A representation theoretic classification of multiprojective spaces

Given a positive integer $n$ and a partition $(n_1,\ldots,n_r)$ of $n$, one can consider the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. These multiprojective spaces are ubiquitous, not only in the realm of algebraic geometry but also in many other branches of mathematics. It is known that these multiprojective spaces corresponding to distinct partitions are not isomorphic. The available classification techniques of these spaces are mostly algebro-geometric in nature. In this paper, we use a decomposition of tensor products of irreducible representations of simple Lie algebras to classify these multiprojective spaces.