SimplicialModuleMap ^ ZZ -- the n-fold composition

Operator: ^
Usage:

f^n
Inputs:
- f, a Map of Simplicial Modules, whose source and target are the same simplicial module
- n, an integer,
Outputs:
- a Map of Simplicial Modules, the composition of $f$ with itself $n$ times.

Description

A simplicial module map $f : C \to C$ can be composed with itself. This method produces these new maps of simplicial modules.

The face/degeneracy map always composes with itself to give the zero map.

i1 : S = ZZ/101[a..c];

i2 : C = simplicialModule(freeResolution coker matrix{{a^2, b^2, c^2}}, Degeneracy => true)

      1      4      10      20
o2 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o2 : SimplicialModule

i3 : f = dd^C

               1                      4
o3 = (0, 0) : S  <------------------ S  : (1, 0)
                    | 1 a2 b2 c2 |

               1                   4
     (0, 1) : S  <--------------- S  : (1, 1)
                    | 1 0 0 0 |

               4                                            10
     (1, 0) : S  <---------------------------------------- S   : (2, 0)
                    {0} | 1 a2 b2 c2 0 0 0 0   0   0   |
                    {2} | 0 0  0  0  1 0 0 -b2 -c2 0   |
                    {2} | 0 0  0  0  0 1 0 a2  0   -c2 |
                    {2} | 0 0  0  0  0 0 1 0   a2  b2  |

               4                                   10
     (1, 1) : S  <------------------------------- S   : (2, 1)
                    {0} | 1 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 1 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 1 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 1 0 0 0 |

               4                                   10
     (1, 2) : S  <------------------------------- S   : (2, 2)
                    {0} | 1 0 0 0 0 0 0 0 0 0 |
                    {2} | 0 1 0 0 0 0 0 0 0 0 |
                    {2} | 0 0 1 0 0 0 0 0 0 0 |
                    {2} | 0 0 0 1 0 0 0 0 0 0 |

               10                                                                        20
     (2, 0) : S   <-------------------------------------------------------------------- S   : (3, 0)
                     {0} | 1 a2 b2 c2 0 0 0 0 0 0 0   0   0   0   0   0   0 0 0 0   |
                     {2} | 0 0  0  0  1 0 0 0 0 0 -b2 -c2 0   0   0   0   0 0 0 0   |
                     {2} | 0 0  0  0  0 1 0 0 0 0 a2  0   -c2 0   0   0   0 0 0 0   |
                     {2} | 0 0  0  0  0 0 1 0 0 0 0   a2  b2  0   0   0   0 0 0 0   |
                     {2} | 0 0  0  0  0 0 0 1 0 0 0   0   0   -b2 -c2 0   0 0 0 0   |
                     {2} | 0 0  0  0  0 0 0 0 1 0 0   0   0   a2  0   -c2 0 0 0 0   |
                     {2} | 0 0  0  0  0 0 0 0 0 1 0   0   0   0   a2  b2  0 0 0 0   |
                     {4} | 0 0  0  0  0 0 0 0 0 0 0   0   0   0   0   0   1 0 0 c2  |
                     {4} | 0 0  0  0  0 0 0 0 0 0 0   0   0   0   0   0   0 1 0 -b2 |
                     {4} | 0 0  0  0  0 0 0 0 0 0 0   0   0   0   0   0   0 0 1 a2  |

               10                                                       20
     (2, 1) : S   <--------------------------------------------------- S   : (3, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |

               10                                                       20
     (2, 2) : S   <--------------------------------------------------- S   : (3, 2)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 |

               10                                                       20
     (2, 3) : S   <--------------------------------------------------- S   : (3, 3)
                     {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |

o3 : SimplicialModuleMap

i4 : f^2

o4 = 0

o4 : SimplicialModuleMap

i5 : assert(source f == target f)

i6 : assert(degree f == -1)

i7 : assert(degree f^2 == -2)

i8 : g = randomSimplicialMap(C, C)

          1              1
o8 = 0 : S  <---------- S  : 0
               | 24 |

          4                              4
     1 : S  <-------------------------- S  : 1
               {0} | 24 0   0   0   |
               {2} | 0  -36 19  -29 |
               {2} | 0  -30 19  -8  |
               {2} | 0  -29 -10 -22 |

          10                                                      10
     2 : S   <-------------------------------------------------- S   : 2
                {0} | 24 0   0   0   0   0   0   0   0   0   |
                {2} | 0  -36 19  -29 0   0   0   0   0   0   |
                {2} | 0  -30 19  -8  0   0   0   0   0   0   |
                {2} | 0  -29 -10 -22 0   0   0   0   0   0   |
                {2} | 0  0   0   0   -36 19  -29 0   0   0   |
                {2} | 0  0   0   0   -30 19  -8  0   0   0   |
                {2} | 0  0   0   0   -29 -10 -22 0   0   0   |
                {4} | 0  0   0   0   0   0   0   -29 -16 34  |
                {4} | 0  0   0   0   0   0   0   -24 39  19  |
                {4} | 0  0   0   0   0   0   0   -38 21  -47 |

          20                                                                                              20
     3 : S   <------------------------------------------------------------------------------------------ S   : 3
                {0} | 24 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  -36 19  -29 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  -30 19  -8  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  -29 -10 -22 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   -36 19  -29 0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   -30 19  -8  0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   -29 -10 -22 0   0   0   0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   -36 19  -29 0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   -30 19  -8  0   0   0   0   0   0   0   0   0   0   |
                {2} | 0  0   0   0   0   0   0   -29 -10 -22 0   0   0   0   0   0   0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   -29 -16 34  0   0   0   0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   -24 39  19  0   0   0   0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   -38 21  -47 0   0   0   0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -29 -16 34  0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -24 39  19  0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   -38 21  -47 0   0   0   0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -29 -16 34  0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -24 39  19  0   |
                {4} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -38 21  -47 0   |
                {6} | 0  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   -39 |

o8 : SimplicialModuleMap

i9 : g^2

          1               1
o9 = 0 : S  <----------- S  : 0
               | -30 |

          4                               4
     1 : S  <--------------------------- S  : 1
               {0} | -30 0   0   0   |
               {2} | 0   -49 -33 15  |
               {2} | 0   35  -28 -15 |
               {2} | 0   -38 -16 -9  |

          10                                                      10
     2 : S   <-------------------------------------------------- S   : 2
                {0} | -30 0   0   0   0   0   0   0   0   0  |
                {2} | 0   -49 -33 15  0   0   0   0   0   0  |
                {2} | 0   35  -28 -15 0   0   0   0   0   0  |
                {2} | 0   -38 -16 -9  0   0   0   0   0   0  |
                {2} | 0   0   0   0   -49 -33 15  0   0   0  |
                {2} | 0   0   0   0   35  -28 -15 0   0   0  |
                {2} | 0   0   0   0   -38 -16 -9  0   0   0  |
                {4} | 0   0   0   0   0   0   0   34  49  41 |
                {4} | 0   0   0   0   0   0   0   48  -19 42 |
                {4} | 0   0   0   0   0   0   0   -40 36  3  |

          20                                                                                          20
     3 : S   <-------------------------------------------------------------------------------------- S   : 3
                {0} | -30 0   0   0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   -49 -33 15  0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   35  -28 -15 0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   -38 -16 -9  0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   0   0   0   -49 -33 15  0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   0   0   0   35  -28 -15 0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   0   0   0   -38 -16 -9  0   0   0   0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   0   0   0   0   0   0   -49 -33 15  0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   0   0   0   0   0   0   35  -28 -15 0   0   0  0   0   0  0   0   0  0 |
                {2} | 0   0   0   0   0   0   0   -38 -16 -9  0   0   0  0   0   0  0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   34  49  41 0   0   0  0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   48  -19 42 0   0   0  0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   -40 36  3  0   0   0  0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  34  49  41 0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  48  -19 42 0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  -40 36  3  0   0   0  0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   0   0  34  49  41 0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   0   0  48  -19 42 0 |
                {4} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   0   0  -40 36  3  0 |
                {6} | 0   0   0   0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  6 |

o9 : SimplicialModuleMap

i10 : g^3

           1               1
o10 = 0 : S  <----------- S  : 0
                | -13 |

           4                              4
      1 : S  <-------------------------- S  : 1
                {0} | -13 0   0   0  |
                {2} | 0   -4  9   42 |
                {2} | 0   15  -20 44 |
                {2} | 0   -12 -27 14 |

           10                                                     10
      2 : S   <------------------------------------------------- S   : 2
                 {0} | -13 0   0   0  0   0   0  0   0   0   |
                 {2} | 0   -4  9   42 0   0   0  0   0   0   |
                 {2} | 0   15  -20 44 0   0   0  0   0   0   |
                 {2} | 0   -12 -27 14 0   0   0  0   0   0   |
                 {2} | 0   0   0   0  -4  9   42 0   0   0   |
                 {2} | 0   0   0   0  15  -20 44 0   0   0   |
                 {2} | 0   0   0   0  -12 -27 14 0   0   0   |
                 {4} | 0   0   0   0  0   0   0  17  6   -42 |
                 {4} | 0   0   0   0  0   0   0  -7  -21 4   |
                 {4} | 0   0   0   0  0   0   0  -20 -14 -9  |

           20                                                                                            20
      3 : S   <---------------------------------------------------------------------------------------- S   : 3
                 {0} | -13 0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   -4  9   42 0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   15  -20 44 0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   -12 -27 14 0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  -4  9   42 0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  15  -20 44 0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  -12 -27 14 0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  0   0   0  -4  9   42 0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  0   0   0  15  -20 44 0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  0   0   0  -12 -27 14 0   0   0   0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  17  6   -42 0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  -7  -21 4   0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  -20 -14 -9  0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   17  6   -42 0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -7  -21 4   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -20 -14 -9  0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   17  6   -42 0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   -7  -21 4   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   -20 -14 -9  0   |
                 {6} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   -32 |

o10 : SimplicialModuleMap

The zero-th power returns the identity map

i11 : f^0 == id_(C[1])

o11 = false

i12 : g^0 == id_C

o12 = true

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

i13 : h = randomSimplicialMap(C, C)

           1               1
o13 = 0 : S  <----------- S  : 0
                | -18 |

           4                              4
      1 : S  <-------------------------- S  : 1
                {0} | -18 0   0   0  |
                {2} | 0   -13 -28 2  |
                {2} | 0   -43 -47 16 |
                {2} | 0   -15 38  22 |

           10                                                     10
      2 : S   <------------------------------------------------- S   : 2
                 {0} | -18 0   0   0  0   0   0  0   0   0   |
                 {2} | 0   -13 -28 2  0   0   0  0   0   0   |
                 {2} | 0   -43 -47 16 0   0   0  0   0   0   |
                 {2} | 0   -15 38  22 0   0   0  0   0   0   |
                 {2} | 0   0   0   0  -13 -28 2  0   0   0   |
                 {2} | 0   0   0   0  -43 -47 16 0   0   0   |
                 {2} | 0   0   0   0  -15 38  22 0   0   0   |
                 {4} | 0   0   0   0  0   0   0  45  -47 -16 |
                 {4} | 0   0   0   0  0   0   0  -34 47  7   |
                 {4} | 0   0   0   0  0   0   0  -48 19  15  |

           20                                                                                            20
      3 : S   <---------------------------------------------------------------------------------------- S   : 3
                 {0} | -18 0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   -13 -28 2  0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   -43 -47 16 0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   -15 38  22 0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  -13 -28 2  0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  -43 -47 16 0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  -15 38  22 0   0   0  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  0   0   0  -13 -28 2  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  0   0   0  -43 -47 16 0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0   0   0   0  0   0   0  -15 38  22 0   0   0   0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  45  -47 -16 0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  -34 47  7   0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  -48 19  15  0   0   0   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   45  -47 -16 0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -34 47  7   0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   -48 19  15  0   0   0   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   45  -47 -16 0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   -34 47  7   0   |
                 {4} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   -48 19  15  0   |
                 {6} | 0   0   0   0  0   0   0  0   0   0  0   0   0   0   0   0   0   0   0   -23 |

o13 : SimplicialModuleMap

i14 : h^-1

           1              1
o14 = 0 : S  <---------- S  : 0
                | 28 |

           4                            4
      1 : S  <------------------------ S  : 1
                {0} | 28 0  0  0   |
                {2} | 0  35 -7 -44 |
                {2} | 0  13 -5 30  |
                {2} | 0  6  36 33  |

           10                                                  10
      2 : S   <---------------------------------------------- S   : 2
                 {0} | 28 0  0  0   0  0  0   0   0   0   |
                 {2} | 0  35 -7 -44 0  0  0   0   0   0   |
                 {2} | 0  13 -5 30  0  0  0   0   0   0   |
                 {2} | 0  6  36 33  0  0  0   0   0   0   |
                 {2} | 0  0  0  0   35 -7 -44 0   0   0   |
                 {2} | 0  0  0  0   13 -5 30  0   0   0   |
                 {2} | 0  0  0  0   6  36 33  0   0   0   |
                 {4} | 0  0  0  0   0  0  0   2   18  -13 |
                 {4} | 0  0  0  0   0  0  0   -34 -48 40  |
                 {4} | 0  0  0  0   0  0  0   36  -23 29  |

           20                                                                                        20
      3 : S   <------------------------------------------------------------------------------------ S   : 3
                 {0} | 28 0  0  0   0  0  0   0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  35 -7 -44 0  0  0   0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  13 -5 30  0  0  0   0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  6  36 33  0  0  0   0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  0  0  0   35 -7 -44 0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  0  0  0   13 -5 30  0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  0  0  0   6  36 33  0  0  0   0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  0  0  0   0  0  0   35 -7 -44 0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  0  0  0   0  0  0   13 -5 30  0   0   0   0   0   0   0   0   0   0   |
                 {2} | 0  0  0  0   0  0  0   6  36 33  0   0   0   0   0   0   0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   2   18  -13 0   0   0   0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   -34 -48 40  0   0   0   0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   36  -23 29  0   0   0   0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   2   18  -13 0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   -34 -48 40  0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   36  -23 29  0   0   0   0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0   2   18  -13 0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0   -34 -48 40  0   |
                 {4} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0   36  -23 29  0   |
                 {6} | 0  0  0  0   0  0  0   0  0  0   0   0   0   0   0   0   0   0   0   -22 |

o14 : SimplicialModuleMap

i15 : assert(h * h^-1 == id_C)

i16 : h^-4

           1               1
o16 = 0 : S  <----------- S  : 0
                | -30 |

           4                               4
      1 : S  <--------------------------- S  : 1
                {0} | -30 0   0   0   |
                {2} | 0   -30 -39 -6  |
                {2} | 0   -9  -15 -23 |
                {2} | 0   0   -49 39  |

           10                                                      10
      2 : S   <-------------------------------------------------- S   : 2
                 {0} | -30 0   0   0   0   0   0   0  0   0   |
                 {2} | 0   -30 -39 -6  0   0   0   0  0   0   |
                 {2} | 0   -9  -15 -23 0   0   0   0  0   0   |
                 {2} | 0   0   -49 39  0   0   0   0  0   0   |
                 {2} | 0   0   0   0   -30 -39 -6  0  0   0   |
                 {2} | 0   0   0   0   -9  -15 -23 0  0   0   |
                 {2} | 0   0   0   0   0   -49 39  0  0   0   |
                 {4} | 0   0   0   0   0   0   0   40 -37 -41 |
                 {4} | 0   0   0   0   0   0   0   38 -13 -2  |
                 {4} | 0   0   0   0   0   0   0   15 7   42  |

           20                                                                                           20
      3 : S   <--------------------------------------------------------------------------------------- S   : 3
                 {0} | -30 0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   -30 -39 -6  0   0   0   0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   -9  -15 -23 0   0   0   0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   -49 39  0   0   0   0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   0   0   -30 -39 -6  0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   0   0   -9  -15 -23 0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   0   0   0   -49 39  0   0   0   0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   0   0   0   0   0   -30 -39 -6  0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   0   0   0   0   0   -9  -15 -23 0  0   0   0  0   0   0  0   0   0  |
                 {2} | 0   0   0   0   0   0   0   0   -49 39  0  0   0   0  0   0   0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   40 -37 -41 0  0   0   0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   38 -13 -2  0  0   0   0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   15 7   42  0  0   0   0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   40 -37 -41 0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   38 -13 -2  0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   15 7   42  0  0   0   0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   40 -37 -41 0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   38 -13 -2  0  |
                 {4} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   15 7   42  0  |
                 {6} | 0   0   0   0   0   0   0   0   0   0   0  0   0   0  0   0   0  0   0   37 |

o16 : SimplicialModuleMap

i17 : assert(h^-4 * h^4 == id_C)

Ways to use this method:

SimplicialModuleMap ^ ZZ -- the n-fold composition

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2025:0.

SimplicialModuleMap ^ ZZ -- the n-fold composition

Description

See also

Ways to use this method: