nLab Levi-Civita symbol

Idea

For $n \in \mathbb{N}$, and $\sigma \in Sym(n)$ a permutation of $n$ elements, the Levi-Civita symbol or antisymmetric symbol

$$\epsilon_{\sigma(1), \sigma(2), \cdots, \sigma(n)} \;\in\; \{\pm 1\}$$

is (tensor-style notation for) the signature of the permutation.

Together with the Kronecker delta-notation, the Levi-Civita symbol is convenient in expressions subject to Einstein summation convention, then serving essentially as a combinatorial form of string diagram/Penrose notation in tensor categories.

Related concepts

• signature of a permutation

• Einstein summation convention

• Kronecker delta

References

See also

• Wikipedia, Levi-Civita symbol