Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.OfFieldLabel

Given a charge spectrum $x$ and a field label $F$ in an $SU(5)$ Grand Unified Theory, the function $\text{ofFieldLabel}(x, F)$ returns the finite set of charges in $\mathcal{Z}$ associated with that label. For labels representing the $\bar{\mathbf{5}}$ (Higgs and matter) and $\mathbf{10}$ (matter) representations, it returns the corresponding set of charges defined in the spectrum $x$. For labels representing the conjugate representations, it returns the set of negated charges $\{ -z \mid z \in S \}$, where $S$ is the set of charges associated with the non-conjugated field label.

For any field label $F$ in an $SU(5)$ Grand Unified Theory, the finite set of charges in $\mathcal{Z}$ associated with that label in the empty charge spectrum $\emptyset$ is the empty set $\emptyset$.

For any two $SU(5)$ charge spectra $x, y \in \text{ChargeSpectrum } \mathcal{Z}$, if $x \subseteq y$, then for any field label $F \in \text{FieldLabel}$, the finite set of charges associated with $F$ in $x$ is a subset of those in $y$, i.e., $\text{ofFieldLabel}(x, F) \subseteq \text{ofFieldLabel}(y, F)$.

In an $SU(5)$ Grand Unified Theory with a charge spectrum $y$, a charge $x \in \mathcal{Z}$ is a member of the set of charges associated with the field label $\text{fiveHd}$ if and only if its negative $-x$ is a member of the set of charges associated with the conjugate field label $\text{fiveBarHd}$.

In an $SU(5)$ Grand Unified Theory with a charge spectrum $y$, a charge $x \in \mathcal{Z}$ is an element of the set of charges associated with the up-type Higgs field label $\text{fiveHu}$ if and only if its negative $-x$ is an element of the set of charges associated with the conjugate up-type Higgs field label $\text{fiveBarHu}$.

In an $SU(5)$ Grand Unified Theory with a charge spectrum $y$, a charge $x \in \mathcal{Z}$ is an element of the set of charges associated with the matter field label $\text{fiveMatter}$ if and only if its negative $-x$ is an element of the set of charges associated with the conjugate matter field label $\text{fiveBarMatter}$.

Let $x$ and $y$ be two charge spectra over a charge type $\mathcal{Z}$ in an $SU(5)$ Grand Unified Theory. If for every field label $F$, the finite set of charges associated with $x$ for that label is equal to the finite set of charges associated with $y$ for that label (i.e., $\text{ofFieldLabel}(x, F) = \text{ofFieldLabel}(y, F)$ for all $F$), then the charge spectra $x$ and $y$ are equal ($x = y$).