Abstract
AbstractLet K be a compact Lie group and V a finite-dimensional representation of K. The orbitope of a vector $$x\in V$$
x
∈
V
is the convex hull $${\mathscr {O}}_x$$
O
x
of the orbit Kx in V. We show that if V is polar then $${\mathscr {O}}_x$$
O
x
is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope $${\mathscr {O}}_x^o$$
O
x
o
, which is the convex set polar to $${\mathscr {O}}_x$$
O
x
. We prove that $${\mathscr {O}}_x^o$$
O
x
o
is the convex hull of finitely many K-orbits, and we identify the cases in which $${\mathscr {O}}_x^o$$
O
x
o
is itself an orbitope. In these cases one has $${\mathscr {O}}_x^o=c\cdot {\mathscr {O}}_x$$
O
x
o
=
c
·
O
x
with $$c>0$$
c
>
0
. Moreover we show that if x has “rational coefficients” then $${\mathscr {O}}_x^o$$
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 algebras can be described in terms of conditions on singular values and Ky Fan matrix norms.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Reference26 articles.
1. Biliotti, L., Ghigi, A., Heinzner, P.: Polar orbitopes. Commun. Anal. Geom. 21, 579–606 (2013)
2. Biliotti, L., Ghigi, A., Heinzner, P.: Coadjoint orbitopes. Osaka J. Math. 51, 935–968 (2014)
3. Borel, A., Tits, J.: Groupes réductifs. Publ. Math. I. H. E. S. 27, 55–151 (1965)
4. Bump, D.: Lie Groups, 2nd edn. Graduate Texts in Mathematics, vol. 225. Springer, New York (2013)
5. Casselman, W.A.: Geometric rationality of Satake compactifications. In: Algebraic Groups and Lie Groups, Australian Mathematical Society Lecture Series, vol. 9. Cambridge, pp. 81–103 (1997)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献