Generalized Kronecker and permanent
deltas, their spinor and tensor equivalents
— Reference Formulae
R L Agacy
L
1
42 Brighton Street, Gulliver, Townsville, QLD 4812, AUSTRALIA
Abstract
T h e work is essentially divided into two parts. T h e purpose of t h e first part, applicable
t o n-dimensional space, is t o (i) introduce a generalized permanent delta (gpd), a
symmetrizer, on a n equal footing with t h e generalized Kronecker delta (gKd), (ii)
derive combinatorial formulae for each separately, and then in combination, which
then leads us t o (iii) provide a comprehensive listing of these formulae. For t h e second
part, applicable t o spinors in t h e mathematical language of General Relativity, t h e
purpose is t o (i) provide formulae for combined g K d / g p d spinors, (ii) obtain and
tabulate spinor equivalents of gKd and gpd tensors and (iii) derive and exhibit tensor
equivalents of gKd and gpd spinors.
1
Introduction
T h e generalized Kronecker delta (gKd) is well known as an alternating function or
antisymmetrizer,
eg S^Xabc = d\X[def\. In contrast, although symmetric tensors are
seen constantly, there does not appear t o be employment of any symmetrizer, for eg
X(abc), in analogy with t h e antisymmetrizer. However, it is exactly t h e combination of
both types of symmetrizers, treated equally, that give us the flexibility t o describe any
type of tensor symmetry. We define such a 'permanent' symmetrizer as a generalized
permanent delta (gpd) below. Our purpose is t o restore an imbalance between the
gKd and gpd and provide a consolidated reference of combinatorial formulae for them,
most of which is not in t h e literature.
T h e work divides into two Parts: t h e first applicable t o tensors in n-dimensions,
t h e second applicable t o 2-component spinors used in t h e mathematical language of
General Relativity. In P a r t I, section 2 gives definitions, illustrations and tables of
combinatorial formulae of t h e gKd and gpd. T h e large section 3 derives an array of
formulae for b o t h a single gKd and products of such, which are collected together in
Appendix A. A similarly large section 4 does the same for gpd's — the formulae here
are not in t h e literature anywhere — they are tabulated in Appendix B. In section
5 simplifying formulae for combined gKd's and gpd's are derived, and presented in
Appendix C.
In P a r t IE, in section 5 we determine the spinor equivalents of all gKd's and gpd's
u p t o 8 (ie 4 up, 4 down) indices, t h e most useful for the tensor/spinor languages of
Relativity — these are listed in Appendix E, I and I I respectively. In section 6, in
reverse, we derive t h e tensor equivalents of gKd and gpd spinors and sums of products
of t h e m — t h e results are collated in Appendix F .
Conventions are as follows. In Part I, general indices range from 1 , . . . , n . B o l d e d
i n d i c e s are t o b e regarded as fixed. An index set {k} stands for {fcj,... , k } (p < n)
and an index set {ki} stands for {fcn,... , &i } etc, all indices ranging over 1 , . . . , n.
T h e index summation convention is understood. All index sets are permutations of
each other. In P a r t II, two-component spinor indices, are in capital lower case roman.
p
p
PART I
2
Definitions and illustrations of the gKd and gpd
Complete symmetry of (a tensor's) indices, as opposed t o total antisymmetry, is
manifested by all positive signs in any p-linear expression. Whereas t h e gKd's an­
tisymmetry comes about through a determinant (interchanges of rows/columns or
indices changes t h e sign), total positive or pure or permanent symmetry, as we term
it, comes about through the use of a permanent
Then in complete analogy t o t h e
gKd we introduce t h e generalized
permanent
delta or gpd. This is defined, like
t h e gKd determinant of a matrix, except that we take all positive signs. W e use t h e
kernel letter 7r t o denote a gpd a n d double vertical lines for t h e permanent of t h e
defining matrix. T h e gKd and gpd are completely complementary t o each other and
are defined, for p ( < n ) distinct indices, respectively by,
ft
3p
7T,
31-3p
31 -3v
4
3p
where t h e first is a determinant and t h e second a permanent. T h e gKd has value + 1
(-1) depending on whether ( j i , . . . , j ) is a n even (odd) permutation of ( t i , . . . ,i ).
T h e g p d has t h e permanent value + 1 for any permutation of t h e index sets. Note
t h a t 7r£ =<5". T h e gpd is a complete symmetrizer (permanent symmetrizer) and now
one easily sees t h a t ir^Xabe = 3!X(«je/).
In terms of bracketed and parenthesized notation, t h e gKd and gpd are
p
p
™n-3p
v
3\-3p
n
P-^Uin
3Y
P
A simple b u t important interaction between t h e gKd and g p d is t h a t in any
expression containing t h e m as a product, where there is a summation on a pair of
indices between them, such expression vanishes. For
,.o...6..
7T...o...b...
= 6 ..a...b..
...6...0..
..0...6..
7T...0...6...
= 0.
From this it also obviously follows t h a t t h e product of any gKd and any gpd with
two or more common contracted indices vanishes.
If A = [a*-] is a n n x n matrix its d e t e r m i n a n t and p e r m a n e n t are:
detA=- aV
n!
T
a'j3n
3 1
3n
in'
l—3n
SI«1—*n
T h e Riemann tensor Robed a n d Lanczos tensor Ly* are Young tableau tensors in
their algebraic symmetries and obey t h e partial symmetries and antisymmetries as
determined by t h e tableau representing t h e m (Agacy ). And so here t h e interplay of
symbols for both permanent symmetries (ir) a n d antisymmetries (6) manifests itself
in t h e one-line definition of each of t h e Riemann and Lanczos tensors in respect of
their algebraic symmetries
1
1
12
1
— o
ab cd V
v
n
qs
"fpk Limn-
^-efgh
3
3.1
Generalized Kronecker 8
Single generalized Kronecker 6
In this section many of t h e derived formulae have been presented (but without deriva­
tion) long ago in Veblen . In t h e following definitions of numerical symbols t h e range
of indices is from 1 t o n . I n any set of p indices
. . . , i } , it will b e understood
t h a t for p < n t h e indices are all distinct
T h e generalized
Kronecker
6 (gKd) may b e defined by
2
p
%
• • •
6
i
n-3p
T h e interchange of any two superscripts (or subscripts) interchanges the correspond­
ing rows (or columns) of t h e determinant thus changing its sign. Additionally if any
particular superscript (or subscript) is not contained in t h e set of subscripts (or su­
perscripts) then t h a t particular row (or column) consists of zeros, t h e determinant
vanishing. Hence only if t h e set of superscripts a n d subscripts are permutations of
each other does one obtain a non-zero result. T h e gKd can then equally well b e
described a s
+1
—1
0
if i\,. •. , i is a n even permutation of j \ , . . . , j
if i\,... , i is an odd permutation of j % , . . . ,j
if t i , . . . , i is not a permutation of j i , . . . , j
P
p
p
p
p
p
Unless specifically stated or obvious from t h e context we will in future take it t o
be t h e case t h a t t h e sets of superscripted a n d subscripted indices are permutations
of each other.
Note t h a t t h e indices i , j range from 1 t o n b u t t h a t there are p (< n) indices.
W h e n p = n ( = dim V ) , and only then, we will also write
P
s
^l.-.tn — °ii...in
a
n
a
€
"~ ° 1
n
and refer t o e a s t h e alternating
or permutation
symbol (and in General Rela­
tivity (GR) t h e Levi-Civita
symbol as well). I t takes t h e value + 1 when t h e set
{ » i , . . . ,i } is a n even permutation of { 1 , . . . , n } a n d —1 when
,i } is a n
odd permutation of { 1 , . . . , n } . T h e gKd
is a tensor (transforms like a tensor)
n
n
1
n
whereas
and e* •* are not
We modify Einstein's summation convention by indicating fixed, repeated indices
by b o l d t y p e so t h a t there is no summation over these indices, which will occur as
matching superscripts and subscripts. This notation makes clearer the derivation of
most of t h e following formulae. Unbolding or unfixing means summations go ahead.
In considering a product of t h e two gKd's
0
h-3p
°ki...k
p
there is no summation over t h e k ' s . For brevity we sometimes denote a set of summed,
repeated indices {ki,...
, k } by {k} and a set of fixed, repeated indices by { k } . Clearly
for a non-zero result above t h e sets {i}, {j} and {k} must all be permutations of each
other.
As t h e set {k} is a fixed permutation of the set {i} and t h e set {j} a permutation
of {k} then t h e resultant value of t h e product is just t h e value of a generalized
Kronecker 6 as t h e product of those permutations. Our permutation
multiplication
convention is from right t o left
p
/ki...k \
(i ...i \ _
p
x
v
/ii...i \
p
V i i \ k i . . . k j
\ji...jp)
Thus
k i . . . k , cti...tp
cki..Jtp
_
°h-ip
°k!...kp -
cti-tp
'ii... j -
c
c
P
In particular we have a result used often
Formula k l .
0
il-in -
°l-..n
°jl...jp
-
6
e
3i-3n-
Formula k 2 .
r
ffi-h-ih
3l-jp-l3p
_
u
(
Z-/^
I \P+s fii. eh
-i—ip
'
3p 3\-3p-\
v
v
8=1
where a caret ( i ) over an index indicates its omission from t h e list.
Imagine t h e gKd as a p x p matrix with t h e indices
. . . ,i signifying rows and
t h e indices j i , . . . , j signifying columns. Expand t h e g K d by t h e last column j .
Note t h a t in t h e determinantal expansion, t h e sign associated with t h e element in
row 1 a n d column p is ( — l ) , and so on. One then has
p
p
p
p+1
CfcS = ( - i r ^ i & t .
2
+
(-ir ^CA+• • • + ( - i )
and so we have t h e result
p
r«i...ip_i»p
3\-3p-\3p
i
v-^ / — 1 V + * tf? X«i- «-H>
Z~J^
'
3p
3\-3p-\
8=1
p + p
^::fcl
Now consider t h e expansion of t h e gKd ^ " ^ j ^
Formula 2). W e get
by its last (p 4- l )
t h
row (see
fcSK - ( - i r ^ c t * + ( - i r ^ ^ ^ + • • •+(-ij^&itt
3
1
where a caret ( j ) over a n index means t h a t it is excluded from t h a t set. There are
p + 1 terms in t h e sum. Then if we let j
=i
so t h a t there is a summation, we
get t h e single common index formula for a gKd
p + i
p + i
Using this result, now for two repeated indices, gives
fcttStt - (— * - UfcttS - (« - p - 1 ) ( « - p)t.%•
Continuing t o a d d repeated summed indices leads t o t h e most general result for a
single g K d
Formula k3
*i!:::tfev:<: - ( » - « + 1 )
•••(» - p ) # : J
=
o <
P
<
9
< „
which clearly gives t h e correct result for p = q. This formula is equation (8.1) of
Veblen .
A particular case of Formula 3, when q = n, is
Formula k 4
2
c*l...*pip+i...i,» .M...i i +i...i ,
=
p p
..
n£
.
—
Another particular case of Formula 3 , when p — 0, should b e noted
Formula k5
2
This formula is equation (8.2) of Veblen .
A useful specialization, when q = 1, is t h e simple result
Formula k6
$ = n.
Another specialization of Formula 3, when p = 0, q = n, is
Formula k7
2
This formula is equation (8.3) of Veblen .
3.2
Products of gKd's
Consider t h e following product of two gKd's
gil-ipip+i-U
£Jp+i-jq
Jl—ipjp+l—Jq
kp+l...k '
q
Note t h a t t h e fixed set J = { j + i , • • • , j } must take on t h e values of t h e specific
set K = {kp+i,... ,k }. Unbolding t h e indices of J so t h a t there are (q — p)\ per­
mutations of it a n d hence (q — p)l summations, gives t h e key result — (q — p) is t h e
number of common indices
Formula k 8
p
q
q
C*l...tptp+l...t,
°h-3pjp+l~3
cjp+1-..jq
_
/
\i
°fcp+l-.fc, ~ ~ W
q
ctl...tptp+l...t,
P>'
° h - 3 p h + l - V
2
This formula is equation (8.6) of Veblen . Letting t h e indices k i,...
,k = i
, i ,
and using Formula k 3 for (q — p) common indices on t h e R.H.S. gKd, results in t h e
first partial common indices product
formula
Formula k 9
p+
lplp
"3l-3p3p+l-3q
"tp+l-tq
+ 1
q
p
+
i
q
...t,
V
V«
V)- ji... 3pip+l—*q
(n - p)\
.tp
(n-q)
2
This formula is equation (8.7) of Veblen .
A second formula for t h e product of two gKd's comes from unbolding two sets of
indices in t h e obvious relation (a right t o left multiplication of two permutations on
reversing t h e order of t h e two L.H.S. terms)
ctl...ipip+1-..lq
rJl--JpJp+l---Jq
Jl--Jpjp+1---Jq
*l...fcplp+l...iq
_ _ £>l...tpi
p +
l...iq
_
cil...ip
m
fcl...fcpip+l...l
kl...kp
q
Some care is required now. Consider unbolding t h e set { i + i , . . . , i } . Each index
ranges over not n, b u t (n — p) distinct values, values complementary t o t h e values
of {*!,... , i } = {hi,... ,k }. There are (,!*) ways of selecting a set of values for
i p + i , . . . ,i in a n y order from t h e set of (n — p) values. B u t there a r e (q — p)\
permutations of these for t h e summation. Unbolding thus introduces t h e factor (q —
P)' (g-p)
(nlg)i —
equivalent of selecting (q — p) ordered values for ip+i,... ,i
from a set of n — p values. An alternative idea is t o consider t h e product of t h e two
gKd's by focussing purely on t h e summation of i i , . . . ,i with (n — p) for n, and
(q — p) for q (see k5) a s
= ^J^Z^lp^
t o obtain t h e factor. This gives t h e
intermediate result
p
p
q
p
q
=
q
p +
fii...ipip+ ...i
1
oJi..JpJp+i..J
q
ji..jp+iJp+i...j
fci...j^,«p+i...t,
q
q
_
q
(n
^
— p)\
n
_
rfi...tp
°fci...* p
Now unbolding t h e set { j i , . . . , j } introduces q\ permutations and we get
q
Formula klO
i ...i i ..,i
h-jpip+i
f 1
u
r p+1
q
n
ch-jpif+i-U
—in *i -kpip+i ...t,
1
\ ( ~ P) - ch-ip
H• ^ _qy
ki ...ftp •
v
n
A specialization of this result, when p = g, is
Formula k l l
Equating t h e indices {k\,...
gives
Formula k l 2
,k ) = ( i i , . . . , i ) a n d then employing Formula 5
p
p
n !
fci&"*'=p\
A specialization of Formula 10, when q = n , is
°ii...ipip+i...in °fci...fc t i...t„ P
n
!
l
p+
n
P)-%...fcp-
Two further specializations of this last can b e applied here.
First, when p — 0 we get
Formula k l 3
Second, when p = n w e get
Equating t h e indices (fci,... , k ) = («i,.. - , i ) confirms the first specialization.
n
n
We now examine product formulae involving the alternating symbol.
For t h e first, it is easy t o see
Formula k l 4
e
« l . . . * p t p . . . t „ gip+l.-jn
_
+ 1
*p+l»-*n
/
>
_
\j
ii...i j ...jn
e
p p+1
* '
as there are (n — p)\ permutations of the common indices.
Secondly, unbolding the (n — p) indices in
<
c»l...tpl +l...i
p
°l...p,p+l...n
n
cl...p,p+l...n
°h...j l + ...l
p p
l
u
cM...tpip+i...i
~ °Ji...ipip+i...i
n
B
which then involves (n—p)\ summations, reproduces an earlier result Formula 4. This
formula is equation (8.5) of Veblen
2
Formula k l 5
h...i i ,+i...i
e
p f
,.
n
1
..
= (n — vV S*. "*?
2
This formula is equation (8.4) of Veblen .
A little consideration is now given t o products of more than two gKd's. For a
compact notation we use {A;} t o stand for {k\,... , kp} (p < n ) and {ki} t o stand for
{kn,...
,k\ } etc.
p
T h e earlier relation o^"'*^ 6^.'"^ can be abbreviated t o
Then more generally, a relation of t h e type
%...ap °bi...bp-
0
°ji...j
p
— h-jp-
can b e compacted t o
Formula k ! 6
W
%} ~ %} I" °ii...J )'
fi
{k }
P
2
Each of t h e sets {k,} = { k n , . . . , k } is a permutation of {i} =
is t h e set {j} = {ji,..,j }.
I p
... ,i } as also
p
p
By now having familiarity with unbolding fixed repeated indices and counting t h e
summations we should be able t o see t h e result
°ki...k °h...3p
p
P-°3\-3P
or briefly
From this we have for a product of three 5's
and more generally
Formula k l 7
All combinatorial formulae for t h e gKd and alternating symbol are collected to­
gether and numbered as above, for reference in Appendix A.
4
Generalized Permanent 8
There are notable differences between gKd and gpd combinatorial formulae. For
example 6jj = n — n whereas 7Ty = n + n. In analogy with t h e gKd it is possible
t o define a symmetric
(or permanental)
symbol p in terms of a gpd by p* -** =
^.'".n ^
Pn...»« =
Like t h e alternating symbol e, t h e symmetric symbol p
is not a tensor. Unlike t h e alternating symbol which satisfies
=
it
is not t h e case t h a t
= p '"**Pji.„j - For example, with n = 2, 7r|j = 6, but
p**'py = 2. In general p* "'* p« ...» = n!. As it is not possible t o match a product
of contracted p's with t h e gpd it is convenient t o define a new symbol 7r, like a
factorial function, t o accommodate such a product. Formally we define a
factorial
symmetric
symbol (also tensor) by
Formula p i
2
2
1
11
n
1
n
1
n
&£&-<p+i)&S
<-*«s-i)
and t h a t it is symmetric in all its upper indices and all its lower indices.
We specify t h a t t h e factorial symmetric symbol 7r be linked t o t h e symmetric
symbol p by t h e relation
Formula p2
0*1 —in
7r
t
,
il-i»=P
i
Ph-Jn-
Symmetry in upper and lower indices is consistent.
T h e first equation shows t h e factorial nature of the symbol
Formula p 3
oti...ip
T h e kernel letter 7r in n is used t o indicate permanent symmetries like t h e gpd;
however, 7r is not a permanent t o be evaluated from a matrix.
T h e symmetric symbol p is useful for total symmetrizations, eg of functions, as in
ffi-^ffa,... , * J
or
pM-Zifo) • • • / „ ( x
i n
)
but does not play any role in Relativity and is not presented in any Relativity for­
mulae, being otherwise included for completeness.
4.1
Single generalized permanent 6
It is now time t o establish formulae for t h e gpd corresponding t o those for t h e gKd.
Proceeding in a similar manner t o t h e gKd, and fortunately because we don't have
t o worry about signs with regard t o a gpd, we can expand t h e gpd by t h e column j
t o get
p
Formula p 4
p
il-ip-lh
3l-3p-l3p
_
V " * ci.
Jp
n
ti...t....ip
31-Jp-l
*
«=1
Let j
p
= ip a n d also change p t o p + 1 t o obtain
T h e result can b e extended beyond one common index, so t h a t more generally we
have
Formula p5
*i-«p»p+i-«« _
'31 -ipip+i•••»«
(w + q — l)t M - i p
(n-fp—1)!
A particular case, when q = n , is
Formula p6
h...i i +i...i
p p
_
n
"ii-ipip+i-•*»
(2n
1)!
(n + p -
ii...«p
1)!
Another particular case, when p = 0, is
Formula p 7
=
(n + q - 1)!
(n - 1)!
A specialization, when g = 1 is t h e simple result
Formula p 8
ttJ =
n =
Another specialization of Formula 7, when q = n, is
Formula p 9
(n-1)!"
4.2
Products of gpd's
T h e reasoning for a product of two gpd's is identical with t h a t for gKd's. Thus we
have t h e analogous result t o Formula k l l
Formula plO
1T
7r
3i-3p ki...k
p
-
P-
^...kp-
Equating t h e indices ( k i , . . . , k ) =
Formula p l l
. . . , i ) a n d using Formula p 7 gives
p
p
e3l-3v
- ^ =*i—*p
p!
(
n
+
p
:.?
)
!
A specialization of this t o t h e case p = n gives
Formula p l 2
j ^ <ti~.t»
i:=n!
3i...jn
(
7 r
n
!
f( n"- 1^) !.
Arguing in exactly t h e same manner as for t h e derivation of Formula k8, b u t for
t h e g p d now, gives t h e analogous formula
Formula p l 3
Specializing t o (fej,... , k ) = (ii,...
Formula p l 4
, i ) gives
p
p
ti...tp» i...t, ^ p + i . . . i p _
, i
p+
"31«.3pip+l-j«"ip+l
•*«
(
n
n
+ g
+p
*i...*V.
_
1)J
31-ip'
Following t h e more involved reasoning for t h e gKd leading t o Formula klO pro­
duces t h e analogous result for gpd's
Formula p l 5
h...ipi i...i
ii-ipip+i-iq
h-ipip+\-U"fci...fc *> i...t
J)+
_
q
| (n + q — 1)!
ii...*p
« • ^
p _ -g j "a*
•
n
P
+
4
n
A specialization of this result, when (fci,... , fc ) = ( i j , . . . , i ) is
p
1
t
* "«
9
p
(n-1)!
a result already obtained as Formula p l l .
T h e further specialization, when q = n, gives
_*i...**_n...j» _
"h-in^i-in
2
n
n
\ (
'
(
n
n
1
1
~ )
_ x ) !
confirming again Formula p l 2 .
Finally for a product formula of t h e symmetric symbol with a gpd, it is easy t o
see
Formula p l 6
pti...»pip l...t„ Jp+l"in _
+
7 r
£
n
_pjjp*l...tpjp+l-Jn
as there a r e (n — p)\ possible permutations of t h e common indices.
For a product of p's with repeated indices we have
Formula p l 7
.
.
.
n\
0*1—H>
W h e n there is summation on all indices we get t h e specialization
Formula p l 8
Formulae for products of several gpd's are obtained by t h e same reasoning as for
gKd's. Thus we have
Formula p l 9
Unbolding t h e bolded indices so t h a t summations occur gives u s
Formula p 2 0
All combinatorial formulae for t h e gpd a n d symmetric symbol are collected to­
gether a n d numbered as above, for reference in Appendix B.
5
Combined gKd and gpd
Here we derive formulae for combined products of gKd's a n d gpd's where a t least one
factor is a gpd or symmetric symbol and another is a gKd or alternating symbol. We
saw earlier t h e useful result t h a t t h e product of any gKd a n d any gpd with two or
more summed common indices vanishes. Consequently for a combined product we
can have at most one matching index pair for superscript and subscript a n d another
matching index pair for subscript a n d superscript. Consider a combined product of
a gpd a n d a gKd with (necessarily) at most one superscript matching a subscript as
i i ^ « . ^ ^ 2 » . i * * - Expanding t h e gpd by its (first) row i\ leads t o
m
x
Formula c l
*ita-..tV rfcifca.-.k, _ r r t ! ia...»V
_
~
P
V
•
riiia.-.ip
,
, c*i « . • • » >
l cfcika-*,
*a
o*ik .»*«
*h..-5-~iv i.h...i
'
2
1
0
q
8=1
T h e unexpanded product can, a t t h e same time, also have a t most one gpd sub­
script matching a gKd superscript as in ^l^.'.^ii^."!*'1
For t h e combined product P* "^^^"^,
i\ we have
Formula c 2
expanding t h e gKd b y its first column
* 1 «.»n£fcl*2...kn _ * l — * n
"
*li2—in
"
0
1
a+1
5
E(- ) * fc^»
0
fcn
.8=1
Consider t h e converse relation ^ "'* T^^".'j
i\ is a summation with all positive signs
Formula c 2
il
n
n
n
t
c
5.1
*l...tn_klka...kn
*u'a...in
/l
• T h e expansion of t h e gpd b y column
n
\ ^ ,k«*3—*n- kl— * « — * n
Z-/
»
•>» '
8=1
r
_
Idempotent and orthogonal relations
We now enter a certain class of identities combining gKd's and gpd's whose origin
stems from orthogonal idempotents in t h e group algebra of t h e symmetric group S .
T h e derivation of, rather t h a n t h e background to, these identities is what concerns us
here. T h e orthogonal identities are visually obvious by having blanks (dots) in gKd's
and gpd's, eg 7r^, where arbitrary free indices could b e placed.
In order t o prove t h e identities it is necessary t o call on some simpler identities,
which, although established in generality earlier a n d listed in tables a t t h e end, is
pertinent t o b e stated here for specialized values.
Expansion of a gKd or gpd as a sum of terms comes about by considering it as
an expansion of a determinant or permanent by a row or column.
T h e basic formulae below can all b e inverted, that is, all superscripts lowered and
subscripts raised. Reference t o a basic formula implies either its use or its inversion.
n
(1)
tJ
6 ir- = 0 (repetition of two indices in a gKd a n d gpd)
T h e fourth t e r m S^S^S^S^^
= 46£6&ir%ir$ using (2). Interchange dummy
indices i, k a n d j , I, a n d as a gpd is totally index symmetric, t h e value of t h e term
>
is unchanged; its index rearrangement becomes 46^^7rf/7r?/ which is t h e new first
term. Hence t h e s u m of t h e first a n d fourth terms in X is %&%&Mtfk tfi
•
Interchange of *, k a n d j , I in t h e third term S^SQS^Sj^n^itji
which does not
change its value, converts it t o t h e second term, so t h a t X includes t h e expression
2^c^M^^r«7r«
A certain facility is acquired after some manipulations with gKd's
and gpd's. W e see t h a t we have
1
t
Interchange of indices i,k a n d j , I converts t h e third and fourth terms t o t h e second
and first terms respectively, so t h a t
=
- «<f < - *S<f< + * t f "Si)
Adding this t o t h e sum of t h e first a n d fourth terms previously leads t o X
12$&«8*«*$fc so t h a t
^ « < <T jg ft 7# TTjf
=
12«««.
This result is recorded as
F o r m u l a 13
XP9 JCr«
~ma c*j ski
ef
ah _
nff^fTi^f
gh
Derivation of Formula i4
+ CP«!*S
=
6£?<f + M2?*g
16
(using (2))
(using (5) and (4))
=
Derivation of Formula i2
= 6 ^ <
9
=
- (flrt? -
tfifiKg
" 2#r£* +M},r£
(using (3))
(using (5))
This result is recorded as
Formula i2
Derivation o f Formula i3
We have
This result is t h e most complicated we encounter.
=
-
Write t h e R.H.S. asX = 6%S£
nfi, where Y are t h e four terms in parenthesis.
They can b e expanded and then compacted
Y = nff^JL +
- n # j « , - *J#ii*
Now reconstruct X a s a sum of four terms, which will b e treated individually.
T h e first term is
T h e fourth t e r m S^S^S^S^iif
= 4$*jE&it%'ii$ using (2). Interchange dummy
indices i, k a n d j , I, and as a gpd is totally index symmetric, the value of the term
is unchanged; its index rearrangement becomes 45jj,5^7T^7r^j which is t h e new first
h
term. Hence t h e sum of t h e first and fourth terms in X is
BS^Sj^ir^i^.
7 1
Interchange of i, k a n d j , I in t h e third t e r m ^ ^ w ^ i ^ p i * * ^ i * which does not
change its value, converts it t o t h e second term, so t h a t X includes the expression
^oc^M^fw^rg^i* "Rji- A certain facility is acquired after some manipulations with gKd's
and gpd's. We see t h a t we have
Interchange of indices i, k and j , I converts the third and fourth terms t o t h e second
and first t e r m s respectively, so t h a t
+ <f *£ + <M? -
"«
«
Adding this t o t h e sum of t h e first and fourth terms previously leads t o X
1 2 ^ S «
so t h a t
*™ « C <f *ff = 124*^ <
« «
This result is recorded as
Formula i3
n
°ab °cd pr
Kqs °tu °uw ™ik ^jl
~
^"ab^cd^pr^a
•
D e r i v a t i o n o f F o r m u l a i4
= 6«X
;
+ Cr<C<
= ME}*J + 2£f
(using (2))
(using (5) and (4))
=
This result is recorded as
Formula i4
cpvwtu cqefnh
°abe "pd°tvw qu
n
_ o cpc/^jh
~ °°ol>c dp7r
Derivation of Formula o l
5
27r
^^^7= « ST = 0
(using (5), then obvious).
This result is recorded as
Formula o l
Derivation of Formula o 2
O f f * =- 2 ^ * 5 - 0
(using (4), then obvious).
This result is recorded as
Formula o 2
Derivation of Formula o3
«2<'«Si = «S«gS-0
(using(7)).
This result is recorded as
Formula o 3
Derivation of Formula o 4
= 2 « * ( ^ + ^ . )
= 0
(obvious).
This result is recorded as
Formula o 4
Derivation of Formula o5
= OT(2<>$ - 2nSJ^)
= W ( «
+
= 0 (obvious).
This result is recorded as
Formula o5
Derivation of Formula 0 6
= 0
(obvious).
This result is recorded as
Formula 0 6
Derivation of Formula o7
= 0
(obvious).
"
(using (5))
-
This result is recorded as
Formula o 7
Derivation of Formula 0 8
/ctvuw
— l°o6cd
= 0
ctwuv
°o6cd
—
cuvtw 1 cuuitw\_*i_fcl
°abcd + °obcd J^tv^uv)
—
(obvious).
This result is recorded as
Formula 0 8
Derivation of Formula o9
=
= 0.
( ™ * « (8))
This result is recorded as
Formula o 9
9h
?!. %i n
9
= 0.
P
Derivation of Formula m l
Depending on t h e manner of verification one may b e led t o other unobvious iden­
tities. We give a few. T h e first is
= 2 [ M ^ - « , ' +
= 2[(w
-
TO€f
« f )
- (W
-
=
+ («Stf-<«+nS«2)]
=
=
2 ( « - « + * M )
2<C
/
+ (W
-
This result is recorded as
Formula m l
a
7V
°abc dqr ps
~
zo
obc
"dq •
Derivation of Formula m 2
For t h e second, consider ^S^it^.n^.
Interchanging dummy indices p,q and r,s
will not change its value. However t h e expression now becomes
S ^S^ir {'n ^.
This result is recorded as
Formula m 2
P
t
q
Derivation of Formula m 3
For t h e third, consider S^S^S^T^TTQ.
p
We have
= 0
(obvious).
This result is recorded as
Formula m 3
All combinatorial formulae for t h e combined gKd a n d gpd are collected together
and numbered as above, for reference in Appendix C.
P A R T II
In this part we first examine t h e 2-component gKd a n d gpd spinors which can be
useful for t h e mathematical spinor algebra of General Relativity. Secondly we find
and tabulate t h e spinor equivalents of gKd's u p t o four double-index (4up, 4down)
indices. Thirdly, t h e other way round, we derive a n d tabulate t h e tensor equivalents
of gKd and g p d spinors.
In t h e first situation since any product of a gKd and a gpd with two (or more)
summed indices vanishes, a non-zero result can only b e obtained if there is at most
one summed index (up a n d down) a n d a different summed index (down and up). For
convenience, products of gKd's are simplified; similarly for gpd's. All these results,
easily obtained are listed in Appendix D.
6
Spinor equivalents of gKd and gpd tensors
T h e spinor equivalent of t h e basic Kronecker 6, 6% = ir% is SgSg,. From this we can
construct t h e spinor equivalent of, for example, t h e gKd A 2 = 6%% and express t h e
result in terms of spinor gKd's and gpd's, eg
8&$
Sfrg
6§6g b%b%
A = ^ = =
2
1
_
rcAB—A'B'
,
— 21rCLl^C'D' +
-ABcA'B'-i
*CD°C<D<\'
Similarly we can easily construct t h e spinor equivalent of t h e tensor gpd II2 = 7T^.
One has
n
2
=
7$
1
_
tcAB cA'B'
— ~2[°CD°C'D'
J 1
=
sgsg
ftsg,
,
7r
^AB-A'B'-i
CD C'£)'J7r
Spinor equivalents of all gKd's a n d gpd's u p t o four double-index gKd's a n d gpd's
are collected together for reference in Appendix E.
7
Tensor equivalents of gKd and gpd spinors
Whilst most attention focuses on spinor equivalents of tensors, it is also of interest
t o exhibit t h e tensor equivalents of certain numerical spinors. T h e various derived
formulae are collected together in a table a t t h e end.
Multiplying t h e fundamental spinor identity
£
£
CD EF
by e
A E
e
B F
£
+ £CE FD
"I" &CF&DE
=
0
we obtain t h e most useful result for t h e numerical gKd spinor
£
£cd
—
o .
CD
3,4
T h e following spinor equivalents are well-known , in mixed-mode
form
&b
e
a b
= e <*i(6£6*6*6§:
ab
a i
-
cd
6*6*6*6%)
and in covariant form as
9ab <=>
Sated
£ab£a'b>
i(£AC£BD£A'D'£B'C'
~
CD
£Ad£bc£A'C'£b'D')-
CD
Multiplying t h e first of this latter pair by e e '
6
Sab£
£A'B'
g** gives
— °AB°A'B'
Q
9ab-
Here we have t h e tensor equivalent of t h e product of two gKd spinors. In using
spinors here, it is convenient t o employ t h e abbreviations
a'
= $g
it
= *#g.
We may therefore write t h e above tensor equivalent as
AA '
sf*9cd
(a A A ' spinor).
We go on t o obtain t h e tensor equivalents of An', HA', I H T spinors. Expansion
of a A n ' is
#g
*$& = W&g
-
+
= 6^6*6^16*,
— 6^6*6^,6*,
— kAcA' cBcB'
= 0 0 ,0 0
C
C
D
D I
s&sg)
+ 6^6*6^,6*,
JiAcA'cBcB'.cAcBcA'cB'
— 0 0 , 0 0 , + O O b ,0 i
D
D
C
C
c
D
D
C
—
6^6*6^16*,
—
cAcBcA'cB'
OpOcO&Ojy.
T h e first two terms comprise A2 t o give t h e tensor equivalent
6c6c<6*6*y
-
6p6o'6*6*,
6$
and t h e last two terms, with a factor i, form the mixed alternating tensor. Combining,
we have t h e tensor equivalent of a A n '
ab
An' =
6**rt:*; &6%-ie .
i
cd
Taking t h e conjugate gives
AB
TTA' _ -*1 1 A = TT
CD
RA'B'
d , ,
c
D
O
cab , • at
d
+ IE
cdc d
Adding these confirms t h e result for A . Subtracting t h e m gives a 'new look' beautiful
appearance for t h e spinor equivalent of t h e mixed alternating tensor, easily remem­
bered,
2
L
2°A<n' —
W
i
)
~
2
D
d
'
e
°c'D')
cd-
We now have tensor equivalents for A A ' , A l l ' , IIA'. To determine t h e remaining
product, a I I I I ' , expand it directly
Now we have t h e spinor equivalent for n
2
T h e R.H.S. is t h e s u m of t h e first a n d fourth terms in t h e I I I I ' equation preceding it.
We substitute for this s u m and write (with abuse of language by mixing tensor and
spinor notations)
1
-AB-A'B
„ab _ cA cB cA' cB' , cA cB cA' cB'
C D C ' D ' ~ cd—
% " D £ ) ' C ' + D C°C' D'-
7 I
7 I
n
6
d
d
d
/i\
\ )
d
l
Having reached here we still do not know any tensor equivalent of t h e last two terms
in t h e above. Holding this latest nn' equation in abeyance we proceed on another
tack. Using t h e 2-index g K d a n d g p d we can isolate a product of two single 5's
Taking t h e conjugate a n d then interchanging indices A',B', gives
* l XA'B' , ^A'B'\
2\~ C'D'
+ C'D')
d
lr
_
cA'cB'
°D'°C'-
Multiplying these last two brings in t h e AA', An', nA',
A
nn'
1
* l X & JsA'B
, cAB —A'B'
^AB cA'B' , —AB ^.A'B'\
-£\—O
0 , i -f- 0
TT , i
— TT
Oi i
-f- licD ^C'D')
CD
c
D
C
D
c D
CD
c D
_
cAcBcA'cB'
~ °C D D'°C'
•
0
0
Except for t h e last nn' term in parenthesis, we have previously obtained tensor
equivalents for all other terms, AA', An' — nA'. Thus
\(-g*g«
- 216°" * + nn') = 6£6Sfi£sg.
(2)
Take t h e conjugate a n d get
al
\(-9 '9 i
+
a
ab
2ie
cd
+ UIl')
=6*6*8*6%.
Add these two, a n d then substitute for t h e two R.H.S terms in (1) t o get
i(-p*0«i+nir) = n f f - i £ .
Solving for II IT we finally obtain its tensor equivalent
nir
=
^ : ^ 2 ^ - ^ c d .
B
If we then use this result back in (2) we find t h a t t h e spinor 8^6*6*,8 .'
equivalent
I
has t h e tensor
We tabulate in Appendix F , t h e spinor/tensor equivalents of t h e numerical tensors/spinors we have come across. Mixed mode spinor and tensor indices can be
lowered by t h e £43, £a'B'> 9ab- T h e covariant form of t h e mixed mode spinors is in
the line directly below each of them.
7.1
Appendix A.
gKd combinatorial formulae
Definition formulae
Single g K d formulae
^
= n = i
c f c t
(
0
* * * °il~ip*P+l-*,
^
4
K
(n-ff)l A~4>'
= jftjj,
c
a
r
e
t
d
e
n
o
t
e
s
o
* * °il...i i i...t
P
k 6 . < = n,
p +
B
k7.
m
i
-
t
t
e
d
^
I™
= e
P)'°ii...3
P
"
"
^
=
nl
Product of gKd's
°ii
K
A
U
•ipip+i-i,
C
*
-
0
3\...jpip+l...J,
W
*i...*pi -l...t,
PJ- (n-,)i
-
lH
y-(n-g)!%...fcp
!
k 1 2
kn. C:£ <::t=*> CX>
kl4.
*"
L U
1
t
j
fc
'
kl6.
+1
e* • V>«j>+i"*»S .» "i
d
{
k
j
}
t
d
{
k
a
}
n
•
d
m
1,
=
Jl-Jp»p+l-«n
- 6
- **£ t i = p ^
(n-p)\e* -Wp+i-in
v
V*
m
r)' j\...j
v
{- 0 ^ ),
h
p
kl7. 6
{ki}
d
{ f c 2 }
-
(J>!j 0 ^
7.2
gpd combinatorial formulae
Appendix B .
Definition formulae
oil-in
-
f
oh-in
,
Single g p d formulae
o»l...*p
P - ^...ip = P
3
.
_c
!
U...ip^ip
™
ii-ipip+i-t,
_
~
ii...*....^
(n+g-l)l
(caret denotes omitted index)
U-tp
-
"ii-ipVn-** ~~ (n+p-i)l ^ i i - V
P°'
f
P *
^ii...*,
(n-l)!
—
'
ii.-.tptp+i.-tn
7l
P ° * ii.«j>vn-»«
P°*
7 1 —
-
°t)
t f
P
il-ip
J^tv^l}^
_
—
J'i- ip
(n+p-i)!
'
^ . . . i n
::
-
(n-l)l
P r o d u c t o f gpd's
p l U . 7rjj...j„7r^ ...fc
1
1
tt* "*" 7 r ? "
nl2
P
A t f
A O
7r
jl-.3pjp+l..j, % « .
ki...kpip ...i
+1
q
A
P
1
7r
7r
- ii...ip ii...J
w
W
1
Pi' (n+p-l)J
1
1
= jjajlj,
1
3l
Ipitf.
ll 9 7
7 TT
W
{ k i }
77 T
T
j
p+1
{ k l }
{ k j }
n
... 7
< nT-
{kr>
0 }
!
(n-l)!
n
h - 3 p
— ? • (n+p-l)l ^fci.-fcp
1
il
P
...jpfcp+i
p* ' " ^ ' " ^ T r j ^ J ' j = ( n - p ) ^ " ^ " *
P I T . P "^ "^P ... pi ...i
_
p
1
W
lr
* ^Si-ipip+i-iq
pl6.
1
^ k p + i -
q
A
2n
= n> ( ~ )
i B
- ^h-ipjp+\:.j
P **
P
1
- P ! 7Tfci...Jfcp)
I>
- 7 T 0} r—
*
' TT*
Jl—ip''
{ i }
1
1
= n!
P l 8 . p^Pir...^
{
}
{
l }
{ f c p }
I>20 - *
7T
7 T{ i } —
- fljlV
P
{ *'! } 7^T{ f *e } • • • ^
IPO 7T
/
U
{
7.3
Appendix C.
7.3.1
C
A
P r o d u c t s of gpd's, gKd's, alternating a n d symmetric symbols
'
c2.
Combined gKd, gpd formulae
^j\h-3p°i\h.-l
~ L>*=1
q
P
7.3.2
^
6*£±
n
0
3l-3.-3p 3sh
= c = i (-i)
a +
V-
Iq
i a
Identities w i t h combined gKd's a n d gpd's
o 1
11. ASwjag-nS =
12.
^ *3 5 -
i4.
«T * 5 C l <
tt^
1
= 8 8%<K'£
=
8«2f<
=
-
o2.
= 0
% TTjlp
4
o - ^ « ? ^ « / h « . =
o5.
ml-
m2.
m3.
=
2
S
% <
=
Q i r ^= 0
0
0
fi^^TrrjTr^^Tr-^O
<*' ^
^
^
f
r =
0
o7. ^ ^ ^ C ^ ^ - =
h
0 8 . 6^ 6%6^-n^
w
o9.
# * $
f
i i * « 0
= 0
0
7.3.3
A l t e r n a t i n g a n d s y m m e t r i c s y m b o l p r o d u c t formulae
c
£ ^ 0 tabcd ~ °abcd
efBh
p Pabcd
efad
p Pabcd
= Kobcd
=
= 2!#
/> "pa6cd =
e/C
12*2
e
e
abcd
= 3!^
P^Pabed
£
Cabal
= 4!
P ^ Poked = 4!
=
24*1
7.4
Appendix D .
7lB =
gAB
°CD
*B
fiCE .
FG
f*CD ^FG '
cAB
°CD
£
^
A
B
B
Spinor Product Formulae
$D ^ F G
—
~AB
CD
n
^FG
„CE _ _ B qr-AE I ~A
"FG ~ "D "FG + "D
AB
CD
CD"FG
26 B
n
F
-BE
"FG
AB
FG
= 27T
B —-AE
~A —-BE
— TV TTpQ — II 71 p
~CE
"FG
D
D
G
£ c d
e
A
7.5
B
Appendix E.
Spinor equivalents for the gKd and gpd
Reference formulae for t h e spinor equivalents of t h e gKd and gpd are given below. It
should b e mentioned t h a t there is a good deal of interplay between t h e gKd and gpd
in such specifications, there being a variety of ways t o express expansions of some
gKd's, gpd's (and also their spinor equivalents). Simple examples at t h e end of t h e
section exhibit application of t h e gKd and gpd.
7.5.1
I. Spinor equivalents of the gKd's
_
A = Sf =
2
d
6*6$
l
CD^C'D'
% % «J
A s = 6£ =
f
6* 61 6)
6* 61 6}
—
°d°ef
°e°df
+
°f°de
d
* ^D^D'i^EF^E'F'
^F^F'i^DE^D'E'
+
+
7 r
BF^E'F')
—
^E^E'i^DF^D'F'
^DF^D'F')
^DE^D'E')]'
W e can expand t h e gKd with 4 (upper/lower) indices by a Laplace expansion of its
first two rows a n d complementary minors. T h e result is
A4 = «
61
6\
61
6i
=
61
6)
6)
6)
6- 61
6) 61
61 81
6\ 6%
T h e R . H . S . can b e w r i t t e n as t h e sum/difference of permanents
6%
VH
6f
6°*
°gh
f
T h e spinor equivalent of 8^^
gab
"eg
fid
"eg
6%
6f
1
T
is
7r
~(6EG E'G> + ^EG^E'G^i^FH^F'H'
^FH^F'H')
7r
~K^£ff E'if' + ^EH^E'H^i^FG^F'G'
+(6pQWpiQ,
C
7 I
^i^GH^G'H'
S
+
+ ^G8P,Q,)(8EJJ'K ,,JJ,
~(^FH F'H'
of g K d ' s , if desired
*FG F'G>)
+T^EH^E'H')
^H^F'Hdi^EG^E'G'
"^EG^E'G')
1
~^ ^H^G'H'^EF^E'F'
+ "^EF^E'F
)]•
7.5.2 II. Spinor equivalents of the gpd's
Iii
=
^<5£
A
n =7r£
2
6 6%
=
6%6%
9
$ 3 9
3
i^^O'C^SF^B'F' +
7 r
£F
7 r
A
A
B
B
B ' F ' ) + S 6 ,(6 p8 ,p,
)
)
^CF^D'F')
Since p e r m a n e n t s only involve positive signs, t h e following Laplace expansion is also
clear
61
6
l
6
l
04 = 7 $ $ =
•7 *
T h e R.H.S. can b e written as t h e sum of permanents of permanents, if desired
+
+
T h e spinor equivalent of T T ^ ^ is
^JSH^E'irX^G^F'G'
•'"(^FG^F'G'
7 r
7 r
7 r
N
FG F'G')(^BH^B'H'
r
r
7 r
,
7 r
GH G'H')(^EF^B'F' +
JR
' EH' E'H')
LT
+(^FH^F'H' +' FH' F'H')(^BG^B G'
"K^Gff^G'tf' +
7 r
FG F'G')
7R
' EG E'G')
7 r
7 r
BF E'F')]
7.6 Appendix F. Spinor^Tensor equivalents
T h e covariant
f o r m o f t h e m i x e d m o d e s p i n o r s is in t h e line d i r e c t l y b e l o w e a c h o f
them.
Spinor
Tensor
6$
sab
0afc
CABGcD
— CAC&BD
£ac£bd
+
~
|
EAD&BC
£ad£bc
mo
+*3S-
e
€AC BD£A'C'£B'D'
MSS #
=
+nA')
^AD^BC^A'D^B'C'
5
1
SACSBDeA'C'SB'D
+
£ac£bd — <7ad<7bc
i(AA'+nn')
ffac&d +
^AD^BC^A'D'SB'C'
9ad9bc
£cd °* = £ ° * erf
*{£ac€bdGa'D'£b'c
1
—
eADGBceA'cea'D )
£abcd
^ & & = AA'
Raided
^AC^BD&A'C'^B'jy
—
^AC^BD^A'iy^B'C'
GAD€BC£A'D'£b'C'
= SAB^CDi^A'C'SB'D'
^ g ^
1
— {SACeBD
(sacSbd
+
—
—eAcSBDSA'D'SB'C
1
SAD^BC^A'C^B'D
1
SA'D'Sb'C )
= n A '
sacSbdSa'c'Sb'd
"^cd^c'd
1
~
+
+
=
^ad^bcSa'd'Sb'c'
<7ac<7bd — 9ad9bc + *So6ed
€Ad£bc£A'C'£b'D'
£ADSBc)eA'B'£C'D'
2*3 - g^ga
nn'
+ sadSbc)
x (ex'c'fiB'B' 4- e x ' z y e B ' c )
29ae9bd + 29ad9bc — 9ab9cd
^(SocPbd + ffodSbc — gab9cd — i£abcd)
1
R.L. Agacy, Generalized Kronecker, permanent delta and Young tableaux
applications to tensors and spinors; Lanczos-Zund spinor classification
and general spinor factorizations, PhD Thesis, London University (1997).
2
0 . Veblen, Invariants
versity Press, (1927).
of Quadratic Differential Forms, Cambridge Uni­
3
R~ Penrose and W. Rindler, Spinors and space-time, Cambridge Univer­
sity Press, Vol. 1, (1984).
4
F . Trautman, F. Pirani, and H. Bondi, Lectures on General
Brandeis Summer Institute in Theoretical Physics, (1964).
Relativity,
PHYSICS AUXILIARY
PUBLICATION SERVICE
Document No:
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Journal Reference: Journal of Mathematical Physics
Vol. 40, No. 4 - April 1999 [p.2055-2063]
Title: Generalized Kronecker delta and permanent
deltas, their spinor and tensor equivilents and
applications.
P A P S Title: Reference Formulae
Authors: R. L. Agacy
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