diagLexInitSE -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from SE corner

Synopsis

Usage:

diagLexInitSE w

diagLexInitSE A
Inputs:
- w, a list, or A is a Matrix
Optional inputs:
- CoefficientRing => a ring, default value QQ
- Variable => a symbol, default value z
Outputs:
- a monomial ideal,

Description

Given a partial alternating sign matrix or a permutation in 1-line notation, return the diagonal initial ideal of the corresponding ASM ideal or Schubert determinantal ideal with respect to lexicographic order, where the variables are ordered reading from right to left and bottom-to-top (starting in the southeast corner).

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

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

o1 = monomialIdeal(z   z   )
                    2,2 1,1

                       ZZ
o1 : MonomialIdeal of ----[z   , z   , z   , z   , z   , z   , z   , z   , z   ]
                      3001  3,3   3,2   3,1   2,3   2,2   2,1   1,3   1,2   1,1

i2 : diagLexInitSE(matrix{{0,0,0,1},{0,1,0,0},{1,-1,1,0},{0,1,0,0}}, Variable => c)

o2 = monomialIdeal (c   c   , c   , c   , c   , c   )
                     3,1 2,2   2,1   1,3   1,2   1,1

o2 : MonomialIdeal of QQ[c   , c   , c   , c   , c   , c   , c   , c   , c   , c   , c   , c   , c   , c   , c   , c   ]
                          4,4   4,3   4,2   4,1   3,4   3,3   3,2   3,1   2,4   2,3   2,2   2,1   1,4   1,3   1,2   1,1

Ways to use diagLexInitSE:

diagLexInitSE(List)
diagLexInitSE(Matrix)

For the programmer

The object diagLexInitSE is a method function with options.