next | previous | forward | backward | up | index | toc

# isSchubertCM -- whether an ASM variety is Cohen-Macaulay

## Synopsis

• Usage:
isSchubertCM A
isSchubertCM w
• Inputs:
• A, , or w is a List
• Outputs:

## 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 .