Physlib.Relativity.Tensors.Contraction.Basis

Given a multi-index $b \in \text{ComponentIdx}(c)$ for a tensor with $n+2$ indices (where $c: \{0, \dots, n+1\} \to C$ defines the index structure), and two specific index positions $i, j \in \{0, \dots, n+1\}$, the function `dropPair` constructs a reduced multi-index of length $n$. This resulting multi-index belongs to the index structure $c \circ \text{dropPairEmb}_{i,j}$, which corresponds to the original structure with the $i$-th and $j$-th entries removed. For each $m \in \{0, \dots, n-1\}$, the value of the new multi-index is $b(\text{dropPairEmb}_{i,j}(m))$, effectively extracting the components of $b$ at all positions except for $i$ and $j$.

Given a multi-index $b \in \text{ComponentIdx}(c \circ \text{dropPairEmb}_{i,j})$ for a tensor with $n$ indices (where the index structure $c$ has had two positions $i, j \in \{0, \dots, n+1\}$ removed), the function `DropPairSection` defines the finite set of all multi-indices $b' \in \text{ComponentIdx}(c)$ that map to $b$ under the `dropPair` operation. Mathematically, it is the fiber of the projection map: $$\text{DropPairSection}(b) = \{ b' \in \text{ComponentIdx}(c) \mid \text{dropPair}_{i,j}(b') = b \}.$$ This set consists of all multi-indices of length $n+2$ that match $b$ at all positions other than $i$ and $j$.

Let $n$ be a natural number and $c: \{0, \dots, n+1\} \to C$ define the index structure of a tensor with $n+2$ indices. For any two index positions $i, j \in \{0, \dots, n+1\}$, let $b'$ be a multi-index for the structure $c$, and $b$ be a reduced multi-index for the structure $c \circ \text{dropPairEmb}_{i,j}$ (where positions $i$ and $j$ are skipped). Then $b'$ belongs to the fiber $\text{DropPairSection}(b)$ if and only if for all $m \in \{0, \dots, n-1\}$, the component of $b'$ at the position $\text{dropPairEmb}_{i,j}(m)$ is equal to the component of $b$ at position $m$. $$b' \in \text{DropPairSection}(b) \iff \forall m \in \{0, \dots, n-1\}, b'(\text{dropPairEmb}_{i,j}(m)) = b(m)$$

Let $n$ be a natural number and $c: \{0, \dots, n+1\} \to C$ define the index structure of a tensor with $n+2$ indices. For any multi-index $b \in \text{ComponentIdx}(c)$ and any two index positions $i, j \in \{0, \dots, n+1\}$, let $\text{dropPair}_{i,j}(b)$ denote the reduced multi-index of length $n$ obtained by skipping the components at positions $i$ and $j$. Then $b$ belongs to the fiber $\text{DropPairSection}(\text{dropPair}_{i,j}(b))$, which consists of all multi-indices $b'$ that satisfy $\text{dropPair}_{i,j}(b') = \text{dropPair}_{i,j}(b)$.

Given a sequence of index types $c : \{0, \dots, n+1\} \to C$ and two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a multi-index (of type `ComponentIdx`) corresponding to the indices of $c$ after positions $i$ and $j$ have been removed (via `dropPairEmb`). Given a pair $x = (x_1, x_2)$ representing the specific coordinate values for the $i$-th and $j$-th indices, where $x_1 \in \{0, \dots, d_i - 1\}$ and $x_2 \in \{0, \dots, d_j - 1\}$, this function constructs a full multi-index $B \in \text{ComponentIdx}(c)$. The components of the resulting multi-index are defined as: $$B_m = \begin{cases} x_1 & \text{if } m = i \\ x_2 & \text{if } m = j \\ b_{f(i, j, m)} & \text{otherwise} \end{cases}$$ where $f(i, j, m)$ is the preimage of $m$ under the embedding that skips $i$ and $j$, effectively mapping the $m$-th position in the full sequence to its corresponding position in the reduced sequence $b$.

Let $n$ be a natural number and let $c: \{0, \dots, n+1\} \to C$ define a sequence of index types for a tensor. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a multi-index of type `ComponentIdx` corresponding to the structure $c$ after indices $i$ and $j$ have been removed. Given a pair of coordinate values $x = (x_1, x_2)$ where $x_1 \in \{0, \dots, d_i-1\}$ and $x_2 \in \{0, \dots, d_j-1\}$ (with $d_k$ being the dimension of the $k$-th representation space), the function `ofFin` constructs a full multi-index $B \in \text{ComponentIdx}(c)$ by inserting $x_1$ at position $i$, $x_2$ at position $j$, and the elements of $b$ at the remaining positions. This theorem states that the $i$-th component of the resulting multi-index $B$ is equal to $x_1$.

Let $n$ be a natural number and let $c: \{0, \dots, n+1\} \to C$ define a sequence of index types for a tensor. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a multi-index of type `ComponentIdx` corresponding to the structure $c$ after indices $i$ and $j$ have been removed. Given a pair of coordinate values $x = (x_1, x_2)$ where $x_1 \in \{0, \dots, d_i-1\}$ and $x_2 \in \{0, \dots, d_j-1\}$ (with $d_k$ being the dimension of the $k$-th representation space), the function `ofFin` constructs a full multi-index $B \in \text{ComponentIdx}(c)$ by inserting $x_1$ at position $i$, $x_2$ at position $j$, and the elements of $b$ at the remaining positions. This theorem states that the $j$-th component of the resulting multi-index $B$ is equal to $x_2$.

Let $n \in \mathbb{N}$ and let $c: \{0, \dots, n+1\} \to C$ define the index structure of a tensor. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a reduced multi-index corresponding to the structure $c$ after indices $i$ and $j$ have been removed. Given a pair of coordinate values $x = (x_1, x_2)$ (where $x_1$ and $x_2$ are valid coordinates for the $i$-th and $j$-th index types, respectively), let $B = \text{ofFin}(i, j, b, x)$ be the full multi-index constructed by inserting $x_1$ at position $i$, $x_2$ at position $j$, and the components of $b$ at all other positions. This theorem states that $B$ is an element of $\text{DropPairSection}(b)$, which is the set of all multi-indices $b'$ such that dropping the $i$-th and $j$-th components of $b'$ results in $b$.

Let $n \in \mathbb{N}$ and let $c: \{0, \dots, n+1\} \to C$ define the index structure of a tensor. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a reduced multi-index (an element of `ComponentIdx`) corresponding to the structure $c$ after the positions $i$ and $j$ have been removed. This definition provides an equivalence (bijection): $$\{0, \dots, d_i - 1\} \times \{0, \dots, d_j - 1\} \simeq \text{DropPairSection}(b)$$ where $d_k = \text{S.repDim}(c_k)$ is the dimension of the representation space associated with the $k$-th index, and $\text{DropPairSection}(b)$ is the set of all full multi-indices $B \in \text{ComponentIdx}(c)$ that reduce to $b$ when the $i$-th and $j$-th components are removed. The bijection is defined such that a pair $(x_1, x_2)$ is mapped to the multi-index $B$ formed by inserting $x_1$ at position $i$, $x_2$ at position $j$, and the components of $b$ in the remaining $n$ positions. The inverse map extracts the values at positions $i$ and $j$ from a given multi-index in the section.

Let $n \in \mathbb{N}$ and let $c: \{0, \dots, n+1\} \to C$ define the sequence of index types for a tensor. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a reduced multi-index corresponding to the structure $c$ after the positions $i$ and $j$ have been removed. Given a pair of index values $x = (x_1, x_2)$, where $x_1 \in \{0, \dots, d_i-1\}$ and $x_2 \in \{0, \dots, d_j-1\}$ (with $d_k$ being the dimension of the $k$-th representation space), the equivalence `ofFinEquiv` constructs a full multi-index $B$ by inserting $x_1$ at position $i$, $x_2$ at position $j$, and the elements of $b$ at the remaining positions. This theorem states that the $i$-th component of the resulting multi-index $B$ is equal to $x_1$.

Let $n \in \mathbb{N}$ and let $c: \{0, \dots, n+1\} \to C$ define the index structure of a tensor. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $b$ be a reduced multi-index (an element of `ComponentIdx`) corresponding to the structure $c$ after the positions $i$ and $j$ have been removed. Given a pair of index values $x = (x_1, x_2)$ with $x_1 \in \{0, \dots, d_i - 1\}$ and $x_2 \in \{0, \dots, d_j - 1\}$ (where $d_k$ is the dimension of the $k$-th representation space), the equivalence `ofFinEquiv` reconstructs a full multi-index $B$ by inserting $x_1$ at position $i$, $x_2$ at position $j$, and the components of $b$ in the remaining positions. This theorem states that the $j$-th component of the resulting multi-index $B$ is equal to $x_2$.

Let $S$ be a tensor species and $c: \{0, \dots, n+1\} \to C$ be a sequence of index types. Let $b = (b_0, \dots, b_{n+1})$ be a multi-index in $\text{ComponentIdx}(c)$, and let $e_b = e_{b_0}^{(c_0)} \otimes \dots \otimes e_{b_{n+1}}^{(c_{n+1})}$ be the corresponding pure basis vector. For any two distinct indices $i, j \in \{0, \dots, n+1\}$, let $\text{dropPair}_{i,j}(e_b)$ be the pure tensor obtained by removing the $i$-th and $j$-th components of $e_b$. Then $$\text{dropPair}_{i,j}(e_b) = e_{b \circ \phi}$$ where $\phi: \{0, \dots, n-1\} \to \{0, \dots, n+1\}$ is the embedding (defined by `dropPairEmb`) that skips the indices $i$ and $j$, and $e_{b \circ \phi}$ is the basis vector in the reduced tensor product space associated with the multi-index restricted to the remaining $n$ positions.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n+1\} \to C$ be a sequence of index colors. Let $t \in S.\text{Tensor}(c)$ be a tensor of rank $n+2$. Suppose $i, j \in \{0, \dots, n+1\}$ are distinct indices such that the color at $j$ is the dual of the color at $i$, i.e., $c(j) = \tau(c(i))$. Let $\text{contrT}_{i,j}(t)$ be the tensor of rank $n$ obtained by contracting $t$ at indices $i$ and $j$. For any multi-index $b$ for the contracted tensor space, the $b$-th component of the contracted tensor is given by: $$[\text{contrT}_{i,j}(t)]_b = \sum_{b' \in \text{DropPairSection}(b)} t_{b'} \cdot \text{contr}(e_{b'_i}^{(c_i)} \otimes e_{b'_j}^{(c_j)})$$ where: - $t_{b'}$ is the component of the original tensor $t$ at the multi-index $b'$. - $\text{DropPairSection}(b)$ is the set of all multi-indices $b'$ of length $n+2$ that reduce to $b$ when the $i$-th and $j$-th indices are removed. - $e_{b'_i}^{(c_i)}$ and $e_{b'_j}^{(c_j)}$ are the basis vectors of the representation spaces corresponding to colors $c(i)$ and $c(j)$ at the specific indices specified by $b'$. - $\text{contr}$ is the natural pairing (contraction morphism) $V \otimes V^* \to k$ associated with the tensor species.

Let $S$ be a tensor species over a field $k$ and $c: \{0, \dots, n+1\} \to C$ be a sequence of index colors. Let $t \in S.\text{Tensor}(c)$ be a tensor of rank $n+2$. Suppose $i, j \in \{0, \dots, n+1\}$ are distinct indices such that the color at $j$ is the dual of the color at $i$, i.e., $c(j) = \tau(c(i))$. Let $\text{contrT}_{i,j}(t)$ be the tensor of rank $n$ obtained by contracting $t$ at indices $i$ and $j$. For any multi-index $b$ of the contracted tensor space, the $b$-th component of the contracted tensor is given by the double sum over the basis indices of the representation spaces at $i$ and $j$: $$[\text{contrT}_{i,j}(t)]_b = \sum_{x_1=0}^{d_i-1} \sum_{x_2=0}^{d_j-1} t_{B(b, x_1, x_2)} \cdot \text{contr}(e_{x_1}^{(c_i)} \otimes e_{x_2}^{(c_j)})$$ where: - $d_i$ and $d_j$ are the dimensions of the representation spaces $V_{c_i}$ and $V_{c_j}$ respectively. - $B(b, x_1, x_2)$ is the multi-index of length $n+2$ formed by inserting the index $x_1$ at position $i$, the index $x_2$ at position $j$, and the indices of $b$ at the remaining $n$ positions. - $t_{B(b, x_1, x_2)}$ is the component of the original tensor $t$ at multi-index $B$. - $e_{x_1}^{(c_i)}$ and $e_{x_2}^{(c_j)}$ are the $x_1$-th and $x_2$-th basis vectors of the spaces $V_{c_i}$ and $V_{c_j}$. - $\text{contr}: V_{c_i} \otimes V_{c_j} \to k$ is the contraction morphism (pairing) defined for the tensor species.