The H-polynomial of a Group Embedding

Details The H-polynomial of a Group Embedding The H-polynomial of a Group Embedding

2009-06-08

arxiv.org/abs/0906.1574v1

Mathematics

Algebraic Geometry

This is a survey of published and soon-to-be published work of the author

Scientific paper

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.

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