Gil Kalai’s blog

## Billerafest

Posted on June 12, 2008 by

I am unable to attend the conference taking place now at Cornell, but I send my warmest greetings to Lou from Jerusalem. The titles and abstracts of the lectures can be found here. Let me tell you about two theorems by Lou.

The first is the famous g-theorem: The g-theorem is a complete description of f-vectors (= vecors of face numbers) of simplicial d-polytopes. This characterization was proposed by Peter McMullen in 1970, and it was settled in two works. Billera and Carl Lee proved the sufficiency part of McMullen’s conjecture, namely for every sequence of numbers which satisfies McMullen’c conjecture they constructed a simplicial d-polytope P whose f-vector is the given sequence. Richard Stanley proved the necessity part based on the hard Lefschetz theorem in algebraic geometry. The assertion of the g-conjecture (the necessity part) for triangulations of spheres is open, and this is probably the one single problem I spent the most time on trying to solve.

The second theorem is a beautiful theorem by Margaret Bayer and Billera. Consider general d-polytopes. for a set $S \subset$ {0,1,2,…,d-1}, $S=${ $i_1,i_2,\dots,i_k$} , $i_1<i_2< \dots$, define the flag number $f_S$ as the number of chains of faces $F_1 \subset F_2 \subset \dots F_k$, where $\dim F_j=i_j$.  Bayer and Billera proved that the affine dimension of flag numbers of d-polytopes is $c_d-1$ where $c_d$ is the dth Fibonacci number. ($c_1=1$, $c_2=2$, $c_3=3$, $c_4=5$, etc.) The harder part of this theorem was to construct $c_d$ d-polytopes whose sequences of flag numbers are affinely independent.  The construction is simple: It is based on polytopes expressed by words of the form PBBPBPBBBPBP  where you start with a point, and P stands for “take a pyramid” and B stands for “take a bipyramid.” And the word starts with a P (to the left) and has no two consecutive B’s.

Let’s practice the notions of f-vectors and flag vectors on the 24-cell   . (The figure is a 3-dimensional projection into one of the facets of the polytope.)

This  4-polytope has 24 octahedral facets. It is self dual.
So $f_0=f_3=24$. And $f_1=f_2=96$. And $f_{02}=288$, $f_{03}=192$, etc.
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