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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

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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

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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

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Polytopes f-vectors and cd-index of Weight Polytopes

Section 1

Introduction to Polytopes

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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 Rr.

Examples of polytopes in R3

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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 Rd.

  • 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.

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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) := (f0, . . . , fr), where fi is the number of i-dimensional faces of P. Define its f-polynomial as fP (t) = r

i=0 fiti.

Example: A cube has 8 vertices, 12 edges and 6 faces. f(P) = (8, 12, 6, 1) fP (t) = 8 + 12t + 6t2 + t3

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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 hP (t) = fP (t − 1) = r

i=0 fi(t − 1)i.

Assume hP (t) = r

i=0 hiti, then define its h-vector as

h(P) := (h0, h1, . . . , hr). Example: A cube has fP (t) = 8+12t+6t2+t3. Replace t with t − 1. hP (t) = fP (t−1) = 1+3t+3t2 +t3 h(P) = (1, 3, 3, 1)

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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 hP (t) = fP (t − 1) = r

i=0 fi(t − 1)i.

Assume hP (t) = r

i=0 hiti, then define its h-vector as

h(P) := (h0, h1, . . . , hr). Example: A cube has fP (t) = 8+12t+6t2+t3. Replace t with t − 1. hP (t) = fP (t−1) = 1+3t+3t2 +t3 h(P) = (1, 3, 3, 1) Is this always symmetric?

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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.

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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. Example: Polytope Empty Face 4 vertices 4 edges The Square Face Poset *Note: A Face Poset is graded.

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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 PS of P is PS = {x ∈ P|ρ(x) ∈ S} ∪ {ˆ 0, ˆ 1}, where ρ is the rank function. 1 2 3

S={1}

− − − − →

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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 PS. Based on that, define the flag h-vector β(S) as: β(S) =

  • T⊆S

(−1)#(S−T)α(T)

  • r,

α(S) =

  • T⊆S

β(T).

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Polytopes f-vectors and cd-index of Weight Polytopes

Example of Flag Vectors

1 2 3 ρ Face Poset S α(S) β(S) ∅ 1 1 {1} 4 3 {2} 4 3 {1, 2} 8 1 Table of Flag Vectors

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Polytopes f-vectors and cd-index of Weight Polytopes

Example of Flag Vectors

1 2 3 ρ Face Poset S α(S) β(S) ∅ 1 1 {1} 4 3 {2} 4 3 {1, 2} 8 1 Table of Flag Vectors β(S) = β( ¯ S)

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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 ΦP (a, b) =

  • S⊆[n]

β(S)uS. Here uS = unun−1 · · · u1, where ui =

  • a,

if i / ∈ S b, if i ∈ S.

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Polytopes f-vectors and cd-index of Weight Polytopes

Example of ab-index

S α(S) β(S) uS ∅ 1 1 a2 {1} 4 3 ab {2} 4 3 ba {1, 2} 8 1 b2 Table of Flag Vectors ΦP (a, b) = a2 + 3ab + 3ba + b2.

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Polytopes f-vectors and cd-index of Weight Polytopes

cd-index

Theorem (cd-index) For any polytope P, there exists a polynomial ΨP (c, d) in the non-commuting variables c and d such that ΦP (a, b) = ΨP (a + b, ab + ba). ΨP (c, d) is also called the cd-index of polytope P.

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Polytopes f-vectors and cd-index of Weight Polytopes

Example of cd-index

S α(S) β(S) uS ∅ 1 1 a2 {1} 4 3 ab {2} 4 3 ba {1, 2} 8 1 b2 Table of Flag Vectors ΦP (a, b) = a2 + 3ab + 3ba + b2 = (a + b)2 + 2(ab + ba). Replace a + b → c, ab + ba → d. ΨP (c, d) = c2 + 2d.

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Polytopes f-vectors and cd-index of Weight Polytopes

Summary

Methods to describe a polytope:

  • f-polynomial/h-polynomial;
  • face poset;
  • cd-index.

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Section 2

Coxeter Group and Weight Polytopes

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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 ⊂ GLn(R) generated by reflections, i.e. elements w such that w2 = 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 In = {s, t | s2 = t2 = e, (st)n = e}.

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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 ∼ = s1, . . . , sn | s2

i = e, (sisj)mij = e

for some mij ∈ {2, 3, 4, . . . } ∪ {∞}. If W is finite, then W is called a Finite Coxeter Group. S = {s1, s2, . . . , sn} is called the Generating Set of W.

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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.

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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 mij = 3, si and sj are connected with no label and if mij > 3, si and sj are connected with label mij. Example: The dihedral group In has Coxeter diagram

n

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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, An, Bn, Dn, and In, as well as a finite collection of exceptional cases. The Coxeter diagrams look as follows: We will focus our energies on types An, Bn, Dn.

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytopes

Definition (Weight Polytope) Given finite Coxeter group W, λ ∈ Rn, we define the Weight Polytope Pλ to be the convex hull of {w · λ | w ∈ W}.

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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 Pλ is only dependent on W, S and J(λ).

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 1

Coxeter Group W = An = symmetric group Sn+1 Vector λ λ = (0, . . . , 0

n zeros

, 1)

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 1

Coxeter Group W = An = symmetric group Sn+1

(12) (23) (34) (45)

· · ·

(n, n + 1)

Vector λ λ = (0, . . . , 0

n zeros

, 1)

1 2 3

· · ·

n − 1 n J(λ)

Polytope Name: Simplex Vertices: Set of vectors with n zeros and 1 one

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 2

Coxeter Group W = Bn = signed permutation group Vector λ λ = (1, 1, . . . , 1

  • n ones

)

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 2

Coxeter Group W = Bn = signed permutation group

(−1) (12) (23) (34)

· · ·

(n − 1, n) 4

Vector λ λ = (1, 1, . . . , 1

  • n ones

)

1 2 3 4

· · ·

n 4 J(λ)

Polytope Name: HyperCube Vertices: Set of vectors with 1 and −1

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 3

Coxeter Group W = Bn = signed permutation group Vector λ λ = (0, . . . , 0

n−1 zeros

, 1)

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 3

Coxeter Group W = Bn = signed permutation group

(−1) (12) (23) (34)

· · ·

(n − 1, n) 4

Vector λ λ = (0, . . . , 0

n−1 zeros

, 1)

1 2 3

· · ·

n − 1 n 4 J(λ)

Polytope Name: HyperOctahedron Vertices: Set of vectors with n − 1 zeros and one ±1

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 4

Coxeter Group W = An = symmetric group Sn+1 Vector λ λ = (0, . . . , 0

k zeros

, 1, . . . , 1

n−k+1 ones

)

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Weight Polytope Example 4

Coxeter Group W = An = symmetric group Sn+1

(12) (23) (34) (45)

· · ·

(n, n + 1)

Vector λ λ = (0, . . . , 0

k zeros

, 1, . . . , 1

n−k+1 ones

)

1

· · ·

k − 1 k k + 1

· · ·

n J(λ) J(λ)

Polytope Name: HyperSimplex Vertices: Set of vectors with k zeros and n − k + 1 ones

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Other Examples

Example 5 Name: Dodecahedron Coxeter Group: W = H3

1 2 3 5 J(λ)

Example 6 Name: Truncated Cube Coxeter Group: W = B3

1 2 3 4 J(λ)

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Weight Polytopes f-vectors and cd-index of Weight Polytopes

Recall Summary

Methods to describe a polytope:

  • f-polynomial/h-polynomial;
  • face poset;
  • cd-index.

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f-polynomial f-vectors and cd-index of Weight Polytopes

Section 3

f-polynomials of Simple Weight Polytopes

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f-polynomial f-vectors and cd-index of Weight Polytopes

Renner’s Classfication of Simple Polytopes

Theorem (Renner) A type An or Bn weight polytope is simple iff its Coxeter diagram has one of the following structures. · · · · · · · · · · · · · · ·

≥ 2 points

· · · · · ·

4

≤ n points

· · · · · ·

4

≤ n − 3 points Gao, McDonald 33 / 51

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f-polynomial f-vectors and cd-index of Weight Polytopes

Renner’s Classfication of Simple Polytopes

Theorem (Renner) A type An or Bn weight polytope is simple iff its Coxeter diagram has one of the following structures. · · · · · · · · · · · · · · ·

≥ 2 points

· · · · · ·

4

≤ n points

· · · · · ·

4

≤ n − 3 points

What are their f-polynomials?

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f-polynomial f-vectors and cd-index of Weight Polytopes

Case 1

Theorem (Golubitsky) Denote Fn,k(t) as the f-polynomial for the f polytope of · · · · · ·

n points k points

Then,

  • n≥k≥0

Fn,k(t) · xn+1yk (n + 1)! = exy y − 1 ·

  • y + etxy − t − 1

t + 1 − etx

  • − 1.

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f-polynomial f-vectors and cd-index of Weight Polytopes

Case 2

Theorem Denote Fn,a,b(t) as the f-polynomial for the f polytope of · · · · · · · · ·

n points a points b points

Then,

  • a,b≥0
  • n>a+b

Fn,a,b(t) · xn+1yazb (n + 1)! = 1 y2 − y

  • x +

(xy − exy + 1)(xz − exz) y +

  • tz + (t + 1)exz − t − e(t+1)xz

ty+(t+1)e(xy)−t−e((t+1)xy)

  • t−e(tx)+1
  • y

− e(xy)

  • t(y − 1)z

+ e(xy+xz) ty +

  • ze(txy) − ye(txz)

e(xy+xz) t(y − z)y

  • .

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f-polynomial f-vectors and cd-index of Weight Polytopes

Case 3

Theorem Denote Fn,k(t) as the f-polynomial for the f polytope of · · · · · ·

4 n points k points

Then,

  • n>k≥0

Fn,k(t) · xnyk n! = 1 y − 1

  • e(t+2)xy + etx ·
  • e2(t+1)xy − (t + 1) e2xy + t − ty
  • (t + 1 − e2tx)y
  • .

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f-polynomial f-vectors and cd-index of Weight Polytopes

Case 4

Theorem Denote Fn,k(t) as the f-polynomial for the f polytope of · · · · · ·

4 n points k points

Then,

  • n−2>k≥0

Fn,k(t) xn+1yk (n + 1)! = 1 y2 − y

  • xy

+

  • y + (t + 1)e(2 xy)

t − e(2 (t+1)xy) t − 1

  • (t + 1)tx − te(tx)

t − e(2 tx) + 1 + 1

  • − x −
  • (t + 1)xy + 1

t + 1

  • e(2 xy) − e(2 (t+1)xy)

t

− e((t+2)xy) y

  • .

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f-polynomial f-vectors and cd-index of Weight Polytopes

Ingredients of the Proof

Definition (J-minimal subset) For a Coxeter diagram Γ = (W, S) and subset J ⊆ S, a J-minimal subset is a subset X ⊆ S such that no connected component of X on the Coxeter diagram lies entirely in J. Example:

1 2 3 J

∅ All six J-minimal subsets Not J-minimal

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f-polynomial f-vectors and cd-index of Weight Polytopes

Ingredients of the Proof

Theorem (Renner, Maxwell) Consider the action of W on {faces of Pλ}, then there is a bijection f : {J(λ)-minimal sets} → {orbits of the action}. If X is J(λ)-minimal, then all faces in f(X) are called X-type

  • face. All X-type face has dimension |X|, and the number of

X-type face is |W| |WX∗|, where WX∗ ⊆ W is the subgroup generated by {s ∈ S|s ∈ X or s and X are not connected}.

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Example of Renner/Maxwell

1 2 3 4

J

X Face WX∗ |W|/|WX∗| ∅ Vertices {3} 48/2 = 24

4

Long Edges {1, 3} 48/4 = 12

4

Triangle Edges {2} 48/2 = 24

4

Octagons {1, 2} 48/8 = 6

4

Triangles {2, 3} 48/6 = 8

4

Truncated Cube {1, 2, 3} 48/48 = 1

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However... Renner only proved the case where W is a Weyl Group (a special type of Coxeter Group that forms a lattice). Is this true for general finite Coxeter Group ?

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However... Renner only proved the case where W is a Weyl Group (a special type of Coxeter Group that forms a lattice). Is this true for general finite Coxeter Group ? Answer: Yes!

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Face Poset f-vectors and cd-index of Weight Polytopes

Section 4

Face Poset of General Weight Polytopes: Maxwell implies Renner

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Face Poset f-vectors and cd-index of Weight Polytopes

Maxwell

Theorem (Maxwell) Given Coxeter System (W, S), and vector λ with stablizer J. The face poset of polytope Pλ is isomorphic to the poset L(W, J) = {gWXWJ|g ∈ W, X ⊆ S is J-minimal}

  • rdered by inclusion.

Here WX is the subgroup generated by elements in X.

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Face Poset f-vectors and cd-index of Weight Polytopes

What does Maxwell Imply?

Corollary

  • All faces are labelled by some J-minimal set X;
  • A X face lies inside a Y face if and only if X ⊆ Y ;
  • If X ⊆ Y , the number of X face inside a Y face is

equal to |WY | |WX| · |WY ∩(X∗\X)|. Take Y = S the entire set, the number of X-face is |W| |WX∗|, the same as Renner.

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cd-index f-vectors and cd-index of Weight Polytopes

Section 5

A glimpse on the cd-index of Weight Polytopes

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cd-index f-vectors and cd-index of Weight Polytopes

cd-index for simplices

Theorem (Stanley) If ΨP denotes the cd-index for a poset P then

2ΨP = 2Ψˆ

0ˆ 1 =

  • ˆ

0<x<ˆ 1 ρ(x,ˆ 1)=2j−1

(c2 − 2d)j−1cΨˆ

0x −

  • ˆ

0<x<ˆ 1 ρ(x,ˆ 1)=2j

(c2 − 2d)jΨˆ

0x

+

  • 2(c2 − 2d)k−1

if ρ(ˆ 0, ˆ 1) = 2k − 1 if ρ(ˆ 0, ˆ 1) = 2k.

Corollary (Stanley) if Ψn(c, d) is the cd-index for the n-simplex then

  • n≥1

Ψn(c, d)xn (n + 1)! = 2 sinh((a − b)x) a − b ·

  • 1 − c sinh((a − b)x)

a − b + cosh((a − b)x))−1

  • Gao, McDonald

46 / 51

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cd-index f-vectors and cd-index of Weight Polytopes

cd-index for hypersimplices

Theorem Denote Ψn,k as the cd-index for the hypersimplex · · ·

k

· · ·

n points Then,

  • n≥k≥1

Ψn,k(c, d) yk (n + 1)! = (1 − s(y + 1) · c + c(y + 1))−1 · c(y + 1) − c(y) − c(1) + 1 c2 − 2d · c − s(y + 1) + y + 1 y − 1 · (s(y) − s(1))

  • where c(x) = cosh((a − b)x) and s(x) = sinh((a − b)x)/(a − b)

Idea of Proof Combine Stanley’s Method with Renner/Maxwell’s formula.

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Summary f-vectors and cd-index of Weight Polytopes

Section 6

Summary

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Summary f-vectors and cd-index of Weight Polytopes

What have we done?

f-polynomial Face Poset cd-index General Weight Polytopes

  • Maxwell

(we rewrote ) Weyl Group Weight Polytopes

  • (some done by

Golubitsky) Renner Hypersimplex

  • Renner
  • Simplex

Known Known Stanley

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Acknowledgement We would like to thank Vic and Andy for their continual guidance and support!

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Summary f-vectors and cd-index of Weight Polytopes

The End!

Thank You!

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