findWeights -- finds the possible weight vectors for the maximal cones

Synopsis

• Usage:

  L = findWeights E

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
• Outputs:
  • L, a list

Description

The list L contains a list for each maximal cone $\sigma$ of the underlying fan. For each maximal cone $\sigma$ this list contains all matrices of possible weight vectors, that induce the filtrations on the rays of this cone (modulo permutations, but yet not all permutations). This means that for one of these matrices $M$ multiplied with the matrix $R$ of rays of this cone (the rays are the rows) gives the matrix of filtrations of these rays (where for each filtration the entries may be permuted).

i1 : E = tangentBundle projectiveSpaceFan 3

o1 = {dimension of the variety => 3 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 3

o1 : ToricVectorBundleKlyachko

i2 : findWeights E

o2 = {{| 1  1 1  |, | 1  1  1 |}, {| -1 0 0  |, | -1 0  0 |}, {| -1 0  0  |,
       | -1 0 0  |  | -1 0  0 |    | 1  1 1  |  | 1  1  1 |    | 0  -1 0  | 
       | 0  0 -1 |  | 0  -1 0 |    | 0  0 -1 |  | 0  -1 0 |    | 0  0  -1 | 
     ------------------------------------------------------------------------
     | -1 0  0  |}, {| -1 0  0 |, | -1 0 0  |}}
     | 0  0  -1 |    | 0  -1 0 |  | 0  0 -1 |
     | 0  -1 0  |    | 1  1  1 |  | 1  1 1  |

o2 : List

See also

• filtration -- the filtration matrices of the vector bundle
• existsDecomposition -- checks if a list of matrices of weight vectors for each maximal cone admits a decomposition
• isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle