Papers › On reductive subgroups of reductive groups having invariants in almost all representations
On reductive subgroups of reductive groups having invariants in almost all representations
Valdemar Tsanov, Yana Staneva
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
Let G and G̃ be connected complex reductive Lie groups, G semisimple. Let Λ^+ be the monoid of dominant weights for a positive root system Δ^+, and let l(w) be the length of a Weyl group element w. Let V_λ denote an irreducible G-module of highest weight λ∈Λ^+. For any closed embedding ι:G̃⊂G, we consider Property (A): ∀λ∈Λ^+,∃q∈ℕ such that V_(qλ)^(G̃)0. A necessary condition for (A) is for G to have no simple factors to which G projects surjectively. We show that this condition is sufficient if G̃ is of type A₁ or E₈. We define and study an integral invariant of a root system, ℓ_G=min{ℓ^λ:λ∈Λ^+∖{0}}, where ℓ^λ=min{l(w):wλ∉Cone(Δ^+)}. We derive the following sufficient condition for (A), independent of ι: ℓ_G - #Δ̃^+ > 0 ⟹ (A). We compute ℓ_G and related data for all simple G, except E₈, where we obtain lower and upper bounds. We consider a stronger property (A-k) defined in terms of Geometric Invariant Theory, related to extreme values of codimensions of unstable loci, and derive a sufficient condition in the form ℓ_G - #Δ̃^+ > k. The invariant ℓ_G proves too week to handle G=SLₙ and we employ a companion ℓ_Gˢᵈ to infer (A-k) for a larger class of subgroups. We derive corollaries on Mori-theoretic properties of GIT-quotients.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections