Physlib.Relativity.Tensors.Contraction.CrossToEnd

Let $S$ be a tensor species over a ring $k$. Given two color sequences $c_A: \text{Fin}(n_A+1) \to C$ and $c_B: \text{Fin}(n_B+1) \to C$, let $i \in \text{Fin}(n_A+1)$ and $j \in \text{Fin}(n_B+1)$ be indices such that the color at $c_A(i)$ is dual to the color at $c_B(j)$ (i.e., $S.\tau(c_A(i)) = c_B(j)$). The cross contraction `crossToEnd` is the $k$-bilinear map: $$\text{Tensor}_S(c_A) \times \text{Tensor}_S(c_B) \to \text{Tensor}_S(c_{out})$$ which takes tensors $t_1$ and $t_2$ and returns the contraction of the $i$-th slot of $t_1$ with the $j$-th slot of $t_2$. The resulting color sequence $c_{out}$ is the concatenation of $c_A$ (omitting index $i$) and $c_B$ (omitting index $j$), specifically $c_{out} = (c_A \circ \text{succAbove}_i) \oplus (c_B \circ \text{succAbove}_j)$, where $\text{succAbove}_k$ is the order-preserving map that skips index $k$.

Let $S$ be a tensor species over a ring $k$. Let $c_A, c_B : \text{Fin } 2 \to C$ be color sequences for rank-2 tensors. If the last slot of $c_A$ is dual to the first slot of $c_B$, such that $h : S.\tau(c_A(1)) = c_B(0)$, then for any tensors $A \in \text{Tensor}_S(c_A)$ and $B \in \text{Tensor}_S(c_B)$, the cross contraction of the second slot of $A$ with the first slot of $B$ is equivalent to the contraction of their tensor product $A \otimes B$ at the middle indices 1 and 2, up to a color-recasting reindexing: $$\text{crossToEnd}(1, 0, h)(A, B) = \text{permT}(\text{id}, \phi)(\text{contrT}_{1,2}(A \otimes B))$$ where $\text{contrT}_{1,2}$ denotes the contraction of the slots at indices 1 and 2 of the rank-4 product tensor, and $\text{permT}(\text{id}, \phi)$ is the reindexing map associated with the identity permutation used to match the resulting color sequences.

Let $S$ be a tensor species over a ring $k$ and $G$ be a group acting on the tensors. Given two tensors $t_1 \in \text{Tensor}_S(c_A)$ and $t_2 \in \text{Tensor}_S(c_B)$, and indices $i \in \text{Fin}(n_A + 1)$ and $j \in \text{Fin}(n_B + 1)$ such that the color of the $i$-th slot of $t_1$ is dual to the color of the $j$-th slot of $t_2$ (i.e., $S.\tau(c_A(i)) = c_B(j)$), the slot-addressed cross contraction is equivariant under the action of $G$. That is, for any group element $g \in G$: $$\text{crossToEnd}_{i, j}(g \cdot t_1, g \cdot t_2) = g \cdot \text{crossToEnd}_{i, j}(t_1, t_2)$$ This property follows from the fact that the underlying operations of tensor product (`prodT`), single-tensor contraction (`contrT`), and permutation (`permT`) are all $G$-equivariant.

Let $S$ be a tensor species over a ring $k$. Let $t_A$ be a tensor of rank $n_A + 1$ with color sequence $c_A$, and let $t_B$ and $t_C$ be tensors of rank 2 with color sequences $c_B$ and $c_C$. Given a slot $i_A$ in $t_A$, suppose that the $i_A$-th slot of $t_A$ is dual to the $0$-th slot of $t_B$ (i.e., $S.\tau(c_A(i_A)) = c_B(0)$), and the $1$-st slot of $t_B$ is dual to the $0$-th slot of $t_C$ (i.e., $S.\tau(c_B(1)) = c_C(0)$). Then the result of first contracting $t_A$ and $t_B$ at slots $i_A$ and $0$, and then contracting the result with $t_C$ at the slot corresponding to $t_B$'s survivor and $t_C$'s $0$-th slot, is equal to the result of first contracting $t_B$ and $t_C$ at slots $1$ and $0$, and then contracting $t_A$ with that result at slot $i_A$ and the slot corresponding to $t_B$'s survivor. This equality holds up to a color reindexing $\text{permT}(\text{id})$. Symbolically, if $\cdot_{i,j}$ denotes the `crossToEnd` contraction at slots $i$ and $j$, then: $$(t_A \cdot_{i_A, 0} t_B) \cdot_{\text{last}, 0} t_C = \text{permT}(\text{id}) (t_A \cdot_{i_A, 0} (t_B \cdot_{1, 0} t_C))$$ where $\text{last}$ refers to the index in the intermediate product where the survivor slot of the rank-two tensor $t_B$ resides.

Let $S$ be a tensor species over a ring $k$. Consider two tensors $t_1$ and $t_2$ of rank $n_A + 1$ and $n_B + 1$ with color sequences $c_A$ and $c_B$, respectively. Let $\sigma$ be a reindexing (a color-preserving bijection) of the indices of $t_1$, and let $i$ and $j$ be indices such that the $i$-th slot of the reindexed tensor $\text{permT}(\sigma, t_1)$ is dual to the $j$-th slot of $t_2$ (i.e., $S.\tau(c_A(\sigma(i))) = c_B(j)$). If $\sigma': \text{Fin}(n_A) \to \text{Fin}(n_A)$ is a reindexing of the surviving indices of $t_1$ that satisfies the consistency condition $(\text{succAbove}_{\sigma(i)} \circ \sigma') = (\sigma \circ \text{succAbove}_i)$, then contracting the $i$-th slot of the permuted tensor $\text{permT}(\sigma, t_1)$ with the $j$-th slot of $t_2$ is equivalent to first contracting the $\sigma(i)$-th slot of the original tensor $t_1$ with the $j$-th slot of $t_2$, and then reindexing the resulting survivors. Specifically: $$\text{crossToEnd}_i^j(\text{permT}_\sigma(t_1), t_2) = \text{permT}_{\sigma' \oplus \text{id}}(\text{crossToEnd}_{\sigma(i)}^j(t_1, t_2))$$ where $\sigma' \oplus \text{id}$ is the reindexing on the $n_A + n_B$ surviving indices that applies $\sigma'$ to the slots originating from $t_1$ and the identity map to the slots originating from $t_2$.

Let $S$ be a tensor species. Let $t_1$ be a tensor of rank $n_A + 1$ with index colors $c_A$, and let $t_2$ be a tensor of rank $n_B + 1$ with index colors $c_B$. Let $\sigma$ be a reindexing (permutation) of the indices of $t_2$ such that the permuted tensor $t_2'$ has colors $c_B'$. Suppose we contract the $i$-th slot of $t_1$ with the $j$-th slot of the permuted tensor $t_2'$. This theorem states that this is equivalent to contracting the $i$-th slot of $t_1$ with the $\sigma(j)$-th slot of the original tensor $t_2$, and then reindexing the resulting survivors. The reindexing of the final tensor is defined by the identity map on the indices originating from $t_1$ and the map $\sigma'$ on the indices originating from $t_2$, where $\sigma'$ is the permutation of the $n_B$ surviving slots of $t_2$ that satisfies $(\sigma(j)).\text{succAbove} \circ \sigma' = \sigma \circ j.\text{succAbove}$.