关于表示和割线簇的动量象
On momentum images of representations and secant varieties
摘要：
设$K$是连通紧半单群，$V_λ$是具有最高权的不可约酉表示$λ$ .我们研究了动量映射$μ:\mathbb P(V_λ)\to\mathfrak k^*$。动量像与固定外尔腔的交点$μ(\mathbb P(V_λ))^+=μ(\mathbb P(V_λ))\cap{\mathfrak t}^+$是一个凸多面体，称为$V_λ$的动量多面体。我们构造了一个仿射有理多面体凸锥$Υ_λ$，其顶点为$λ$，使得$μ(\mathbb P(V_λ))^+\subsetΥ_λ\cap {\mathfrak t}^+$ .我们发现，平等持有一类代表，包括那些经常最高的权重。对于这些情况下，我们得到一个完整的组合描述的动量多面体，在$λ$。我们也给出了关于$||μ||^2$临界点的一些结果。也就是说，我们考虑的存在性问题的临界点的原像Kirwan的候选人的临界值。同时，我们考虑了唯一复轨道$\mathbb X\subset\mathbb P(V_λ)$的割线簇，证明了割线簇的动量像与$V_λ$上$K$不变多项式的次数之间的关系.
英文摘要：
Let $K$ be a connected compact semisimple group and $V_λ$ be an irreducible unitary representation with highest weight $λ$. We study the momentum map $μ:\mathbb P(V_λ)\to\mathfrak k^*$. The intersection $μ(\mathbb P(V_λ))^+=μ(\mathbb P(V_λ))\cap{\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_λ$. We construct an affine rational polyhedral convex cone $Υ_λ$ with vertex $λ$, such that $μ(\mathbb P(V_λ))^+\subsetΥ_λ\cap {\mathfrak t}^+$. 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 $λ$. We also present some results on the critical points of $||μ||^2$. 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 $\mathbb X\subset\mathbb P(V_λ)$, and prove a relation between the momentum images of the secant varieties and the degrees of $K$-invariant polynomials on $V_λ$.
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