Mathematics Colloquia and Seminars

Let ${\Phi} \subset \ZZ^n$ denote one of the classical irreducible root systems $\Ab_{n-1}$, $\Bb_n$, $\Cb_n$ and $\Db_n$, and write $\Phi^{(+)}$ for the configuration consisting of all positive roots of $\Phi$ together with the origin of $\RR^{n}$. By constructing an explicit unimodular and flag triangulation, Gelfand, Graev and Postnikov showed that the normalized volume of the convex hull of $\Ab_{n-1}^{(+)}$ is equal to the Catalan number. On the other hand, in her dissertation, W. Fong computed the normalized volume of the convex hull of each of the configurations $\Bb_n^{(+)}$, $\Cb_n^{(+)}$ and $\Db_n^{(+)}$ by using its natural triangulation. However, the triangulations which Fong used are, in general, neither unimodular nor flag. In my talk, via the theory of Gr\"obner bases of toric ideals, the existence of unimodular and flag triangulations of certain subconfigurations of $\Ab_{n-1}^{(+)}$, $\Bb_n^{(+)}$, $\Cb_n^{(+)}$ and $\Db_n^{(+)}$ will be discussed. This is a joint work with Hidefumi Ohsugi.