Physlib.Mathematics.LeviCivita.Basic

For a finite index set $\iota$ and a map $g: \iota \to \iota$, the Levi-Civita symbol $\epsilon_g$ is the integer defined as the generalized Kronecker delta of $g$ and the identity map $\text{id}_\iota$. It is given by the determinant of the matrix whose $(i, j)$-th entry is the Kronecker delta $\delta_{g(i), j}$: $$\epsilon_g = \det(\delta_{g(i), j})_{i, j \in \iota}$$ The value $\epsilon_g$ is equal to the sign of the permutation $\text{sgn}(g)$ if $g$ is a permutation, and $0$ otherwise. In the specific case where $\iota = \{0, 1, \dots, d-1\}$, this corresponds to the components of the $d$-dimensional Levi-Civita tensor $\epsilon_{g(0) g(1) \dots g(d-1)}$.

Let $\iota$ be a finite index set. For any map $g : \iota \to \iota$, the Levi-Civita symbol $\epsilon_g$ is equal to the determinant of the matrix whose $(i, j)$-th entry is the Kronecker delta $\delta_{g(i), j}$: $$\epsilon_g = \det \left( \delta_{g(i), j} \right)_{i, j \in \iota}$$ where $\delta_{g(i), j}$ is $1$ if $g(i) = j$ and $0$ otherwise.

For any finite index set $\iota$, let $\text{id} : \iota \to \iota$ be the identity map. The Levi-Civita symbol of the identity map is $1$: $$\epsilon_{\text{id}} = 1$$ This corresponds to the normalization of the symbol in $d$ dimensions, $\epsilon_{0 1 \dots (d-1)} = 1$.

For a finite index set $\iota$ and a permutation $\sigma$ of $\iota$, the Levi-Civita symbol $\epsilon_\sigma$ is equal to the sign of the permutation $\text{sgn}(\sigma)$, considered as an integer in $\mathbb{Z}$.

Let $\iota$ be a finite index set and $g: \iota \to \iota$ be a map. For any two distinct indices $i, j \in \iota$, the Levi-Civita symbol $\epsilon_g$ is antisymmetric under the transposition of these indices: $$\epsilon_{g \circ \tau_{i,j}} = -\epsilon_g$$ where $\tau_{i,j}$ is the permutation that swaps $i$ and $j$ and leaves all other elements of $\iota$ fixed.

Let $\iota$ be a finite index set and $g: \iota \to \iota$ be a map. For any two distinct elements $i, j \in \iota$, let $\tau_{i, j}$ denote the transposition (swap) of $i$ and $j$. The Levi-Civita symbol $\epsilon_g$ is antisymmetric under post-composition with this transposition: $$\epsilon_{\tau_{i, j} \circ g} = -\epsilon_g$$ This means that swapping two distinct index values wherever they occur in the map $g$ negates the value of the Levi-Civita symbol.

Let $\iota$ be a finite index set and $g: \iota \to \iota$ be a map. If there exist two distinct indices $i, j \in \iota$ such that $i \neq j$ and $g(i) = g(j)$, then the Levi-Civita symbol $\epsilon_g$ vanishes (i.e., $\epsilon_g = 0$).

Let $\iota$ be a finite index set and $g : \iota \to \iota$ be a map. If $g$ is not injective, then the Levi-Civita symbol $\epsilon_g$ is equal to $0$.

Let $\iota$ be a finite index set and $g: \iota \to \iota$ be a map. The Levi-Civita symbol $\epsilon_g$ is equal to $0$ if and only if $g$ is not an injective function (which, for a map from a finite set to itself, is equivalent to $g$ having at least one repeated index).