Okounkov bodies for ample line bundles with applications to multiplicities for group representations (Q330638)

Let \(X\) be a complex normal projective variety of dimension \(n\) and let \(L\) be a big line bundle on \(X\). Via the choice of a flag in \(X\) one can associate to \(L\) a convex body in \(\mathbb{R}^n\), the so called \textit{Okounkov body}, encoding the information of the order of vanishing of the sections of the powers of \(L\) along the flag. A natural question is to know when this body is a rational polytope, that is, the convex hull of a finite set of points in \(\mathbb{Q}^n\). The main result of this paper (see Thm. 3.3) states that when \(L\) is ample there exists a flag providing an Okounkov body which is a rational polytope. (This result is also proved in [\textit{D. Anderson} et al, Int. Math. Res. Not. 2014, No. 9, 2343--2355 (2014; Zbl 1316.14013)] under weaker assumptions, see the Introduction of the paper under review.) Moreover, when \(X\) is a homogeneous surface the choice of the flag to get a rational polytope is proved to be generic. Applications to the asymptotic study of group representations are also provided.