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
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 7 If 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 .
- , .
- 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 .