Algebra Seminar

Speaker: Letitia Banu (Western)

"Betti numbers of a rationally smooth toric variety"

Time: 12:30

Room: MC 108

Consider an irreducible representation of a semisimple algebraic group with $\lambda$ its highest weight and look at the action of the Weyl group $W$ on the rational vector space spanned by the roots. Take the convex hull of the $W$-orbit of $\lambda$ and obtain the polytope $P_{\lambda}= {\textrm{Conv}}(W.\lambda)$. We are interested in describing the Betti numbers of the toric variety $X(J)$ associated to the polytope $P_{\lambda}$ when the Weyl group is the $n$ symmetric group and $X(J)$ is a rationally smooth variety which doesn't depend on the highest weight $\lambda$ but on the set of reflections that fix $\lambda$ called $J$. The main result is a recursion formula for the Betti numbers of $X(J)$ in terms of Eulerian polynomials. The theory of algebraic monoids developed by Renner and Putcha is effectively used in our computations.