Physlib.Relativity.Tensors.TensorSpecies.Basic

For a tensor species $S$ and any color $c$ in its set of colors $C$, the representation dimension $\text{repDim}(c)$ is non-zero, i.e., $\text{repDim}(c) \neq 0$.

For any tensor species $S$ and any color $c \in C$, applying the duality map $\tau$ twice results in the original color, i.e., $\tau(\tau(c)) = c$.

Let $S$ be a tensor species and $C$ be the collection of its colors. For any colors $c, c_1 \in C$ such that $c = c_1$, let $h: c_1 \to c$ be the morphism induced by this equality in the discrete category of colors. For any index $i$ in the finite set $\{0, \dots, \text{repDim}(c)-1\}$, the $i$-th basis vector of the representation space corresponding to color $c$ is the image of the $i$-th basis vector of the representation space corresponding to color $c_1$ under the linear map $S.FD(h)$ induced by the representation functor. That is, $$S.\text{basis}(c, i) = S.FD(h)(S.\text{basis}(c_1, i'))$$ where $i'$ is the index $i$ cast to the dimension of the representation space for $c_1$.

Let $S$ be a tensor species and $C$ be the collection of its colors. For any colors $c, c_1 \in C$ such that $c = c_1$, let $h: c \to c_1$ be the isomorphism in the discrete category of colors induced by this equality. For any vector $t$ in the representation space $S.FD(c)$ and any index $i$ in the finite set $\{0, \dots, \text{repDim}(c)-1\}$, the $i$-th coordinate of $t$ with respect to the basis $S.\text{basis}(c)$ is equal to the $i$-th coordinate of the image of $t$ under the linear map $S.FD(h)$ with respect to the basis $S.\text{basis}(c_1)$. That is, $$(S.\text{basis } c).\text{repr}(t)_i = (S.\text{basis } c_1).\text{repr}(S.FD(h)(t))_{i'}$$ where $i'$ is the index $i$ cast to the dimension of the representation space for color $c_1$.

Let $S$ be a tensor species and let $c, c_1$ be objects in the category of colors $C$. If $c = c_1$, let $h$ be the proof of this equality. The linear map induced by the base representation functor $S.FD$ for the morphism $h$ maps the $i$-th basis vector of the representation space corresponding to $c$ to the $i$-th basis vector of the representation space corresponding to $c_1$. That is, $$S.FD(h)(e_i^{(c)}) = e_i^{(c_1)}$$ where $e_i^{(c)}$ denotes the $i$-th basis element for color $c$, and the index $i$ is cast appropriately between the two identical dimensions.

Let $S$ be a tensor species with a category of colors $C$ and a group $G$. For any colors $c, c_1 \in C$ such that $c = c_1$, let $\rho^{(c)}(g)$ be the linear operator representing $g \in G$ on the representation space $S.FD(c)$. For any indices $i, b \in \{0, \dots, \text{repDim}(c)-1\}$, the $i$-th component of the vector $\rho^{(c)}(g) e_b^{(c)}$ relative to the basis $S.\text{basis}(c)$ is equal to the $i'$-th component of the vector $\rho^{(c_1)}(g) e_{b'}^{(c_1)}$ relative to the basis $S.\text{basis}(c_1)$, where $e_k^{(c)}$ denotes the $k$-th basis vector for color $c$, and $i', b'$ are the indices $i$ and $b$ cast to the dimension of the representation space for color $c_1$. In terms of matrix elements, this states: $$[ \rho^{(c)}(g) ]^i_b = [ \rho^{(c_1)}(g) ]^{i'}_{b'}$$

For a given tensor species $S$, the functor $F: \text{OverColor}(C) \to \text{Rep}_k(G)$ is the lift of the base representation functor $S.FD$ to the category of "colored" objects. It maps collections of indices (objects in $\text{OverColor}(C)$) to their corresponding representation spaces of the group $G$ over the field $k$.

For a given tensor species $S$, the representation functor $F$ is defined as the lift of the base representation functor $S.FD$ to the category of colored objects $\text{OverColor}(C)$.

For a given tensor species $S$, the representation functor $F: \text{OverColor}(C) \to \text{Rep}_k(G)$, which maps collections of tensor indices to their corresponding representation spaces, is a monoidal functor. This means that $F$ preserves the monoidal structure between the category of colored indices and the category of representations, naturally commuting with the tensor product operation.

For a given tensor species $S$, the representation functor $F: \text{OverColor}(C) \to \text{Rep}_k(G)$, which maps collections of tensor indices to their corresponding representation spaces, is a lax-braided monoidal functor. This implies that $F$ preserves the braiding structure (the commutativity of the tensor product) between the category of colored indices $\text{OverColor}(C)$ and the category of representations $\text{Rep}_k(G)$.

For a given tensor species $S$, the representation functor $F: \text{OverColor}(C) \to \text{Rep}_k(G)$, which maps collections of tensor indices to their corresponding representation spaces, is a braided monoidal functor. This means that $F$ preserves the braiding structure—the natural isomorphism that commutes the tensor product—between the category of colored indices $\text{OverColor}(C)$ and the category of representations $\text{Rep}_k(G)$.

For a tensor species $S$ over a field $k$, this function maps an element $v$ of the underlying vector space of the monoidal unit object $\mathbb{1}$ in the category of representations $\text{Rep}(k, G)$ (evaluated at a specific color or index $c$) to its corresponding scalar value in the field $k$.

Let $S$ be a tensor species over a field $k$, with a set of colors $C$ and a group $G$. Let $\mathbb{1}$ denote the monoidal unit in the functor category $\text{Discrete } C \leadsto \text{Rep}(k, G)$. For any color $c \in C$, let $v$ be an element of the underlying vector space of the representation obtained by evaluating the monoidal unit at $c$. Then, the result of casting $v$ to the field $k$ is equal to $v$ itself, i.e., $S.\text{castToField}(v) = v$.

For a tensor species $S$ over a field $k$, let $c: \text{Fin } 0 \to C$ be the unique map from the empty set of indices to the set of colors $C$. The function is the $k$-linear map from the vector space $(S.F(\text{OverColor.mk } c)).V$ to the field $k$, defined by the canonical isomorphism between the tensor product of an empty family of vector spaces and the base field $k$.

Let $S$ be a tensor species over a field $k$ and $C$ be a set of colors. Let $c: \text{Fin } 0 \to C$ be the unique map from the empty set of indices to $C$. For an empty family of vectors $x$ indexed by $\text{Fin } 0$, where each $x_i$ belongs to the representation space associated with color $c(i)$, the linear map $\text{castFin0ToField}$ applied to the tensor product of this family $\bigotimes_{i \in \text{Fin } 0} x_i$ is equal to $1$.

Let $S$ be a tensor species over a field $k$ with a set of index colors $C$ and a duality map $\tau: C \to C$. Let $V_c$ denote the vector space associated with a color $c \in C$. For any two colors $c, c' \in C$ such that $c = c'$, and for any vectors $x \in V_c$ and $y \in V_{\tau(c)}$, the contraction of the tensor product $x \otimes y$ using the contraction map for color $c$ is equal to the contraction using the map for color $c'$ applied to the images of $x$ and $y$ under the canonical isomorphisms $V_c \cong V_{c'}$ and $V_{\tau(c)} \cong V_{\tau(c')}$ induced by the equality of the colors.

Given a tensor species $S$ over a field $k$, a set of index colors $C$, and a group $G$, let $n \in \mathbb{N}$ be a natural number and $c : \{0, \dots, n-1\} \to C$ be a mapping that assigns a color to each index position. For any tensor belonging to the representation space $S.F(\text{OverColor.mk } c)$, this function returns the total number of indices $n$ associated with that tensor.