Physlib.Relativity.Tensors.Contraction.Basic

For a tensor species $S$ over a ring $k$, let $c: \{0, \dots, n+1\} \to C$ be a color sequence of length $n+2$. Given two distinct indices $i, j \in \{0, \dots, n+1\}$ such that the color at $j$ is the dual of the color at $i$ (i.e., $S.\tau(c(i)) = c(j)$), the contraction map `contrT` is the $k$-linear map $$\text{Tensor}_S(c) \to \text{Tensor}_S(c \circ \text{dropPairEmb}_{i,j})$$ defined by lifting the multilinear contraction map of pure tensors to the tensor product. This map reduces the rank of the tensor from $n+2$ to $n$ by contracting the $i$-th and $j$-th indices using the natural pairing of dual colors, where the resulting tensor's color sequence is obtained by skipping the indices $i$ and $j$ in $c$.

For a tensor species $S$ over a ring $k$ and a color sequence $c: \{0, \dots, n+1\} \to C$, let $p$ be a pure tensor. Given two distinct indices $i, j \in \{0, \dots, n+1\}$ such that the color at $j$ is the dual of the color at $i$ ($S.\tau(c(i)) = c(j)$), the $k$-linear contraction map $\text{contrT}_{i,j}$ applied to the image of $p$ in the tensor product space is equal to the contraction of the pure tensor $\text{contrP}_{i,j}(p)$. That is, $$\text{contrT}_{i,j}(\text{toTensor}(p)) = \text{contrP}_{i,j}(p)$$ where $\text{toTensor}(p)$ is the tensor product $\bigotimes_{k=0}^{n+1} p_k$ and $\text{contrP}_{i,j}(p)$ is the contraction defined by the pairing of the $i$-th and $j$-th components of $p$.

Let $S$ be a tensor species over a ring $k$ and $G$ be a group. For a sequence of colors $c: \{0, \dots, n+1\} \to C$, let $t \in S.\text{Tensor}(c)$ be a tensor. Given two distinct indices $i, j \in \{0, \dots, n+1\}$ such that the color at $j$ is the dual of the color at $i$ (i.e., $S.\tau(c(i)) = c(j)$), the tensor contraction map $\text{contrT}_{i,j}$ is equivariant with respect to the group action of $G$. That is, for any $g \in G$: $$\text{contrT}_{i,j}(g \cdot t) = g \cdot \text{contrT}_{i,j}(t)$$ where the action on the left is on the tensor space of rank $n+2$ and the action on the right is on the contracted tensor space of rank $n$.

Let $S$ be a tensor species over a ring $k$. Let $c: \text{Fin}(n+2) \to C$ and $c_1: \text{Fin}(n_1+2) \to C$ be color sequences, and let $\sigma: \text{Fin}(n_1+2) \to \text{Fin}(n+2)$ be a map satisfying the permutation condition $c \circ \sigma = c_1$ (which implies $\sigma$ is a bijection). For any tensor $t \in \text{Tensor}_S(c)$ and any two distinct indices $i, j \in \text{Fin}(n_1+2)$ such that the color at index $j$ is the dual of the color at index $i$ (i.e., $S.\tau(c_1(i)) = c_1(j)$), the contraction of the permuted tensor at indices $i$ and $j$ is equal to the permutation of the contraction of $t$ at indices $\sigma(i)$ and $\sigma(j)$ by the induced map $\sigma'$: $$\text{contrT}_{i, j}(\text{permT}(\sigma, t)) = \text{permT}(\sigma', \text{contrT}_{\sigma(i), \sigma(j)}(t))$$ where $\sigma' = \text{dropPairOfMap}(i, j, \sigma)$ is the bijection between the reduced index sets $\text{Fin}(n_1)$ and $\text{Fin}(n)$ induced by $\sigma$ after removing indices $\{i, j\}$ from the domain and $\{\sigma(i), \sigma(j)\}$ from the codomain.

For a tensor species $S$ over a ring $k$ and a tensor $t$ with rank $n+2$ and color sequence $c: \{0, \dots, n+1\} \to C$, let $i, j \in \{0, \dots, n+1\}$ be distinct indices such that the color at $j$ is the dual of the color at $i$ (i.e., $S.\tau(c(i)) = c(j)$). The $k$-linear contraction of $t$ at indices $i$ and $j$ is equal to the contraction of $t$ at indices $j$ and $i$ up to an identity permutation: $$\text{contrT}_{i, j}(t) = \text{permT}(\text{id}, h, \text{contrT}_{j, i}(t))$$ where $\text{permT}$ is the $k$-linear map that reorders indices, $\text{id}$ is the identity map on $\{0, \dots, n-1\}$, and $h$ is the permutation condition ensuring that the resulting color sequences $c \circ \text{dropPairEmb}_{i, j}$ and $c \circ \text{dropPairEmb}_{j, i}$ are equivalent.

Let $S$ be a tensor species over a ring $k$. Let $t$ be a tensor of rank $n+4$ associated with a color sequence $c: \text{Fin}(n+4) \to C$. Suppose we perform two successive contractions. The first contraction is at indices $i_1, j_1 \in \text{Fin}(n+4)$ where $i_1 \neq j_1$ and $S.\tau(c(i_1)) = c(j_1)$. This results in a tensor $\text{contrT}_{i_1, j_1}(t)$ of rank $n+2$. We then perform a second contraction at indices $i_2, j_2 \in \text{Fin}(n+2)$ where $i_2 \neq j_2$ and the colors of the corresponding indices in the reduced tensor are duals. To express the commutativity, we define the corresponding indices for the reverse order: - Let $i_2' = \text{dropPairEmb}_{i_1, j_1}(i_2)$ and $j_2' = \text{dropPairEmb}_{i_1, j_1}(j_2)$ be the indices in the original rank $n+4$ tensor that correspond to $i_2$ and $j_2$ after $i_1$ and $j_1$ have been removed. - Let $i_1' = \text{dropPairEmbPre}_{i_2', j_2'}(i_1)$ and $j_1' = \text{dropPairEmbPre}_{i_2', j_2'}(j_1)$ be the indices in the rank $n+2$ tensor (obtained after contracting $i_2'$ and $j_2'$) that correspond to the original indices $i_1$ and $j_1$. Then the result of the sequential contractions is independent of the order, up to a canonical identification: $$\text{contrT}_{i_2, j_2}(\text{contrT}_{i_1, j_1}(t)) = \text{permT}(\text{id}, h, \text{contrT}_{i_1', j_1'}(\text{contrT}_{i_2', j_2'}(t)))$$ where $\text{permT}(\text{id}, h, \cdot)$ is the canonical isomorphism (reindexing) between the resulting tensor spaces.