# Triangulations of Convex Polytopes

## ABSTRACT

Let $K$ be a finite set of vectors in $\mathbb R^n$. Let ${\operatorname{conv}^*}(K)$ be the convex hull of $\{O\}\cup K$. Given any linear order $\prec$ on $K$, say $K= \{ v_1, v_2, v_3, \ldots, v_m\}$ with $v_1\prec v_2 \prec v_3 \prec \cdots \prec v_m$, we define the notion of a \emph{$\Delta_\prec$-basis} on the linear matroid $\mathcal{M}(K)$ whose underlying set of vectors is $K$ and present a local triangulation of ${\operatorname{conv}^*}(K)$ determined by $\prec$. We will specifically look at the polytopes ${\operatorname{conv}^*}(A_n^+)$, where $A_n^+$ is the set of positive roots of the root system $A_n$, and polytopes similarly associated to other root systems. We will talk about their volumes, shellabilities, and Ehrhart polynomials and, while doing so, introduce the C_n anaolog of Catalan numbers and Narayana numbers.

Speaker's Contact Info: wungkum(at-sign)math.mit.edu