cohenMacaulayASMsList -- lists all Cohen-Macaulay ASMs of a fixed size which are not permutation matrices

Synopsis

Usage:

cohenMacaulayASMsList n
Inputs:
- n, an integer,

Description

For $1 \leq n \leq 6$, this function lists all $n\times n$ alternating sign matrices $A$ which are not permutation matrices such that the corresponding ASM variety is Cohen-Macaulay. By a theorem of Fulton [Ful92], permutation matrices always have Cohen-Macaulay Schubert determinantal ideals.

[Ful92] William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas , Duke Math J. 65 (1992): 381-420.

i1 : cohenMacaulayASMsList(4)

o1 = {| 0 1  0 0 |, | 1 0 0  0 |, | 0 1  0  0 |, | 0 0 1  0 |, | 0 1 0  0 |,
      | 1 -1 1 0 |  | 0 0 1  0 |  | 1 -1 1  0 |  | 1 0 0  0 |  | 0 0 1  0 | 
      | 0 1  0 0 |  | 0 1 -1 1 |  | 0 1  -1 1 |  | 0 1 -1 1 |  | 1 0 -1 1 | 
      | 0 0  0 1 |  | 0 0 1  0 |  | 0 0  1  0 |  | 0 0 1  0 |  | 0 0 1  0 | 
     ------------------------------------------------------------------------
     | 0 0 1  0 |, | 0 1  0 0 |, | 0 0 1  0 |, | 0 1  0 0 |, | 0 0  1 0 |, |
     | 0 1 0  0 |  | 1 -1 0 1 |  | 0 1 -1 1 |  | 1 -1 1 0 |  | 0 1  0 0 |  |
     | 1 0 -1 1 |  | 0 1  0 0 |  | 1 0 0  0 |  | 0 0  0 1 |  | 1 -1 0 1 |  |
     | 0 0 1  0 |  | 0 0  1 0 |  | 0 0 1  0 |  | 0 1  0 0 |  | 0 1  0 0 |  |
     ------------------------------------------------------------------------
     0 1  0 0 |, | 0 0 1  0 |, | 0 1  0 0 |, | 0 0  0 1 |, | 0 0 1  0 |}
     1 -1 0 1 |  | 1 0 -1 1 |  | 0 0  0 1 |  | 0 1  0 0 |  | 0 1 -1 1 |
     0 0  1 0 |  | 0 0 1  0 |  | 1 -1 1 0 |  | 1 -1 1 0 |  | 0 0 1  0 |
     0 1  0 0 |  | 0 1 0  0 |  | 0 1  0 0 |  | 0 1  0 0 |  | 1 0 0  0 |

o1 : List

Ways to use cohenMacaulayASMsList :

cohenMacaulayASMsList(ZZ)

For the programmer

The object cohenMacaulayASMsList is a method function.