Physlib.Mathematics.KroneckerDelta.Contraction

Let $\alpha$ be a finite type. For any maps $\mu, \nu : \alpha \to \alpha$, the generalized Kronecker delta $\delta^\mu_\nu$ is defined as the determinant of the matrix whose $(i, j)$ entry is the Kronecker delta $\delta_{\mu(i), \nu(j)}$. The product of two such deltas where the bottom index is the identity map $\text{id}$ on $\alpha$ satisfies: $$\delta^\mu_{\text{id}} \cdot \delta^\nu_{\text{id}} = \delta^\mu_\nu$$ This is the abstract functional form of the identity relating the product of two Levi-Civita symbols to a generalized Kronecker delta: $$\epsilon^{\mu_1 \dots \mu_n} \epsilon_{\nu_1 \dots \nu_n} = \delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n}$$

Let $\alpha$ be a finite set. For any two sequences of indices $\mu, \nu$ of length $n$ in $\alpha$, the generalized Kronecker delta $\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n}$ satisfies the following contraction identity when summing over a shared index $a \in \alpha$ appended to the end of both sequences: $$\sum_{a \in \alpha} \delta^{\mu_1 \dots \mu_n a}_{\nu_1 \dots \nu_n a} = (|\alpha| - n) \delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n}$$ where $|\alpha|$ is the cardinality of the set $\alpha$.

For any natural number $k$, let $h$ be a sequence of $k$ indices $(h_1, \dots, h_k)$ where each $h_i$ takes values in a 4-dimensional set (represented by $\text{Fin } 4$). The total contraction of the generalized Kronecker delta over all such sequences is given by the product: $$\sum_{h \in (\text{Fin } 4)^k} \delta^{h_1 \dots h_k}_{h_1 \dots h_k} = \prod_{j=0}^{k-1} (4 - j)$$ where the generalized Kronecker delta $\delta^{h_1 \dots h_k}_{h_1 \dots h_k}$ is defined as the determinant of the $k \times k$ matrix whose $(i, j)$-th entry is the standard Kronecker delta $\delta_{h_i h_j}$.

Let $\sigma, \tau \in \{0, 1, 2, 3\}$ be indices in a 4-dimensional space. For any natural number $k$, summing the generalized Kronecker delta over $k$ shared indices $h = (h_1, \dots, h_k) \in \{0, 1, 2, 3\}^k$ yields: $$\sum_{h \in \{0, 1, 2, 3\}^k} \delta^{\sigma h_1 \dots h_k}_{\tau h_1 \dots h_k} = \left( \prod_{j=0}^{k-1} (3 - j) \right) \delta_{\sigma \tau}$$ where $\delta_{\sigma \tau}$ is the standard Kronecker delta, and the generalized Kronecker delta is defined as the determinant of the matrix of Kronecker deltas of the respective indices.

For any indices $\rho, \sigma, \tau, \omega \in \{0, 1, 2, 3\}$ and any natural number $k$, the sum of the $(k+2)$-indexed generalized Kronecker delta over $k$ shared indices $h_1, \dots, h_k$ (where each $h_i \in \{0, 1, 2, 3\}$) is given by: $$\sum_{h_1, \dots, h_k = 0}^3 \delta^{\rho \sigma h_1 \dots h_k}_{\tau \omega h_1 \dots h_k} = \left( \prod_{j=0}^{k-1} (2 - j) \right) \delta^{\rho \sigma}_{\tau \omega}$$ where $\delta^{\rho \sigma}_{\tau \omega}$ is the $2 \times 2$ generalized Kronecker delta, defined as $\delta^\rho_\tau \delta^\sigma_\omega - \delta^\rho_\omega \delta^\sigma_\tau$.

The sum over all mappings $g: \text{Fin } 4 \to \text{Fin } 4$ of the square of the generalized Kronecker delta $\delta^{g}_{\text{id}}$ is equal to $24$. Mathematically, this is expressed as: $$\sum_{g : \text{Fin } 4 \to \text{Fin } 4} \left( \delta^{g_0 g_1 g_2 g_3}_{0 1 2 3} \right)^2 = 24$$ where the generalized Kronecker delta $\delta^{g_0 g_1 g_2 g_3}_{0 1 2 3}$ is defined as the determinant of the $4 \times 4$ matrix whose $(i, j)$-th entry is the Kronecker delta $\delta_{g_i, j}$.

For any indices $\sigma, \tau \in \{0, 1, 2, 3\}$, let $\delta^{\mu_1 \mu_2 \mu_3 \mu_4}_{\nu_1 \nu_2 \nu_3 \nu_4}$ denote the generalized Kronecker delta and $\delta_{\sigma \tau}$ denote the standard Kronecker delta. Summing over the three shared indices $h_1, h_2, h_3 \in \{0, 1, 2, 3\}$, the following identity holds: $$\sum_{h_1, h_2, h_3 = 0}^{3} \delta^{\sigma h_1 h_2 h_3}_{0 1 2 3} \delta^{\tau h_1 h_2 h_3}_{0 1 2 3} = 6 \delta_{\sigma \tau}$$ Here, the lower indices $(0, 1, 2, 3)$ correspond to the identity sequence on $\text{Fin } 4$.

For any indices $\rho, \sigma, \tau, \omega$ in the set $\{0, 1, 2, 3\}$, the sum over all possible sequences $h = (h_0, h_1)$ where $h_i \in \{0, 1, 2, 3\}$ of the product of two generalized Kronecker deltas is given by: $$\sum_{h_0, h_1=0}^3 \delta^{\rho, \sigma, h_0, h_1}_{0, 1, 2, 3} \delta^{\tau, \omega, h_0, h_1}_{0, 1, 2, 3} = 2 (\delta_{\rho\tau} \delta_{\sigma\omega} - \delta_{\rho\omega} \delta_{\sigma\tau})$$ where $\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n}$ is the generalized Kronecker delta (defined as the determinant of the $n \times n$ matrix with entries $\delta_{\mu_i \nu_j}$) and $\delta_{ij}$ is the standard Kronecker delta which equals $1$ if $i=j$ and $0$ otherwise.