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\chapter{Supplementary Material}
\section{Projective representations}
\label{sec:proj_rep}
Let $G$ be a group, and $c:\,G\times G \to \mathbb{C}$ a function that fulfills the following conditions (for arbitrary $x,y,z \in G$):
\begin{itemize}
    \item $\abs{c(x,y)} = 1$ 
    \item $c(1_G,x) = c(x,1_G) = 1$
    \item $c(x,y)^{-1} = c(y,x)^*$
    \item $c(x,y)c(xy,z) = c(x,yz)c(y,z)$
\end{itemize}
Then $c$ is called a \textit{standard factor system} or a \textit{co-cycle} of $G$.\\
A projective representation of dimension $n$ of $G$ is given by a $n$ dimensional vector space $V$ and a mapping $\rho:G \to GL(V)$, that fulfills the condition
\begin{align}
    \rho(x)\cdot\rho(y) = c(x,y)\rho(xy)\quad \forall x,y,\in G.
\end{align}
If $c(x,y) = 1$ for all $x,y$ in $G$ then this projective representation is called a \textit{true representation} or simply a \textit{representation}~\cite{10.1063/1.523181}. The matrix representation of $\rho$ is often written as $d^{\rho}$. A subspace $U\subset V$ is called an invariant subspace of $\rho$, if $\rho(U) \subset U$. If $V$ and $\{0\}$ are the only possible choices for $U$, then $\rho$ is called an irreducible representation or an \textit{irrep}~\cite{scherer2015symmetrien}. \\
A representation is called \textit{faithful} if it is injective. 
\section{Central Group Extensions}
\label{sec:central_extension}
It is possible to write a group $G$ as a direct product of a normal subgroup $N$ and the quotient group $G/N$ as
\begin{align}
    G = N \times G/N.
\end{align}
A group extension proceeds the other way around, a group $G$ is constructed out of two groups $N$ and $Q$, where $N$ is a normal subgroup of $G$ and $Q \sim G/N$. One often calls $G$ a group extension of $N$ by $Q$. If $N$ is in the center of $G$, i.e., each element of $N$ commutes with every element of $G$, the group extension is called a central extension.\\
In this thesis mainly irreducible matrix groups are considered. The center of those groups is always proportional to the identity. For example every central extension of $SU(N)$ is a subgroup of $U(n)=\mathbb{C}\rtimes SU(n)$.
\section{Generelization of Wigner's Little Group Procedure}
\label{sec:wigner}
In this section the method to compute irreps of semi-direct products is given, i.e., the way to construct the irreps of $G = N \rtimes H$ out of the irreps of $N$ and $H$. This method consists of four steps which will be explained in detail:
\begin{enumerate}
    \item At first one has to decompose the representations of $N$ into $H$-orbits. In order to do this one defines the following equivalence relation:
    \begin{align}
        \gamma \sim \gamma' \Longleftrightarrow \exists h \in H \textrm{ and }U \in SU(V):\,d^{\gamma}(hnh^{-1}) = Ud^{\gamma'}(n)U^{\dagger}.
    \end{align}
    This means that two irreps of $N$ are in the same orbits iff they can be transformed into each other by conjugation with $h$ up to equivalence. If $N$ is an abelian group, all of its irreps are one dimensional and the transformation $U$ can be left out.
    \item Next one has to select a representative $\gamma$ for each orbit and define its little group $H(\gamma)$ in the following way:
    \begin{align}
        H(\gamma) = \lbrace h \in H: d^{\gamma}(hnh^{-1}) =  U d^{\gamma}(n)U^{\dagger}\textrm{ for some $U \in SU(V)$}\rbrace.
    \end{align}
    Again, if $N$ is an abelian group the unitary transformation can be left out. The stability group of $\gamma$ is then given by
    \begin{align}
        L(\gamma) = N \rtimes H(\gamma) \subset G.
    \end{align}
    \item The next task is to find the \textit{allowable} irreps of $L(\gamma)$. If $N$ is an abelian group it is rather simple: The irreps are given by $d^{\gamma} \otimes d^{\delta}$, where $\delta$ is an irrep of $H(\gamma)$. For the non-abelian case we have
    \begin{align}
        d^{(\gamma)}(hnh^{-1}) = U(h) d^{(\gamma)}(n)U^{-1}(h).
    \end{align}
    By Schur's Lemma, $U(h)$ is unique up to a phase, and therefore these transformations generate a projective representation of $H$. If the phases can be chosen such that the $U(h)$ give a true representation, then the irreps are given by
    \begin{align}
      (0,h)\cdot(n,1) = (hnh^{-1},h) \mapsto U(h)\cdot d^{\gamma}(n) \otimes d^{\delta}(h).
    \end{align}
    If the unitary matrices cannot be chosen in this way, one uses projective representations of $H(\gamma)$ such that they cancel. In order to do this one considers the central extension $\Tilde H(\gamma)$ of $H(\gamma)$ and takes its faithful representations $\tilde \delta$ instead of $\delta$~\cite{10.1063/1.523181,KILCUP1987493,janssen2004theory}.
\end{enumerate}
\section{Grassmann algebra}
\label{sec:Grassmann}
While scalar fields can be described by complex numbers and vector fields are usually described by real four-component vectors, fermions have to be described by a type of numbers that anticommute in order to satisfy the spin-statistics theorem. The type of numbers needed in this case is called the Grassmann algebra. Its construction is explained in the following section. \\
The first ingredient is a vector space $V$ whose dimension $N$ corresponds to two times the number of degrees of freedoms of the fermion field. In the context of staggered fermions in lattice gauge theory we have $\mathrm{dim}(V) = 2N_c \times L^3 \times T$, in a continuous theory this vector space has infinite dimension. Next a basis $\lbrace \eta_i \rbrace_{1 \leq i \leq N}\subset V$ is chosen, which is used to construct a basis of the exterior product of $V$ with itself $V_2 \coloneqq V \wedge V$
\begin{align}
    \lbrace \eta_i \wedge \eta_j \rbrace_{1 \leq i \leq j \leq N}\subset V_2 \quad (\eta_j \wedge \eta_i = -\eta_i\wedge \eta_j),
\end{align}
as well as the basis $n$-fold exterior product $V_n$
\begin{align}
    \lbrace \eta_{i_1} \wedge \eta_{i_2} \wedge ... \wedge \eta_{i_n} \rbrace_{1 \leq i_1 \leq i_2 \leq ... \leq i_n} \subset V_n.
\end{align}
The Grassmann algebra over $V$ is thus defined as
\begin{align}
    \Lambda(V) = V_0 \oplus V_1 \oplus ... \oplus V_N,
\end{align}
where $V_0$ is defined to be the field of complex or real numbers and $V_1 \coloneqq V$. The wedge product symbol is often omitted, so one writes $vw$ instead of $v \wedge w$. The Grassmann algebra is an anticommutative algebra with unity of dimension $2^{2N}$, where usually half of its generators are labeled by $(\eta_i)_i$ and half by $(\eta_i^\dagger)_i$. The dagger does not mean that the second set is the adjoint of the first set, but it plays a similar role as the complex conjugate in the case of pseudo-fermions.\\
Due to the fact that the square of an element of the Grassmann algebra is zero, it can be concluded that each Taylor expansion only has a finite number of terms. In other words, every analytical function on the Grassmann algebra is a polynomial. Integration and differentiation on $\Lambda(V)$ are defined via
\begin{align}
    \int \mathrm{d} \eta_i \,\eta_j = \delta_{ij},\\
    \frac{\partial}{\partial \eta_i}\eta_j = \delta_{ij}.
\end{align}
The most important integral in the context of Grassmann numbers, the Gaussian integral on $\Lambda(V)$, is
\begin{align}
    \int \prod_{i = 1}^N\left[ \mathrm{d}\eta_i^{\dagger} \mathrm{d}\eta_i \right]\, \exp\left(-\sum_{kl} \eta_k^\dagger M^{kl} \eta_l \right) = \det(M),
\end{align}
whenever it is a normal matrix, i.e., it commutes with its adjoint and therefore is diagonalizable. This can be derived using the one dimensional integral
\begin{align}
    \int \mathrm{d}\eta^{\dagger} \mathrm{d}\eta\, \exp\left(-\lambda \eta^\dagger \eta\right) = \int \mathrm{d}\eta^{\dagger} \mathrm{d}\eta\, \left(1- \lambda \eta^\dagger \eta\right) = \lambda
\end{align}
and a suitable coordinate transformation for the Grassmann algebra.
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