Return to Colloquia & Seminar listing
Unimodular and flag triangulations arising from classical root systems
Algebra & Discrete Mathematics| Speaker: | Takayuki Hibi, Osaka University Japan |
| Location: | 693 Kerr |
| Start time: | Thu, Mar 6 2003, 12:10PM |
Description
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.