论文标题
极性轨道的光谱体表示
Spectrahedral representation of polar orbitopes
论文作者
论文摘要
让$ k $是一个紧凑的谎言组，$ v $ $ k $的有限维度表示。 v $中的向量$ x \的轨道是$ v $中的轨道$ kx $的凸壳$ \ mathscr o_x $。我们表明，如果$ v $是极性的，则$ \ mathscr o_x $是频谱，我们产生明确的线性矩阵不等式表示。我们还考虑coorbitope $ \ mathscr o_x^o $，它是凸的polar to $ \ mathscr o_x $。我们证明$ \ Mathscr o_x^o $是有限的$ k $ -orbits的凸壳，我们确定了$ \ mathscr o_x^o $本身就是轨道的案例。在这些情况下，有$ \ mathscr o_x^o = c \ cdot \ mathscr o_x $，带有$ c> 0 $。此外，我们表明，如果$ x $具有“理性系数”，那么$ \ mathscr o_x^o $ $又是频谱。这为许多新的谱系轨道提供了许多新家庭。从经典的半圣像谎言中得出的所有极性轨道都可以用奇异值和ky风扇矩阵规范的条件来描述。
Let $K$ be a compact Lie group and $V$ a finite-dimensional representation of $K$. The orbitope of a vector $x\in V$ is the convex hull $\mathscr O_x$ of the orbit $Kx$ in $V$. We show that if $V$ is polar then $\mathscr O_x$ is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope $\mathscr O_x^o$, which is the convex set polar to $\mathscr O_x$. We prove that $\mathscr O_x^o$ is the convex hull of finitely many $K$-orbits, and we identify the cases in which $\mathscr O_x^o$ is itself an orbitope. In these cases one has $\mathscr O_x^o=c\cdot\mathscr O_x$ with $c>0$. Moreover we show that if $x$ has "rational coefficients" then $\mathscr O_x^o$ is again a spectrahedron. This provides many new families of doubly spectrahedral orbitopes. All polar orbitopes that are derived from classical semisimple Lie can be described in terms of conditions on singular values and Ky Fan matrix norms.
