论文信息 - Spectrahedral representation of polar orbitopes

Spectrahedral representation of polar orbitopes

Let K be a compact Lie group and V a finite-dimensional representation of K. The orbitope of a vector $$x\in V$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> is the convex hull $${\mathscr {O}}_x$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> of the orbit Kx in V. We show that if V is polar then $${\mathscr {O}}_x$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope $${\mathscr {O}}_x^o$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msubsup> </mml:math>, which is the convex set polar to $${\mathscr {O}}_x$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math>. We prove that $${\mathscr {O}}_x^o$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msubsup> </mml:math> is the convex hull of finitely many K-orbits, and we identify the cases in which $${\mathscr {O}}_x^o$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msubsup> </mml:math> is itself an orbitope. In these cases one has $${\mathscr {O}}_x^o=c\cdot {\mathscr {O}}_x$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:mo>·</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> </mml:math> with $$c>0$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math>. Moreover we show that if x has “rational coefficients” then $${\mathscr {O}}_x^o$$<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>x</mml:mi> <mml:mi>o</mml:mi> </mml:msubsup> </mml:math> 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.

Claus Scheiderer | Tim Kobert

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