Cubical Dehn-Sommerville equations
max hlavacek
UC Berkeley

August 2, 2018

What are the classical Dehn-Sommerville Equations?
Definition
A polytope is simplicial if all ofits faces are simplices.

Theorem (Dehn, Sommerville,1905-1927)
Let P be a simplicial
d-dimensional polytope with
face-vector (f−1 , . . . , fd ). Thenfor 0 ≤ j ≤ d:
fj−1 =dX
k=j(−1)d−k kfk−1jHow else can we formulate this?

I Recall from Katharina’s problem sheet that we can define anh − vector for simplicial polytopes:dX

fi−1 z i (1 − z)d−i =i=0dX

hk z i .

i=0

I The Dehn-Sommerville equations can be stated as asymmetry in the h-vector:
hi = hd−iDehn-Sommerville Equations for cubical polytopes
Definition
A polytope is cubical if all of its
faces are combinatoriallyequivalent to cubes.

Theorem
Let P be a cubical d-dimensional
polytope with face-vector(f−1 , . . . , fd ). Then for0 ≤ j ≤ d, we have:dX
d−k k−j k − 1
fj−1 =(−1)
2
fk−1
j −1k=jQuestions!!!

I There are generalizations of Dehn-Sommerville equations forsimplicial polytopes! Can these generalizations be extended as
far in the cubical case?
I There are many proofs of Dehn-Sommerville. Do these proof

techniques work well in the cubical case?
I Can we express the cubical Dehn-Sommerville as a symmetry

relation of some cubical h-vector?

How far can one generalize Dehn-Sommerville?
Definition
An (abstract) simplicial complex is a nonempty collection Γ ofsubsets of a finite set V such that:
I if σ ∈ Γ and σ 0 ⊂ σ, then σ 0 ∈ Γ.

We call the σ faces of Γ.
13Maximal Faces

1024

98
576

11

{1,2,3,4}{4,5,6,7}{5,8}{4,7,9}{9,10}{10,11}{9,11}

How far can one generalize Dehn-Sommerville?
Definition
An (abstract) simplicial complex is a nonempty collection Γ ofsubsets of a finite set V such that:
I if σ ∈ Γ and σ 0 ⊂ σ, then σ 0 ∈ Γ.

We call the σ faces of Γ.
13

Maximal Faces

1024

98
57

11

{1,2,3,4}{4,5,6,7}{5,8}{4,7,9}{9,10}{10,11}{9,11}

6Note: Every (abstract) simplical complex Γ can be realized as theface poset of a complex of unimodular simplices!

How far can one generalize Dehn-Sommerville?
TheoremLet Γ be a d-dimensional Eularian simplicial complex. Then for0 ≤ i ≤ d + 1,
hi = hd−i+1
which is equivalent tofj−1 =d+1
X

(−1)

k=j

for 0 ≤ j ≤ d + 1.

d+1−k kfk−1jHow far can one generalize Dehn-Sommerville?
TheoremLet Γ be a d-dimensional Eularian simplicial complex. Then for0 ≤ i ≤ d + 1,
hi = hd−i+1
which is equivalent tofj−1 =d+1Xd+1−k(−1)

k=j kfk−1
jfor 0 ≤ j ≤ d + 1.
Proofs outlined in Beck-Sanyal’s new book:
I uses the idea of self-reciprocol complexes.
I uses the idea of chain partition functions

Generalized form for cubical complexes?
Definition
A cubical complex Γ is a nonempty collection of subsets on a finite
set V closed under intersection such that:I {v } ∈ V for all v ∈ V
I For every face σ ∈ Γ the interval [∅, σ] is isomorphic to thelattice of faces of a cube.

Generalized form for cubical complexes?
Definition
A cubical complex Γ is a nonempty collection of subsets on a finite
set V closed under intersection such that:I {v } ∈ V for all v ∈ VI For every face σ ∈ Γ the interval [∅, σ] is isomorphic to thelattice of faces of a cube.

Theorem (Adin, 1995)
Let Γ be a d-dimensional Eularian cubical complex. Then for1 ≤ j ≤ d + 1,
fj−1 =d+1
X

d−k+1 k−j

(−1)

k=j2


k −1
fk−1
j −1

Generalized form for cubical complexes?
Definition
A cubical complex Γ is a nonempty collection of subsets on a finite
set V closed under intersection such that:I {v } ∈ V for all v ∈ VI For every face σ ∈ Γ the interval [∅, σ] is isomorphic to thelattice of faces of a cube.

Theorem (Adin, 1995)
Let Γ be a d-dimensional Eularian cubical complex. Then for1 ≤ j ≤ d + 1,
fj−1 =d+1
X

d−k+1 k−j

(−1)

k=j2



k −1
fk−1
j −1

Proof: uses flags, defines a long and a short cubical h-vector!!!

What is a self-reciprocal complex?

Let K be a complex of lattice polytopes in Rd . We can extend thedefinition of the ehrhart polynomial:
ehrK = |nK ∩ Zd |This always agrees with a polynomial of degree dim(K).

Definition
K is self-reciprocal if for all n > 0:(−1)dim(K) ehrK (−n) = ehrK (n)In terms of h∗ -vector
Recall from Katharina’s lectures:

Definition

EhrK (z) := 1 +Xn≥1ehrK (n)z n =∗h0∗ + h1∗ z + · · · + hd+1z d+1(1 − z)d+1∗
We call (h0∗ , . . . hd+1) the h∗ −vector of K.

In terms of h∗ -vector
Recall from Katharina’s lectures:

Definition

EhrK (z) := 1 +Xn≥1ehrK (n)z n =∗h0∗ + h1∗ z + · · · + hd+1z d+1(1 − z)d+1∗
We call (h0∗ , . . . hd+1) the h∗ −vector of K.

In some cases, you can determine whether a complex isself-reciprocal from the h∗ -vector!

In terms of h∗ -vector
Recall from Katharina’s lectures:

Definition

EhrK (z) := 1 +Xn≥1ehrK (n)z n =∗h0∗ + h1∗ z + · · · + hd+1z d+1(1 − z)d+1∗We call (h0∗ , . . . hd+1) the h∗ −vector of K.

In some cases, you can determine whether a complex isself-reciprocal from the h∗ -vector!

TheoremLet χ(K) = 1 − (−1)d+1 . Then K is self-reciprocal if and only if:∗
hd+1−i(K) = hi∗ (K)What are examples of self-reciprocal complexes?
Proposition
The boundary complex (all proper faces) of any lattice polytope isself-reciprocal.
Proof uses Ehrhart-Mcdonald reciprocity.

What are examples of self-reciprocal complexes?
Proposition
The boundary complex (all proper faces) of any lattice polytope isself-reciprocal.
Proof uses Ehrhart-Mcdonald reciprocity.

Definition
A graded poset Π with 0̂, 1̂ is Eularian if:µΠ (x, y ) = (−1)`(x,y )What are examples of self-reciprocal complexes?
Proposition
The boundary complex (all proper faces) of any lattice polytope isself-reciprocal.
Proof uses Ehrhart-Mcdonald reciprocity.

Definition
A graded poset Π with 0̂, 1̂ is Eularian if:µΠ (x, y ) = (−1)`(x,y )Proposition
Eularian complexes of lattice polytopes are self reciprocal.
Again, use Ehrhart-Mcdonald reciprocity.

What are examples of self-reciprocal complexes?
Proposition
The boundary complex (all proper faces) of any lattice polytope isself-reciprocal.
Proof uses Ehrhart-Mcdonald reciprocity.

Definition
A graded poset Π with 0̂, 1̂ is Eularian if:µΠ (x, y ) = (−1)`(x,y )Proposition
Eularian complexes of lattice polytopes are self reciprocal.
Again, use Ehrhart-Mcdonald reciprocity.
In all these cases, χ(K) = 1 − (−1)d+1 so the h∗ vector issymmetric!

How do we use this to prove generalized Dehn-Somerville?

I Start with an Eularian simplicial complex Γ.

How do we use this to prove generalized Dehn-Somerville?

I Start with an Eularian simplicial complex Γ.
I There is a complex of unimodular simplices that has the same

face lattice. Call this R[Γ].

How do we use this to prove generalized Dehn-Somerville?

I Start with an Eularian simplicial complex Γ.
I There is a complex of unimodular simplices that has the same

face lattice. Call this R[Γ].
I This is self-reciprocal!

How do we use this to prove generalized Dehn-Somerville?

I Start with an Eularian simplicial complex Γ.
I There is a complex of unimodular simplices that has the same

face lattice. Call this R[Γ].
I This is self-reciprocal!
I Since all faces of R[Γ] are unimodular, the h∗ -vector and h−vector are the same.

How do we use this to prove generalized Dehn-Somerville?

I Start with an Eularian simplicial complex Γ.
I There is a complex of unimodular simplices that has the same

face lattice. Call this R[Γ].
I This is self-reciprocal!
I Since all faces of R[Γ] are unimodular, the h∗ -vector and h−vector are the same.
I Since the h∗ −vector is symmetric, so is the h vector!

How do we use this to prove generalized Dehn-Somerville?

I Start with an Eularian simplicial complex Γ.
I There is a complex of unimodular simplices that has the same

face lattice. Call this R[Γ].
I This is self-reciprocal!
I Since all faces of R[Γ] are unimodular, the h∗ -vector and h−vector are the same.
I Since the h∗ −vector is symmetric, so is the h vector!
I Generalized Dehn-Sommerville!

Can we use this method for the cubical case?

Can we use this method for the cubical case?

I don’t think so...

Can we use this method for the cubical case?

I don’t think so...
13522461What are chain partitions of a poset?
Let Π be a finite poset with 0̂, 1̂. Let φ : Π → Z+ be anorder-preserving map.

Definition
A (Π, φ)−chain partition of n is of the form:n = φ(c1 ) + · · · + φ(cm )for some multichain 0̂ ≺ c1  · · ·  cm ≺ 1̂What are chain partitions of a poset?
Let Π be a finite poset with 0̂, 1̂. Let φ : Π → Z+ be anorder-preserving map.

Definition
A (Π, φ)−chain partition of n is of the form:n = φ(c1 ) + · · · + φ(cm )for some multichain 0̂ ≺ c1  · · ·  cm ≺ 1̂ We let cpΠ,φ (n) be thenumber of (Π, φ)- chain partitions of n.

What are chain partitions of a poset?
Let Π be a finite poset with 0̂, 1̂. Let φ : Π → Z+ be anorder-preserving map.

Definition
A (Π, φ)−chain partition of n is of the form:n = φ(c1 ) + · · · + φ(cm )for some multichain 0̂ ≺ c1  · · ·  cm ≺ 1̂ We let cpΠ,φ (n) be thenumber of (Π, φ)- chain partitions of n.

TheoremLet Π be Eulerian of rank d + 1, and let φ be ranked. Then,(−1)d cpΠ,φ (−n) = cpΠ,φ (n − φ1 − · · · − φd )Chain partitions of simplicial complexes?
Let Γ be a simplicial complex on V .
Consider: multisubset represented as a ∈ ZV
≥0 , where av is themultiplicity of v .
supp(a) := {v ∈ V |av > 0}

Chain partitions of simplicial complexes?
Let Γ be a simplicial complex on V .
Consider: multisubset represented as a ∈ ZV
≥0 , where av is themultiplicity of v .
supp(a) := {v ∈ V |av > 0}{multisubsets awith fixed support{ multichainssupp(a)=σ ⊇ ...⊇ σ ⊃ 0}}mA(2, 3, 2)
A B C1

B
⊇

BC

⊇
ACB

From chain partitions to Dehn-Sommerville
Theorem (Beck, Sanyal)
Let Γ be a simplicial complex of dimension d − 1 with φ(σ) = |σ|.
Then: dXncpΓ,φ =fk−1kk=0

From chain partitions to Dehn-Sommerville
Theorem (Beck, Sanyal)
Let Γ be a simplicial complex of dimension d − 1 with φ(σ) = |σ|.
Then: dXncpΓ,φ =fk−1kk=0

I This is a polynomial! So the series CPΓ,φ can be written as:CPΓ,φ (z) =h0 + · · · + hd−1 z d−1(1 − z)dFrom chain partitions to Dehn-Sommerville
Theorem (Beck, Sanyal)
Let Γ be a simplicial complex of dimension d − 1 with φ(σ) = |σ|.
Then: dXncpΓ,φ =fk−1kk=0

I This is a polynomial! So the series CPΓ,φ can be written as:CPΓ,φ (z) =h0 + · · · + hd−1 z d−1(1 − z)dI You can show that the hi are indeed the h− vector of Γ.

From chain partitions to Dehn-Sommerville
Theorem (Beck, Sanyal)
Let Γ be a simplicial complex of dimension d − 1 with φ(σ) = |σ|.
Then:
 dX
ncpΓ,φ =fk−1
kk=0

I This is a polynomial! So the series CPΓ,φ can be written as:CPΓ,φ (z) =h0 + · · · + hd−1 z d−1(1 − z)dI You can show that the hi are indeed the h− vector of Γ.
I The proposition a few slides ago implies:

hi = hd−1−iFrom chain partitions to Dehn-Sommerville
Theorem (Beck, Sanyal)
Let Γ be a simplicial complex of dimension d − 1 with φ(σ) = |σ|.
Then:
 dX
ncpΓ,φ =fk−1
kk=0

I This is a polynomial! So the series CPΓ,φ can be written as:CPΓ,φ (z) =h0 + · · · + hd−1 z d−1(1 − z)dI You can show that the hi are indeed the h− vector of Γ.
I The proposition a few slides ago implies:

hi = hd−1−iI Dehn Sommerville!!

Can you do this for the cubical case?

Can you do this for the cubical case?

I hope so!

Thank you all!!! :) :)