Generalized Kronecker and permanent deltas, their spinor and tensor 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 spinors in the mathematical language of General Relativity, the purpose is to (i) provide formulae for combined gKd/gpd 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 The 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 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-dimensions, 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 determinant (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 matrix, 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).