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# wellformedBundleFiltrations -- produces the same toric vector bundle, but where the filtration steps are stored in matrices with ascending entries.

- Usage:
    `F = wellformedBundleFiltrations E`
- Inputs:
  - E, a vector bundle on a toric variety (Klyachko's description)
- Outputs:
  - F, a vector bundle on a toric variety (Klyachko's description)

## Description

A toric vector bundle in Klyachko's description as used in ToricVectorBundles is given by ascending filtrations. Unfortunately, the filtration steps are not always stored in an ascending way by certain methods, like dual(ToricVectorBundle) or tensor(ToricVectorBundle,ToricVectorBundle) (and maybe others). This may cause problems in some methods of PositivityToricBundles, for example in toricChernCharacter. This method ensures that the filtration steps are stored in an ascending way and therefore, that the methods of this package can be used safely.

    i1 : T = tangentBundle projectiveSpaceFan 2

    o1 = {dimension of the variety => 2 }
          number of affine charts => 3
          number of rays => 3
          rank of the vector bundle => 2

    o1 : ToricVectorBundleKlyachko

    i2 : E = T ** (dual T)

    o2 = {dimension of the variety => 2 }
          number of affine charts => 3
          number of rays => 3
          rank of the vector bundle => 4

    o2 : ToricVectorBundleKlyachko

    i3 : details E

    o3 = HashTable{| -1 | => (| 0 1  0 1  |, | 0 -1 1 0 |)}
                   | -1 |     | 1 -1 1 -1 |
                              | 0 1  0 0  |
                              | 1 -1 0 0  |
                   | 0 | => (| 0 0 0 1 |, | 0 -1 1 0 |)
                   | 1 |     | 0 0 1 0 |
                             | 0 1 0 0 |
                             | 1 0 0 0 |
                   | 1 | => (| 1 0 0 0 |, | 0 -1 1 0 |)
                   | 0 |     | 0 1 0 0 |
                             | 0 0 1 0 |
                             | 0 0 0 1 |

    o3 : HashTable

    i4 : toricChernCharacter E

    o4 = HashTable{| -1 0 | => {| 2  |, | -1 |, | -1 |, | 2  |}}
                   | -1 1 |     | -1 |  | 1  |  | 0  |  | -1 |
                   | 1 -1 | => {| -1 |, | 1  |, | 0  |, | -1 |}
                   | 0 -1 |     | 2  |  | -1 |  | -1 |  | 2  |
                   | 1 0 | => {| -1 |, | 0 |, | 1 |, | -1 |}
                   | 0 1 |     | -1 |  | 1 |  | 0 |  | -1 |

    o4 : HashTable

Note that the filtration steps of E are neither ascending nor descending. The method toricChernCharacter produces nonsense here: $\mathcal{O}_{\mathbb P^2}$ is a direct summand of E, so 0 has to appear in the toric Chern character.

    i5 : F = wellformedBundleFiltrations E

    o5 = {dimension of the variety => 2 }
          number of affine charts => 3
          number of rays => 3
          rank of the vector bundle => 4

    o5 : ToricVectorBundleKlyachko

    i6 : details F

    o6 = HashTable{| -1 | => (| 1  0 1  0 |, | -1 0 0 1 |)}
                   | -1 |     | -1 1 -1 1 |
                              | 1  0 0  0 |
                              | -1 1 0  0 |
                   | 0 | => (| 0 0 1 0 |, | -1 0 0 1 |)
                   | 1 |     | 0 0 0 1 |
                             | 1 0 0 0 |
                             | 0 1 0 0 |
                   | 1 | => (| 0 1 0 0 |, | -1 0 0 1 |)
                   | 0 |     | 1 0 0 0 |
                             | 0 0 0 1 |
                             | 0 0 1 0 |

    o6 : HashTable

    i7 : toricChernCharacter F

    o7 = HashTable{| -1 0 | => {0, | 0 |, | 0  |, 0}}
                   | -1 1 |        | 1 |  | -1 |
                   | 1 -1 | => {0, | 1 |, | -1 |, 0}
                   | 0 -1 |        | 0 |  | 0  |
                   | 1 0 | => {0, | -1 |, | 1  |, 0}
                   | 0 1 |        | 1  |  | -1 |

    o7 : HashTable

By passing to ascending filtration steps, the result becomes correct.

## Caveat

This method works for any toric reflexive sheaf.

## See also

- dual(ToricVectorBundle) -- the dual bundle of a toric vector bundle
- tensor(ToricVectorBundle,ToricVectorBundle) -- the tensor product of two toric vector bundles

## Ways to use wellformedBundleFiltrations:

- `wellformedBundleFiltrations(ToricVectorBundleKlyachko)`

## For the programmer

The object wellformedBundleFiltrations is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PositivityToricBundles.m2:1183:0.