multidimensionalMatrix(RingElement) -- make a multidimensional matrix from a multilinear form

Synopsis

Function: multidimensionalMatrix
Usage:

multidimensionalMatrix F
Inputs:
- F, a ring element, a homogeneous polynomial of multidegree $(1,\ldots,1)$ in a polynomial ring $R$ with the $\mathbb{Z}^n$-grading where the degree of each variable is a standard basis vector; in other words, $R$ is the homogeneous coordinate ring of a product of $n$ projective spaces.
Outputs:
- a multidimensional matrix, the $n$-dimensional matrix having as entries the coefficients of $F$.

Description

If $F$ is a multilinear form on $\mathbb{P}^{k_1}\times\mathbb{P}^{k_2}\times\cdots\times\mathbb{P}^{k_n}$, then the shape of the corresponding matrix is $(k_1+1)\times(k_2+1)\times\cdots\times(k_n+1)$. You can use permute to rearrange the dimensions.

i1 : R = ZZ[x_1..x_3,y_1..y_4,z_1..z_2,Degrees=>{3:{1,0,0},4:{0,1,0},2:{0,0,1}}];

i2 : F = random({1,1,1},R)

o2 = 8x y z  + 8x y z  + 6x y z  + 3x y z  + 5x y z  + 6x y z  + 8x y z  +
       1 1 1     2 1 1     3 1 1     1 2 1     2 2 1     3 2 1     1 3 1  
     ------------------------------------------------------------------------
     8x y z  + 6x y z  + 3x y z  + 2x y z  + 3x y z  + x y z  + 8x y z  +
       2 3 1     3 3 1     1 4 1     2 4 1     3 4 1    1 1 2     2 1 2  
     ------------------------------------------------------------------------
     3x y z  + 7x y z  + 7x y z  + 8x y z  + 3x y z  + 5x y z  + 9x y z  +
       3 1 2     1 2 2     2 2 2     3 2 2     1 3 2     2 3 2     3 3 2  
     ------------------------------------------------------------------------
     7x y z  + 3x y z  + 7x y z
       1 4 2     2 4 2     3 4 2

o2 : R

i3 : M = multidimensionalMatrix F

o3 = {{{8, 1}, {3, 7}, {8, 3}, {3, 7}}, {{8, 8}, {5, 7}, {8, 5}, {2, 3}},
     ------------------------------------------------------------------------
     {{6, 3}, {6, 8}, {6, 9}, {3, 7}}}

o3 : 3-dimensional matrix of shape 3 x 4 x 2 over ZZ

The inverse operation can be obtained using the command "!" as follows:

i4 : F' = M!

o4 = 8x0 x1 x2  + 8x0 x1 x2  + 6x0 x1 x2  + 3x0 x1 x2  + 5x0 x1 x2  +
        0  0  0      1  0  0      2  0  0      0  1  0      1  1  0  
     ------------------------------------------------------------------------
     6x0 x1 x2  + 8x0 x1 x2  + 8x0 x1 x2  + 6x0 x1 x2  + 3x0 x1 x2  +
        2  1  0      0  2  0      1  2  0      2  2  0      0  3  0  
     ------------------------------------------------------------------------
     2x0 x1 x2  + 3x0 x1 x2  + x0 x1 x2  + 8x0 x1 x2  + 3x0 x1 x2  +
        1  3  0      2  3  0     0  0  1      1  0  1      2  0  1  
     ------------------------------------------------------------------------
     7x0 x1 x2  + 7x0 x1 x2  + 8x0 x1 x2  + 3x0 x1 x2  + 5x0 x1 x2  +
        0  1  1      1  1  1      2  1  1      0  2  1      1  2  1  
     ------------------------------------------------------------------------
     9x0 x1 x2  + 7x0 x1 x2  + 3x0 x1 x2  + 7x0 x1 x2
        2  2  1      0  3  1      1  3  1      2  3  1

o4 : ZZ[x0 ..x0 , x1 ..x1 , x2 ..x2 ]
          0    2    0    3    0    1

i5 : assert(M === multidimensionalMatrix F')

i6 : assert(sub(F',vars ring F) === F)

Ways to use this method: