schubertDeterminantalIdeal -- compute an alternating sign matrix ideal (for example, a Schubert determinantal ideal)

Synopsis

Usage:

schubertDeterminantalIdeal w

schubertDeterminantalIdeal A
Inputs:
- w, a list, or A is a Matrix
Optional inputs:
- CoefficientRing => ..., default value QQ
- Variable => ..., default value z
Outputs:
- an ideal,

Description

Given a permutation in 1-line notation or, more generally, a partial alternating sign matrix, outputs the associated alternating sign matrix ideal (which is called a Schubert determinantal ideal in the case of a permutation). (The convention throughout this package is that the permutation matrix of a permutation $w$ has 1's in positions $(i,w(i))$.)

This function computes over the coefficient field of rational numbers unless an alternative is specified.

i1 : schubertDeterminantalIdeal({1,3,2},CoefficientRing=>ZZ/3001)

o1 = ideal(- z   z    + z   z   )
              1,2 2,1    1,1 2,2

               ZZ
o1 : Ideal of ----[z   ..z   ]
              3001  1,1   3,3

i2 : schubertDeterminantalIdeal(matrix{{0,0,0,1},{0,1,0,0},{1,-1,1,0},{0,1,0,0}}, Variable => y)

o2 = ideal (y   , y   , y   , y   , - y   y    + y   y   , - y   y    +
             1,1   1,2   1,3   2,1     1,2 2,1    1,1 2,2     1,2 3,1  
     ------------------------------------------------------------------------
     y   y   , - y   y    + y   y   )
      1,1 3,2     2,2 3,1    2,1 3,2

o2 : Ideal of QQ[y   ..y   ]
                  1,1   4,4

Ways to use schubertDeterminantalIdeal:

schubertDeterminantalIdeal(List)
schubertDeterminantalIdeal(Matrix)

For the programmer

The object schubertDeterminantalIdeal is a method function with options.