Statistics of identities of the classical vector analysis in R3

B. Fiedler 
Eichelbaumstr. 13, D-04249 Leipzig
email: Bernd.Fiedler.RoschStr.Leipzig@t-online.de

May, 2001

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>
<a,b><c,d,x>
<a,c><b,d,x>
<a,d><b,c,d>
<a,x><b,c,d>
<b,c><a,d,x>
<b,d><a,c,x>
<b,x><a,c,d>
<c,d><a,b,x>
<c,x><a,b,d>
<d,x><a,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

<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>
 =   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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