schubertDecompose -- finds the decomposition of an ASM ideal into Schubert determinantal ideals

Synopsis

Usage:

schubertDecompose I
Inputs:
- I, an ideal, or A is a Matrix
Outputs:
- a list,

Description

Given an ASM ideal $I_A$, it can be decomposed into Schubert determinantal ideals as $I_A = I_{w_1} \cap ... \cap I_{w_k}$, where the $w_i$ are permutations. As output, each element in the list is the permutation associated to a prime component in the Schubert decomposition of the antidiagonal initial ideal of $I$.

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 : netList schubertDecompose J

     +-+-+-+-+-+
o3 = |4|1|2|5|3|
     +-+-+-+-+-+
     |3|1|4|5|2|
     +-+-+-+-+-+

If the ASM ideal for an ASM $A$ has not het been computed, one may also give the ASM $A$ as input.

i4 : A = matrix{{0,0,0,1},{0,1,0,0},{1,-1,1,0},{0,1,0,0}};

              4       4
o4 : Matrix ZZ  <-- ZZ

i5 : netList schubertDecompose A

     +-+-+-+-+
o5 = |4|3|1|2|
     +-+-+-+-+
     |4|2|3|1|
     +-+-+-+-+

Ways to use schubertDecompose:

schubertDecompose(Ideal)
schubertDecompose(Matrix)

For the programmer

The object schubertDecompose is a method function.