Essays/Complete Tensor

The complete tensor (or complete skew tensor) of order n is the array C with shape n$n such that for row i in (#:i.)n$n , the atom (<i){C is

•  0 if i is not a permutation
•  1 if i is an even permutation
•  _1 if i is an odd permutation

Since the monad C.!.2 computes the parity of a permutation, the complete tensor of order n obtains by applying C.!.2 to (#:i.)n$n .

For example, for n=:3

   C.!.2 ] 0 0 1
0
   C.!.2 ] 0 1 2
1
   C.!.2 ] 0 2 1
_1

   ] i=: (#: i.) 3$3
0 0 0
0 0 1
0 0 2

0 1 0
0 1 1
0 1 2

0 2 0
0 2 1
0 2 2


1 0 0
1 0 1
1 0 2

1 1 0
1 1 1
1 1 2

1 2 0
1 2 1
1 2 2


2 0 0
2 0 1
2 0 2

2 1 0
2 1 1
2 1 2

2 2 0
2 2 1
2 2 2
   t=: ,"2 ] 2 1 1": i  NB. a more compact display
   (C.!.2 i) ; t
┌────────┬────────────┐
│ 0  0  0│ 000 001 002│
│ 0  0  1│ 010 011 012│
│ 0 _1  0│ 020 021 022│
│        │            │
│ 0  0 _1│ 100 101 102│
│ 0  0  0│ 110 111 112│
│ 1  0  0│ 120 121 122│
│        │            │
│ 0  1  0│ 200 201 202│
│_1  0  0│ 210 211 212│
│ 0  0  0│ 220 221 222│
└────────┴────────────┘

   CT=: C.!.2 @ (#:i.) @ $~

   CT 3
 0  0  0
 0  0  1
 0 _1  0

 0  0 _1
 0  0  0
 1  0  0

 0  1  0
_1  0  0
 0  0  0

The complete tensor can be also be represented as a sparse array.

   perm=: i.@! A. i.  NB. all permutations of i.n
   CT1=: 3 : '(C.!.2 p) (<"1 p=. perm y)} 1$.$~y'

   CT1 3
0 1 2 │  1
0 2 1 │ _1
1 0 2 │ _1
1 2 0 │  1
2 0 1 │  1
2 1 0 │ _1

   (CT -: CT1)"0 >: i.6
1 1 1 1 1 1

Cross Product

The complete tensor is used in the generalized cross product:

   ip   =: +/ .*
   cross=: [ ip CT@#@[ ip ]

   'x y'=: _40 + 2 3 ?.@$ 100
   x
6 15 39
   y
12 14 _1
   ] c=. x cross y
561 _474 96

   x ip c
0
   c ip x
0
   y ip c
0
   c ip y
0

   'x y'=: _40 + 2 4 ?.@$ 100
   x
6 15 39 12
   y
14 _1 20 17
   ] c=. x cross y
   0  423 _267  339
_423    0  _66  426
 267   66    0 _216
_339 _426  216    0

   x ip c
0 0 0 0
   c ip x
0 0 0 0
   y ip c
0 0 0 0
   c ip y
0 0 0 0

Determinant

   ] m=: _40 + 3 3 ?@$ 100
15 _35 15
33  18 10
 0 _35 29

   *// m
0 _17325  14355
0  _9450   7830
0  _5250   4350

0  40425 _33495
0  22050 _18270
0  12250 _10150

0 _17325  14355
0  _9450   7830
0  _5250   4350

   (CT 3) * *// m
0      0     0
0      0  7830
0   5250     0

0      0 33495
0      0     0
0      0     0

0 _17325     0
0      0     0
0      0     0

   +/ , (CT 3) * *// m
29250
   -/ .* m
29250

   det=: +/ @ , @ (CT@# * *//)
   det m
29250

Collected Definitions

CT   =: C.!.2 @ (#:i.) @ $~

perm =: i.@! A. i.
CT1  =: 3 : '(C.!.2 p) (<"1 p=. perm y)} 1$.$~y'

ip   =: +/ .*
cross=: [ ip CT@#@[ ip ]

det  =: +/ @ , @ (CT@# * *//)

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Contributed by Roger Hui. The complete tensor and its applications are described in the dictionary entry for C. and in the Parity & Symmetry section of J Phrases.