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Partial convex hulls of coadjoint orbits and degrees of invariants

Valdemar V. Tsanov

Abstract

We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical invariants of a coadjoint orbit of a semisimple connected compact Lie group. It is shown that the orbits, where any one of these invariants does not exceed a given number $r$, form, upon intersection with a fixed Weyl chamber, a rational convex polyhedral cone in that chamber, related to the Littlewood-Richardson cone of the $r$-fold diagonal embedding of $K$. The numerical invariants are shown to provide lower bounds for degrees of invariant polynomials on irreducible unitary representations.

Partial convex hulls of coadjoint orbits and degrees of invariants

Abstract

We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical invariants of a coadjoint orbit of a semisimple connected compact Lie group. It is shown that the orbits, where any one of these invariants does not exceed a given number , form, upon intersection with a fixed Weyl chamber, a rational convex polyhedral cone in that chamber, related to the Littlewood-Richardson cone of the -fold diagonal embedding of . The numerical invariants are shown to provide lower bounds for degrees of invariant polynomials on irreducible unitary representations.
Paper Structure (3 sections, 10 theorems, 43 equations)

This paper contains 3 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction and main results
  2. The momentum map and the Littlewood-Richardson cone
  3. Coadjoint orbitopes and partial convex hulls

Key Result

Theorem 1.1

The inequality ${\bf r}_0(\lambda)\leq d_1(q\lambda)$ holds for all $\lambda\in\Lambda^+\setminus\{0\}$ and $q\in {\mathbb Q}_{>0}$ such that $q\lambda\in\Lambda^+$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • ...and 12 more