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# filtration -- the filtration matrices of the vector bundle

- Usage:
    `f = filtration E`
- Inputs:
  - E, a vector bundle on a toric variety (Klyachko's description)
- Outputs:
  - f, a hash table

## Description

For each ray of the fan there is a filtration matrix. If the bundle has rank $k$ then this is a one row matrix over ZZ with $k$ entries. This defines the filtration on the corresponding base matrix (see base) such that the $j$-th filtration is generated by all columns of the base matrix for which the entry in the same column of the filtration matrix is less or equal to $j$.

    i1 : E = tangentBundle hirzebruchFan 2

    o1 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 2

    o1 : ToricVectorBundleKlyachko

    i2 : filtration E

    o2 = HashTable{| -1 | => | -1 0 |}
                   | 2  |
                   | 0  | => | -1 0 |
                   | -1 |
                   | 0 | => | -1 0 |
                   | 1 |
                   | 1 | => | -1 0 |
                   | 0 |

    o2 : HashTable

So in this example for each ray the first column of the basis appears at -1 and the second at 0.

## See also

- addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description
- base -- the basis matrices for the rays
- isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle

## Ways to use filtration:

- `filtration(ToricVectorBundleKlyachko)`

## For the programmer

The object filtration is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2923:0.