2024/01/23 by Arkadev Ghosh, Ghosh, Arkadev, S. Senthamarai Kannan +1 · 1 citation
Mathematics · #14M15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2401.12527
openalex publication_date 2024/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a simple algebraic group of adjoint type of rank n over ℂ. Let T be a maximal torus of G, and B be a Borel subgroup of G containing T. Let W=NG(T)/T be the Weyl group of G. Let S=\α1,…,αn\ be the set of simple roots of G relative to (B,T). Let λs be the one parameter subgroup of T dual to αs. In this paper, we give a criterion for Schubert varieties admitting semistable points for the λs-linearized line bundles L(χ) associated to every dominant character χ of T. If ωr is a minuscule fundamental weight and mωr∈ X(T), then we prove that there is a unique minimal dimensional Schubert variety X(ws,r) in G/P_S∖\αr\ such that X(ws,r)ss_λs(L(mωr))≠ ϕ. Further, we prove that if G=PSL(n,ℂ), and n\nmid rs, m=(n)/((rs,n)), and p=\lfloor(rs)/(n)\rfloor then the GIT quotient of the minimal dimensional Schubert variety X(ws,r) is isomorphic to the projective space ℙ(M(s-p, r-p)), where M(s-p, r-p) is the (s-p)× (r-p)-matrices with complex numbers as entries.