antiDiagInit -- compute the (unique) antidiagonal initial ideal of an ASM ideal

Let $Z = (z_{i,j})$ be a generic matrix and $R=k[Z]$ a polynomial ring in the entries of $Z$ over the field $k$. We call a term order on $R$ antidiagonal if the lead term of the determinant of each submatrix $Z'$ of $Z$ is the product of terms along the antidiagonal of $Z'$.

This method computes the antidiagonal initial ideal of an ASM ideal by directly forming the ideal of the lead terms of the Fulton generators.

• [KM05]: Knutson and Miller, Gröbner geometry of Schubert polynomials (see arXiv:0110058).

tells us that the Fulton generators of each Schubert determinantal ideal form a Gröbner basis. For an extension to ASM ideals, see

• [KW]: Klein and Weigant, Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials (see arXiv:2108.08370).

• [Wei]: Weigandt, Prism tableaux for alternating sign matrix varieties (see arXiv:1708.07236).

• [Knu]: Knutson, Frobenius splitting, point-counting, and degeneration (see arXiv:0911.4941).

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

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

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

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

i2 : antiDiagInit(matrix{{0,0,0,1},{0,1,0,0},{1,-1,1,0},{0,1,0,0}}, Variable => t)

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

o2 : MonomialIdeal of QQ[t   ..t   ]
                          1,1   4,4