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flattenedGenericTensor -- Make a generic tensor of given format

Synopsis

Description

Given a list $L = \{a, b_1,\dots, b_n\}$ of positive integers with $ a= sum_i b_i, $ and a field (or ring of integers) kk, the script creates a polynomial ring $S$ over $kk$ with $a\times b_1\times\cdots\times b_n$ variables, and a generic map $$ f: A \to B_1\otimes\cdots \otimes B_n $$ of LabeledModules over $S$, where $A$ is a free LabeledModule of rank $a$ and $B_i$ is a free LabeledModule of rank $b_i$. We think of $f$ as representing a tensor of type $(a,b_1,\dots,b_n)$ made from the elementary symmetric functions.

The format of $F$ is the one required by tensorComplex1, namely $f: A \to B_1\otimes \cdots \otimes B_n$, with $a = rank A, b_i = rank B_i$.

i1 : kk = ZZ/101

o1 = kk

o1 : QuotientRing
 
i2 : f = flattenedGenericTensor({5,2,1,2},kk)

o2 = | x_(0,0,0,0) x_(1,0,0,0) x_(2,0,0,0) x_(3,0,0,0) x_(4,0,0,0) |
     | x_(0,0,0,1) x_(1,0,0,1) x_(2,0,0,1) x_(3,0,0,1) x_(4,0,0,1) |
     | x_(0,1,0,0) x_(1,1,0,0) x_(2,1,0,0) x_(3,1,0,0) x_(4,1,0,0) |
     | x_(0,1,0,1) x_(1,1,0,1) x_(2,1,0,1) x_(3,1,0,1) x_(4,1,0,1) |

                                    4                              5
o2 : Matrix (kk[x       ..x       ])  <--- (kk[x       ..x       ])
                 0,0,0,0   4,1,0,1              0,0,0,0   4,1,0,1
 
i3 : numgens ring f

o3 = 20
 
i4 : betti matrix f

            0 1
o4 = total: 4 5
        -1: . 5
         0: 4 .

o4 : BettiTally
 
i5 : S = ring f

o5 = S

o5 : PolynomialRing
 
i6 : tensorComplex1 f

o6 = | -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0)                                               0                                                                                            -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0)                                               0                                                                                            -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0)                                               0                                                                                            -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0)                                               0                                                                                            -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0)                                               0                                                                                            -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0)                                               0                                                                                            -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0)                                               0                                                                                            -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0)                                               0                                                                                            -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0)                                               0                                                                                            -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0)                                               0                                                                                            |
     | -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0)                                               -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0)                                               -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0)                                               -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0)                                               -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0)                                               -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0)                                               -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0)                                               -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0)                                               -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0)                                               -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0)                                               |
     | -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1)                                               -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1)                                               -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1)                                               -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1)                                               -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1)                                               -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1)                                               -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1)                                               -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1)                                               -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1)                                               -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1)                                               -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) |
     | 0                                                                                            -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1)                                               0                                                                                            -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1)                                               0                                                                                            -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1)                                               0                                                                                            -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1)                                               0                                                                                            -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1)                                               0                                                                                            -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1)                                               0                                                                                            -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1)                                               0                                                                                            -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1)                                               0                                                                                            -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1)                                               0                                                                                            -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1)                                               |

             4       20
o6 : Matrix S  <--- S

See also

Ways to use flattenedGenericTensor :

For the programmer

The object flattenedGenericTensor is a method function.