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# isIntersectionOfSchubertDeterminantalIdeals -- whether an ideal is the intersection of Schubert determinantal ideals

- Usage:
    `isIntersectionOfSchubertDeterminantalIdeals I`
- Inputs:
  - I, an ideal,
- Outputs:
  - a Boolean value,

## Description

Checks if the input ideal $I$ can be written as $I = I_{w_1} \cap ... \cap I_{w_k}$, where each $I_{w_i}$ is a Schubert determinantal ideal.

This function computes the antidiagonal initial ideal of $I$ (using the default term order in Macaulay2, which is antidiagonal), finds the primes in the decomposition of $I$, reads a permutation from each such prime, and checks if $I$ is the intersection of the Schubert determinantal ideals of those permutations.

The following theorems combine to guarantee that, if $I$ can be written as the intersection of Schubert determinantal ideals, it is exactly the intersection of the Schubert determinantal ideals found by the algorithm described above.

- [KM05, Theorem B]: Knutson and Miller, Gröbner geometry of Schubert polynomials (see arXiv:0110058).

- [Wei, Proposition 5.4]: Weigandt, Prism tableaux for alternating sign matrix varieties (see arXiv:1708.07236).

- [BB93, Theorem 3.7]: Bergeron and Billey, RC-graphs and Schubert polynomials.

    i1 : A = matrix{{0,0,1,0,0},{1,0,0,0,0},{0,1,-1,1,0},{0,0,0,0,1},{0,0,1,0,0}};

                  5       5
    o1 : Matrix ZZ  <-- ZZ

    i2 : J = schubertDeterminantalIdeal A;

    o2 : Ideal of QQ[z   ..z   ]
                      1,1   5,5

    i3 : isIntersectionOfSchubertDeterminantalIdeals J

    o3 = true

## Ways to use isIntersectionOfSchubertDeterminantalIdeals:

- `isIntersectionOfSchubertDeterminantalIdeals(Ideal)`

## For the programmer

The object isIntersectionOfSchubertDeterminantalIdeals is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixSchubert/MatrixSchubertConstructionsDOC.m2:1067:0.