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Permutation Symbol


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

 epsilon_(ijk)={0   for i=j,j=k, or k=i; +1   for (i,j,k) in {(1,2,3),(2,3,1),(3,1,2)}; -1   for (i,j,k) in {(1,3,2),(3,2,1),(2,1,3)}.
(1)

The permutation symbol is implemented in the Wolfram Language as Signature[list].

There are several common notations for the symbol, the first of which uses the usual Greek epsilon character epsilon_(ijk) (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 epsilon_(ijk) (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 e_(ijk) (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 permutation symbol satisfies

delta_(ij)epsilon_(ijk)=0
(2)
epsilon_(ipq)epsilon_(jpq)=2delta_(ij)
(3)
epsilon_(ijk)epsilon_(ijk)=6
(4)
epsilon_(ijk)epsilon_(pqk)=delta_(ip)delta_(jq)-delta_(iq)delta_(jp),
(5)

where delta_(ij) is the Kronecker delta (Arfken 1985, p. 136).

The symbol can be defined as the scalar triple product of unit vectors in a right-handed coordinate system,

 epsilon_(ijk)=x_i^^·(x_j^^xx_k^^).
(6)

The symbol can be generalized to an arbitrary number of elements, in which case the permutation symbol is (-1)^(i(p)), where i(p) is the number of transpositions of pairs of elements (i.e., permutation inversions) that must be composed to build up the permutation p (Skiena 1990). This type of symbol arises in computation of determinants of nxn matrices. The number of permutations on n symbols having signature -1 is n!/2, which is also the number of permutations having signature +1.


See also

Even Permutation, Odd Permutation, Permutation, Permutation Cycle, Permutation Inversion, Permutation Tensor, Transposition

Related Wolfram sites

http://functions.wolfram.com/IntegerFunctions/Signature/

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 132-133 and 136, 1985.Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. New York: Dover, 1989.Chandrasekhar, S. The Mathematical Theory of Black Holes. Oxford, England: Clarendon Press, 1998.Chou, P. C. and Pagano, N. J. "The Alternating Tensor." §8.7 in Elasticity: Tensor, Dyadic, and Engineering Approaches. New York: Dover, pp. 182-186, 1992.Evett, A. A. "Permutation Symbol Approach to Elementary Vector Analysis." Amer. J. Phys. 34, 503-507, 1966.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Green, A. E. and Zerna, W. Theoretical Elasticity, 2nd ed. New York: Dover, 1992.Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, 1987.Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 69-74, 1988.Landau, L. D. and Lifschitz, E. M. Theory of Elasticity, 3rd rev. enl. ed. Oxford, England: Pergamon Press, 1986.Lightman, A. P.; Price, R. H.; and Teukolsky, S. Problem Book in Relativity and Gravitation, 2nd pr. Princeton, NJ: Princeton University Press, 1979.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, 1973.Skiena, S. "Signature." §1.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 38, 1972.

Referenced on Wolfram|Alpha

Permutation Symbol

Cite this as:

Weisstein, Eric W. "Permutation Symbol." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PermutationSymbol.html

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