Physlib.Relativity.Tensors.ComplexTensor.Weyl.Two

This definition establishes a $\mathbb{C}$-linear isomorphism between the vector space of the tensor product of two left-handed Weyl fermions, $(L \otimes L).V$, and the space of $2 \times 2$ complex matrices, $\text{Mat}_{2 \times 2}(\mathbb{C})$. The equivalence is constructed by mapping the tensor product of the standard bases for each left-handed fermion space to the corresponding entries of a $2 \times 2$ matrix.

For any $2 \times 2$ complex matrix $M \in \text{Mat}_{2 \times 2}(\mathbb{C})$, the inverse of the linear isomorphism $\text{leftLeftToMatrix}$ (which identifies the vector space of the tensor product of two left-handed Weyl fermions $(L \otimes L).V$ with $2 \times 2$ matrices) maps $M$ to: $$\text{leftLeftToMatrix.symm}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (e_i \otimes e_j)$$ where $e_i$ and $e_j$ denote the standard basis vectors (`leftBasis`) of the left-handed Weyl fermion vector space, $M_{ij}$ represents the entry of the matrix $M$ at row $i$ and column $j$, and $\otimes$ denotes the tensor product over $\mathbb{C}$.

This definition establishes a linear equivalence (isomorphism) between the vector space of the tensor product of two alternative left-handed Weyl fermions, denoted $V_{L'} \otimes V_{L'}$, and the space of $2 \times 2$ complex matrices $\text{Mat}_{2 \times 2}(\mathbb{C})$. The equivalence is constructed by mapping the tensor product basis (formed by `altLeftBasis`) to the standard matrix basis.

For any $2 \times 2$ complex matrix $M \in \text{Mat}_{2 \times 2}(\mathbb{C})$, the image of $M$ under the inverse of the linear equivalence `altLeftaltLeftToMatrix` is given by the sum $$\text{altLeftaltLeftToMatrix}^{-1}(M) = \sum_{i=0}^1 \sum_{j=0}^1 M_{ij} (e'_i \otimes e'_j)$$ where $M_{ij}$ are the entries of the matrix $M$, and $e'_k$ denotes the $k$-th basis vector of the alternative left-handed Weyl fermion space (represented by `altLeftBasis`).

This definition establishes a linear isomorphism between the vector space associated with the tensor product of a left-handed Weyl fermion $V_L$ and an alternate left-handed Weyl fermion $V_{\tilde{L}}$ and the space of $2 \times 2$ complex matrices $M_{2 \times 2}(\mathbb{C})$. Specifically, for a tensor expressed in the basis $\{e_i \otimes f_j\}_{i,j \in \{0,1\}}$, where $\{e_i\}$ is the standard basis for the left-handed spinors (`leftBasis`) and $\{f_j\}$ is the standard basis for the alternate left-handed spinors (`altLeftBasis`), the map identifies the tensor with a matrix whose $(i, j)$-th entry is the coefficient of the basis element $e_i \otimes f_j$.

For any $2 \times 2$ complex matrix $M \in M_{2 \times 2}(\mathbb{C})$, the inverse of the linear isomorphism $\text{leftAltLeftToMatrix} : (V_L \otimes V_{\tilde{L}}) \cong M_{2 \times 2}(\mathbb{C})$ maps $M$ to the tensor sum $$\text{leftAltLeftToMatrix}^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (e_i \otimes f_j)$$ where $\{e_i\}$ is the standard basis for the left-handed Weyl spinor space $V_L$ (`leftBasis`) and $\{f_j\}$ is the standard basis for the alternate left-handed Weyl spinor space $V_{\tilde{L}}$ (`altLeftBasis`).

The definition `altLeftLeftToMatrix` provides a $\mathbb{C}$-linear equivalence between the vector space of the tensor product of an alternative left-handed Weyl fermion and a left-handed Weyl fermion, denoted $(altLeftHanded \otimes leftHanded).V$, and the space of $2 \times 2$ complex matrices $M_{2 \times 2}(\mathbb{C})$. Under this equivalence, a matrix $M$ corresponds to the tensor $\sum_{i,j} M_{ij} (e'_i \otimes e_j)$, where $\{e'_i\}$ and $\{e_j\}$ are the standard bases for the alternative left-handed and left-handed fermion spaces, respectively.

Let $M \in M_{2 \times 2}(\mathbb{C})$ be a $2 \times 2$ complex matrix. The inverse of the $\mathbb{C}$-linear equivalence `altLeftLeftToMatrix` maps $M$ to a tensor in the product space of an alternative left-handed Weyl fermion and a left-handed Weyl fermion, expressed in the standard bases as: $$\text{altLeftLeftToMatrix}^{-1}(M) = \sum_{i,j} M_{ij} (e'_i \otimes e_j)$$ where $\{e'_i\}_{i \in \{0,1\}}$ is the standard basis for the alternative left-handed fermion space (`altLeftBasis`) and $\{e_j\}_{j \in \{0,1\}}$ is the standard basis for the left-handed fermion space (`leftBasis`).

The complex linear equivalence between the vector space of the tensor product of two right-handed Weyl fermions and the space of $2 \times 2$ complex matrices.

For any $2 \times 2$ complex matrix $M \in M_{2 \times 2}(\mathbb{C})$, the inverse of the linear equivalence `rightRightToMatrix` (which maps the tensor product of two right-handed Weyl fermions to matrices) maps $M$ back to the tensor product space according to the expansion: $$\text{rightRightToMatrix}^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (b_i \otimes b_j)$$ where $M_{ij}$ represents the entry of the matrix $M$ at row $i$ and column $j$, and $b_k$ represents the $k$-th element of the standard basis for right-handed Weyl fermions (`rightBasis`).

The complex linear equivalence between the vector space of the tensor product of two alternative right-handed Weyl fermions and the space of $2 \times 2$ complex matrices $M_{2 \times 2}(\mathbb{C})$. This equivalence is established by representing elements of the tensor product in the basis formed by the tensor product of `altRightBasis` with itself.

For any $2 \times 2$ complex matrix $M \in M_{2 \times 2}(\mathbb{C})$, the inverse of the linear equivalence $\text{altRightAltRightToMatrix}$ (which maps the tensor product of alternative right-handed Weyl fermions to matrices) can be expanded as: $$\text{altRightAltRightToMatrix}^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (\tilde{e}_i \otimes \tilde{e}_j)$$ where $\tilde{e}_k$ denotes the $k$-th element of the standard basis for the alternative right-handed Weyl fermion space (`altRightBasis`).

This definition establishes a $\mathbb{C}$-linear equivalence between the vector space of the tensor product of a right-handed Weyl fermion and an alternative right-handed Weyl fermion, denoted as $(V_R \otimes V_{\text{alt}R})$, and the space of $2 \times 2$ complex matrices, $\text{Mat}_{2 \times 2}(\mathbb{C})$. The isomorphism is defined by mapping the tensor product of the canonical bases $\{e_i \otimes \tilde{e}_j\}$ to the corresponding matrix entries $M_{ij}$.

For any $2 \times 2$ complex matrix $M \in \text{Mat}_{2 \times 2}(\mathbb{C})$, the inverse of the linear equivalence $\text{rightAltRightToMatrix}$ (which maps elements of the tensor product space $V_R \otimes V_{\text{alt}R}$ to matrices) maps $M$ back to its representation in the tensor product basis: $$\text{rightAltRightToMatrix}^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (e_i \otimes \tilde{e}_j)$$ where $\{e_i\}$ is the standard basis for the right-handed Weyl fermion space (`rightBasis`) and $\{\tilde{e}_j\}$ is the standard basis for the alternative right-handed Weyl fermion space (`altRightBasis`).

The linear equivalence `altRightRightToMatrix` is an isomorphism between the vector space $V$ associated with the tensor product of an alternate right-handed Weyl fermion and a right-handed Weyl fermion, denoted as $(\text{altRightHanded} \otimes \text{rightHanded}).V$, and the space of $2 \times 2$ complex matrices $M_{2 \times 2}(\mathbb{C})$. This isomorphism is constructed by mapping the tensor product basis $\{e_i \otimes f_j\}$—where $e_i$ is the $i$-th basis vector of the alternate right-handed fermion space (`altRightBasis`) and $f_j$ is the $j$-th basis vector of the right-handed fermion space (`rightBasis`)—to the corresponding matrix components $M_{ij}$.

For any $2 \times 2$ complex matrix $M \in M_{2 \times 2}(\mathbb{C})$, the inverse of the linear isomorphism `altRightRightToMatrix` maps $M$ back to the tensor product space of an alternate right-handed and a right-handed Weyl fermion according to the formula: $$\text{altRightRightToMatrix}^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} \cdot (e_i \otimes f_j)$$ where $e_i$ are the basis vectors of the alternate right-handed Weyl fermion space (`altRightBasis`) and $f_j$ are the basis vectors of the right-handed Weyl fermion space (`rightBasis`).

This is a $\mathbb{C}$-linear equivalence between the vector space $V$ of the tensor product of an alternating left-handed Weyl fermion and an alternating right-handed Weyl fermion, denoted by $(\text{altLeftHanded} \otimes \text{altRightHanded}).V$, and the space of $2 \times 2$ matrices over the complex numbers, $\text{Mat}_{2 \times 2}(\mathbb{C})$.

For any $2 \times 2$ complex matrix $M \in \text{Mat}_{2 \times 2}(\mathbb{C})$, the inverse of the linear equivalence between the tensor product of alternating left-handed and right-handed Weyl fermions and $2 \times 2$ matrices, denoted $\text{altLeftAltRightToMatrix}^{-1}$, maps $M$ to the sum: $$\text{altLeftAltRightToMatrix}^{-1}(M) = \sum_{i, j} M_{ij} (e^L_i \otimes e^R_j)$$ where $M_{ij}$ are the entries of the matrix $M$, $e^L_i$ are the basis elements of the alternating left-handed Weyl fermion space (`altLeftBasis`), and $e^R_j$ are the basis elements of the alternating right-handed Weyl fermion space (`altRightBasis`).

This definition provides a $\mathbb{C}$-linear equivalence between the tensor product of the vector spaces associated with left-handed and right-handed Weyl fermions, denoted $V_L \otimes V_R$, and the space of $2 \times 2$ complex matrices $\text{Mat}_{2 \times 2}(\mathbb{C})$.

Let $V_L$ and $V_R$ be the vector spaces associated with left-handed and right-handed Weyl fermions, respectively, and let $\{e^L_i\}$ and $\{e^R_j\}$ be their standard bases. Let $\Phi : V_L \otimes V_R \cong \text{Mat}_{2 \times 2}(\mathbb{C})$ be the linear equivalence `leftRightToMatrix`. For any $2 \times 2$ complex matrix $M$, the inverse mapping $\Phi^{-1}$ (denoted as `leftRightToMatrix.symm`) is given by: $$\Phi^{-1}(M) = \sum_{i, j \in \{0, 1\}} M_{ij} (e^L_i \otimes e^R_j)$$ where $M_{ij}$ are the entries of the matrix $M$, and $\otimes$ denotes the tensor product over $\mathbb{C}$.

Let $L$ denote the vector space of left-handed Weyl fermions. Let $\Phi: L \otimes L \xrightarrow{\cong} \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{C}$-linear isomorphism (defined as `leftLeftToMatrix`) that maps the tensor product of two left-handed fermions to the space of $2 \times 2$ complex matrices. For any vector $v \in L \otimes L$ and any transformation $M \in SL(2, \mathbb{C})$, the action of $M$ on the tensor product space, given by the induced representation $\rho(M) \otimes \rho(M)$, is equivalent to the matrix transformation: $$\Phi((\rho(M) \otimes \rho(M)) v) = M \Phi(v) M^\intercal$$ where $M$ is the matrix representation of the group element and $M^\intercal$ is its transpose.

Let $V_{L'}$ denote the vector space of alternative left-handed Weyl fermions. For any element $v \in V_{L'} \otimes V_{L'}$ in the tensor product space and any matrix $M \in SL(2, \mathbb{C})$, the action of $M$ on $v$ (defined by the tensor product of the representations $\rho(M) \otimes \rho(M)$) corresponds to the following matrix transformation: $$\Phi((\rho(M) \otimes \rho(M)) v) = (M^{-1})^T \Phi(v) M^{-1}$$ where $\Phi: V_{L'} \otimes V_{L'} \to \text{Mat}_{2 \times 2}(\mathbb{C})$ is the linear isomorphism `altLeftaltLeftToMatrix`, and $M^{-1}$ and $M^T$ denote the matrix inverse and transpose of $M$, respectively.

Let $V_L$ and $V_{\tilde{L}}$ be the vector spaces associated with left-handed Weyl spinors and alternate left-handed Weyl spinors, respectively. Let $\Phi: V_L \otimes V_{\tilde{L}} \to M_{2 \times 2}(\mathbb{C})$ be the linear isomorphism `leftAltLeftToMatrix` which maps a tensor $v$ to a $2 \times 2$ complex matrix. For any group element $M \in SL(2, \mathbb{C})$ and any tensor $v \in V_L \otimes V_{\tilde{L}}$, the representation of $SL(2, \mathbb{C})$ on the tensor product space satisfies: $$\Phi((\rho_L(M) \otimes \rho_{\tilde{L}}(M))v) = M \Phi(v) M^{-1}$$ where $\rho_L$ and $\rho_{\tilde{L}}$ are the group representations on $V_L$ and $V_{\tilde{L}}$, and $M$ is the matrix representation of the group element.

Let $V_{\text{altL}}$ and $V_{\text{L}}$ be the vector spaces associated with the alternative left-handed and left-handed Weyl fermion representations, respectively. Let $\rho_{\text{altL}}$ and $\rho_{\text{L}}$ be the corresponding representations of $SL(2, \mathbb{C})$ on these spaces. For any element $v \in V_{\text{altL}} \otimes V_{\text{L}}$ and any group element $M \in SL(2, \mathbb{C})$, the $\mathbb{C}$-linear equivalence $f: V_{\text{altL}} \otimes V_{\text{L}} \to M_{2 \times 2}(\mathbb{C})$ (defined by `altLeftLeftToMatrix`) satisfies: $$f((\rho_{\text{altL}}(M) \otimes \rho_{\text{L}}(M))v) = (M^{-1})^T f(v) M^T$$ where $M^{-1}$ is the matrix inverse and $M^T$ is the matrix transpose of the matrix representation of $M$.

Let $V_R \otimes V_R$ be the vector space of the tensor product of two right-handed Weyl fermions. Let $\Phi : V_R \otimes V_R \cong \text{Mat}(2, \mathbb{C})$ be the complex linear equivalence `rightRightToMatrix` that maps a tensor to a $2 \times 2$ complex matrix. For any tensor $v \in V_R \otimes V_R$ and any group element $M \in SL(2, \mathbb{C})$, the action of $M$ on the tensor product, denoted by $\rho(M) \otimes \rho(M)$, satisfies the identity: $$\Phi((\rho(M) \otimes \rho(M)) v) = \bar{M} \Phi(v) \bar{M}^{\text{T}}$$ where $\bar{M}$ is the complex conjugate of the matrix $M$ and $\bar{M}^{\text{T}}$ is its transpose.

Let $V_{\text{altRight}}$ be the vector space associated with alternative right-handed Weyl fermions, and let $\rho$ be the representation of $SL(2, \mathbb{C})$ on $V_{\text{altRight}}$. Let $\Phi : V_{\text{altRight}} \otimes V_{\text{altRight}} \xrightarrow{\cong} M_{2 \times 2}(\mathbb{C})$ be the complex linear equivalence that maps an element of the tensor product to its corresponding $2 \times 2$ complex matrix representation. For any vector $v \in V_{\text{altRight}} \otimes V_{\text{altRight}}$ and any matrix $M \in SL(2, \mathbb{C})$, the matrix representation of the transformed vector under the induced group action is given by: $$\Phi((\rho(M) \otimes \rho(M)) v) = (M^{-1})^\dagger \Phi(v) ((M^{-1})^\dagger)^T$$ where $(M^{-1})^\dagger$ denotes the conjugate transpose of the inverse of $M$, and $T$ denotes the transpose.

Let $V_R$ be the vector space of right-handed Weyl fermions and $V_{\text{alt}R}$ be the vector space of alternative right-handed Weyl fermions. Let $\Phi: V_R \otimes V_{\text{alt}R} \cong \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{C}$-linear equivalence that maps the tensor product of these spaces to $2 \times 2$ complex matrices. For any vector $v \in V_R \otimes V_{\text{alt}R}$ and any matrix $M \in SL(2, \mathbb{C})$, the group action of $SL(2, \mathbb{C})$ on the tensor product is given by: $$\Phi\left( (\rho_R(M) \otimes \rho_{\text{alt}R}(M)) v \right) = \bar{M} \cdot \Phi(v) \cdot ((M^{-1})^\dagger)^T$$ where $\rho_R$ and $\rho_{\text{alt}R}$ are the representations of $SL(2, \mathbb{C})$ on $V_R$ and $V_{\text{alt}R}$ respectively, $\bar{M}$ denotes the entry-wise complex conjugate of $M$, $M^{-1}$ is the matrix inverse, and $\dagger$ and $T$ denote the conjugate transpose and transpose operations.

Let $V$ be the vector space associated with the tensor product of an alternate right-handed Weyl fermion and a right-handed Weyl fermion, denoted as $\text{altRightHanded} \otimes \text{rightHanded}$. Let $\Phi: V \to M_{2 \times 2}(\mathbb{C})$ be the linear isomorphism `altRightRightToMatrix`. For any $M \in SL(2, \mathbb{C})$ and any $v \in V$, the group action of $SL(2, \mathbb{C})$ on $v$ through the tensor product of representations $\rho_{\text{altRight}} \otimes \rho_{\text{right}}$ corresponds to the matrix transformation: $$\Phi((\rho_{\text{altRight}}(M) \otimes \rho_{\text{right}}(M)) v) = (M^{-1})^\dagger \Phi(v) \overline{M}^\top$$ where $(M^{-1})^\dagger$ denotes the conjugate transpose of the inverse of $M$, and $\overline{M}^\top$ denotes the transpose of the complex conjugate of $M$.

Let $V$ be the vector space corresponding to the tensor product of an alternating left-handed Weyl fermion and an alternating right-handed Weyl fermion, denoted by $(\text{altLeftHanded} \otimes \text{altRightHanded}).V$. Let $\Phi: V \to \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{C}$-linear equivalence `altLeftAltRightToMatrix` which maps vectors in this space to $2 \times 2$ complex matrices. For any vector $v \in V$ and any group element $M \in SL(2, \mathbb{C})$, the action of $M$ on $v$ through the representation $\rho$ satisfies: \[ \Phi(\rho(M)v) = (M^{-1})^T \Phi(v) \overline{M^{-1}} \] where $(M^{-1})^T$ is the transpose of the inverse of $M$, and $\overline{M^{-1}}$ is the complex conjugate of the inverse of $M$.

Let $V_L$ and $V_R$ be the vector spaces associated with left-handed and right-handed Weyl fermions, respectively. Let $\phi : V_L \otimes V_R \cong \text{Mat}_{2 \times 2}(\mathbb{C})$ be the linear equivalence `leftRightToMatrix`. For any tensor $v \in V_L \otimes V_R$ and any matrix $M \in SL(2, \mathbb{C})$, the action of the group on the tensor product satisfies: $$\phi\left( (\rho_L(M) \otimes \rho_R(M)) v \right) = M \phi(v) M^\dagger$$ where $\rho_L(M)$ and $\rho_R(M)$ are the representations of $SL(2, \mathbb{C})$ on $V_L$ and $V_R$ respectively, and $M^\dagger$ denotes the conjugate transpose (Hermitian conjugate) of the matrix $M$.

Let $L$ be the vector space of left-handed Weyl fermions. Let $\Phi: L \otimes L \cong \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{C}$-linear isomorphism (defined as `leftLeftToMatrix`) that identifies the tensor product of two left-handed fermions with the space of $2 \times 2$ complex matrices. For any matrix $v \in \text{Mat}_{2 \times 2}(\mathbb{C})$ and any transformation $M \in SL(2, \mathbb{C})$, the induced representation $\rho_L(M) \otimes \rho_L(M)$ acting on the tensor $\Phi^{-1}(v)$ satisfies: $$(\rho_L(M) \otimes \rho_L(M))(\Phi^{-1}(v)) = \Phi^{-1}(M v M^\intercal)$$ where $\Phi^{-1}$ is the inverse isomorphism `leftLeftToMatrix.symm`, $\rho_L(M)$ is the representation of $SL(2, \mathbb{C})$ on $L$, and $M^\intercal$ denotes the transpose of the matrix $M$.

Let $V_{L'}$ be the vector space of alternative left-handed Weyl fermions. Let $\Phi: V_{L'} \otimes V_{L'} \to \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{C}$-linear isomorphism `altLeftaltLeftToMatrix` that identifies the tensor product with the space of $2 \times 2$ complex matrices. For any matrix $v \in \text{Mat}_{2 \times 2}(\mathbb{C})$ and any transformation $M \in SL(2, \mathbb{C})$, the induced representation $\rho_{L'}(M) \otimes \rho_{L'}(M)$ acting on the tensor $\Phi^{-1}(v)$ satisfies: $$(\rho_{L'}(M) \otimes \rho_{L'}(M))(\Phi^{-1}(v)) = \Phi^{-1}((M^{-1})^T v M^{-1})$$ where $\Phi^{-1}$ is the inverse isomorphism `altLeftaltLeftToMatrix.symm`, $\rho_{L'}$ is the representation of $SL(2, \mathbb{C})$ on $V_{L'}$, and $(M^{-1})^T$ denotes the transpose of the inverse matrix of $M$.

Let $V_L$ and $V_{\tilde{L}}$ be the vector spaces associated with left-handed Weyl spinors and alternate left-handed Weyl spinors, respectively. Let $\Phi: V_L \otimes V_{\tilde{L}} \to M_{2 \times 2}(\mathbb{C})$ be the linear isomorphism `leftAltLeftToMatrix` that identifies a tensor with a $2 \times 2$ complex matrix. For any matrix $v \in M_{2 \times 2}(\mathbb{C})$ and any group element $M \in SL(2, \mathbb{C})$, the representation of $SL(2, \mathbb{C})$ on the tensor product space satisfies: $$(\rho_L(M) \otimes \rho_{\tilde{L}}(M))(\Phi^{-1}(v)) = \Phi^{-1}(M v M^{-1})$$ where $\Phi^{-1}$ is the inverse isomorphism (`leftAltLeftToMatrix.symm`), and $\rho_L$ and $\rho_{\tilde{L}}$ are the group representations on $V_L$ and $V_{\tilde{L}}$.

Let $V_{\text{altL}}$ and $V_{\text{L}}$ be the vector spaces associated with the alternative left-handed and left-handed Weyl fermion representations, respectively. Let $\Phi: V_{\text{altL}} \otimes V_{\text{L}} \to M_{2 \times 2}(\mathbb{C})$ be the complex linear equivalence `altLeftLeftToMatrix`. For any matrix $v \in M_{2 \times 2}(\mathbb{C})$ and any group element $M \in SL(2, \mathbb{C})$, the representation of $SL(2, \mathbb{C})$ on the tensor product space satisfies: $$(\rho_{\text{altL}}(M) \otimes \rho_{\text{L}}(M))(\Phi^{-1}(v)) = \Phi^{-1}((M^{-1})^T v M^T)$$ where $\Phi^{-1}$ is the inverse isomorphism `altLeftLeftToMatrix.symm`, $\rho_{\text{altL}}$ and $\rho_{\text{L}}$ are the representations of $SL(2, \mathbb{C})$ on $V_{\text{altL}}$ and $V_{\text{L}}$, respectively, $(M^{-1})^T$ is the transpose of the inverse matrix of $M$, and $M^T$ is the transpose matrix of $M$.

Let $V_R$ be the vector space of right-handed Weyl spinors. Let $\Phi: V_R \otimes V_R \to M_{2 \times 2}(\mathbb{C})$ be the complex linear equivalence `rightRightToMatrix` that identifies a tensor with a $2 \times 2$ complex matrix. For any matrix $v \in M_{2 \times 2}(\mathbb{C})$ and any group element $M \in SL(2, \mathbb{C})$, the representation of $SL(2, \mathbb{C})$ on the tensor product space satisfies: $$(\rho_R(M) \otimes \rho_R(M))(\Phi^{-1}(v)) = \Phi^{-1}(\bar{M} v \bar{M}^{\text{T}})$$ where $\Phi^{-1}$ is the inverse isomorphism `rightRightToMatrix.symm`, $\rho_R$ is the group representation on $V_R$, $\bar{M}$ is the complex conjugate of the matrix $M$, and $\bar{M}^{\text{T}}$ is its transpose.

Let $V_{\text{altRight}}$ be the vector space associated with alternative right-handed Weyl fermions, and let $\rho_{\text{altRight}}$ be the representation of $SL(2, \mathbb{C})$ on $V_{\text{altRight}}$. Let $\Phi: V_{\text{altRight}} \otimes V_{\text{altRight}} \to M_{2 \times 2}(\mathbb{C})$ be the complex linear equivalence `altRightAltRightToMatrix`. For any matrix $v \in M_{2 \times 2}(\mathbb{C})$ and any group element $M \in SL(2, \mathbb{C})$, the representation of $SL(2, \mathbb{C})$ on the tensor product space satisfies: $$(\rho_{\text{altRight}}(M) \otimes \rho_{\text{altRight}}(M))(\Phi^{-1}(v)) = \Phi^{-1}\left((M^{-1})^\dagger v ((M^{-1})^\dagger)^T\right)$$ where $\Phi^{-1}$ is the inverse isomorphism `altRightAltRightToMatrix.symm`, $(M^{-1})^\dagger$ denotes the conjugate transpose of the inverse of $M$, and $T$ denotes the transpose.

Let $V_R$ and $V_{\text{alt}R}$ be the vector spaces of right-handed and alternative right-handed Weyl fermions, respectively. Let $\Phi: V_R \otimes V_{\text{alt}R} \xrightarrow{\cong} \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{C}$-linear equivalence `rightAltRightToMatrix` that identifies the tensor product space with $2 \times 2$ complex matrices. For any matrix $v \in \text{Mat}_{2 \times 2}(\mathbb{C})$ and any group element $M \in SL(2, \mathbb{C})$, the group action on the pre-image of $v$ under $\Phi$ satisfies: $$(\rho_R(M) \otimes \rho_{\text{alt}R}(M)) (\Phi^{-1}(v)) = \Phi^{-1} \left( \bar{M} v ((M^{-1})^\dagger)^T \right)$$ where $\Phi^{-1}$ is the inverse map `rightAltRightToMatrix.symm`, $\rho_R$ and $\rho_{\text{alt}R}$ are the group representations on $V_R$ and $V_{\text{alt}R}$, $\bar{M}$ denotes the entry-wise complex conjugate of $M$, $M^{-1}$ is the matrix inverse, and $\dagger$ and $T$ denote the conjugate transpose and transpose operations, respectively.