Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion

Abstract

Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the "relative" Schubert calculus of the flag varieties of the two groups. Contents 1. Introduction 1 2. Subgroups and Weyl chambers 3 3. Main results 13 4. Semistability 20 5. Examples 24 Appendix A. Flag varieties 30 Appendix B. Notation 32 References 33

A Geometric Description of Feasible Singular Values in the Tensor Train Format

“…For each specific n 1 and n 2 , a finite set of linear inequalities can thereby be derived which are equivalent (cf. [1]) to compatibility. Although the two latter solutions are in a certain sense complete (from an algebraic perspective), [4] could for example only conjecture that in the special case n 1 ≤ n 2 , compatibility of (…”

“…. , edge(h (M ) )) ∈ R N ×M h (1) h (2) h (3) a (1) a (2) −a Figure 6: A (4, 2)-hive consisting of two coupled honeycombs (blue for γ, magenta for θ), which are slightly shifted for better visibility, generated by Algorithm 1. Note that some lines have multiplicity 2.…”

“…2. For simplicity, each of these singular values is considered n 2 n 3 n 4 A ({1}) : n 1 sv σ (1) n 3 n 4…”

“…is based on the same entries within A, the question about the feasibility of prescribed TT-singular value arises immediately: Definition 1.1 (TT-feasibility). Let σ = (σ (1) , . .…”

“…this denotes f P (1, w) = λ, f P (1, e) = µ, f P (1, s) = ν for f P = δ P (H) Zero eigenvalues). If the relation a(1) . .…”