The permutation symbol (Evett 1966; Goldstein 1980, p. 172; Aris 1989, p. 16) is a three-index object sometimes called the Levi-Civita symbol (Weinberg 1972, p. 38;
Misner et al. 1973, p. 87; Arfken 1985, p. 132; Chandrasekhar 1998,
p. 68), Levi-Civita density (Goldstein 1980, p. 172), alternating tensor
(Goldstein 1980, p. 172; Landau and Lifshitz 1986, p. 110; Chou and Pagano
1992, p. 182), or signature. It is defined by
There are several common notations for the symbol, the first of which uses the usual Greek epsilon character (Goldstein 1980, p. 172; Griffiths 1987,
p. 139; Jeffreys and Jeffreys 1988, p. 69; Aris 1989, p. 16; Chou
and Pagano 1992, p. 182), the second of which uses the curly variant (Weinberg 1972, p. 38; Misner et al. 1973,
p. 87; Lightman et al. 1979, pp. 19-21 and 183-188; Arfken 1985,
p. 132; Chandrasekhar 1998, p. 68), and the third of which uses a Latin
lower case
(Landau and Lifshitz 1986, p. 110; Green and Zerna 1992, p. 11).
The symbol can also be interpreted as a tensor, in which
case it is called the permutation tensor.
The symbol can be defined as the scalar triple
product of unit vectors in a right-handed coordinate system,
(6)
The symbol can be generalized to an arbitrary number of elements, in which case the permutation symbol is , where is the number of transpositions of pairs of elements (i.e.,
permutation inversions) that must be composed
to build up the permutation (Skiena 1990). This type of symbol arises in computation of
determinants of
matrices. The number of permutations on symbols having signature is , which is also the number of permutations having signature
.
(Goldstein 1980, p. 172; Griffiths 1987,
p. 139; Jeffreys and Jeffreys 1988, p. 69; Aris 1989, p. 16; Chou
and Pagano 1992, p. 182), the second of which uses the curly variant 
(Weinberg 1972, p. 38; Misner et al. 1973,
p. 87; Lightman et al. 1979, pp. 19-21 and 183-188; Arfken 1985,
p. 132; Chandrasekhar 1998, p. 68), and the third of which uses a Latin
lower case 
(Landau and Lifshitz 1986, p. 110; Green and Zerna 1992, p. 11).
is the Kronecker delta (Arfken 1985, p. 136).
, where 
is the number of transpositions of pairs of elements (i.e.,
permutation inversions) that must be composed
to build up the permutation 
(Skiena 1990). This type of symbol arises in computation of
determinants of 
matrices. The number of permutations on 
symbols having signature 
is 
, which is also the number of permutations having signature
.
