| <a,e> | <a,f> | <a,g> | <a,x> |
| <b,e> | <b,f> | <b,g> | <b,x> |
| <c,e> | <c,f> | <c,g> | <c,x> |
| <d,e> | <d,f> | <d,g> | <d,x> |
The following table contains statistics about identities which exist between so-called standard terms of the classical vector analysis in R3. Standard terms formed from k vectors and 1 test vector x are for instance
| k + 1 = 4 | k + 1 = 5 | ||
| <a,b><c,x>
<a,c><b,x> <a,x><b,c> |
|
Examples of identities between standard terms are for instance
<a,d> b × c + <b,d> c × a + <c,d> a × b = <a,b,c> d (an identity which does not contain a test vector x)
and
|
|
|
= | 0 . |
More details can be found on the page of PERMS notebooks where I present the text of a talk (given at SLC46, Lyon, 2001) and the Mathematica notebooks of the calculation of the following table.
| k + 1 | l | blocks | # ideals | dim | terms | # ident | # summands |
| 4 | (4)
(2,2) |
(4)
(2,2) |
1
1 |
1
2 |
3 | 0 | --- |
| 5 | (3,1,1) | (3,1,1) | 1 | 6 | 10 | 4 | 4 |
| 6 | (6)
(4,2) (2,2,2) |
(6)
(4,2) (2,2,2) |
1
1 1 |
1
9 5 |
15 | 0 | --- |
| 7 | (5,1,1)
(3,3,1) |
(5,1,1)
(3,3,1) |
1
1 |
15
21 |
105 | 69 | 4, 10, 12, 14, 16, 18 |
| 8 | (8)
(6,2) (4,4) (4,2,2) |
(8)
(6,2) (4,4) (4,2,2) |
1
1 1 1 |
1
20 14 56 |
105 | 14 | 24, 36, 40, 44, 50, 52 |
| 9 | (7,1,1)
(5,3,1) (3,3,3) |
(7,1,1)
(5,3,1) (3,3,3) |
1
1 1 |
28
162 42 |
1260 | 1028 | ??? |
| k | number of vectors which are used to form standard expressions (k vectors + 1 test vector x) |
| l | partitions which are grouping partitions for a given k |
| blocks | partitions which denote minimal two-sided ideals of the group ring in which the characterizing spaces W have non-trivial projections |
| # ideals | number of minimal right ideals which form the projection of W in the two-sided ideal denoted by 'blocks' |
| dim | dimension of the minimal right ideals from the column '# ideals' |
| terms | number of standard terms for a given k |
| # ident | number of linearly independent identities for a given k |
| # summands | number of summands which I found in a set of linearly independent identities |
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| B. Fiedler, 04.09.2022 |