TY - JOUR

T1 - Some conditions for descent of line bundles to GIT quotients (G/B × G/B × G/B)//G

AU - Bushek, Nathaniel

N1 - Publisher Copyright:
© 2017 Elsevier B.V.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2017/12

Y1 - 2017/12

N2 - We consider the descent of line bundles to GIT quotients of products of flag varieties. Let G be a simple, connected, algebraic group over C. We fix a Borel subgroup B and consider the diagonal action of G on the projective variety X=G/B×G/B×G/B. For any triple (λ,μ,ν) of dominant regular characters there is a G-equivariant line bundle L on X. Then, L is said to descend to the GIT quotient π:Xss(L)→X(L)//G if there exists a line bundle Lˆ on X(L)//G such that L|Xss(L)≅π⁎Lˆ. Let Q be the root lattice, Λ the weight lattice, and d the least common multiple of the coefficients of the highest root θ of the Lie algebra g of G written in terms of simple roots. We show that L descends if λ,μ,ν∈dΛ and λ+μ+ν∈Γ, where Γ is a fixed sublattice of Q depending only on the type of g. Moreover, L never descends if λ+μ+ν∉Q.

AB - We consider the descent of line bundles to GIT quotients of products of flag varieties. Let G be a simple, connected, algebraic group over C. We fix a Borel subgroup B and consider the diagonal action of G on the projective variety X=G/B×G/B×G/B. For any triple (λ,μ,ν) of dominant regular characters there is a G-equivariant line bundle L on X. Then, L is said to descend to the GIT quotient π:Xss(L)→X(L)//G if there exists a line bundle Lˆ on X(L)//G such that L|Xss(L)≅π⁎Lˆ. Let Q be the root lattice, Λ the weight lattice, and d the least common multiple of the coefficients of the highest root θ of the Lie algebra g of G written in terms of simple roots. We show that L descends if λ,μ,ν∈dΛ and λ+μ+ν∈Γ, where Γ is a fixed sublattice of Q depending only on the type of g. Moreover, L never descends if λ+μ+ν∉Q.

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U2 - 10.1016/j.difgeo.2017.09.002

DO - 10.1016/j.difgeo.2017.09.002

M3 - Article

AN - SCOPUS:85029529315

VL - 55

SP - 2

EP - 12

JO - Differential Geometry and its Applications

JF - Differential Geometry and its Applications

SN - 0926-2245

ER -