Betti numbers of toric varieties and eulerian polynomials

TL;DR

This paper explores the Betti numbers of certain toric varieties linked to generalized permutohedra, deriving recurrence relations for their $h$-vectors and Poincaré polynomials using combinatorics and algebraic monoid theory.

Contribution

It introduces new recurrence formulas for the $h$-vectors and Betti numbers of a family of rationally smooth toric varieties associated with generalized permutohedra.

Findings

Derived recurrences for $h$-vectors of generalized permutohedra.

Established relations between Betti numbers and Eulerian polynomials.

Connected toric varieties to algebraic monoid theory.

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 paper 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 non-generic weight, namely a weight fixed by only the simple reflections $J=\{s_{n},s_{n-1},s_{n-2} \cdots,s_{n-k+2},s_{n-k+1}\}$ 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 Poincar\'e polynomial of $X(J)$ in terms of the Eulerian polynomials.