# On momentum images of representations and secant varieties

Research paper by **Elitza Hristova, Tomasz Maciazek, Valdemar V. Tsanov**

Indexed on: **28 Apr '15**Published on: **28 Apr '15**Published in: **Mathematics - Representation Theory**

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#### 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 $V_\lambda$.