MultigradedBettiTally -- the class of all multigraded Betti tallies

Description

A multigraded Betti tally is a special type of BettiTally that is printed as a diagram of the multigraded Betti numbers. The class was created so that the method multigraded(BettiTally) could return something that both prints nicely and from which information could be extracted. The content of a multigraded Betti tally is identical to the Betti tally from which it was constructed.

i1 : B = new BettiTally from {(0, {0, 0}, 0) => 1, (1, {0, 2}, 2) => 1, (1, {1, 1}, 2) => 2, (1, {2, 0}, 2) => 1, (2, {1, 2}, 3) => 2, (2, {2, 1}, 3) => 2, (3, {2, 2}, 4) => 1}

            0 1 2 3
o1 = total: 1 4 4 1
         0: 1 . . .
         1: . 4 4 1

o1 : BettiTally

i2 : B = multigraded B

        0             1               2       3
o2 = 0: 1             .               .       .
     2: . a^2+2*a*b+b^2               .       .
     3: .             . 2*a^2*b+2*a*b^2       .
     4: .             .               . a^2*b^2

o2 : MultigradedBettiTally

i3 : peek B

o3 = MultigradedBettiTally{(0, {0, 0}, 0) => 1}
                           (1, {0, 2}, 2) => 1
                           (1, {1, 1}, 2) => 2
                           (1, {2, 0}, 2) => 1
                           (2, {1, 2}, 3) => 2
                           (2, {2, 1}, 3) => 2
                           (3, {2, 2}, 4) => 1

By default the data is presented as a table of polynomials where each column corresponds to a given homological degree appearing as the top entry and each monomial in the other entries represents the multidegree of a given summand.

When compactMatrixForm is set to false, the entries represent the multidegree of summands ordered by the total weight. The number of summands corresponding to a given multidegree appears to the left of the multidegree.

i4 : compactMatrixForm = false

o4 = false

i5 : B

            0        1        2        3
o5 = 1:{0, 0} 1:{0, 2} 2:{1, 2} 1:{2, 2}
              1:{2, 0} 2:{2, 1}         
              2:{1, 1}                  

o5 : MultigradedBettiTally

For convenience, various operations on BettiTally such as direct sum (++), tensor product (**), pdim and degree shifting (numbers in brackets or parentheses) are extended to work with multigraded Betti tables. These operations mimic the corresponding operations on chain complexes.

i6 : compactMatrixForm = true

o6 = true

i7 : B(-1,-1)

          0                     1                   2       3
o7 = 0: a*b                     .                   .       .
     2:   . a^3*b+2*a^2*b^2+a*b^3                   .       .
     3:   .                     . 2*a^3*b^2+2*a^2*b^3       .
     4:   .                     .                   . a^3*b^3

o7 : MultigradedBettiTally

i8 : B[1]

        -1             0               1       2
o8 = 0:  1             .               .       .
     2:  . a^2+2*a*b+b^2               .       .
     3:  .             . 2*a^2*b+2*a*b^2       .
     4:  .             .               . a^2*b^2

o8 : MultigradedBettiTally

i9 : B[1] ++ B

        -1             0               1               2       3
o9 = 0:  1             1               .               .       .
     2:  . a^2+2*a*b+b^2   a^2+2*a*b+b^2               .       .
     3:  .             . 2*a^2*b+2*a*b^2 2*a^2*b+2*a*b^2       .
     4:  .             .               .         a^2*b^2 a^2*b^2

o9 : MultigradedBettiTally

i10 : B ** B

         0                 1                                 2                                     3                              4                   5       6
o10 = 0: 1                 .                                 .                                     .                              .                   .       .
      2: . 2*a^2+4*a*b+2*b^2                                 .                                     .                              .                   .       .
      3: .                 .                   4*a^2*b+4*a*b^2                                     .                              .                   .       .
      4: .                 . a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4                             2*a^2*b^2                              .                   .       .
      5: .                 .                                 . 4*a^4*b+12*a^3*b^2+12*a^2*b^3+4*a*b^4                              .                   .       .
      6: .                 .                                 .                                     . 6*a^4*b^2+12*a^3*b^3+6*a^2*b^4                   .       .
      7: .                 .                                 .                                     .                              . 4*a^4*b^3+4*a^3*b^4       .
      8: .                 .                                 .                                     .                              .                   . a^4*b^4

o10 : MultigradedBettiTally

i11 : compactMatrixForm = false

o11 = false

i12 : B ** B

             0        1        2         3         4        5        6
o12 = 1:{0, 0} 2:{2, 0} 4:{3, 1} 12:{3, 2}  6:{4, 2} 4:{4, 3} 1:{4, 4}
               4:{1, 1} 4:{2, 1}  2:{2, 2} 12:{3, 3} 4:{3, 4}         
               2:{0, 2} 6:{2, 2} 12:{2, 3}  6:{2, 4}                  
                        4:{1, 2}  4:{1, 4}                            
                        4:{1, 3}  4:{4, 1}                            
                        1:{4, 0}                                      
                        1:{0, 4}                                      

o12 : MultigradedBettiTally

Contributors

This feature was implemented by Mahrud Sayrafi based on earlier work by Gregory G. Smith.

Functions and methods returning a multigraded Betti tally :

multigraded(BettiTally) -- see multigraded -- convert a Betti tally into a multigraded Betti tally

Methods that use a multigraded Betti tally :

MultigradedBettiTally Sequence
MultigradedBettiTally List (missing documentation)

For the programmer

The object MultigradedBettiTally is a type, with ancestor classes BettiTally < VirtualTally < HashTable < Thing.