The H-polynomial of a Group Embedding

TL;DR

This paper introduces the H-polynomial as a unifying invariant for various algebraic varieties with group actions, generalizing known polynomials like the Poincaré and h-vector, and provides explicit descriptions for certain projective varieties.

Contribution

It defines the H-polynomial for a broad class of algebraic varieties with solvable group actions and describes its computation for specific projective G×G-varieties using cellular decompositions.

Findings

H-polynomial unifies Poincaré polynomial and h-vector concepts.

Explicit combinatorial description of H-polynomials for certain varieties.

Application to the wonderful compactification of semisimple groups.

Abstract

The Poincar\'e polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the $H$-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety $X$ where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the $H$-polynomials of certain projective $G\times G$-varieties $X$, where $G$ is a semisimple group and $B$ is a Borel subgroup of $G$. This description is made possible by finding an appropriate cellular decomposition for $X$ and then describing the cells combinatorially in terms of the underlying monoid of $B\times B$-orbits. The most familiar example here is the wonderful compactification of a semisimple group of adjoint type.