Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.

The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis: $${\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}}$$ where each index i₁, i₂, ..., i_n takes values 1, 2, ..., n. There are nⁿ indexed values of ε_i1i2...in, which can be arranged into an n-dimensional array. The key defining property of the symbol is total antisymmetry in the indices. When any two indices are interchanged, equal or not, the symbol is negated: $${\displaystyle \varepsilon _{\dots i_{p}\dots i_{q}\dots }=-\varepsilon _{\dots i_{q}\dots i_{p}\dots }.}$$

If any two indices are equal, the symbol is zero. When all indices are unequal, we have: $${\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}=(-1)^{p}\varepsilon _{1\,2\,\dots n},}$$ where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i₁, i₂, ..., i_n into the order 1, 2, ..., n, and the factor (−1)^p is called the sign, or signature of the permutation. The value ε_{1 2 ... n} must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε_{1 2 ... n} = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.

The term "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, ${\displaystyle \mathbb {R} ^{3}}$ or Minkowski space. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density.

The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation.