dual(ZZdFactorizationMap,RingElement) -- the dual of a map of ZZ/d-graded factorizations

Function: dual
Usage:

h = dual(f,t)
Inputs:
- f, a map of ZZ/d-graded factorizations,
- t, a ring element, optional input specifying the symbol used to represent the distinguished root of unity, required only if the underlying ring does not have a specified root of unity and the period is > 2
Outputs:
- h, a map of ZZ/d-graded factorizations,

Description

The map $f : C \to D$ of ZZ/d-graded factorizations over the ring $S$ induces the map $h = Hom(f, S^1) : Hom(D, S^1) \to Hom(C,S^1)$ defined by $\phi \mapsto \phi f$. If the period of the factorization is >2, then there must be a root of unity adjoined in order for the dual to be defined.

The dual is a factorization of the negative of the potential.

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

o1 = S

o1 : PolynomialRing

i2 : C = linearMF(a^3+b^3, t)

     /   S[t]   \3     /   S[t]   \3     /   S[t]   \3     /   S[t]   \3
o2 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
     | 2        |      | 2        |      | 2        |      | 2        |
     \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/
                                                            
     0                 1                 2                 0

o2 : ZZdFactorization

i3 : St = ring C;

i4 : D = (koszulComplex {a,b})**St

       1       2       1
o4 = St  <-- St  <-- St
                      
     0       1       2

o4 : Complex

i5 : f = randomFactorizationMap(C,D**C, Cycle => true)

           3                                                                                                                                                                                                                                                                                                                                                                                                                    12
o5 = 0 : St  <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St   : 0
                {0, 2} | 0 0 0 (21a+47b)t+29a-7b -14bt+41b          (-4a-8b)t+21a-47b  (39a-10b-27c)t+15a-4b+8c    (-10a-29b-8c)t+29a-19b-19c (46a+27b+50c)t-2a-20b-4c    (49a2+33ab-36b2-44ac+40bc+26c2-7a-35b+19c)t+48a2+36ab+11b2+35ac-38bc+33c2-a-3b+11c (7a2-39ab+b2-23ac+41bc-17c2+18a+44b+34c)t+7a2+15ab+39b2-23ac+43bc-17c2+46a-45b+50c    (34a+50b+50c)t-47a2-39ab-13b2-18ac-43bc-15c2+37a+34b+19c               |
                {0, 2} | 0 0 0 (9a+6b)t+7a-46b   (-35a+47b)t-25a-7b -44bt+27b          (-48a-47b+47c)t-46a-27b-50c (39a-10b-27c)t+15a-4b+8c   (-29a+19b+19c)t-39a-10b+11c (7a2-ab-13b2-16ac-43bc-15c2-19a-38b-39c)t-19a+25b+16c                              (49a2+33ab-36b2-44ac+40bc+26c2-31a-18b+8c)t+48a2+36ab+11b2+35ac-38bc+33c2-29a+19b+19c (-7a2-15ab-39b2+23ac-43bc+17c2-40a-27b)t+31a+9b-20c                    |
                {0, 2} | 0 0 0 -30bt+11b         (-5a+2b)t-28a-8b   (-33a+47b)t+31a-7b (39a+10b-11c)t+10a+29b+8c   (2a+20b+4c)t+48a+47b-47c   (39a-10b-27c)t+15a-4b+8c    (22a-15b-34c)t-7a2+39ab-b2+23ac-41bc+17c2-16a-38b-30c                              (-7a2+ab+13b2+16ac+43bc+15c2-20a-19b+47c)t+47a2+39ab+13b2+18ac+43bc+15c2-40a+9b+4c    (49a2+33ab-36b2-44ac+40bc+26c2)t+48a2+36ab+11b2+35ac-38bc+33c2+19a-47b |

           3                                                                                                                                                                                                                                                                                                                                                                                                                    12
     1 : St  <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St   : 1
                {0, 2} | 0 0 0 (-37a+47b)t+33a-47b -41bt+46b         (-23a-8b)t+5a-47b  (-24a+6b+35c)t-39a+10b+27c  (-29a+19b+19c)t-39a-10b+11c (46a+27b+50c)t-2a-20b-4c   (-48a2-36ab-11b2-35ac+38bc-33c2+20a-44b-11c)t+a2-3ab-47b2+22ac-23bc-7c2-9a-25b+8c     (7a2-39ab+b2-23ac+41bc-17c2+18a-43b+34c)t+7a2+15ab+39b2-23ac+43bc-17c2-15a+15b+50c (-47a2-39ab-13b2-18ac-43bc-15c2-22a-8b-31c)t+47a2-38ab-2ac+35a-3b-50c |
                {0, 2} | 0 0 0 (25a+6b)t+4a-46b    (8a+47b)t-21a-47b -27bt+30b          (-48a-47b+47c)t-46a-27b-50c (-24a+6b+35c)t-39a+10b+27c  (39a+10b-11c)t+10a+29b+8c  (47a2+39ab+13b2+18ac+43bc+15c2+2a+17b-46c)t+47a2+39ab+13b2+18ac+43bc+15c2+12a-14b+39c (-48a2-36ab-11b2-35ac+38bc-33c2-9a-26b-19c)t+a2-3ab-47b2+22ac-23bc-7c2+23a+10b-11c (-7a2-15ab-39b2+23ac-43bc+17c2)t-30a-21b-20c                          |
                {0, 2} | 0 0 0 -11bt-41b           (-2a+2b)t-9a-8b   (10a+47b)t+35a-47b (-10a-29b-8c)t+29a-19b-19c  (2a+20b+4c)t+48a+47b-47c    (-24a+6b+35c)t-39a+10b+27c (22a+45b-34c)t-7a2+39ab-b2+23ac-41bc+17c2-16a+33b-30c                                 (-43a+38b-43c)t+7a2-ab-13b2-16ac-43bc-15c2+15a+21b-47c                             (-48a2-36ab-11b2-35ac+38bc-33c2)t+a2-3ab-47b2+22ac-23bc-7c2-41a-7b    |

           3                                                                                                                                                                                                                                                                                                                                                                                                                        12
     2 : St  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St   : 2
                {0, 2} | 0 0 0 (25a+7b)t-10a-47b -46bt+14b          (-7a-8b)t+2a-47b  (-15a+4b-8c)t+24a-6b-35c    (39a+10b-11c)t+10a+29b+8c (46a+27b+50c)t-2a-20b-4c   (-a2+3ab+47b2-22ac+23bc+7c2+31a-22b-8c)t-49a2-33ab+36b2+44ac-40bc-26c2-31a+28b-19c (7a2-39ab+b2-23ac+41bc-17c2+18a-2b+34c)t+7a2+15ab+39b2-23ac+43bc-17c2-15a-31b+50c  (-7a2+ab+13b2+16ac+43bc+15c2-39a+13b-19c)t-7a2+ab+13b2+16ac+43bc+15c2+4a-46b+31c |
                {0, 2} | 0 0 0 (28a+6b)t+23a-46b (-31a+7b)t+37a-47b -30bt+44b         (-48a-47b+47c)t-46a-27b-50c (-15a+4b-8c)t+24a-6b-35c  (-10a-29b-8c)t+29a-19b-19c (-47a2+38ab+2ac+42a+24b-16c)t+7a2-ab-13b2-16ac-43bc-15c2+5a+32b+46c                (-a2+3ab+47b2-22ac+23bc+7c2-20a+44b+11c)t-49a2-33ab+36b2+44ac-40bc-26c2-10a-29b-8c (-7a2-15ab-39b2+23ac-43bc+17c2-40a+30b)t+31a+36b-20c                             |
                {0, 2} | 0 0 0 41bt+30b          (-21a+2b)t-25a-8b  (-29a+7b)t-8a-47b (-29a+19b+19c)t-39a-10b+11c (2a+20b+4c)t+48a+47b-47c  (-15a+4b-8c)t+24a-6b-35c   (-18a-26b-34c)t-7a2+39ab-b2+23ac-41bc+17c2-16a+22b-30c                             (-47a2-39ab-13b2-18ac-43bc-15c2-36a-7b-4c)t-49a-36b+43c                            (-a2+3ab+47b2-22ac+23bc+7c2+16a)t-49a2-33ab+36b2+44ac-40bc-26c2-22a-47b          |

o5 : ZZdFactorizationMap

i6 : h = dual f

           12                                                                                                                                                                                                                                                                                3
o6 = 0 : St   <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St  : 0
                 {0, -2} | 0                                                                                  0                                                                                     0                                                                                  |
                 {0, -2} | 0                                                                                  0                                                                                     0                                                                                  |
                 {0, -2} | 0                                                                                  0                                                                                     0                                                                                  |
                 {0, -3} | (21a+47b)t+29a-7b                                                                  (9a+6b)t+7a-46b                                                                       -30bt+11b                                                                          |
                 {0, -3} | -14bt+41b                                                                          (-35a+47b)t-25a-7b                                                                    (-5a+2b)t-28a-8b                                                                   |
                 {0, -3} | (-4a-8b)t+21a-47b                                                                  -44bt+27b                                                                             (-33a+47b)t+31a-7b                                                                 |
                 {0, -3} | (39a-10b-27c)t+15a-4b+8c                                                           (-48a-47b+47c)t-46a-27b-50c                                                           (39a+10b-11c)t+10a+29b+8c                                                          |
                 {0, -3} | (-10a-29b-8c)t+29a-19b-19c                                                         (39a-10b-27c)t+15a-4b+8c                                                              (2a+20b+4c)t+48a+47b-47c                                                           |
                 {0, -3} | (46a+27b+50c)t-2a-20b-4c                                                           (-29a+19b+19c)t-39a-10b+11c                                                           (39a-10b-27c)t+15a-4b+8c                                                           |
                 {0, -4} | (49a2+33ab-36b2-44ac+40bc+26c2-7a-35b+19c)t+48a2+36ab+11b2+35ac-38bc+33c2-a-3b+11c (7a2-ab-13b2-16ac-43bc-15c2-19a-38b-39c)t-19a+25b+16c                                 (22a-15b-34c)t-7a2+39ab-b2+23ac-41bc+17c2-16a-38b-30c                              |
                 {0, -4} | (7a2-39ab+b2-23ac+41bc-17c2+18a+44b+34c)t+7a2+15ab+39b2-23ac+43bc-17c2+46a-45b+50c (49a2+33ab-36b2-44ac+40bc+26c2-31a-18b+8c)t+48a2+36ab+11b2+35ac-38bc+33c2-29a+19b+19c (-7a2+ab+13b2+16ac+43bc+15c2-20a-19b+47c)t+47a2+39ab+13b2+18ac+43bc+15c2-40a+9b+4c |
                 {0, -4} | (34a+50b+50c)t-47a2-39ab-13b2-18ac-43bc-15c2+37a+34b+19c                           (-7a2-15ab-39b2+23ac-43bc+17c2-40a-27b)t+31a+9b-20c                                   (49a2+33ab-36b2-44ac+40bc+26c2)t+48a2+36ab+11b2+35ac-38bc+33c2+19a-47b             |

           12                                                                                                                                                                                                                                                                  3
     1 : St   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St  : 1
                 {0, -2} | 0                                                                                  0                                                                                  0                                                                       |
                 {0, -2} | 0                                                                                  0                                                                                  0                                                                       |
                 {0, -2} | 0                                                                                  0                                                                                  0                                                                       |
                 {0, -3} | (25a+7b)t-10a-47b                                                                  (28a+6b)t+23a-46b                                                                  41bt+30b                                                                |
                 {0, -3} | -46bt+14b                                                                          (-31a+7b)t+37a-47b                                                                 (-21a+2b)t-25a-8b                                                       |
                 {0, -3} | (-7a-8b)t+2a-47b                                                                   -30bt+44b                                                                          (-29a+7b)t-8a-47b                                                       |
                 {0, -3} | (-15a+4b-8c)t+24a-6b-35c                                                           (-48a-47b+47c)t-46a-27b-50c                                                        (-29a+19b+19c)t-39a-10b+11c                                             |
                 {0, -3} | (39a+10b-11c)t+10a+29b+8c                                                          (-15a+4b-8c)t+24a-6b-35c                                                           (2a+20b+4c)t+48a+47b-47c                                                |
                 {0, -3} | (46a+27b+50c)t-2a-20b-4c                                                           (-10a-29b-8c)t+29a-19b-19c                                                         (-15a+4b-8c)t+24a-6b-35c                                                |
                 {0, -4} | (-a2+3ab+47b2-22ac+23bc+7c2+31a-22b-8c)t-49a2-33ab+36b2+44ac-40bc-26c2-31a+28b-19c (-47a2+38ab+2ac+42a+24b-16c)t+7a2-ab-13b2-16ac-43bc-15c2+5a+32b+46c                (-18a-26b-34c)t-7a2+39ab-b2+23ac-41bc+17c2-16a+22b-30c                  |
                 {0, -4} | (7a2-39ab+b2-23ac+41bc-17c2+18a-2b+34c)t+7a2+15ab+39b2-23ac+43bc-17c2-15a-31b+50c  (-a2+3ab+47b2-22ac+23bc+7c2-20a+44b+11c)t-49a2-33ab+36b2+44ac-40bc-26c2-10a-29b-8c (-47a2-39ab-13b2-18ac-43bc-15c2-36a-7b-4c)t-49a-36b+43c                 |
                 {0, -4} | (-7a2+ab+13b2+16ac+43bc+15c2-39a+13b-19c)t-7a2+ab+13b2+16ac+43bc+15c2+4a-46b+31c   (-7a2-15ab-39b2+23ac-43bc+17c2-40a+30b)t+31a+36b-20c                               (-a2+3ab+47b2-22ac+23bc+7c2+16a)t-49a2-33ab+36b2+44ac-40bc-26c2-22a-47b |

           12                                                                                                                                                                                                                                                                3
     2 : St   <----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St  : 2
                 {0, -2} | 0                                                                                  0                                                                                     0                                                                  |
                 {0, -2} | 0                                                                                  0                                                                                     0                                                                  |
                 {0, -2} | 0                                                                                  0                                                                                     0                                                                  |
                 {0, -3} | (-37a+47b)t+33a-47b                                                                (25a+6b)t+4a-46b                                                                      -11bt-41b                                                          |
                 {0, -3} | -41bt+46b                                                                          (8a+47b)t-21a-47b                                                                     (-2a+2b)t-9a-8b                                                    |
                 {0, -3} | (-23a-8b)t+5a-47b                                                                  -27bt+30b                                                                             (10a+47b)t+35a-47b                                                 |
                 {0, -3} | (-24a+6b+35c)t-39a+10b+27c                                                         (-48a-47b+47c)t-46a-27b-50c                                                           (-10a-29b-8c)t+29a-19b-19c                                         |
                 {0, -3} | (-29a+19b+19c)t-39a-10b+11c                                                        (-24a+6b+35c)t-39a+10b+27c                                                            (2a+20b+4c)t+48a+47b-47c                                           |
                 {0, -3} | (46a+27b+50c)t-2a-20b-4c                                                           (39a+10b-11c)t+10a+29b+8c                                                             (-24a+6b+35c)t-39a+10b+27c                                         |
                 {0, -4} | (-48a2-36ab-11b2-35ac+38bc-33c2+20a-44b-11c)t+a2-3ab-47b2+22ac-23bc-7c2-9a-25b+8c  (47a2+39ab+13b2+18ac+43bc+15c2+2a+17b-46c)t+47a2+39ab+13b2+18ac+43bc+15c2+12a-14b+39c (22a+45b-34c)t-7a2+39ab-b2+23ac-41bc+17c2-16a+33b-30c              |
                 {0, -4} | (7a2-39ab+b2-23ac+41bc-17c2+18a-43b+34c)t+7a2+15ab+39b2-23ac+43bc-17c2-15a+15b+50c (-48a2-36ab-11b2-35ac+38bc-33c2-9a-26b-19c)t+a2-3ab-47b2+22ac-23bc-7c2+23a+10b-11c    (-43a+38b-43c)t+7a2-ab-13b2-16ac-43bc-15c2+15a+21b-47c             |
                 {0, -4} | (-47a2-39ab-13b2-18ac-43bc-15c2-22a-8b-31c)t+47a2-38ab-2ac+35a-3b-50c              (-7a2-15ab-39b2+23ac-43bc+17c2)t-30a-21b-20c                                          (-48a2-36ab-11b2-35ac+38bc-33c2)t+a2-3ab-47b2+22ac-23bc-7c2-41a-7b |

o6 : ZZdFactorizationMap

i7 : assert isWellDefined h

This routine is functorial.

i8 : D' = C**D

       12       12       12       12
o8 = St   <-- St   <-- St   <-- St
                                 
     0        1        2        0

o8 : ZZdFactorization

i9 : f' = randomFactorizationMap(D**C, D')

           12                                                                                                                                                                                                      12
o9 = 0 : St   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St   : 0
                 {0, 2} | 15  -10 39  32a2-9ab-20b2-32ac+24bc-30c2   -9a2-35ab+40b2+6ac+3bc-31c2   -13a2+3ab+8b2-41ac+8bc-29c2    5a-37b-33c   12a+3b+9c   -14a-33b-42c 18a-28b+42c 2a-19b+43c   37a+34b+5c  |
                 {0, 2} | -37 30  27  -48a2-15ab+39ac+33bc-49c2      25a2-2ab-49b2-41ac-13bc+4c2   30a2-46ab-18b2+49ac+42bc+23c2  -28a+42b+44c -2a+20b-26c -44a-5b-16c  -11a+8b+42c -28a-50b-29c 16a-31b-49c |
                 {0, 2} | -13 -18 -22 -33a2-19ab-20b2+17ac+44bc-39c2 30a2-47ab-40b2+27ac+37bc-35c2 -28a2+15ab-16b2+18ac-46bc+12c2 30a+4b+22c   33a+16b+10c -35a-39b-4c  49a+5b-38c  -26a-49b+34c -a+43b+48c  |
                 {0, 3} | 0   0   0   36a+9b-39c                     -31a-39b-31c                  -18a+27b-21c                   5            31          -24          -26         31           8           |
                 {0, 3} | 0   0   0   4a+13b-26c                     -48a-29b-48c                  23a-37b-23c                    -20          28          -32          28          28           -40         |
                 {0, 3} | 0   0   0   22a-49b-11c                    30a-37b+47c                   44a-39b+20c                    -13          -6          -23          -33         36           -47         |
                 {0, 3} | 0   0   0   -8a+43b-8c                     -49a+28b-18c                  19a-47c                        -29          21          -18          9           -40          19          |
                 {0, 3} | 0   0   0   36a-3b-22c                     46a+b+40c                     -28a+47b-28c                   15           -30         27           -7          45           -45         |
                 {0, 3} | 0   0   0   -30a+41b+16c                   -22a+10b+7c                   6a-9b-33c                      -4           -4          -45          -46         42           -29         |
                 {0, 4} | 0   0   0   -28                            30                            28                             0            0           0            0           0            0           |
                 {0, 4} | 0   0   0   -6                             13                            -29                            0            0           0            0           0            0           |
                 {0, 4} | 0   0   0   35                             -17                           26                             0            0           0            0           0            0           |

           12                                                                                                                                                                                                      12
     1 : St   <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St   : 1
                 {0, 2} | -44a-42b-17c 24a-24b-47c 43a+5b-21c   -37a+10b+10c -3a+14b-37c  -33a-19b-29c -17 46  33 -26a2-41ab-17b2-17ac-34bc     -49a2-49ab-49b2+20ac-34bc+24c2 -2a2+29ab+44b2+37ac-44bc-24c2 |
                 {0, 2} | -23a-4b-2c   12a-31b+37c 23a-47b+10c  34a-42b+34c  10a+33b+26c  21a+46b+10c  -47 -35 13 39a2-35ab-24b2+44ac+15bc-20c2 -38a2+41ab+10b2+15ac+16bc+16c2 -33a2-6ab-46b2-47ac-27bc+38c2 |
                 {0, 2} | 34a-8b+27c   4a+5b+36c   -15a+47b+15c -44a-39b-40c -18a+49b-41c -16a+4b+45c  -41 -39 32 28a2-37ab-12b2-41ac+45bc+22c2 22a2+17ab-41b2-8ac-36bc+13c2   43a2-33ab+15b2+25ac-42bc-25c2 |
                 {0, 3} | 7            -16         -11          -29          38           -39          0   0   0  -49a-46b+4c                   -42a+21b-50c                   -10a+25b+24c                  |
                 {0, 3} | 38           6           -3           50           14           20           0   0   0  22a+7b+49c                    35a+43c                        -27a-28b-24c                  |
                 {0, 3} | 37           -43         -28          -10          6            25           0   0   0  -20a-45b+36c                  47a-16b+11c                    -30a+43b-25c                  |
                 {0, 3} | -24          -27         -50          -37          50           11           0   0   0  -43a+46b+26c                  -29a-18b-3c                    7a+7b+9c                      |
                 {0, 3} | 25           -50         17           -36          -36          41           0   0   0  -13a+29b+10c                  42a-43b+48c                    37a-4b+30c                    |
                 {0, 3} | -42          -6          -30          32           -40          47           0   0   0  3a-47b+2c                     24a-23b-46c                    -26a-17b-14c                  |
                 {0, 4} | 0            0           0            0            0            0            0   0   0  -50                           -36                            18                            |
                 {0, 4} | 0            0           0            0            0            0            0   0   0  0                             50                             -12                           |
                 {0, 4} | 0            0           0            0            0            0            0   0   0  9                             10                             -12                           |

           12                                                                                                                                                                                                       12
     2 : St   <-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- St   : 2
                 {0, 2} | 2a2-25ab+27b2-46ac-44bc-40c2  -33a2-21ab-43b2+39ac-22bc-15c2 -43a2-14ab+26b2+20ac+36bc+2c2  -4a+11b+c    -21a+18b-33c 22a-41b+3c  -9a-17b-32c  -17a+8b-6c   34a+18b-44c  16 28  -35 |
                 {0, 2} | 20a2+18ab-3b2-32ac+21bc+38c2  28a2+47ab-18b2-20ac+18bc-36c2  -47a2+30ab+13b2-5ac+45bc-46c2  -38a+48b+29c -35a+19b+7c  -6a+7b+17c  -27a+27b-48c -8a-25b-23c  -21a-19b+30c -7 19  50  |
                 {0, 2} | -48a2-33ab+39b2-34ac-6bc-18c2 -29a2-21ab+17b2+2ac+18bc+9c2   -25a2-11ab+27b2+19ac-15bc-30c2 -39a-20b+48c 36a+11b+7c   -18a+8b+25c 41a-39b+22c  -23a+26b+44c 5a-24b-23c   26 -11 48  |
                 {0, 3} | 18a+37b-17c                   -3a+42b+46c                    11a-9b-34c                     -46          19           -31         26           -12          -13          0  0   0   |
                 {0, 3} | 34a+7b-44c                    47a+29b+37c                    44a-28b-42c                    -6           -40          -2          43           3            -44          0  0   0   |
                 {0, 3} | 36a+48b+46c                   -10a+16b-10c                   -32a-38b-7c                    -27          -4           45          26           -26          11           0  0   0   |
                 {0, 3} | 38a+27b-5c                    -47a-27b-c                     21a-37b-10c                    23           49           36          2            6            -45          0  0   0   |
                 {0, 3} | 12a+39b-41c                   44a+43b+12c                    24a-48b+19c                    -45          -37          50          1            -36          7            0  0   0   |
                 {0, 3} | 17a-40b+35c                   9a-25b+16c                     29a-31b-35c                    -18          22           -41         13           34           6            0  0   0   |
                 {0, 4} | 28                            32                             41                             0            0            0           0            0            0            0  0   0   |
                 {0, 4} | 17                            16                             -20                            0            0            0           0            0            0            0  0   0   |
                 {0, 4} | 23                            -43                            43                             0            0            0           0            0            0            0  0   0   |

o9 : ZZdFactorizationMap

Ways to use this method:

dual(ZZdFactorizationMap,RingElement) -- the dual of a map of ZZ/d-graded factorizations
dual(ZZdFactorizationMap,Symbol)

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3712:0.

dual(ZZdFactorizationMap,RingElement) -- the dual of a map of ZZ/d-graded factorizations

Description

See also

Ways to use this method: