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

## Synopsis

• Usage:
antiDiagInit w
antiDiagInit A
• Inputs:
• Optional inputs:
• CoefficientRing => a ring, default value QQ
• Variable => , default value z
• Outputs:

## Description

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

## Ways to use antiDiagInit :

• antiDiagInit(List)
• antiDiagInit(Matrix)

## For the programmer

The object antiDiagInit is .