How many simplices are needed to triangulate a Grassmannian?

We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k({\mathbb{R}}^n)$. In particular, we show that the number of top-dimensional simplices grows exponentially with $n$. More precise estimates are given for $k=2,3,4$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.