On momentum images of representations and secant varieties

TL;DR

This paper investigates the structure of momentum polytopes associated with irreducible representations of compact semisimple groups, providing a combinatorial description for regular weights and exploring critical points and secant varieties.

Contribution

It constructs a convex cone containing the momentum polytope, characterizes cases of equality, and relates secant varieties to invariant polynomial degrees.

Findings

Complete description of momentum polytopes for regular highest weights.

Identification of conditions for equality of the polytope and the convex cone.

Establishment of a link between secant varieties and degrees of invariant polynomials.

Abstract

Let $K$ be a connected compact semisimple group and $V_\lambda$ be an irreducible unitary representation with highest weight $\lambda$. We study the momentum map $\mu:\mathbb P(V_\lambda)\to\mathfrak k^*$. The intersection $\mu(\mathbb P(V_\lambda))^+=\mu(\mathbb P(V_\lambda))\cap{\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_\lambda$. We construct an affine rational polyhedral convex cone $\Upsilon_\lambda$ with vertex $\lambda$, such that $\mu(\mathbb P(V_\lambda))^+\subset\Upsilon_\lambda \cap {\mathfrak t}^+$. We show that equality holds for a class of representations, including those with regular highest weight. For those cases, we obtain a complete combinatorial description of the momentum polytope, in terms of $\lambda$. We also present some results on the critical points of $||\mu||^2$. Namely, we consider the existence problem for critical points in the preimages of Kirwan's candidates for critical values. Also, we consider the secant varieties to the unique complex orbit $\mathbb X\subset\mathbb P(V_\lambda)$, and prove a relation between the momentum images of the secant varieties and the degrees of $K$-invariant polynomials on $V_\lambda$.