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



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 :

For the programmer

The object isIntersectionOfSchubertDeterminantalIdeals is a method function.