Generalized Kronecker and permanent deltas, their and equivalents — Reference Formulae

R L Agacy1

L42 Brighton Street, Gulliver, Townsville, QLD 4812, AUSTRALIA Abstract

The work is essentially divided into two parts. The purpose of the first part, applicable to n-dimensional space, is to (i) introduce a generalized permanent delta (gpd), a symmetrizer, on an equal footing with the generalized Kronecker delta (gKd), (ii) derive combinatorial formulae for each separately, and then in combination, which then leads us to (iii) provide a comprehensive listing of these formulae. For the second part, applicable to in the mathematical language of , the purpose is to (i) provide formulae for combined gKd/gpd spinors, (ii) obtain and tabulate spinor equivalents of gKd and gpd and (iii) derive and exhibit tensor equivalents of gKd and gpd spinors. 1 Introduction

The generalized Kronecker delta (gKd) is well known as an alternating or antisymmetrizer, eg S^Xabc = d\X[def\. In contrast, although symmetric tensors are seen constantly, there does not appear to be employment of any symmetrizer, for eg X(abc), in analogy with the antisymmetrizer. However, it is exactly the combination of both types of symmetrizers, treated equally, that give us the flexibility to describe any type of tensor symmetry. We define such a 'permanent' symmetrizer as a generalized permanent delta (gpd) below. Our purpose is to restore an imbalance between the gKd and gpd and provide a consolidated reference of combinatorial formulae for them, most of which is not in the literature. The work divides into two Parts: the first applicable to tensors in n-, the second applicable to 2-component spinors used in the mathematical language of General Relativity. In Part I, section 2 gives definitions, illustrations and tables of combinatorial formulae of the gKd and gpd. The large section 3 derives an array of formulae for both 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 the 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 Part IE, in section 5 we determine the spinor equivalents of all gKd's and gpd's up to 8 (ie 4 up, 4 down) indices, the most useful for the tensor/spinor languages of Relativity — these are listed in Appendix E, I and II respectively. In section 6, in reverse, we derive the tensor equivalents of gKd and gpd spinors and sums of products of them — the results are collated in Appendix F. Conventions are as follows. In Part I, general indices range from 1,... , n. Bolded

indices are to be regarded as fixed. An index set {k} stands for {fcj,... , kp} (p < n)

and an index set {ki} stands for {fcn,... , &ip} etc, all indices ranging over 1,... , n. The index summation convention is understood. All index sets are permutations of each other. In Part II, two-component spinor indices, are in capital lower case roman.

PART I

2 Definitions and illustrations of the gKd and gpd

Complete symmetry of (a tensor's) indices, as opposed to total antisymmetry, is manifested by all positive signs in any p-linear expression. Whereas the gKd's an­ tisymmetry comes about through a (interchanges of rows/columns or indices changes the sign), total positive or pure or permanent symmetry, as we term it, comes about through the use of a permanent Then in complete analogy to the gKd we introduce the generalized permanent delta or gpd. This is defined, like the gKd determinant of a , except that we take all positive signs. We use the kernel letter 7r to denote a gpd and double vertical lines for the permanent of the defining matrix. The gKd and gpd are completely complementary to each other and are defined, for p (< n) distinct indices, respectively by,

ft 3p

7T, 31 -3v 31-3p

3p 4 where the first is a determinant and the second a permanent. The gKd has value +1

(-1) depending on whether (ji,... ,jp) is an even (odd) permutation of (ti,... ,ip). The gpd has the permanent value +1 for any permutation of the index sets. Note that 7r£ =<5". The gpd is a complete symmetrizer (permanent symmetrizer) and now one easily sees that ir^Xabe = 3!X(«je/). In terms of bracketed and parenthesized notation, the gKd and gpd are

v n 3\-3p ™n-3p P-^Uin 3PY A simple but important interaction between the gKd and gpd is that in any expression containing them as a product, where there is a summation on a pair of indices between them, such expression vanishes. For

,.o...6.. ..a...b.. ...6...0.. ..0...6.. 7T...o...b.. . = 6 7T...0...6.. . = 0.

From this it also obviously follows that the product of any gKd and any gpd with two or more common contracted indices vanishes. If A = [a*-] is an n x n matrix its determinant and permanent are:

l—3n 3n detA=-TaV SI n! 31 a'j3n «1—*n in'

The Riemann tensor Robed and Lanczos tensor Ly* are Young tableau tensors in their algebraic symmetries and obey the partial symmetries and antisymmetries as determined by the tableau representing them (Agacy1). And so here the interplay of symbols for both permanent symmetries (ir) and antisymmetries (6) manifests itself in the one-line definition of each of the Riemann and Lanczos tensors in respect of their algebraic symmetries1

12 ab vcd V nqs ^-efgh 1 — o "fpk Limn- 3 Generalized Kronecker 8 3.1 Single generalized Kronecker 6 In this section many of the derived formulae have been presented (but without deriva­ tion) long ago in Veblen2. In the following definitions of numerical symbols the range of indices is from 1 to n. In any set of p indices ... ,ip}, it will be understood that for p < n the indices are all distinct The generalized Kronecker 6 (gKd) may be defined by

% • • • 6i n-3p

The interchange of any two superscripts (or subscripts) interchanges the correspond­ ing rows (or columns) of the determinant thus changing its sign. Additionally if any particular superscript (or subscript) is not contained in the set of subscripts (or su­ perscripts) then that particular row (or column) consists of zeros, the determinant vanishing. Hence only if the set of superscripts and subscripts are permutations of each other does one obtain a non-zero result. The gKd can then equally well be described as

+1 if i\,. •. , iP is an even permutation of j\,... , jp

—1 if i\,... , ip is an odd permutation of j%,... ,jp

0 if ti,... ,ip is not a permutation of ji,... , jp

Unless specifically stated or obvious from the context we will in future take it to be the case that the sets of superscripted and subscripted indices are permutations of each other.

Note that the indices iP, js range from 1 to n but that there are p (< n) indices. When p = n (= dim V), and only then, we will also write

^l.-.tn — °ii...in ana € "~ °1 n

and refer to e as the alternating or permutation symbol (and in General Rela­ tivity (GR) the Levi-Civita symbol as well). It takes the value +1 when the set

{»i,... ,in} is an even permutation of {1,... ,n} and —1 when ,in} is an odd permutation of {1,... ,n}. The gKd is a tensor (transforms like a tensor) whereas and e*1 •*n are not We modify Einstein's summation convention by indicating fixed, repeated indices by bold type so that there is no summation over these indices, which will occur as matching superscripts and subscripts. This notation makes clearer the derivation of most of the following formulae. Unbolding or unfixing means summations go ahead. In considering a product of the two gKd's

0 h-3p °ki...kp there is no summation over the k's. For brevity we sometimes denote a set of summed, repeated indices {ki,... , kp} by {k} and a set of fixed, repeated indices by {k}. Clearly for a non-zero result above the sets {i}, {j} and {k} must all be permutations of each other. As the set {k} is a fixed permutation of the set {i} and the set {j} a permutation of {k} then the resultant value of the product is just the value of a generalized Kronecker 6 as the product of those permutations. Our permutation multiplication convention is from right to left

/ki...kp\ (ix...iv\ _ /ii...ip\ Vii\ki...kj \ji...jp)

Thus

cki..Jtcki...k,p cti...tp _ cti-tp

c °h-ip °k!...kp - 'ii... jP-

In particular we have a result used often Formula kl.

0il-in - °l-..n °jl...jp - 6 e3i-3n-

Formula k2.

r ffi-h-ih _ ( I \P+s fii. eh -i—ip u3l-jp-l3p Z-/^ ' v3pv3\-3p-\ 8=1 where a caret ( i ) over an index indicates its omission from the list.

Imagine the gKd as a p x p matrix with the indices ... ,ip signifying rows and

the indices ji,... , jp signifying columns. Expand the gKd by the last column jp. Note that in the determinantal expansion, the sign associated with the element in row 1 and column p is (—l)p+1, and so on. One then has

2 p+p CfcS = (-ir^i&t. + (-ir ^CA+• • •+(-i) ^::fcl and so we have the result

p r«i...ip_i»p v-^ / —1V+* tf? X«i-i«-H> 3\-3p-\3p Z~J^ ' 3p 3\-3p-\ 8=1 Now consider the expansion of the gKd ^"^j^ by its last (p 4- l)th row (see Formula 2). We get fcSK - (-ir^ct*+(-ir3^1^^+• • •+(-ij^&itt where a caret ( j ) over an index means that it is excluded from that set. There are p + 1 terms in the sum. Then if we let jp+i = ip+i so that there is a summation, we get the single common index formula for a gKd

Using this result, now for two repeated indices, gives fcttStt - (— * - UfcttS - (« - p -1)(« - p)t. %• Continuing to add repeated summed indices leads to the most general result for a single gKd Formula k3

*i!:::tfev:<: -(»-«+1) •••(» - p)#:J

= o

which clearly gives the correct result for p = q. This formula is equation (8.1) of Veblen2. A particular case of Formula 3, when q = n, is Formula k4

c*l...*pip+i...i,» =.M...ipip+i...in£, .. . —

Another particular case of Formula 3, when p — 0, should be noted Formula k5

This formula is equation (8.2) of Veblen2. A useful specialization, when q = 1, is the simple result Formula k6 $ = n. Another specialization of Formula 3, when p = 0, q = n, is Formula k7

This formula is equation (8.3) of Veblen2. 3.2 Products of gKd's Consider the following product of two gKd's

gil-ipip+i-U £Jp+i-jq

Jl—ipjp+l—Jq kp+l...kq'

Note that the fixed set J = {jp+i, • • • , jq} must take on the values of the specific set K = {kp+i,... ,kq}. Unbolding the indices of J so that there are (q — p)\ per­ mutations of it and hence (q — p)l summations, gives the key result — (q — p) is the number of common indices Formula k8

C*l...tptp+l...t, cjp+1-..jq _ / \i ctl...tptp+l...t,

°h-3pjp+l~3q °fcp+l-.fc, ~~ W P>' °h-3ph+l-V

2 This formula is equation (8.6) of Veblen . Letting the indices kp+i,... , kq = ip+i,iq, and using Formula k3 for (q — p) common indices on the R.H.S. gKd, results in the first partial common indices product formula Formula k9

lplp+1...t, "3l-3p3p+l-3q "tp+l-tq V« V)-Vji... 3pip+l—*q

(n - p)\ .tp (n-q)

This formula is equation (8.7) of Veblen2. A second formula for the product of two gKd's comes from unbolding two sets of indices in the obvious relation (a right to left multiplication of two permutations on reversing the order of the two L.H.S. terms)

ctl...ipip+1-..lq rJl--JpJp+l---Jq __ £>l...tpip+l...iq _ cil...ip

m Jl--Jpjp+1---Jq *l...fcplp+l...iq fcl...fcpip+l...lq kl...kp

Some care is required now. Consider unbolding the set {ip+i,... ,iq}. Each index ranges over not n, but (n — p) distinct values, values complementary to the values

of {*!,... ,ip} = {hi,... ,kp}. There are (,!*) ways of selecting a set of values for

ip+i,... ,iq in any order from the set of (n — p) values. But there are (q — p)\ permutations of these for the summation. Unbolding thus introduces the factor (q —

= P)' (g-p) (nlg)i — equivalent of selecting (q — p) ordered values for ip+i,... ,iq from a set of n — p values. An alternative idea is to consider the product of the two

gKd's by focussing purely on the summation of ip+i,... ,iq with (n — p) for n, and (q — p) for q (see k5) as = ^J^Z^lp^ to obtain the factor. This gives the intermediate result

fii...ipip+1...iq oJi..JpJp+i..Jq _ (n — p)\ rfi...tp

ji..jp+iJp+i...jq fci...j^,«p+i...t, ^n _ °fci...*p-

Now unbolding the set {ji,... ,jq} introduces q\ permutations and we get Formula klO

n 1 fi1...irip+1..,iq ch-jpif+i-U \ ( ~ P) - ch-ip u v h-jpip+i —in *i -kpip+i ...t, H• ^n_qy ki ...ftp •

A specialization of this result, when p = g, is Formula kll

Equating the indices {k\,... ,kp) = (ii,... ,ip) and then employing Formula 5 gives Formula kl2

fci&"*'=p\ n!

A specialization of Formula 10, when q = n, is

n! n °ii...ipip+i...in °fci...fcPtp+i...t„ - l P)-%...fcp-

Two further specializations of this last can be applied here. First, when p — 0 we get Formula kl3

Second, when p = nwe get

Equating the indices (fci,... , kn) = («i,.. - , in) confirms the first specialization. We now examine product formulae involving the alternating symbol. For the first, it is easy to see Formula kl4

e«l...*ptp+1...t„ gip+l.-jn _ / _ \j eii...ipjp+1...jn *p+l»-*n > * '

as there are (n — p)\ permutations of the common indices. Secondly, unbolding the (n — p) indices in

°l...p,p+l...n °h...jplp+l...lu ~ °Ji...ipip+i...iB

which then involves (n—p)\ summations, reproduces an earlier result Formula 4. This formula is equation (8.5) of Veblen2 Formula kl5

1 eh...ipif,+i...in ,. . . = (n — vV S*. "*?

This formula is equation (8.4) of Veblen2.

A little consideration is now given to products of more than two gKd's. For a compact notation we use {A;} to stand for {k\,... , kp} (p < n) and {ki} to stand for

{kn,... ,k\p} etc. The earlier relation o^"'*^ 6^.'"^ can be abbreviated to

Then more generally, a relation of the type

0 %...ap °bi...bp- °ji...jp — h-jp- can be compacted to Formula k!6

fi W {k2} %} ~ %} I" °ii...JP)'

Each of the sets {k,} = {kn,... ,kIp} is a permutation of {i} = ... ,ip} as also

is the set {j} = {ji,..- ,jp}.

By now having familiarity with unbolding fixed repeated indices and counting the summations we should be able to see the result

°ki...kp°h...3p P-°3\-3P or briefly

From this we have for a product of three 5's

and more generally Formula kl7

All combinatorial formulae for the 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 = n2 — n whereas 7Ty = n2 + n. In analogy with the gKd it is possible to define a symmetric (or permanental) symbol p in terms of a gpd by p*1-** = ^.'".n ^ Pn...»« = Like the alternating symbol e, the symmetric symbol p is not a tensor. Unlike the alternating symbol which satisfies = it

11 is not the case that = p '"**Pji.„jn- For example, with n = 2, 7r|j = 6, but

1 n p**'py = 2. In general p* "'* p«1...»n = n!. As it is not possible to match a product of contracted p's with the gpd it is convenient to define a new symbol 7r, like a factorial function, to accommodate such a product. Formally we define a factorial symmetric symbol (also tensor) by Formula pi &£&-

0*1 —in t, i 7r il-i»=P Ph-Jn- Symmetry in upper and lower indices is consistent. The first equation shows the factorial nature of the symbol Formula p3

oti...ip

The kernel letter 7r in n is used to indicate permanent symmetries like the gpd; however, 7r is not a permanent to be evaluated from a matrix. The symmetric symbol p is useful for total symmetrizations, eg of functions, as in

ffi-^ffa,... ,*J or pM-Zifo) • • • /„(xin) 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 to establish formulae for the gpd corresponding to those for the gKd. Proceeding in a similar manner to the gKd, and fortunately because we don't have

to worry about signs with regard to a gpd, we can expand the gpd by the column jp to get Formula p4

p il-ip-lh _ V"* ci. ti...t....ip n3l-3p-l3p Jp 31-Jp-l * «=1

Let jp = ip and also change p to p + 1 to obtain

The result can be extended beyond one common index, so that more generally we have Formula p5

*i-«p»p+i-«« _ (w + q — l)t M-ip '31 -ipip+i•••»« (n-fp—1)!

A particular case, when q = n, is Formula p6

h...ipip+i...in _ (2n 1)! ii...«p "ii-ipip+i-•*» (n + p - 1)!

Another particular case, when p = 0, is Formula p7

= (n + q- 1)! (n - 1)!

A specialization, when g = 1 is the simple result Formula p8

ttJ = n =

Another specialization of Formula 7, when q = n, is Formula p9

(n-1)!"

4.2 Products of gpd's The reasoning for a product of two gpd's is identical with that for gKd's. Thus we have the analogous result to Formula kll Formula plO

1T 7r 3i-3p ki...kp - P- ^...kp- Equating the indices (ki,... , kp) = ... ,ip) and using Formula p7 gives Formula pll

(n+p )! e-^=p!3l-3v *i—*p :.?

A specialization of this to the case p = n gives Formula pl2

j^

Arguing in exactly the same manner as for the derivation of Formula k8, but for the gpd now, gives the analogous formula Formula pl3

Specializing to (fej,... , kp) = (ii,... , ip) gives Formula pl4

n ti...tp»p+i...t, ^p+i...ip _ , i + g *i...*V.

"31«.3pip+l-j«"ip+l •*« (n +p _ 1)J 31-ip'

Following the more involved reasoning for the gKd leading to Formula klO pro­ duces the analogous result for gpd's Formula pl5

h...ipiJ)+i...iq ii-ipip+i-iq _ | (n + q — 1)! ii...*p

n h-ipip+\-U"fci...fcP*>+i...t4 «• ^n p _ -g j "a* •

A specialization of this result, when (fci,... , fcp) = (ij,... ,ip) is

*1"t« 9 (n-1)!

a result already obtained as Formula pll. The further specialization, when q = n, gives

2n 1 1 _*i...**_n...j» _ n\ ( ~ ) n "h-in^i-in ' (n_x)!

confirming again Formula pl2. Finally for a product formula of the symmetric symbol with a gpd, it is easy to see Formula pl6

pti...»pip+l...t„7rJp+l"in _ £n _pjjp*l...tpjp+l-Jn as there are (n — p)\ possible permutations of the common indices. For a product of p's with repeated indices we have Formula pl7

... n\ 0*1—H>

When there is summation on all indices we get the specialization Formula pl8

Formulae for products of several gpd's are obtained by the same reasoning as for gKd's. Thus we have Formula pl9

Unbolding the bolded indices so that summations occur gives us Formula p20

All combinatorial formulae for the gpd and symmetric symbol are collected to­ gether and numbered as above, for reference in Appendix B.

5 Combined gKd and gpd Here we derive formulae for combined products of gKd's and gpd's where at least one factor is a gpd or symmetric symbol and another is a gKd or alternating symbol. We saw earlier the useful result that the product of any gKd and 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 and another matching index pair for subscript and superscript. Consider a combined product of a gpd and a gKd with (necessarily) at most one superscript matching a subscript as m xii^«.^^2».i**- Expanding the gpd by its (first) row i\ leads to Formula cl

*ita-..tV rfcifca.-.k, _ rrt! ia...»V • riiia.-.ip , , c*i «.••»> l cfcika-*,

P

_ V *a o*ik2.»*« 1 0 ~ *h..-5-~iv i.h...iq ' 8=1 The unexpanded product can, at the same time, also have at most one gpd sub­ script matching a gKd superscript as in ^l^.'.^ii^."!*'-

For the combined product P*1"^^^"^, expanding the gKd by its first column i\ we have Formula c2

0*1 «.»n£fcl*2...kn _ 0*l—*n 1 a+1 5 fcn " *li2—in " E(- ) * fc^» .8=1

il n n Consider the converse relation ^ "'* T^^".'jn • The expansion of the gpd by column i\ is a summation with all positive signs Formula c2

n

t*l...tn_klka...kn \ ^ ,k«*3—*n-rkl— *«—*n c /l*u'a...in _ Z-/ » •>» ' 8=1

5.1 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 the group algebra of the symmetric group Sn. The derivation of, rather than the background to, these identities is what concerns us here. The orthogonal identities are visually obvious by having blanks (dots) in gKd's and gpd's, eg 7r^, where arbitrary free indices could be placed. In order to prove the identities it is necessary to call on some simpler identities, which, although established in generality earlier and listed in tables at the end, is pertinent to be 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. The basic formulae below can all be inverted, that is, all superscripts lowered and subscripts raised. Reference to a basic formula implies either its use or its inversion.

(1) 6tJ ir- = 0 (repetition of two indices in a gKd and gpd) The fourth term S^S^S^S^^ = 46£6&ir%ir$ using (2). Interchange dummy indices i, k and j, I, and as a gpd is totally index symmetric, the value of the term is unchanged; its index rearrangement becomes 46^^7rf/7r?/> which is the new first term. Hence the sum of the first and fourth terms in X is %&%&Mtfk1tfi •

Interchange of *, k and j, I in the third term S^SQS^Sj^n^itji t which does not change its value, converts it to the second term, so that X includes the expression 2^c^M^^r«7r« A certain facility is acquired after some manipulations with gKd's and gpd's. We see that we have

Interchange of indices i,k and j, I converts the third and fourth terms to the second and first terms respectively, so that

= - «

Adding this to the sum of the first and fourth terms previously leads to X = 12$&«8*«*$fc so that

^ « <

XP9 JCr« ~ma c*j ski ef ah _ nff^fTi^f gh

Derivation of Formula i4

+ CP«!*S (using (2))

= 6£?

16 Derivation of Formula i2

= 6^<9 - (flrt? - tfifiKg (using (3))

= " 2#r£* +M},r£ (using (5))

This result is recorded as Formula i2

Derivation of Formula i3 This result is the most complicated we encounter. We have

= -

Write the R.H.S. asX = 6%S£ nfi, where Y are the four terms in parenthesis. They can be expanded and then compacted

Y = nff^JL + - n#j«, - *J#ii*

Now reconstruct X as a sum of four terms, which will be treated individually. The first term is The fourth term S^S^S^S^iif = 4$*jE&it%'ii$ using (2). Interchange dummy indices i, k and 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^jh which is the new first term. Hence the sum of the first and fourth terms in X is BS^Sj^ir^i^. Interchange of i, k and j, I in the third term ^^w^i^pi71** ^i* which does not change its value, converts it to the second term, so that 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 that we have

Interchange of indices i, k and j, I converts the third and fourth terms to the second and first terms respectively, so that

" « +

Adding this to the sum of the first and fourth terms previously leads to X = 12^S« so that « « *™ « C

°ab °cd npr Kqs °tu °uw ™ik ^jl ~ ^"ab^cd^pr^a •

Derivation of Formula i4

= 6«X; + Cr

= ME}*J + 2£f (using (5) and (4)) This result is recorded as Formula i4

cpvwtu cqefnh _ o cpc/^jh °abe "pd°tvwnqu ~ °°ol>c7rdp-

Derivation of Formula ol

5 27r ^^^7= « ST = 0 (using (5), then obvious).

This result is recorded as Formula ol

Derivation of Formula o2

Off* = -2^*5-0 (using (4), then obvious).

This result is recorded as Formula o2

Derivation of Formula o3

«2<'«Si = «S«gS-0 (using(7)).

This result is recorded as Formula o3

Derivation of Formula o4

= 2«*(^ + ^.)

= 0 (obvious). This result is recorded as Formula o4

Derivation of Formula o5

= OT(2<>$ - 2nSJ^) (using (5)) = W(« + " - = 0 (obvious).

This result is recorded as Formula o5

Derivation of Formula 06

= 0 (obvious).

This result is recorded as Formula 06

Derivation of Formula o7

= 0 (obvious). This result is recorded as Formula o7

Derivation of Formula 08

/ctvuw ctwuv cuvtw 1 cuuitw\_*i_fcl — l°o6cd — °o6cd — °abcd + °obcd J^tv^uv) = 0 (obvious). This result is recorded as Formula 08

Derivation of Formula o9

= (™*« (8)) = 0. This result is recorded as Formula o9

9h ?!. %i9nP = 0.

Derivation of Formula ml Depending on the manner of verification one may be led to other unobvious iden­ tities. We give a few. The first is

= 2[M^-«,' + «f)

= 2[(w - TO€f - (W - + (W - = + («Stf-<«+nS«2)] = 2(«-«+*M)

= 2

°abcadqr7Vps ~ zoobc "dq •

Derivation of Formula m2 For the second, consider ^S^it^.n^. Interchanging dummy indices p,q and r,s

P t will not change its value. However the expression now becomes S ^S^irq {'np^. This result is recorded as Formula m2

Derivation of Formula m3 For the third, consider S^S^S^T^TTQ. We have

= 0 (obvious).

This result is recorded as Formula m3

All combinatorial formulae for the combined gKd and gpd are collected together and numbered as above, for reference in Appendix C. PART II

In this part we first examine the 2-component gKd and gpd spinors which can be useful for the mathematical spinor algebra of General Relativity. Secondly we find and tabulate the spinor equivalents of gKd's up to four double-index (4up, 4down) indices. Thirdly, the other way round, we derive and tabulate the tensor equivalents of gKd and gpd spinors. In the first situation since any product of a gKd and a gpd with two (or more) summed indices vanishes, a non-zero result can only be obtained if there is at most one summed index (up and down) and 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

The spinor equivalent of the basic Kronecker 6, 6% = ir% is SgSg,. From this we can construct the spinor equivalent of, for example, the gKd A2 = 6%% and express the result in terms of spinor gKd's and gpd's, eg

8&$ Sfrg 1 A = ^== _ rcAB—A'B' , -ABcA'B'-i 2 6§6g b%b% — 21rCLl^C'D' + *CD°C

Similarly we can easily construct the spinor equivalent of the tensor gpd II2 = 7T^. One has

1 J 1 _ tcAB cA'B' , ^AB-A'B'-i n = 7$ = 7r 7r 2 sgsg ftsg, — ~2[°CD°C'D' CD C'£)'J-

Spinor equivalents of all gKd's and gpd's up to four double-index gKd's and 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 to exhibit the tensor equivalents of certain numerical spinors. The various derived formulae are collected together in a table at the end. Multiplying the fundamental spinor identity

£CD£EF + £CE£FD "I" &CF&DE = 0

by eAEeBF we obtain the most useful result for the numerical gKd spinor

£ £cd — oCD. The following spinor equivalents are well-known3,4, in mixed-mode form

&b

ab ab e ai = ecd <*i(6£6*6*6§: - 6*6*6*6%) and in covariant form as

9ab <=> £ab£a'b>

Sated i(£AC£BD£A'D'£B'C' ~ £Ad£bc£A'C'£b'D')-

Multiplying the first of this latter pair by eCDeCD' g** gives

6 Sab£ £A'B' — °AB°A'B' Q 9ab-

Here we have the tensor equivalent of the product of two gKd spinors. In using spinors here, it is convenient to employ the abbreviations

a' = $g it = *#g.

We may therefore write the above tensor equivalent as

sf*9cd A A' (a A A' spinor).

We go on to obtain the tensor equivalents of An', HA', IHT spinors. Expansion of a An' is

#g *$& = W&g - + s&sg) = 6^6*6^16*, — 6^6*6^,6*, + 6^6*6^,6*, — 6^6*6^16*, — kAcA' cBcB' JiAcA'cBcB'.cAcBcA'cB' cAcBcA'cB' = 0C0C,0D0DI — 0D0D,0C0C, + OcODbD,0Ci — OpOcO&Ojy.

The first two terms comprise A2 to give the tensor equivalent

6c6c<6*6*y - 6p6o'6*6*, 6$

and the last two terms, with a factor i, form the mixed alternating tensor. Combining, we have the tensor equivalent of a An'

ab An' = 6**rt:*;i&6%-ie cd.

Taking the conjugate gives

TTA' _ -*-AB RA'B' cab , • at 11A = TTCD dc,D, O dcd + IE cd- Adding these confirms the result for A2. Subtracting them gives a 'new look' beautiful appearance for the spinor equivalent of the mixed alternating tensor, easily remem­ bered,

2°A

We now have tensor equivalents for AA', All', IIA'. To determine the remaining product, a IIII', expand it directly

Now we have the spinor equivalent for n2

The R.H.S. is the sum of the first and fourth terms in the IIII' equation preceding it. We substitute for this sum and write (with abuse of language by mixing tensor and spinor notations)

-AB-A'B1 „ab _ cA cB cA' cB' , cA cB cA' cB' /i\ 7ICD7IC'D' ~ncd— %"D6£)'dC' + dDdC°C'dD'- \l)

Having reached here we still do not know any tensor equivalent of the last two terms in the above. Holding this latest nn' equation in abeyance we proceed on another tack. Using the 2-index gKd and gpd we can isolate a product of two single 5's

Taking the conjugate and then interchanging indices A',B', gives

* l XA'B' , ^A'B'\ _ cA'cB' 2\~dC'D' +lrC'D') °D'°C'-

Multiplying these last two brings in the AA', An', nA', nn'

* l XA& JsA'B1 , cAB —A'B' ^AB cA'B' , —AB ^.A'B'\ _ cAcBcA'cB' 0 0 -£\—OCD 0c,Di -f- 0CD TTc,Di — TTCD OciDi -f- licD ^C'D') ~ °C D D'°C' •

Except for the 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 the conjugate and get

al ab \(-9 '9ai + 2ie cd + UIl') =6*6*8*6%.

Add these two, and then substitute for the two R.H.S terms in (1) to get

i(-p*0«i+nir) = nff-i£.

Solving for II IT we finally obtain its tensor equivalent

nir = ^:^2^-^cd.

B If we then use this result back in (2) we find that the spinor 8^6*6*,8 .'I has the tensor equivalent

We tabulate in Appendix F, the spinor/tensor equivalents of the numerical ten- sors/spinors we have come across. Mixed mode spinor and tensor indices can be lowered by the £43, £a'B'> 9ab- The covariant form of the mixed mode spinors is in the line directly below each of them. 7.1 Appendix A. gKd combinatorial formulae

Definition formulae

Single gKd formulae

^ =n=i 4 cfct (caret denotes omitted ^

0 K *** °il~ip*P+l-*, (n-ff)l A~4>' ** °il...iPip+i...tB - I™ P)'°ii...3P

^ = jftjj, k6. < = n, k7. = e""^= nl

Product of gKd's

°ii •ipip+i-i, - W PJ- (n-,)i

KAU C 0 - * 3\...jpip+l...J, *i...*pilH-l...t, y-(n-g)!%...fcp

kn. C:£ <::t=*>! CX> k12- **£ ti=p^

kl4. e*1 •tV>«j>+i"*»Sj.»+1"in = (n-p)\e*1,-Wp+i-in

LU fc t v *" ' Jl-Jp»p+l-«n V* r)' j\...jv

kl6. d{kj} d{ka} • dm - 6m {- 0h^p), kl7. 6{ki} d{fc2} - (J>!j 0^ 7.2 Appendix B. gpd combinatorial formulae

Definition formulae

oil-in f - oh-in ,

Single gpd formulae

o»l...*p P3- ^...ip = P!

. U...ip^ip ™ ~ ii...*....^ (caret denotes omitted index)

_c ii-ipip+i-t, _ (n+g-l)l U-tp - ii.-.tptp+i.-tn _ J^tv^l}^ il-ip — 7l P°' "ii-ipVn-** ~~ (n+p-i)l ^ii-V P°* ii.«j>vn-»« (n+p-i)! J'i-::ip

f 71 — tf P * ^ii...*, — (n-l)! ' P°* - °t) P ' ^...in - (n-l)l

Product of gpd's

A1 7r 7r _ ! plU. 7rjj...j„7r^1...fcI> -P! 7Tfci...Jfcp) P - ii...ip ii...Jp P (n-l)!

nl2 tt*1"*" 7r?1"iB = n> (2n~1)1

Atf 1 P - ^h-ipjp+\:.jq ^kp+i- w W ...jpfcp+i

A 7r P ** jl-.3pjp+l..j, %«. W Pi' (n+p-l)J nh-3p

AO lr P * ^Si-ipip+i-iq ki...kpip+1...iq — ?• (n+p-l)l ^fci.-fcp

pl6. p*1 '"^'"^Trj^J'j = (n-p)^1"^1"*1

il 1 PIT. P "^ "^P3l...jpip+1...in = jjajlj, Pl8. p^Pir...^ = n!

W {kl} {kr> {i} 1 { } { l} {fcp} { Ill 9 7T 7T ... 2/U0 7T ' 7T * • • • 7T - fljlV 7T pitf. 7T{ki} 7T{kj} 7T0} 0} ' Jl—ip'' P - *{*!} ^{fe} ^{i} — IPO 7.3 Appendix C. Combined gKd, gpd formulae 7.3.1 Products of gpd's, gKd's, alternating and symmetric symbols

CA n 0 ' ^j\h-3p°i\h.-lq ~ L>*=1 3l-3.-3p 3sh Iq

a+ ia c2. P^ 6*£± = c=i (-i) V-

7.3.2 Identities with combined gKd's and gpd's

11. ASwjag-nS = o1- = 0

12. ^ *3 5- tt^1 = 8 8%

i4. «T *5Cl< = 8«2f< o4- ^«?^«/h«. = 0

o5. fi^^TrrjTr^^Tr-^O

ml- = 2S%< <*' ^^^fr = 0 m2. = o7. ^^^C^^- = 0

h m3. Qir^ = 0 08. 6^ 6%6^-n^w = 0

o9. #*$fii*«0 7.3.3 Alternating and symmetric symbol product formulae

c £ ^0 tabcd ~ °abcd pefBhPabcd = Kobcd

pefadPabcd =

= 2!# />e/C"pa6cd = 12*2

e eabcd = 3!^ P^Pabed = 24*1

£ Cabal = 4! P^ Poked = 4! 7.4 Appendix D. Spinor Product Formulae

7lB = *B

gAB fiCE . ~AB „CE _ _B qr-AE I ~A -BE °CD FG $D ^FG — ^FG nCD "FG ~ "D "FG + "D "FG

AB CD AB 26 B n f*CD ^FG ' F CD"FG = 27TFG cAB ~CE B —-AE ~A —-BE

°CD "FG — TVD TTpQ — IID 71 pG

£AB£cd

B ^ eAB

7.5 Appendix E. Spinor equivalents for the gKd and gpd Reference formulae for the spinor equivalents of the gKd and gpd are given below. It should be mentioned that there is a good deal of interplay between the gKd and gpd in such specifications, there being a variety of ways to express expansions of some gKd's, gpd's (and also their spinor equivalents). Simple examples at the end of the section exhibit application of the gKd and gpd.

7.5.1 I. Spinor equivalents of the gKd's

A = Sf = _ l 2 d 6*6$ CD^C'D' % % «J

— As = 6£f = 6* 61 6) °d°ef °e°df + °f°de

6*d 61 6}

* ^D^D'i^EF^E'F' +7rBF^E'F') — ^E^E'i^DF^D'F' ^DF^D'F')

^F^F'i^DE^D'E' + ^DE^D'E')]'

We can expand the gKd with 4 (upper/lower) indices by a Laplace expansion of its first two rows and complementary minors. The result is 61 61 6- 61 6\ 6) 6) 61 A4 = « = 61 6) 61 81 6i 6) 6\ 6%

The R.H.S. can be written as the sum/difference of permanents of gKd's, if desired

gab 6% 6f VH "eg 6% 1 fid T 6ff 6°* °gh "eg

The spinor equivalent of 8^^ is

~(6EG7rE'G> + ^EG^E'G^i^FH^F'H' ^FH^F'H')

~K^£ff7rE'if' + ^EH^E'H^i^FG^F'G' + *FGSF'G>)

+(6pQWpiQ, + ^G8P,Q,)(8EJJ'KC,,JJ, +T^EH^E'H')

~(^FH7IF'H' ^H^F'Hdi^EG^E'G' "^EG^E'G')

^i^GH^G'H' ~^ ^H^G'H'^EF^E'F' + "^EF^E'F1 )]•

7.5.2 II. Spinor equivalents of the gpd's

Iii = ^<5£ 6A6% n =7r£ = 2 6%6% 9 $ 3 9 3

7r 7r A A B B i^^O'C^SF^B'F' + £F B'F') + S 6 ,(6 )p8 ),p, ^CF^D'F')

Since permanents only involve positive signs, the following Laplace expansion is also clear

61 6l 6l 04 = 7$$ = •7 * The R.H.S. can be written as the sum of permanents of permanents, if desired

+ +

The spinor equivalent of TT^^ is

^JSH^E'irX^G^F'G' 7rFG7rF'G')

•'"(^FG^F'G' 7rFG7rF'G')(^BH^B'H' 'NEH'JRE'H')

+(^FH^F'H' +'rFH'rF'H')(^BG^B,G' 'LTEG7RE'G')

"K^Gff^G'tf' + 7rGH7rG'H')(^EF^B'F' + 7rBF7rE'F')] 7.6 Appendix F. Spinor^Tensor equivalents

The covariant form of the mixed mode spinors is in the line directly below each of them.

Spinor Tensor

6$

sab

0afc |

CABGcD — CAC&BD ~ EAD&BC

£ac£bd + £ad£bc mo +*3S = +nA') €ACeBD£A'C'£B'D' - ^AD^BC^A'D^B'C' £ac£bd — <7ad<7bc MSS #5- i(AA'+nn')

1 ffac&d + 9ad9bc SACSBDeA'C'SB'D + ^AD^BC^A'D'SB'C' £cd °* = £°* erf

*{£ac€bdGa'D'£b'c — eADGBceA'cea'D1) £abcd ^&& = AA'

Raided

^AC^BD&A'C'^B'jy — GAD€BC£A'D'£b'C'

^AC^BD^A'iy^B'C' ~ SAD^BC^A'C^B'D1

= SAB^CDi^A'C'SB'D' + SA'D'Sb'C1)

^g^ = nA'

sacSbdSa'c'Sb'd1 — ^ad^bcSa'd'Sb'c' <7ac<7bd — 9ad9bc + *So6ed

—eAcSBDSA'D'SB'C + €Ad£bc£A'C'£b'D'

— {SACeBD + £ADSBc)eA'B'£C'D'

"^cd^c'd1 = nn' 2*3 - g^ga (sacSbd + sadSbc) x (ex'c'fiB'B' 4- ex'zyeB'c) 29ae9bd + 29ad9bc — 9ab9cd

^(SocPbd + ffodSbc — gab9cd — i£abcd) 1R.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). 20. Veblen, Invariants of Quadratic Differential Forms, Cambridge Uni­ versity Press, (1927). 3R~ Penrose and W. Rindler, Spinors and space-time, Cambridge Univer­ sity Press, Vol. 1, (1984). 4F. Trautman, F. Pirani, and H. Bondi, Lectures on General Relativity, Brandeis Summer Institute in Theoretical , (1964). PHYSICS AUXILIARY PUBLICATION SERVICE

Document No: JMAPAQ-40-033903-35

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.

PAPS Title: Reference Formulae

Authors: R. L. Agacy

For further information: e-mail: [email protected] or fax: 516-576-2223 (see http://ojps.aip.org), via the web (http://www.aip.org/pubservs/epaps.html) or from ^tg^a^gjorg^jn^h^directory^

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