isSchubertCM -- whether an ASM variety is Cohen-Macaulay

Synopsis

Usage:

isSchubertCM A

isSchubertCM w
Inputs:
- A, a matrix, or w is a List
Outputs:
- a Boolean value,

Description

Given an alternating sign matrix $A$ (resp. permutation $w$), checks whether $R/I_A$ (resp. $R/I_w$) is Cohen-Macaulay. If the input is a permutation $w$, the output is always true since $I_w$ is a Schubert determinantal ideal, and a theorem of Fulton says $R/I_w$ is always Cohen-Macaulay.

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

o1 = | 0 0  0 1 |
     | 0 1  0 0 |
     | 1 -1 1 0 |
     | 0 1  0 0 |

              4       4
o1 : Matrix ZZ  <-- ZZ

i2 : isSchubertCM A

o2 = true

i3 : w = {1,3,2}

o3 = {1, 3, 2}

o3 : List

i4 : isSchubertCM w
We know from a theorem of Fulton that the quotient by any Schubert determinantal ideal is actually Cohen--Macaulay!

o4 = true

Ways to use isSchubertCM :

isSchubertCM(List)
isSchubertCM(Matrix)

For the programmer

The object isSchubertCM is a method function.