2021/10/21 by Valdemar V. Tsanov, Tsanov, Valdemar, Yana Staneva +1
Chemistry · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Crystal structures of chemical compounds #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2110.11066
openalex publication_date 2021/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G and G be connected complex reductive Lie groups, G\nsemisimple. Let \Λ+ be the monoid of dominant weights for a positive\nroot system \Δ+, and let l(w) be the length of a Weyl group element\nw. Let V_\λ denote an irreducible G-module of highest weight\n\λ\∈\Λ+. For any closed embedding \ι: G\⊂ G, we\nconsider\n Property (A): \∀\λ\∈\Λ+,\∃ q\∈\ℕ such\nthat Vq\λ G\≠0.\n A necessary condition for (A) is for G to have no simple factors to which\nG projects surjectively. We show that this condition is sufficient if nG is of type bf A1 or bf E8.\n We define and study an integral invariant of a root system,\n\ℓG=\min \ℓ^\λ:\λ\∈\Λ+\∖ 0 , where\n\ℓ^\λ=\min l(w):w\λ\∉ rm Cone(\Δ+) . We derive the\nfollowing sufficient condition for (A), independent of \ι:
ellG -\n
#
tilde
Delta+ gt; 0
;
Longrightarrow
; (A). We compute \ℓG and\nrelated data for all simple G, except bf E8, where we obtain lower and\nupper bounds. We consider a stronger property (A-k) defined in terms of\nGeometric Invariant Theory, related to extreme values of codimensions of\nunstable loci, and derive a sufficient condition in the form \ℓG -\n # Δ+ > k. The invariant \ℓG proves too week to handle\nG=SLn and we employ a companion \ℓG rm sd to infer (A-k) for a\nlarger class of subgroups. We derive corollaries on Mori-theoretic properties\nof GIT-quotients.\n