Investigating ASM varieties -- basic functions for alternating sign matrix ideals

Alternating sign matrix (ASM) varieties were introduced by Weigandt [Wei17]. ASM varieties generalize matrix Schubert varieties. This package contains functions for investigating ASM varieties and their defining ideals.

[CV20] Aldo Conca and Matteo Varbaro, Square-free Gröbner degenerations, Inventiones mathematicae, 221(3), pp.713-730.
[Wei17] Anna Weigandt, Prism tableaux for alternating sign matrix varieties, arXiv preprint 1708.07236.

The general method for defining the ideal of an ASM variety is schubertDeterminantalIdeal. The input can be an alternating sign matrix or a partial alternating sign matrix. This package contains functions for checking if a matrix is a partial ASM, extending a partial ASM to an ASM, and computing the rank matrix for an ASM.

i1 : A = matrix{{0,0,0},{0,1,0},{1,-1,0}} --Example 3.15 in [Wei17]

o1 = | 0 0  0 |
     | 0 1  0 |
     | 1 -1 0 |

              3       3
o1 : Matrix ZZ  <-- ZZ

i2 : isPartialASM A

o2 = true

i3 : A' = partialASMToASM A

o3 = | 0 0  0 1 0 |
     | 0 1  0 0 0 |
     | 1 -1 0 0 1 |
     | 0 1  0 0 0 |
     | 0 0  1 0 0 |

              5       5
o3 : Matrix ZZ  <-- ZZ

This package contains functions for investigating the rank matrix, the Rothe diagram, and the essential cells of the Rothe diagram of a partial ASM as defined by Fulton in [Ful92] and Weigandt in [Wei17].

i4 : rotheDiagram A

o4 = {(1, 1), (1, 2), (1, 3), (2, 1), (3, 2), (3, 3)}

o4 : List

i5 : essentialSet A

o5 = {(1, 3), (2, 1), (3, 3)}

o5 : List

i6 : rankTable A

o6 = | 0 0 0 |
     | 0 1 1 |
     | 1 1 1 |

              3       3
o6 : Matrix ZZ  <-- ZZ

i7 : netList fultonGens A

     +---------------------+
o7 = |z                    |
     | 1,1                 |
     +---------------------+
     |z                    |
     | 1,2                 |
     +---------------------+
     |z                    |
     | 1,3                 |
     +---------------------+
     |z                    |
     | 2,1                 |
     +---------------------+
     |- z   z    + z   z   |
     |   1,2 2,1    1,1 2,2|
     +---------------------+
     |- z   z    + z   z   |
     |   1,2 3,1    1,1 3,2|
     +---------------------+
     |- z   z    + z   z   |
     |   2,2 3,1    2,1 3,2|
     +---------------------+
     |- z   z    + z   z   |
     |   1,3 2,1    1,1 2,3|
     +---------------------+
     |- z   z    + z   z   |
     |   1,3 3,1    1,1 3,3|
     +---------------------+
     |- z   z    + z   z   |
     |   2,3 3,1    2,1 3,3|
     +---------------------+
     |- z   z    + z   z   |
     |   1,3 2,2    1,2 2,3|
     +---------------------+
     |- z   z    + z   z   |
     |   1,3 3,2    1,2 3,3|
     +---------------------+
     |- z   z    + z   z   |
     |   2,3 3,2    2,2 3,3|
     +---------------------+

The default presentation given by schubertDeterminantalIdeal is given by the Fulton generators of the ideal. In order to access a minimal generating set, use trim.

i8 : I = schubertDeterminantalIdeal A;

o8 : Ideal of QQ[z   ..z   ]
                  1,1   3,3

i9 : # (I_*)

o9 = 13

i10 : # ((trim I)_*)

o10 = 7

After creating an ASM ideal, the corresponding ASM is stored in the cache table.

i11 : peek I.cache

o11 = CacheTable{ASM => | 0 0  0 |                                                                                                                                                                                                                                                                                                                         }
                        | 0 1  0 |
                        | 1 -1 0 |
                 module => image | z_(1,1) z_(1,2) z_(1,3) z_(2,1) -z_(1,2)z_(2,1)+z_(1,1)z_(2,2) -z_(1,2)z_(3,1)+z_(1,1)z_(3,2) -z_(2,2)z_(3,1)+z_(2,1)z_(3,2) -z_(1,3)z_(2,1)+z_(1,1)z_(2,3) -z_(1,3)z_(3,1)+z_(1,1)z_(3,3) -z_(2,3)z_(3,1)+z_(2,1)z_(3,3) -z_(1,3)z_(2,2)+z_(1,2)z_(2,3) -z_(1,3)z_(3,2)+z_(1,2)z_(3,3) -z_(2,3)z_(3,2)+z_(2,2)z_(3,3) |

This package also contains methods for investigating or using initial ideals of ASM ideals.

i12 : antiDiagInit A

o12 = monomialIdeal (z   , z   , z   , z   , z   z   , z   z   , z   z   )
                      1,1   1,2   1,3   2,1   2,2 3,1   2,3 3,1   2,3 3,2

o12 : MonomialIdeal of QQ[z   ..z   ]
                           1,1   3,3

Every ASM ideal can be written as the intersection of Schubert determinantal ideals. The function schubertDecompose outputs the list of permutations that index the prime components of the ASM variety associated to A.

i13 : schubertDecompose I

o13 = {{4, 5, 1, 2, 3}, {4, 2, 5, 1, 3}}

o13 : List

Given a list of partial ASMs, this package also contains functions for intersecting and adding the ASM ideals associated to the list of ASMs or partial ASMs. Every sum of ASM ideals (equivalently, of ideals defined by partial ASMs) is again an ASM ideal. The function schubertAdd automatically stores the ASM associated to this new ideal in its cache table.

i14 : B = matrix{{0,1,0,0,0},{0,0,0,1,0},{1,-1,1,0,0},{0,0,0,0,1},{0,1,0,0,0}}

o14 = | 0 1  0 0 0 |
      | 0 0  0 1 0 |
      | 1 -1 1 0 0 |
      | 0 0  0 0 1 |
      | 0 1  0 0 0 |

               5       5
o14 : Matrix ZZ  <-- ZZ

i15 : L = {A, B} -- a list of 2 partial ASMs

o15 = {| 0 0  0 |, | 0 1  0 0 0 |}
       | 0 1  0 |  | 0 0  0 1 0 |
       | 1 -1 0 |  | 1 -1 1 0 0 |
                   | 0 0  0 0 1 |
                   | 0 1  0 0 0 |

o15 : List

i16 : J = schubertAdd L

o16 = ideal (z   , z   , z   , z   , - z   z    + z   z   , - z   z    +
              1,1   1,2   1,3   2,1     1,2 2,1    1,1 2,2     1,2 3,1  
      -----------------------------------------------------------------------
      z   z   , - z   z    + z   z   , - z   z    + z   z   , - z   z    +
       1,1 3,2     2,2 3,1    2,1 3,2     1,3 2,1    1,1 2,3     1,3 3,1  
      -----------------------------------------------------------------------
      z   z   , - z   z    + z   z   , - z   z    + z   z   , - z   z    +
       1,1 3,3     2,3 3,1    2,1 3,3     1,3 2,2    1,2 2,3     1,3 3,2  
      -----------------------------------------------------------------------
      z   z   , - z   z    + z   z   , - z   z    + z   z   , - z   z    +
       1,2 3,3     2,3 3,2    2,2 3,3     1,2 4,1    1,1 4,2     2,2 4,1  
      -----------------------------------------------------------------------
      z   z   , - z   z    + z   z   )
       2,1 4,2     3,2 4,1    3,1 4,2

o16 : Ideal of QQ[z   ..z   ]
                   1,1   5,5

i17 : peek J.cache

o17 = CacheTable{ASM => | 0 0  0 1 0 |     }
                        | 0 1  0 0 0 |
                        | 1 -1 0 0 1 |
                        | 0 0  1 0 0 |
                        | 0 1  0 0 0 |
                 rankTable => | 0 0 0 1 1 |
                              | 0 1 1 2 2 |
                              | 1 1 1 2 3 |
                              | 1 1 2 3 4 |
                              | 1 2 3 4 5 |

i18 : K = schubertIntersect L

o18 = ideal (z   , z   , z   z   , z   z   , z   z   , z   z    - z   z   ,
              2,1   1,1   1,2 4,1   2,2 3,1   1,2 3,1   1,3 2,2    1,2 2,3 
      -----------------------------------------------------------------------
      z   z   z    - z   z   z    - z   z   z   , z   z   z    -
       2,3 3,2 4,1    2,2 3,3 4,1    2,3 3,1 4,2   1,3 3,2 4,1  
      -----------------------------------------------------------------------
      z   z   z   )
       1,3 3,1 4,2

o18 : Ideal of QQ[z   ..z   ]
                   1,1   5,5

Although every ASM ideal is an intersection of Schubert determinantal ideals, many intersections of Schubert determinantal ideals are not ASM ideals. The function isASMIdeal determines whether or not an ideal is the ASM ideal of any ASM. If it is, the ASM is stored in its cache table and is accessible via getASM.

i19 : isASMIdeal K

o19 = false

i20 : K' = schubertIntersect {{3, 1, 2}, {2, 3, 1}}

o20 = ideal (z   , z   z   )
              1,1   1,2 2,1

o20 : Ideal of QQ[z   ..z   ]
                   1,1   3,3

i21 : isASMIdeal K'

o21 = true

i22 : getASM K'

o22 = | 0 1  0 |
      | 1 -1 1 |
      | 0 1  0 |

               3       3
o22 : Matrix ZZ  <-- ZZ

Additionally, this package facilitates the investigating homological invariants of ASM ideals efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].

i23 : time schubertRegularity B
 -- used 0.0569678s (cpu); 0.0256449s (thread); 0s (gc)

o23 = 1

i24 : time regularity comodule schubertDeterminantalIdeal B
 -- used 0.0120002s (cpu); 0.0124505s (thread); 0s (gc)

o24 = 1

Functions for investigating ASM varieties

isPartialASM(Matrix) -- whether a matrix is a partial alternating sign matrix
partialASMToASM(Matrix) -- extend a partial alternating sign matrix to an alternating sign matrix
rankTable(Matrix) -- compute a table of rank conditions that determines a Schubert determinantal ideal or, more generally, an alternating sign matrix ideal.
rankTableFromMatrix(Matrix) -- returns the minimal rank table from an arbitrary integer matrix
rankTableToASM(Matrix) -- to find the a partial ASM associated to a given rank table
isASMIdeal(Ideal) -- whether an ideal is an ASM ideal
isASM(Matrix) -- whether a matrix is an ASM
isASMUnion(List) -- whether the union of matrix Schubert varieties is an ASM variety
rotheDiagram(Matrix) -- find the Rothe diagram of a partial alternating sign matrix
augmentedRotheDiagram(Matrix) -- find the Rothe diagram of a partial alternating sign matrix together with the rank table determining the alternating sign matrix variety
essentialSet(Matrix) -- compute the essential set in the Rothe Diagram for a partial alternating sign matrix or a permutation.
augmentedEssentialSet(Matrix) -- find the essential set of a partial alternating sign matrix or a permutation together with the rank conditions determining the alternating sign matrix variety
schubertDeterminantalIdeal(Matrix) -- compute an alternating sign matrix ideal (for example, a Schubert determinantal ideal)
fultonGens(Matrix) -- compute the Fulton generators of an ASM ideal (for example, a Schubert determinantal ideal)
schubertIntersect(List) -- compute the intersection of ASM ideals
schubertAdd(List) -- compute the sum of ASM ideals
schubertRegularity(Matrix) -- compute the Castelnuovo-Mumford regularity of the quotient by a Schubert determinantal ideal or ASM ideal
schubertCodim(Matrix) -- compute the codimension (i.e., height) of a Schubert determinantal ideal or ASM ideal
permSetOfASM(Matrix) -- finds the permutation set of an alternating sign matrix
toOneLineNotation(Matrix) -- converts a permutation to one line notation