Journal of the Ramanujan Mathematical Society
Volume 39, Issue 4, December 2024 pp. 389–408.
GIT quotient of Schubert varieties modulo one dimensional torus
Authors:
Arkadev Ghosh and S. S. Kannan
Author institution:Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Siruseri, Kelambakkam, 603 103, India.
Summary:
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 = N{G}(T)/T be the Weyl group of G. Let S = {α{1}, … , α{n}} be the set
of simple roots of G relative to (B, T). Let λ{s} be the one parameter subgroup of T dual to α{s}. In this paper, we give a
criterion for Schubert varieties admitting semistable points for the λ{s}-linearized line bundles L(χ) associated to every
dominant character χ of T. If ω{r} 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(w{s,r}){ss}
{λs} (L(mωr)) ≠ ϕ. Further, we prove
that if G = PS L(n,ℂ), and n ł rs, m = {n}/{(rs,n)},
and p =
⌊rs/n⌋
then λs\\X(ws,r)ss
λs (L(mω{r})) ≃ (ℙ(M(s − p, r − p)),𝕆(a)) for
some a ∈ ℕ.
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