On Orbifold Criteria for Symplectic Toric Quotients

TL;DR

This paper introduces new criteria for symplectic quotients of torus actions, demonstrating examples where these quotients differ from finite group quotients, and connects representation properties to Gaussian elimination.

Contribution

It defines regular symplectomorphism concepts and provides examples distinguishing symplectic quotients from finite group quotients, linking representation theory to algebraic methods.

Findings

Examples of torus representations with non-equivalent symplectic quotients.

Connection between simplicialness of representations and Gaussian elimination.

Identification of cases where GIT quotients are smooth despite symplectic differences.

Abstract

We introduce the notion of regular symplectomorphism and graded regular symplectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly symplectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicialness of a torus representation to Gaussian elimination.