Let Q be a convex polytope in , and let be the number of i-dimensional faces. How are the related? Of course Euler’s relation states and for convex polytope, the genus is . Thus for , we can parametrize all admissible face vector by two coordinates, and .

Theorem 4(Steinutz 1905) The admissible set forms a cone anchored at . One can build all points from by adding or .

The case of is completely open. some progress: the polytopes are simplicial, i.e., every face is a simplex if and only if the extreme points are in general position.

From -vector we can get -vector, defined by

More explicitly,

Theorem 5(Billera, Stanley) Reference: G. Ziegler. is an h-vector of -dimensional polytope if and only if

- The Dehn-Sommerville condition holds: .
- Set , , , the .
- (see definition below).
Here given any positive integer , , write , for . Then

For example, if , , and , then , and .

For general polytopes, the Euler relation becomes . Polya tried harder problem: count not only faces, but chains of faces. Given and convex, let number of chains of faces of Q with dimensions in . The image of is called a flag-f vector.

- it specializes to , , etc as .
- There is a natural flag h-vector, and analogue of Dehn-Sommerville condition holds.
- Some progress has been made in the flag case.

A natural setting to study is when gives the flag vectors of graded posets. Face lattice of Q is a graded poset, obviously. Recall P is graded poset if for all and all max chains from to have the same length. It’s the same as ranked poset.

Definition 6, the flag f-vector , , with the distance of from ; is the dimension set.

Try to characterize Dehn-Sommerville relation: for example, for every graded poset of rank , .

Tool: (Gil Kalai) convolution: for , , and a ranked poset of rank , we have

Theorem 7If and , and , for every polytope resp. graded poset, then

for every polytope resp graded poset of dimension rank .

Now back to algebra. Recall . , . Given a poset , define

where , and .

Theorem 8

- , .
- .
- .
Ref:

- Luke Billera (Cornell) Flag enumerations and polytopes, Eulerian posets, and coxeter groups. 2010 ICM.
- Khazdan-Lusztig symmetric polynomials.

Last topic (I am very confused about the end of this part, someone please explain to me how the zeta function is related to the algebra : consider which is not only an algebra, but also a cogebra.

For , let , with 1’s. Then , if .

is a quasi-symmetric function, with . If is the incidence Hopf algebra of a graded poset (i.e., elements are generated by intervals ), is a Hopf map, from which one recovers the famous formula

where . Thus we have a map inducing the zeta function

where the number of multichains of length in .