Physlib.Particles.StandardModel.Representations

The function maps an element $g$ of the unitary group $U(1)$ (represented as complex numbers with absolute value 1) to a $2 \times 2$ unitary matrix in $U(2)$. Specifically, it maps $g \in U(1)$ to the scalar matrix $g^3 I_2$, where $I_2$ is the $2 \times 2$ identity matrix. This represents a 2-dimensional representation of $U(1)$ with charge 3.

The group homomorphism from the unitary group $U(1)$ to the unitary group $U(2)$ (the group of $2 \times 2$ complex unitary matrices) representing a 2-dimensional representation with charge 3. Specifically, for an element $g \in U(1)$, the representation maps $g$ to the scalar matrix $g^3 I_2$, where $I_2$ is the $2 \times 2$ identity matrix.

The group homomorphism from the special unitary group $SU(2)$ to the unitary group $U(2)$ that corresponds to the fundamental representation. It maps a $2 \times 2$ complex matrix $g$ with $\det g = 1$ and $g^\dagger g = I$ to itself, viewed as an element of the unitary group of degree 2.

For any element $u_1$ in the unitary group $U(1)$ and any element $g$ in the special unitary group $SU(2)$, the 2-dimensional representation of $u_1$ with charge 3 and the fundamental representation of $g$ commute within the unitary group $U(2)$. That is, $\rho_{U(1)}(u_1) \cdot \rho_{SU(2)}(g) = \rho_{SU(2)}(g) \cdot \rho_{U(1)}(u_1)$, where $\rho_{U(1)}(u_1) = u_1^3 I_2$ and $\rho_{SU(2)}(g)$ is the inclusion of $g$ into $U(2)$.