be self-dual codes found in [Jos]. § 5 Motivation Reference § 0 Thanks

A combina- toric invariant of simple Theorem polytopes Bo Chen The following statements are equivalent. P n is n -colorable. § 1 Simple polytopes § 2 graded each 2 -face of P has even number of vertices. Boolean ring of B 1 ( P ) = m − n + 1 . simple polytopes § 3 n -colorable simple Remark polytopes § 4 Binary The proof of equivalence of the first two statements can be self-dual codes found in [Jos]. § 5 Motivation Reference § 0 Thanks