Physlib.Relativity.Tensors.Contraction.CrossToSlot

Let $n_A$ be a natural number, $c: \{0, \dots, n_A\} \to C$ be a sequence of tensor index types (colors), and $c_M: \{0, 1\} \to C$ be the colors of a rank-2 tensor. For any index $i \in \{0, \dots, n_A\}$ and $j \in \{0, 1\}$, the inverse of the cyclic permutation on the interval from $i$ to $n_A$, denoted by $\sigma = (\text{cycle}(i, n_A))^{-1}$, is a reindexing from a source color sequence to a target color sequence. The source sequence is the concatenation of $c$ (omitting index $i$) and $c_M$ (omitting index $j$). The target sequence is the original sequence $c$ where the $i$-th entry is updated to be the surviving index type of $c_M$ (the entry not at index $j$).

Let $S$ be a tensor species over a ring $k$. Let $c: \{0, \dots, n\} \to C$ be a sequence of colors for a tensor $t$, and let $c_M: \{0, 1\} \to C$ be the color sequence for a rank-2 tensor $M$. Given indices $i \in \{0, \dots, n\}$ and $j \in \{0, 1\}$, such that the color $c(i)$ is dual to the color $c_M(j)$ (i.e., $S.\tau(c(i)) = c_M(j)$), the $k$-linear map `crossToSlot` performs a contraction between the $i$-th slot of $t$ and the $j$-th slot of $M$. In this operation, the surviving index of $M$ (the index $j' \neq j$) is placed back into the original position $i$ of the resulting tensor. The color sequence of the result is $c$ with the $i$-th entry updated to the color $c_M(j')$. This operation corresponds to raising or lowering the $i$-th index of a tensor using a rank-2 tensor such as a metric.

Let $S$ be a tensor species over a ring $k$. Let $t$ be a tensor with color sequence $c: \text{Fin}(n_A + 1) \to C$ and $M$ be a rank-2 tensor with color sequence $c_M: \text{Fin}(2) \to C$. For indices $i \in \text{Fin}(n_A + 1)$ and $j \in \text{Fin}(2)$ such that the color $c(i)$ is dual to $c_M(j)$ (i.e., $S.\tau(c(i)) = c_M(j)$), the result-to-slot contraction $\text{crossToSlot}$ is related to the result-to-end contraction $\text{crossToEnd}$ by a cyclic permutation. Specifically: $$\text{crossToSlot}(i, j, M, t) = \text{permT}\left(\sigma, \text{crossToEnd}(i, j, t, M)\right)$$ where $\sigma = (\text{cycle}(i, n_A))^{-1}$ is the inverse of the cyclic permutation on the interval $[i, n_A]$, which rotates the surviving index from the last slot back into the $i$-th slot.

Let $S$ be a tensor species over a ring $k$. Let $c: \{0, \dots, n\} \to C$ be a color sequence and $i \in \{0, \dots, n\}$ be a fixed index such that the color at that position is $c(i) = e$. Let $d, b \in C$ be colors such that the dual of $d$ is $b$ (i.e., $S.\tau(d) = b$). Given a rank-2 tensor $M'$ with color sequence $(b, e)$, the $k$-linear map `crossToSlotInv` transforms a tensor $t$ from the space $\text{Tensor}(S, c[i \mapsto d])$ (where the $i$-th color of $c$ has been replaced by $d$) to the original space $\text{Tensor}(S, c)$. This is performed by contracting the $i$-th slot of $t$ (color $d$) with the first slot of $M'$ (color $b$), and placing the surviving second slot of $M'$ (color $e$) back into the $i$-th position. This operation serves as the "returning" half of an index-shifting round trip (such as raising then lowering an index), ensuring the result aligns with the original color sequence $c$.

Let $S$ be a tensor species over a ring $k$. Let $t$ be a tensor of rank $n+1$ with color sequence $c: \{0, \dots, n\} \to C$, and let $M$ be a rank-2 tensor with color sequence $c_M: \{0, 1\} \to C$. Suppose $c_M'$ is another color sequence such that the identity map is a reindexing from $c_M$ to $c_M'$ (meaning $c_M(k) = c_M'(k)$ for $k \in \{0, 1\}$). Let $i \in \{0, \dots, n\}$ and $j \in \{0, 1\}$ be indices such that the $i$-th color of $t$ is dual to the $j$-th color of $M$ under the sequence $c_M'$, i.e., $S.\tau(c(i)) = c_M'(j)$. Then, the `crossToSlot` contraction of $t$ with $M$ (after $M$ has been reindexed to $c_M'$ via the identity map) is equal to the identity reindexing of the `crossToSlot` contraction of $t$ with $M$ using its original colors $c_M$.

Let $S$ be a tensor species over a ring $k$ and $G$ be a group acting on the tensors of $S$. Let $t$ be a tensor of rank $n+1$ with color sequence $c: \{0, \dots, n\} \to C$, and $M$ be a rank-2 tensor with color sequence $c_M: \{0, 1\} \to C$. Given indices $i \in \{0, \dots, n\}$ and $j \in \{0, 1\}$ such that the color $c(i)$ is dual to $c_M(j)$ (i.e., $S.\tau(c(i)) = c_M(j)$), the operation `crossToSlot` contracts the $i$-th slot of $t$ with the $j$-th slot of $M$ and places the surviving index of $M$ into the original position $i$. This operation is $G$-equivariant, meaning that for any $g \in G$: $$\text{crossToSlot}(i, j, g \cdot M, g \cdot t) = g \cdot (\text{crossToSlot}(i, j, M, t))$$