f -vectors and cd -index of Weight Polytopes Simple, Seedy Derivations of Generating Functions for Simple Polytopes and cd -indices Jiyang Gao, Vaughan McDonald University of Minnesota - Twin Cities REU 2018 3 August 2018 Gao, McDonald 1 / 51
Outline f -vectors and cd -index of Weight Polytopes 1 Introduction to Polytopes 2 Coxeter Group and Weight Polytopes 3 f -polynomials of Simple Weight Polytopes 4 Face Poset of General Weight Polytopes 5 A glimpse on the cd -index of Weight Polytopes 6 Summary Gao, McDonald 2 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Section 1 Introduction to Polytopes Gao, McDonald 3 / 51
Polytopes f -vectors and cd -index of Weight Polytopes What are polytopes? Definition (Polytope) A polytope is the convex hull of a finite number of points in R r . Examples of polytopes in R 3 Gao, McDonald 4 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Faces of Polytopes • Polytopes have faces . • Faces are polytopes themselves. • Faces have dimensions. It’s the minimal integer d such that the face can live in R d . • A j -dimensional face is called a j -face . • A 0-face is usually called a vertex . A 1-face is usually called an edge . An r -face is the polytope itself. Gao, McDonald 5 / 51
Polytopes f -vectors and cd -index of Weight Polytopes f -vector and f -polynomial Definition ( f -vector and f -polynomial) Define the f -vector of a r -dim Polytope P as f ( P ) := ( f 0 , . . . , f r ), where f i is the number of i -dimensional faces of P . Define its f -polynomial as f P ( t ) = � r i =0 f i t i . Example: A cube has 8 vertices, 12 edges and 6 faces. f ( P ) = (8 , 12 , 6 , 1) f P ( t ) = 8 + 12 t + 6 t 2 + t 3 Gao, McDonald 6 / 51
Polytopes f -vectors and cd -index of Weight Polytopes h -vector and h -polynomial Definition ( h -vector and h -polynomial) Define the h -polynomial of a r -dim Polytope P as h P ( t ) = f P ( t − 1) = � r i =0 f i ( t − 1) i . Assume h P ( t ) = � r i =0 h i t i , then define its h -vector as h ( P ) := ( h 0 , h 1 , . . . , h r ). A cube has f P ( t ) = 8+12 t +6 t 2 + t 3 . Example: Replace t with t − 1. h P ( t ) = f P ( t − 1) = 1+3 t +3 t 2 + t 3 h ( P ) = (1 , 3 , 3 , 1) Gao, McDonald 7 / 51
Polytopes f -vectors and cd -index of Weight Polytopes h -vector and h -polynomial Definition ( h -vector and h -polynomial) Define the h -polynomial of a r -dim Polytope P as h P ( t ) = f P ( t − 1) = � r i =0 f i ( t − 1) i . Assume h P ( t ) = � r i =0 h i t i , then define its h -vector as h ( P ) := ( h 0 , h 1 , . . . , h r ). A cube has f P ( t ) = 8+12 t +6 t 2 + t 3 . Example: Replace t with t − 1. h P ( t ) = f P ( t − 1) = 1+3 t +3 t 2 + t 3 h ( P ) = (1 , 3 , 3 , 1) Is this always symmetric? Gao, McDonald 7 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Dehn-Somerville Equation Definition (Simple Polytope) A r -dimensional polytope is called a simple polytope if and only if each vertex has exactly r incident edges. For example, a cube is a simple polytope. Theorem (Dehn-Sommerville equation) For any simple polytope P , its h -vector is symmetric. Gao, McDonald 8 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Face Poset Definition (Face Poset) The face poset of polytope P is the poset { faces of P } ordered by inclusion of faces. The Square Example: 4 edges 4 vertices Empty Face Polytope Face Poset *Note: A Face Poset is graded. Gao, McDonald 9 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Rank Selected Poset Definition (Rank Selected Poset) Let S ⊆ [ r ] = { 1 , 2 , . . . , r } . The rank-selected poset P S of P is P S = { x ∈ P | ρ ( x ) ∈ S } ∪ { ˆ 0 , ˆ 1 } , where ρ is the rank function. 3 2 S = { 1 } − − − − → 1 0 Gao, McDonald 10 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Flag f -vector and Flag h -vector Definition (Flag f -vector and Flag h -vector) Define the flag f -vector α ( S ) as the number of maximal chains in P S . Based on that, define the flag h -vector β ( S ) as: � � ( − 1) #( S − T ) α ( T ) β ( S ) = or, α ( S ) = β ( T ) . T ⊆ S T ⊆ S Gao, McDonald 11 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Example of Flag Vectors ρ S α ( S ) β ( S ) 3 ∅ 1 1 { 1 } 4 3 2 { 2 } 4 3 { 1 , 2 } 8 1 1 Table of Flag Vectors 0 Face Poset Gao, McDonald 12 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Example of Flag Vectors ρ S α ( S ) β ( S ) 3 ∅ 1 1 { 1 } 4 3 2 { 2 } 4 3 { 1 , 2 } 8 1 1 Table of Flag Vectors 0 Face Poset β ( S ) = β ( ¯ S ) Gao, McDonald 12 / 51
Polytopes f -vectors and cd -index of Weight Polytopes ab -index Definition ( ab -index) Define the ab -index of Polytope P as a polynomial over non-commutative variables a, b as � Φ P ( a, b ) = β ( S ) u S . S ⊆ [ n ] Here u S = u n u n − 1 · · · u 1 , where � a, if i / ∈ S u i = if i ∈ S. b, Gao, McDonald 13 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Example of ab -index S α ( S ) β ( S ) u S a 2 ∅ 1 1 { 1 } 4 3 ab { 2 } 4 3 ba Φ P ( a, b ) = a 2 + 3 ab + 3 ba + b 2 . b 2 { 1 , 2 } 8 1 Table of Flag Vectors Gao, McDonald 14 / 51
Polytopes f -vectors and cd -index of Weight Polytopes cd -index Theorem ( cd -index) For any polytope P , there exists a polynomial Ψ P ( c, d ) in the non-commuting variables c and d such that Φ P ( a, b ) = Ψ P ( a + b, ab + ba ) . Ψ P ( c, d ) is also called the cd-index of polytope P . Gao, McDonald 15 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Example of cd -index Φ P ( a, b ) = a 2 + 3 ab + 3 ba + b 2 S α ( S ) β ( S ) u S a 2 ∅ 1 1 = ( a + b ) 2 + 2( ab + ba ) . { 1 } 4 3 ab { 2 } 4 3 ba Replace a + b → c , ab + ba → d . b 2 { 1 , 2 } 8 1 Ψ P ( c, d ) = c 2 + 2 d. Table of Flag Vectors Gao, McDonald 16 / 51
Polytopes f -vectors and cd -index of Weight Polytopes Summary Methods to describe a polytope: • f -polynomial/ h -polynomial; • face poset; • cd -index. Gao, McDonald 17 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Section 2 Coxeter Group and Weight Polytopes Gao, McDonald 18 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Finite Reflection groups Definition (Finite Reflection Group) A finite reflection group is a finite subgroup W ⊂ GL n ( R ) generated by reflections, i.e. elements w such that w 2 = 1 and they fix a hyperplane H and negate the line perpendicular to H Example: One example of a finite reflection group is the Dihedral Group I n = { s, t | s 2 = t 2 = e, ( st ) n = e } . Gao, McDonald 19 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Coxeter groups Definition (Coxeter Group) A Coxeter Group is a group W of the form W ∼ i = e, ( s i s j ) m ij = e � = � s 1 , . . . , s n | s 2 for some m ij ∈ { 2 , 3 , 4 , . . . } ∪ {∞} . If W is finite, then W is called a Finite Coxeter Group . S = { s 1 , s 2 , . . . , s n } is called the Generating Set of W . Gao, McDonald 20 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Finite Coxeter Groups = Finite Reflection Groups Here is a BIG theorem of Coxeter: Theorem (Coxeter) Finite Coxeter groups = Finite reflection groups. Gao, McDonald 21 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Coxeter Diagram Definition (Coxeter Diagram) Given a Coxeter presentation ( W, S ), we can encapsulate it in the Coxeter Diagram , denoted Γ( W ), a graph with V = S and if m ij = 3, s i and s j are connected with no label and if m ij > 3, s i and s j are connected with label m ij . Example: The dihedral group I n has Coxeter diagram n Gao, McDonald 22 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Finite Coxeter Groups Amazingly, finite Coxeter groups are classified! They come in four infinite families, A n , B n , D n , and I n , as well as a finite collection of exceptional cases. The Coxeter diagrams look as follows: We will focus our energies on types A n , B n , D n . Gao, McDonald 23 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Weight Polytopes Definition (Weight Polytope) Given finite Coxeter group W , λ ∈ R n , we define the Weight Polytope P λ to be the convex hull of { w · λ | w ∈ W } . Gao, McDonald 24 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Weight Polytopes Definition (Stabilizer) Let J ( λ ) = { s ∈ S | s ( λ ) = λ } be the stabilizer of λ . Theorem (Maxwell) The f -vector and face lattice of a weight polytope P λ is only dependent on W , S and J ( λ ) . Gao, McDonald 25 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Weight Polytope Example 1 Coxeter Group W = A n = symmetric group S n +1 Vector λ λ = (0 , . . . , 0 , 1) � �� � n zeros Gao, McDonald 26 / 51
Weight Polytopes f -vectors and cd -index of Weight Polytopes Weight Polytope Example 1 Coxeter Group W = A n = symmetric group S n +1 Polytope · · · Name: Simplex (12) (23) (34) (45) ( n, n + 1) Vector λ λ = (0 , . . . , 0 , 1) � �� � n zeros Vertices: Set of vectors with n zeros and 1 one J ( λ ) · · · n 1 2 3 n − 1 Gao, McDonald 26 / 51
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