next | previous | forward | backward | up | index | toc

# Initial ideals of ASM ideals -- basic functions for investigating initial ideals of ASM varieties

By work of Knutson and Miller [KM05], Weigandt [Wei17], and Knutson [Knu09] the Fulton generators of an ASM ideal form a Gröbner basis with respect to any antidiagonal term order. However, Gröbner bases for ASM ideals with respect to other term orders, including diagonal ones, remain largely mysterious.

Given a permutation or a partial ASM, one may compute its antidiagonal initial ideal. By [KM05] and [Wei17] or [Knu09], the Fulton generators form a Gröbner basis for any ASM ideal with respect to any antidiagonal term order.

 i1 : w = {2,4,5,1,3}; i2 : I = schubertDeterminantalIdeal w; o2 : Ideal of QQ[z ..z ] 1,1 5,5 i3 : inI = antiDiagInit w; o3 : MonomialIdeal of QQ[z ..z ] 1,1 5,5 i4 : (netList sort inI_*, netList sort (trim I)_*) +--------+ +-------------------+ o4 = (|z |, |z |) | 3,1 | | 3,1 | +--------+ +-------------------+ |z | |z | | 2,1 | | 2,1 | +--------+ +-------------------+ |z | |z | | 1,1 | | 1,1 | +--------+ +-------------------+ |z z | |z z - z z | | 2,3 3,2| | 2,3 3,2 2,2 3,3| +--------+ +-------------------+ |z z | |z z - z z | | 1,3 3,2| | 1,3 3,2 1,2 3,3| +--------+ +-------------------+ |z z | |z z - z z | | 1,3 2,2| | 1,3 2,2 1,2 2,3| +--------+ +-------------------+ o4 : Sequence

By work of Conca and Varbaro [CV21], we know that the extremal Betti numbers (which encode regularity and projective dimension) of an ASM ideal and its antidiagonal initial ideal must coincide because the antidiagonal initial ideal is squarefree. For this running example, all of the Betti numbers coincide (not just the extremal ones).

 i5 : (betti res I, betti res inI) 0 1 2 3 4 5 0 1 2 3 4 5 o5 = (total: 1 6 14 16 9 2, total: 1 6 14 16 9 2) 0: 1 3 3 1 . . 0: 1 3 3 1 . . 1: . 3 11 15 9 2 1: . 3 11 15 9 2 o5 : Sequence

By work of Knutson and Miller [KM05], building off of work of Bergeron and Billey [BB93], the prime components of the antidiagonal initial ideal of a Schubert determinantal ideal for a permutation w are in bijection with the pipe dreams associated to w. See [BB93] for a detailed description of pipe dreams, there called RC-graphs.

 i6 : # pipeDreams w == # (decompose inI) o6 = true

To read off an associated prime of the antidiagonal initial ideal from a pipe dream, one reads off the + tiles from the grid. When there is a + in location $(i,j)$, then $z_{i,j}$ is a generator of the associated prime in question.

 i7 : (pipeDreams w)_0 o7 = +/+// +/+// +//// ///// ///// o7 : PipeDream i8 : (decompose inI)_0 o8 = monomialIdeal (z , z , z , z , z ) 1,1 1,3 2,1 2,3 3,1 o8 : MonomialIdeal of QQ[z ..z ] 1,1 5,5

Initial ideals of Schubert determinantal ideals and ASM ideals under diagonal term orders are must less well understood. They have been studied in [KMY09],[HPW22], [Kle23], and [KW21]. This package provides functionality for investigating three diagonal term orders: One which uses lex and orders the variables reading right-to-left across rows starting from the southeast corner diagLexInitSE, one which uses lex and orders the variables reading left-to-right across rows starting from the northwest corner diagLexInitNW, and one which uses revlex and orders the variables smallest to largest reading left-to-right across rows starting from the southwest corner diagRevLexInit.

Two diagonal term orders may give two distinct initial ideals.

 i9 : v = {2,1,4,3,6,5} o9 = {2, 1, 4, 3, 6, 5} o9 : List i10 : diagLexInitSE v o10 = monomialIdeal (z z z z z z , z z z z z , 5,5 4,3 3,4 3,2 2,1 1,3 5,5 4,3 3,4 2,1 1,2 ----------------------------------------------------------------------- 2 z z z , z z z z z z , z ) 3,3 2,1 1,2 5,5 4,3 3,4 3,1 2,3 1,2 1,1 o10 : MonomialIdeal of QQ[z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z ] 6,6 6,5 6,4 6,3 6,2 6,1 5,6 5,5 5,4 5,3 5,2 5,1 4,6 4,5 4,4 4,3 4,2 4,1 3,6 3,5 3,4 3,3 3,2 3,1 2,6 2,5 2,4 2,3 2,2 2,1 1,6 1,5 1,4 1,3 1,2 1,1 i11 : netList (decompose oo) +--------------------------------+ o11 = |monomialIdeal (z , z , z )| | 5,5 3,3 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 4,3 3,3 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 3,4 3,3 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 5,5 2,1 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 4,3 2,1 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 3,4 2,1 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 3,1 2,1 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 2,3 2,1 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 5,5 1,2 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 4,3 1,2 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 3,4 1,2 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 3,2 1,2 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 2,1 1,2 1,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,3 1,2 1,1 | +--------------------------------+ i12 : diagLexInitNW v o12 = monomialIdeal (z , z z z , z z z z z , 1,1 1,2 2,1 3,3 1,2 2,1 3,4 4,3 5,5 ----------------------------------------------------------------------- 2 z z z z z z , z z z z z z ) 1,2 2,3 3,1 3,4 4,3 5,5 1,3 2,1 3,2 3,4 4,3 5,5 o12 : MonomialIdeal of QQ[z ..z ] 1,1 6,6 i13 : netList (decompose oo) +--------------------------------+ o13 = |monomialIdeal (z , z , z )| | 1,1 1,2 1,3 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 1,2 2,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 2,1 2,3 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 2,1 3,1 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 1,2 3,2 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 1,2 3,4 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 2,1 3,4 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 3,3 3,4 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 1,2 4,3 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 2,1 4,3 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 3,3 4,3 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 1,2 5,5 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 2,1 5,5 | +--------------------------------+ |monomialIdeal (z , z , z )| | 1,1 3,3 5,5 | +--------------------------------+

In this example, diagRevLexInit and diagLexInitSE give the same initial ideal. It is unknown if this is the case in general.

 i14 : diagRevLexInit v o14 = monomialIdeal (z , z z z , z z z z z , 1,1 1,2 2,1 3,3 1,2 2,1 3,4 4,3 5,5 ----------------------------------------------------------------------- 2 z z z z z z , z z z z z z ) 1,3 2,1 3,4 3,2 4,3 5,5 1,2 2,3 3,4 3,1 4,3 5,5 o14 : MonomialIdeal of QQ[z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z , z ] 1,6 1,5 1,4 1,3 1,2 1,1 2,6 2,5 2,4 2,3 2,2 2,1 3,6 3,5 3,4 3,3 3,2 3,1 4,6 4,5 4,4 4,3 4,2 4,1 5,6 5,5 5,4 5,3 5,2 5,1 6,6 6,5 6,4 6,3 6,2 6,1

## Functions for studying initial ideals of ASM ideals

• antiDiagInit(Matrix) -- compute the (unique) antidiagonal initial ideal of an ASM ideal
• diagLexInitSE(Matrix) -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from SE corner
• diagLexInitNW(Matrix) -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from NW corner
• diagRevLexInit(Matrix) -- Diagonal initial ideal of an ASM ideal with respect to revlex, ordering variables from NW corner
• pipeDreams(List) -- computes the set of reduced pipe dreams corresponding to a permutation