Permutation Symbol (Levi-Civita Symbol) — Definition, Formula & Examples

The Levi-Civita symbol (also called the permutation symbol) is a function of indices that equals +1 for even permutations, −1 for odd permutations, and 0 whenever any two indices are repeated. It provides a compact way to express determinants and cross products using index notation.

For indices $i, j, k$ each ranging over $\{1, 2, 3\}$ , the Levi-Civita symbol $\varepsilon_{ijk}$ is defined as: $\varepsilon_{ijk} = +1$ if $(i,j,k)$ is an even permutation of $(1,2,3)$ ; $\varepsilon_{ijk} = -1$ if $(i,j,k)$ is an odd permutation of $(1,2,3)$ ; and $\varepsilon_{ijk} = 0$ if any index is repeated. This generalizes to $n$ dimensions, where $\varepsilon_{i_1 i_2 \cdots i_n}$ equals the sign of the permutation $(i_1, \ldots, i_n)$ or zero if any indices coincide.

Key Formula

\varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{if any index is repeated} \end{cases}

Where:

$i, j, k$ = Indices each taking values from {1, 2, 3}
$\varepsilon_{ijk}$ = The Levi-Civita symbol for the index triple (i, j, k)

How It Works

To evaluate

\varepsilon_{ijk}

, check whether any two of

i, j, k

are equal — if so, the value is 0. Otherwise, determine whether the ordering

(i,j,k)

can be reached from

(1,2,3)

by an even or odd number of pairwise swaps. An even number of swaps gives +1; an odd number gives −1. The cyclic orderings

(1,2,3)

(2,3,1)

, and

(3,1,2)

are even permutations, while

(1,3,2)

(3,2,1)

, and

(2,1,3)

are odd permutations. Using this symbol, the determinant of a

3 \times 3

matrix

A

can be written as

\det(A) = \sum_{i,j,k} \varepsilon_{ijk}\, a_{1i}\, a_{2j}\, a_{3k}

, and the cross product component is

(\mathbf{u} \times \mathbf{v})_i = \sum_{j,k} \varepsilon_{ijk}\, u_j\, v_k

Worked Example

Problem: Evaluate

\varepsilon_{312}

and

\varepsilon_{213}

Evaluate ε₃₁₂: Start from (1,2,3). The ordering (3,1,2) is a cyclic shift: move the last element to the front twice. A cyclic permutation of three elements equals two transpositions, which is even.

\varepsilon_{312} = +1

Evaluate ε₂₁₃: Start from (1,2,3) and swap the first two elements to get (2,1,3). That is one transposition, which is odd.

\varepsilon_{213} = -1

Answer:

\varepsilon_{312} = +1

and

\varepsilon_{213} = -1

Why It Matters

The Levi-Civita symbol appears throughout physics and engineering — in Maxwell's equations, fluid dynamics, and general relativity — wherever determinants or cross products must be expressed in index notation. In linear algebra courses, it gives a systematic way to derive the cofactor expansion of a determinant and proves identities involving the cross product and triple scalar product.

Common Mistakes

Mistake: Forgetting that any repeated index makes the symbol zero.

Correction: Before counting transpositions, first check whether two or more indices are the same. If

i = j

j = k

, or

i = k

, the value is immediately 0, regardless of the other index.