Physlib.Relativity.Tensors.Dual

Let $S$ be a tensor species over a ring $k$, and let $c \in C$ be a color with dual color $\tau(c)$. The linear map $\text{fromDualMap}$ transforms a tensor $t$ of shape $(\tau(c))$ into a tensor of shape $(c)$. This is defined by taking the tensor product of the metric tensor $g$ (of shape $(c, c)$) and the tensor $t$, and then contracting the second index of the metric tensor with the index of $t$.

For a tensor species $S$ over a ring $k$ and a color $c \in C$ with dual color $\tau(c)$, let $t \in S.\text{Tensor}(\tau(c))$ be a tensor. Then the application of the map $\text{fromDualMap}$ to $t$ is equal to the contraction of the second index of the metric tensor $g$ (associated with color $c$, having shape $(c, c)$) with the single index of $t$ in their tensor product $g \otimes t$.

Let $S$ be a tensor species over a ring $k$. For any color $c \in C$ with dual color $\tau(c)$, the linear map `toDualMap` transforms a tensor $t$ of shape $(c)$ into a tensor of shape $(\tau(c))$. The result is obtained by taking the tensor product of the metric tensor $g$ (associated with the dual color $\tau(c)$, having shape $(\tau(c), \tau(c))$) and the tensor $t$, and then contracting the second index of the metric tensor with the index of $t$. This construction relies on the property that the dual of the dual color $\tau(\tau(c))$ is the original color $c$.

For a tensor species $S$ over a ring $k$ and a color $c \in C$ with dual color $\tau(c)$, let $t \in S.\text{Tensor}(c)$ be a tensor. The application of the map $\text{toDualMap}$ to $t$ is equal to the contraction of the second index of the metric tensor $g$ (associated with the dual color $\tau(c)$ and having shape $(\tau(c), \tau(c))$) with the single index of $t$ in the tensor product $g \otimes t$.

Let $S$ be a tensor species over a ring $k$ and $c \in C$ be a color with dual $\tau(c)$. For any rank-1 tensor $t \in S.\text{Tensor}(\tau(c))$, the composition of the metric-induced linear maps $\text{fromDualMap} : S.\text{Tensor}(\tau(c)) \to S.\text{Tensor}(c)$ and $\text{toDualMap} : S.\text{Tensor}(c) \to S.\text{Tensor}(\tau(c))$ acts as the identity. That is, $$\text{toDualMap}(\text{fromDualMap}(t)) = t$$

Let $S$ be a tensor species over a ring $k$, and let $c \in C$ be a color with dual color $\tau(c)$. For any tensor $t$ of shape $(\tau(c))$, the result of applying the metric-induced linear map $\text{fromDualMap} : S.\text{Tensor}(\tau(c)) \to S.\text{Tensor}(c)$ to $t$ is equal to the result of applying $\text{toDualMap}$ to $t$ (which, for a tensor of color $\tau(c)$, maps to the double-dual space $S.\text{Tensor}(\tau(\tau(c)))$), subject to the identity permutation $\text{perm}_{\text{id}}$ that identifies the index color $\tau(\tau(c))$ with $c$.

Let $S$ be a tensor species over a ring $k$ and $c \in C$ be a color with dual $\tau(c)$. For any tensor $t \in S.\text{Tensor}(c)$, the application of the metric-induced linear map $\text{toDualMap} : S.\text{Tensor}(c) \to S.\text{Tensor}(\tau(c))$ to $t$ is equivalent to applying the map $\text{fromDualMap}$ to $t$, subject to an identity permutation $\text{perm}_{\text{id}}$ that identifies the index color $c$ with the double-dual color $\tau(\tau(c))$.

Let $S$ be a tensor species over a ring $k$ and $c \in C$ be a color with dual $\tau(c)$. For any rank-1 tensor $t \in S.\text{Tensor}(c)$, the composition of the metric-induced linear maps $\text{toDualMap} : S.\text{Tensor}(c) \to S.\text{Tensor}(\tau(c))$ and $\text{fromDualMap} : S.\text{Tensor}(\tau(c)) \to S.\text{Tensor}(c)$ acts as the identity. That is, $$\text{fromDualMap}(\text{toDualMap}(t)) = t$$

Let $S$ be a tensor species over a ring $k$. For any color $c \in C$ with dual color $\tau(c)$, the term `toDual` is the $k$-linear equivalence between the tensor space $S.\text{Tensor}(c)$ and its dual $S.\text{Tensor}(\tau(c))$. This isomorphism is formed by contracting tensors with the metric tensors, utilizing the linear maps `toDualMap` and `fromDualMap` as inverses of each other.

Let $S$ be a tensor species and $G$ be a group acting on the tensors. For any color $c \in C$, any group element $g \in G$, and any tensor $t \in S.\text{Tensor}([c])$, the dualization map $\text{toDual}: S.\text{Tensor}([c]) \to S.\text{Tensor}([\tau(c)])$ is equivariant with respect to the group action. That is, \[ \text{toDual}(g \cdot t) = g \cdot \text{toDual}(t) \] where $\tau(c)$ is the dual color and $\cdot$ denotes the group action of $G$ on the tensor space.

Let $S$ be a tensor species over a ring $k$ satisfying the strong rank condition. For any color $c \in C$, the representation dimension of its dual color $\tau(c)$ is equal to the representation dimension of $c$: $$\text{repDim}(\tau(c)) = \text{repDim}(c)$$