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ZZdFactorizationMap ^ ZZ -- the n-fold composition

Description

A ZZ/d-graded factorization map $f : C \to C$ can be composed with itself. This method produces these new maps of ZZ/d-graded factorizations.

The differential on a ZZ/d-graded factorization should compose with itself d times to give a scalar multiple of the identity map.

i1 : S = ZZ/101[a..c];
 
i2 : C = koszulMF({a,b,c}, a^3 + b^3 + c^3)

      4      4      4
o2 = S  <-- S  <-- S
                    
     0      1      0

o2 : ZZdFactorization
 
i3 : f = dd^C

          4                        4
o3 = 1 : S  <-------------------- S  : 0
               | c2 0  b  a   |
               | 0  c2 a2 -b2 |
               | b2 a  -c 0   |
               | a2 -b 0  -c  |

          4                         4
     0 : S  <--------------------- S  : 1
               | c  0  b   a   |
               | 0  c  a2  -b2 |
               | b2 a  -c2 0   |
               | a2 -b 0   -c2 |

o3 : ZZdFactorizationMap
 
i4 : f^2

          4                                               4
o4 = 0 : S  <------------------------------------------- S  : 0
               | a3+b3+c3 0        0        0        |
               | 0        a3+b3+c3 0        0        |
               | 0        0        a3+b3+c3 0        |
               | 0        0        0        a3+b3+c3 |

          4                                               4
     1 : S  <------------------------------------------- S  : 1
               | a3+b3+c3 0        0        0        |
               | 0        a3+b3+c3 0        0        |
               | 0        0        a3+b3+c3 0        |
               | 0        0        0        a3+b3+c3 |

o4 : ZZdFactorizationMap
 
i5 : assert(source f == target f)
 
i6 : assert(degree f == -1)
 
i7 : assert(degree f^2 == -2)
 
i8 : K' = linearMF(a^3+b^3+c^3, t)

     /   S[t]   \9     /   S[t]   \9     /   S[t]   \9     /   S[t]   \9
o8 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
     | 2        |      | 2        |      | 2        |      | 2        |
     \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/
                                                            
     0                 1                 2                 0

o8 : ZZdFactorization
 
i9 : g = K'.dd

         /   S[t]   \9                                                          /   S[t]   \9
o9 = 2 : |----------|  <------------------------------------------------------- |----------|  : 0
         | 2        |     {0, 3} | c 0  0     b  a  0     0     0     0     |   | 2        |
         \t  + t + 1/     {0, 3} | 0 c  0     0  bt a     0     0     0     |   \t  + t + 1/
                          {0, 3} | 0 0  c     a  0  -bt-b 0     0     0     |
                          {0, 3} | 0 0  0     ct 0  0     b     a     0     |
                          {0, 3} | 0 0  0     0  ct 0     0     bt    a     |
                          {0, 3} | 0 0  0     0  0  ct    a     0     -bt-b |
                          {0, 3} | b a  0     0  0  0     -ct-c 0     0     |
                          {0, 3} | 0 bt a     0  0  0     0     -ct-c 0     |
                          {0, 3} | a 0  -bt-b 0  0  0     0     0     -ct-c |

         /   S[t]   \9                                                          /   S[t]   \9
     0 : |----------|  <------------------------------------------------------- |----------|  : 1
         | 2        |     {0, 3} | c 0  0     b  a  0     0     0     0     |   | 2        |
         \t  + t + 1/     {0, 3} | 0 c  0     0  bt a     0     0     0     |   \t  + t + 1/
                          {0, 3} | 0 0  c     a  0  -bt-b 0     0     0     |
                          {0, 3} | 0 0  0     ct 0  0     b     a     0     |
                          {0, 3} | 0 0  0     0  ct 0     0     bt    a     |
                          {0, 3} | 0 0  0     0  0  ct    a     0     -bt-b |
                          {0, 3} | b a  0     0  0  0     -ct-c 0     0     |
                          {0, 3} | 0 bt a     0  0  0     0     -ct-c 0     |
                          {0, 3} | a 0  -bt-b 0  0  0     0     0     -ct-c |

         /   S[t]   \9                                                          /   S[t]   \9
     1 : |----------|  <------------------------------------------------------- |----------|  : 2
         | 2        |     {0, 3} | c 0  0     b  a  0     0     0     0     |   | 2        |
         \t  + t + 1/     {0, 3} | 0 c  0     0  bt a     0     0     0     |   \t  + t + 1/
                          {0, 3} | 0 0  c     a  0  -bt-b 0     0     0     |
                          {0, 3} | 0 0  0     ct 0  0     b     a     0     |
                          {0, 3} | 0 0  0     0  ct 0     0     bt    a     |
                          {0, 3} | 0 0  0     0  0  ct    a     0     -bt-b |
                          {0, 3} | b a  0     0  0  0     -ct-c 0     0     |
                          {0, 3} | 0 bt a     0  0  0     0     -ct-c 0     |
                          {0, 3} | a 0  -bt-b 0  0  0     0     0     -ct-c |

o9 : ZZdFactorizationMap
 
i10 : g^3

          /   S[t]   \9                                                                                                  /   S[t]   \9
o10 = 0 : |----------|  <----------------------------------------------------------------------------------------------- |----------|  : 0
          | 2        |     {0, 3} | a3+b3+c3 0        0        0        0        0        0        0        0        |   | 2        |
          \t  + t + 1/     {0, 3} | 0        a3+b3+c3 0        0        0        0        0        0        0        |   \t  + t + 1/
                           {0, 3} | 0        0        a3+b3+c3 0        0        0        0        0        0        |
                           {0, 3} | 0        0        0        a3+b3+c3 0        0        0        0        0        |
                           {0, 3} | 0        0        0        0        a3+b3+c3 0        0        0        0        |
                           {0, 3} | 0        0        0        0        0        a3+b3+c3 0        0        0        |
                           {0, 3} | 0        0        0        0        0        0        a3+b3+c3 0        0        |
                           {0, 3} | 0        0        0        0        0        0        0        a3+b3+c3 0        |
                           {0, 3} | 0        0        0        0        0        0        0        0        a3+b3+c3 |

          /   S[t]   \9                                                                                                  /   S[t]   \9
      1 : |----------|  <----------------------------------------------------------------------------------------------- |----------|  : 1
          | 2        |     {0, 3} | a3+b3+c3 0        0        0        0        0        0        0        0        |   | 2        |
          \t  + t + 1/     {0, 3} | 0        a3+b3+c3 0        0        0        0        0        0        0        |   \t  + t + 1/
                           {0, 3} | 0        0        a3+b3+c3 0        0        0        0        0        0        |
                           {0, 3} | 0        0        0        a3+b3+c3 0        0        0        0        0        |
                           {0, 3} | 0        0        0        0        a3+b3+c3 0        0        0        0        |
                           {0, 3} | 0        0        0        0        0        a3+b3+c3 0        0        0        |
                           {0, 3} | 0        0        0        0        0        0        a3+b3+c3 0        0        |
                           {0, 3} | 0        0        0        0        0        0        0        a3+b3+c3 0        |
                           {0, 3} | 0        0        0        0        0        0        0        0        a3+b3+c3 |

          /   S[t]   \9                                                                                                  /   S[t]   \9
      2 : |----------|  <----------------------------------------------------------------------------------------------- |----------|  : 2
          | 2        |     {0, 3} | a3+b3+c3 0        0        0        0        0        0        0        0        |   | 2        |
          \t  + t + 1/     {0, 3} | 0        a3+b3+c3 0        0        0        0        0        0        0        |   \t  + t + 1/
                           {0, 3} | 0        0        a3+b3+c3 0        0        0        0        0        0        |
                           {0, 3} | 0        0        0        a3+b3+c3 0        0        0        0        0        |
                           {0, 3} | 0        0        0        0        a3+b3+c3 0        0        0        0        |
                           {0, 3} | 0        0        0        0        0        a3+b3+c3 0        0        0        |
                           {0, 3} | 0        0        0        0        0        0        a3+b3+c3 0        0        |
                           {0, 3} | 0        0        0        0        0        0        0        a3+b3+c3 0        |
                           {0, 3} | 0        0        0        0        0        0        0        0        a3+b3+c3 |

o10 : ZZdFactorizationMap
 
i11 : g = randomFactorizationMap(C, C, Degree => -1)

           4                           4
o11 = 1 : S  <----------------------- S  : 0
                | 24  19  -8  -38 |
                | -36 19  -22 -16 |
                | -30 -10 -29 39  |
                | -29 -29 -24 21  |

           4                           4
      0 : S  <----------------------- S  : 1
                | 34  -18 -28 16  |
                | 19  -13 -47 22  |
                | -47 -43 38  45  |
                | -39 -15 2   -34 |

o11 : ZZdFactorizationMap
 
i12 : g^2

           4                          4
o12 = 0 : S  <---------------------- S  : 0
                | 22  19 47  -43 |
                | -21 47 -41 34  |
                | -5  39 49  -48 |
                | 25  41 -14 -25 |

           4                          4
      1 : S  <---------------------- S  : 1
                | 5   33  -26 17 |
                | -13 -29 -46 2  |
                | 46  19  -17 2  |
                | -16 0   -8  33 |

o12 : ZZdFactorizationMap
 
i13 : g^3

           4                           4
o13 = 1 : S  <----------------------- S  : 0
                | 27  -16 -16 39  |
                | 34  8   8   14  |
                | -37 34  -38 -47 |
                | 10  31  -28 -21 |

           4                          4
      0 : S  <---------------------- S  : 1
                | -29 1  -11 4   |
                | -28 10 20  20  |
                | 39  14 -28 -31 |
                | 30  -6 23  7   |

o13 : ZZdFactorizationMap

The zero-th power returns the identity map

i14 : f^0 == id_C

o14 = true
 
i15 : g^0 == id_C

o15 = true

When $n$ is negative, the result is the $n$-fold power of the inverse complex map, if it exists.

i16 : h = randomFactorizationMap(C, C)

           4                          4
o16 = 0 : S  <---------------------- S  : 0
                | -48 -16 39  48 |
                | -47 7   43  36 |
                | 47  15  -17 35 |
                | 19  -23 -11 11 |

           4                           4
      1 : S  <----------------------- S  : 1
                | -38 46  22  2   |
                | 33  -28 -47 29  |
                | 40  1   -23 -47 |
                | 11  -3  -7  15  |

o16 : ZZdFactorizationMap
 
i17 : h^-1

           4                           4
o17 = 0 : S  <----------------------- S  : 0
                | 6   33  45  44  |
                | -18 -12 5   -45 |
                | -42 10  -26 -33 |
                | 11  -17 -29 -1  |

           4                           4
      1 : S  <----------------------- S  : 1
                | 42  -12 4   -17 |
                | 29  3   -21 39  |
                | 48  32  -7  31  |
                | -43 -43 30  -46 |

o17 : ZZdFactorizationMap
 
i18 : assert(h * h^-1 == id_C)
 
i19 : h^-4

           4                          4
o19 = 0 : S  <---------------------- S  : 0
                | -26 19  5   14 |
                | -33 15  -9  -2 |
                | 3   -21 -47 20 |
                | 31  20  29  49 |

           4                          4
      1 : S  <---------------------- S  : 1
                | -11 44 -29 -40 |
                | 37  30 37  -10 |
                | 13  35 -35 40  |
                | 43  -2 -32 -12 |

o19 : ZZdFactorizationMap
 
i20 : assert(h^-4 * h^4 == id_C)

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3137:0.