A representation theoretic classification of multiprojective spaces

TL;DR

This paper introduces a novel classification of multiprojective spaces using tensor product decompositions of irreducible representations of simple Lie algebras, offering a representation-theoretic perspective beyond traditional algebraic geometry methods.

Contribution

The paper develops a new classification approach for multiprojective spaces based on representation theory, providing an alternative to existing algebro-geometric techniques.

Findings

Multiprojective spaces are classified via tensor product decompositions.

Distinct partitions correspond to non-isomorphic multiprojective spaces.

Representation theory offers a new perspective for classifying algebraic varieties.

Abstract

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.