If
E is in Klyachko's description then the data in
E defines an equivariant toric vector on the toric variety if and only if for each maximal cone exists a decomposition into torus eigenspaces of the bundle. See
Sam Payne's Moduli of toric vector bundles, Compositio Math. 144, 2008. Section 2.3. This uses the two functions
findWeights and
existsDecomposition.
E = toricVectorBundle(2,pp1ProductFan 2) |
E = addBase(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}}) |
isVectorBundle E |
F = toricVectorBundle(1,normalFan crossPolytope 3) |
F = addFiltration(F,apply({2,1,1,2,2,1,1,2}, i -> matrix {{i}})) |
isVectorBundle F |
If
E is in Kaneyama's description then data in
E defines an equivariant toric vector bundle on the toric variety if and only if it satisfies the regularity and the cocycle condition (See
cocycleCheck and
regCheck).
E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama") |
isVectorBundle E |
E = addBaseChange(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}}) |
isVectorBundle E |