Betti Numbers of Toric Varieties and Eulerian Polynomials

Abstract

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n−1, s _n−2,…, s _n−k+2, s _n−k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.

Key Words:

• Betti numbers
• Linear algebraic monaids
• Simple polytope
• Toric variety

2010 Mathematics Subject Classification:

• 14xx
• 05xx

ACKNOWLEDGMENTS

I would like to thank my advisor, Lex Renner, for his supervision during my graduate studies and for bringing to my attention this problem, and Nicole Lemire for helpful conversations. I also thank Oleg Golubitsky, for helping me with LaTeX. I am very grateful to the anonymous referee for his valuable contributions to the article. The referee provided the statement of Corollary 2 identifying the generating function and suggested me to include Figures 1 and 2 for a better understanding of Corollary 1.

Notes

Communicated by V. Gould.