A {\it dominating set} for a graph $G = (V,E)$ is a subset $S \subseteq V$ such that every vertex $v\in V$ is either in $S$ or has a neighbor $u\in S$. The vertex $u \in S$ is said to \textit{cover}  the vertex $v \in V$ if either $u=v$ or $uv\in E$. A dominating set $S$ is a \textit{minimal dominating set} if no proper subset $S'\subset S$ is a dominating set. The \textit{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \textbf{$\gamma$-sets}.

Dominating sets, in particular $\gamma$-sets, have been extensively researched. We recommend to the reader Haynes et al.'s book \cite{Hay2} and several other research works \cite{FB, BR, EC, GD, MH, FH, Hay1, ML} on variants and applications of dominating sets.
In this literature we found various versions of $\gamma$-set such as the perfect dominating set, the double dominating set, the strong dominating set, and the restrained dominating set. In general, each such type of $\gamma$-set has some specified properties. Here, we list five of the most commonly studied $\gamma$-sets along with their properties. Later in Example \ref{Ex}, we illustrate each such $\gamma$-set.

\begin{enumerate}
	\item \textit{Perfect $\gamma$-set} requires that no two vertices in a $\gamma$-set cover the same vertex (outside the $\gamma$-set). \item \textit{Connected $\gamma$-set} requires that the graph induced by the $\gamma$-set be connected.
	\item \textit{Total $\gamma$-set} forbids isolated vertices in the subgraph induced by the $\gamma$-set.
	\item \textit{Independent $\gamma$-set} requires all vertices in a $\gamma$-set to be isolated vertices in the subgraph they induce.
	\item \textit{clique $\gamma$ set} requires that the vertices in a $\gamma$-set induce a clique.
\end{enumerate}

Because a given graph may have multiple $\gamma$-sets, this article addresses a natural but fundamental question:\textit{ How many  $\gamma$-sets does a given graph have?} To answer this question, we introduce the notion of \textbf{dominion}.

The \textit{dominion (number)} of  a graph $G=(V,E)$, denoted by $\zeta(G)$, is the number of its $\gamma$-sets. In other words,

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a  complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
	\centering
	\begin{tikzpicture}[scale=1]
	\clip(0,-1) rectangle (4,4);
	\draw [line width=2pt] (2.62,1)-- (2,2);
	\draw [line width=2pt] (2,2)-- (1.98,2.84);
	\draw [line width=2pt] (2.62,1)-- (3,0.34);
	\draw [line width=2pt] (2,2)-- (1.48,1.02);
	\draw [line width=2pt] (1.48,1.02)-- (2.62,1);
	\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
	\begin{scriptsize}
	\draw [fill=black] (2,2) circle (2.5pt);
	\draw[color=black] (2.2,2.1) node {$u$};
	\draw [fill=black] (2.62,1) circle (2.5pt);
	\draw[color=black] (2.92,1) node {$v$};
	\draw [fill=black] (1.98,2.84) circle (2.5pt);
	\draw[color=black] (1.94,3.1) node {$x$};
	\draw [fill=black] (3,0.34) circle (2.5pt);
	\draw[color=black] (3.2,0.15) node {$y$};
	\draw [fill=black] (1.48,1.02) circle (2.5pt);
	\draw[color=black] (1.14,1) node {$w$};
	\draw [fill=black] (1.06,0.34) circle (2.5pt);
	\draw[color=black] (0.9,0.15) node {$z$};
	\end{scriptsize}
	\end{tikzpicture}
	\caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\end{figure}

Consider the unicyclic graph $G$ as shown in Figure \ref{fig1}. It is easy to see that $\gamma(G)=3$. We list all the $\gamma$-sets with properties which are among the five commonly studied, as  previously mentioned.

\begin{itemize}

	\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
	\item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
	 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
	\item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are  $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\displaystyle \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

A {\it dominating set} for a graph $G = (V,E)$ is a subset $S \subseteq V$ such that every vertex $v\in V$ is either in $S$ or has a neighbor $u\in S$. The vertex $u \in S$ is said to \textit{cover} the vertex $v \in V$ if either $u=v$ or $uv\in E$. A dominating set $S$ is a \textit{minimal dominating set} if no proper subset $S'\subset S$ is a dominating set. The \textit{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \textbf{$\gamma$-sets}.

\begin{enumerate}
 \item \textit{Perfect $\gamma$-set} requires that no two vertices in a $\gamma$-set cover the same vertex (outside the $\gamma$-set). \item \textit{Connected $\gamma$-set} requires that the graph induced by the $\gamma$-set be connected.
 \item \textit{Total $\gamma$-set} forbids isolated vertices in the subgraph induced by the $\gamma$-set.
 \item \textit{Independent $\gamma$-set} requires all vertices in a $\gamma$-set to be isolated vertices in the subgraph they induce.
 \item \textit{clique $\gamma$ set} requires that the vertices in a $\gamma$-set induce a clique.
\end{enumerate}

The \textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\zeta(G)$, is the number of its $\gamma$-sets. In other words,

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
 \centering
 \begin{tikzpicture}[scale=1]
 \clip(0,-1) rectangle (4,4);
 \draw [line width=2pt] (2.62,1)-- (2,2);
 \draw [line width=2pt] (2,2)-- (1.98,2.84);
 \draw [line width=2pt] (2.62,1)-- (3,0.34);
 \draw [line width=2pt] (2,2)-- (1.48,1.02);
 \draw [line width=2pt] (1.48,1.02)-- (2.62,1);
 \draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
 \begin{scriptsize}
 \draw [fill=black] (2,2) circle (2.5pt);
 \draw[color=black] (2.2,2.1) node {$u$};
 \draw [fill=black] (2.62,1) circle (2.5pt);
 \draw[color=black] (2.92,1) node {$v$};
 \draw [fill=black] (1.98,2.84) circle (2.5pt);
 \draw[color=black] (1.94,3.1) node {$x$};
 \draw [fill=black] (3,0.34) circle (2.5pt);
 \draw[color=black] (3.2,0.15) node {$y$};
 \draw [fill=black] (1.48,1.02) circle (2.5pt);
 \draw[color=black] (1.14,1) node {$w$};
 \draw [fill=black] (1.06,0.34) circle (2.5pt);
 \draw[color=black] (0.9,0.15) node {$z$};
 \end{scriptsize}
 \end{tikzpicture}
 \caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\begin{itemize}

\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
 \item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
 \item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\displaystyle \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

\label{fig1}

\end{figure}

Consider the unicyclic graph $G$ as shown in Figure \ref{fig1}. It is easy to see that $\gamma(G)=3$. We list all the $\gamma$-sets with properties which are among the five

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
 \centering
 \begin{tikzpicture}[scale=1]
 \clip(0,-1) rectangle (4,4);
 \draw [line width=2pt] (2.62,1)-- (2,2);
 \draw [line width=2pt] (2,2)-- (1.98,2.84);
 \draw [line width=2pt] (2.62,1)-- (3,0.34);
 \draw [line width=2pt] (2,2)-- (1.48,1.02);
 \draw [line width=2pt] (1.48,1.02)-- (2.62,1);
 \draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
 \begin{scriptsize}
 \draw [fill=black] (2,2) circle (2.5pt);
 \draw[color=black] (2.2,2.1) node {$u$};
 \draw [fill=black] (2.62,1) circle (2.5pt);
 \draw[color=black] (2.92,1) node {$v$};
 \draw [fill=black] (1.98,2.84) circle (2.5pt);
 \draw[color=black] (1.94,3.1) node {$x$};
 \draw [fill=black] (3,0.34) circle (2.5pt);
 \draw[color=black] (3.2,0.15) node {$y$};
 \draw [fill=black] (1.48,1.02) circle (2.5pt);
 \draw[color=black] (1.14,1) node {$w$};
 \draw [fill=black] (1.06,0.34) circle (2.5pt);
 \draw[color=black] (0.9,0.15) node {$z$};
 \end{scriptsize}
 \end{tikzpicture}
 \caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
 \item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
 \item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\ds \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

\begin{proof}
For each vertex on the cycle, and for each dominating set $S\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives $\ds \sum_{i=0}^n{n\choose i}$. Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$. This gives $\zeta(G)=2^n$.
\end{proof}

\begin{claim}{\label{c4}}
 When $n=3k, k\ge 1$, $\zeta(P_n)=1$ and the unique $\gamma$-set in $P_n$ contains $v_{n-1}$. Moreover, when $n=3k+2, k\ge 1$, there is only one $\gamma$-set in $P_n$ containing $v_{n}$.
\end{claim}
\begin{proof}
 When $n=3k, k\ge 1$, we have $\zeta(P_n)=k$ and the conclusion follows by the pigeon-hold argument alluded in Claim \ref{c1}. Each of the sets of $3$ consecutive vertices $\{v_{3r+1},v_{3r+2},v_{3r+3}\}$, $r=0,\ldots,k-1$, must contain at least one vertex of any $\gamma$-set $S$, and since $\gamma=k$, each of those sets must contain exactly one element of $S$. Since $S$ is dominating, the element of $S$ in $\{v_1,v_2,v_3\}$ must be $v_1$ or $v_2$. If $v_1$ then the ``next'' element of $S$ must be $v_4$, which forces $S=\{v_1,v_4,\ldots,v_{3k-2}\}$, leaving $v_{3k}$ uncovered. Therefore, the first element of $S$ must be $v_2$, which forces $S=\{v_2,v_5,\ldots,v_{3k-1}, v_{3k+2}\}$.

\begin{proposition}{\label{Prop2}}
 For any connected graph $G$ on $n$ vertices, $1\le \zeta(G)\le {n\choose \gamma(G)}$.

\end{proof}
\begin{corollary}
For any connected graph $G$ with $\gamma(G)=1$, $\zeta(\underbrace{G\vee \ldots \vee G}_r)=r\zeta(G)$, $r\ge 1$.
\end{corollary}
\begin{proof}
From Part (a) of Theorem \ref{T4} (when $\gamma(G_1)=1=\gamma(G_2)$), the result follows by induction on $r\ge 1$.
\end{proof}
We note that in the case when $G=K_1$, it is clear that $\underbrace{G\vee \ldots \vee G}_r:=K_r$ and $\zeta(K_r)=r=r\zeta(K_1)$. Here, we present a more general result for the special case when $G=\overline{K_{m}}$, the complement of a complete graph on $m\ge 1$ vertices. Note that $\overline{K_{m_1}} \vee \ldots \vee \overline{K_{m_k}}$ is a complete $k$-partite graph often written as $K(m_1,m_2,\ldots, m_k)$, with $m_i\ge 1$ and $k\ge 2$.

\end{theorem}
\begin{proof}
Let $V_i$ denote each disjoint partite (vertex) set of $G$, with $|V_i|=m_i$, $i=1,\ldots, k$. If, for some $j=1,\ldots, r$, $m_j=1$ and $m_i>1$ for $i>r$, then it is clear that $\gamma(G)=1$ and the $\gamma$-sets in $G$ are the singleton subsets of $V_1\cup\ldots \cup V_r$. It follows that $\zeta(G)=r$.
Assume $m_i>1$, for $i=1,\ldots, k$. It is clear that $\gamma(G)=2$. Any vertex $v\in V_i$ covers all vertices $v'\in V(G)\setminus V_i$ except those in $V_i$. Thus, for each $v\in V_i$ and $v'\in V_j$, $i\ne j$, $\{v,v'\}$ is a $\gamma$-set of $G$. It follows that there are $\ds \sum_{i\ne j}^{{k}}m_im_j$ total such sets. Further, if $|V_i|=2$ for $i=1,\ldots, r$, then each such sets, $V_i$, is an additional $\gamma$-set, giving the result.
\end{proof}

\begin{proof}
For each vertex on the cycle, and for each dominating set $S\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives  $\displaystyle \sum_{i=0}^n{n\choose i}$.  Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$. This gives $\zeta(G)=2^n$.
\end{proof}

For each vertex on the cycle and each dominating set $S\subseteq V(G)$, "either the vertex or its leaf neighbor must be in $S$." Hence all $\gamma$-sets can be formed by choosing a subset of the $n$ leaves, and then including "the cycle neighbors of unchosen leafs," yielding $\sum_{i=0}^n \binom{n}{i}$. Conversely, "every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$," so $\zeta(G)=\sum_{i=0}^n \binom{n}{i}=2^n$.

If $G$ is a sun graph on $2n\ge 6$ vertices, then $\displaystyle \zeta(G)=\sum_{i=0}^n{n\choose i}=2^n. $,

Question: Let b3(G) denote the domination number of a graph G, i.e. the minimum size of a dominating set, and let b6(G) be the number of b3-sets of G, where a b3-set is a dominating set of cardinality b3(G). A sun graph on 2n vertices (with 2n \ge 6) is the graph obtained from a cycle C_n by attaching exactly one pendant leaf to each cycle vertex. Which quantitative statement holds for b6(G)?

If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\sum_{i=0}^{n}\binom{n}{i}=2^n\).

Choices:
B. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\binom{2n}{n}\).
C. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)\ge 2^n\).
D. If G is a sun graph on 2n \ge 6 vertices, then every \(\gamma\)-set is obtained by choosing exactly one vertex from each leaf-cycle pair, and hence \(\zeta(G)=n\).
E. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\sum_{i=1}^{n}\binom{n}{i}=2^n-1\).

independent binary choices at n cycle-leaf pairs replaced by unrestricted n-subset count from 2n vertices

replaced exact enumeration by the weaker lower bound only

local one-from-each-pair structure kept but global count collapsed from 2 choices per pair to one canonical choice

excluded one valid endpoint case in the binomial sum


\begin{theorem}\label{theorem:main}
	Let $\lambda\in DP_n$ and $(\rho_\lambda,V_\lambda)$ be the corresponding irreducible spin representation of $\tilde S_n$.
	Let $g\in \tilde S_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and let $k=\lcm(\mu)$.
	Then the minimal polynomial $p_\lambda(g)$ of $\rho_\lambda(g)$ has $k$ distinct roots except in the following cases:

	\noindent For $n\leq 8$, $(\lambda,\mu)$ is one of the pairs listed in Table~\ref{table:exceptional_cases_1_8}.

	For $n>8$, the following four exceptions occur:
\begin{itemize}
		\item $\lambda=(n)$ and $\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/3}\pm 1}$.
		\item $\lambda=(n)$ and $\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/5}\pm 1}$
		\item $\lambda=(n)$ and $\mu$ contains $3$ and $5$ and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}$.
		\item $\lambda=(n-1,1)$ and $\mu$ contains $3$ and $5$, and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)}{(x^{k/15}\pm1)}$.
	\end{itemize}
\end{theorem}

Let $G$ denote a double cover of either the symmetric group or the alternating group.
In this article, we determine the minimal polynomials of elements of $G$ in its irreducible complex spin representations.
We find that (see Theorems~\ref{theorem:main} and~\ref{theorem:main_alt}), for most elements $g\in G$, and most irreducible spin representations $\rho$ of $G$, the minimal polynomial of $\rho(g)$ is $x^k\pm 1$, where $k$ is the order of $g$ modulo the center of $G$. The exceptions fall into four infinite families, and finitely many sporadic examples.

The question of finding minimal polynomials of elements in representations for a given group goes back to the famous work of Hall and Higman~\cite{Hall_Higman}, as well as Shult~\cite{Shult}.
The general problem was formulated explicitly by Tiep and Zalesskii: given an element $g$ of a finite group $G$ and an irreducible representation $\rho$ of $g$, determine the eigenvalues of $\rho(g)$~\cite{Tiep_Zalesski}.
In his survey articles~\cite{survey_Zalesski_algebraic_Chevalley_2009,survey_Zalesski_eigenvalue_1_2025}, Zalesskii discusses the importance of this problem and its applications to arithmetic and algebraic geometry, topology, game theory, combinatorics, group theory, etc.
The only groups $G$ where this problem is solved for all $g\in G$ and every irreducible complex representation $\rho$ of $G$ are the symmetric groups and the alternating groups, due to the work of Swanson~\cite{Swanson} (for longest cycles in symmetric groups), Yang--Staroletov~\cite{Yang_Staroletov} and Velmurugan~\cite{vel_dec_2024} (for certain families of permutations in symmetric and alternating groups), Amrutha--Prasad--Velmurugan~\cite{p2024cyclic,Inv_vectors,fpsac2024} (for the occurrence of the eigenvalue $1$ for all permutations in all representations of symmetric and alternating groups), and finally completed by Staroletov~\cite{Staroletov_all_eigenvalues}.

Let $\tilde S_n$ and $\tilde A_n$ denote the double covers of $S_n$ and $A_n$, respectively.
To each partition $\lambda$ in $DP_n$, the partitions of $n$ with distinct parts, there is an associated irreducible spin representation $(\rho_\lambda,V_\lambda)$ of $\tilde S_n$.
When $\lambda$ has an odd number of even parts, then $V_\lambda$ can be twisted by the sign character of $\tilde S_n$ to obtain a non-isomorphic representation.
In this case, we sometimes denote $V_\lambda$ by $V_\lambda^+$ and its twist by $V_\lambda^-$.
It suffices to determine the minimal polynomial of $g\in G$ in $V_\lambda$ for $\lambda\in DP_n$.
We now state our main theorem for $\tilde S_n$:

\noindent For $n\leq 10$ and $((\rho,V),g)$ is one of the pairs listed in Tables~\ref{table:exceptional_cases_1_8_alt} and \ref{table:exceptional_cases_9_10_alt}.

In this section we shall prove the following theorem.
\begin{theorem}
 \label{theorem:cycle_double_S_n}
 Let $(\phi_{\lambda}, V_\lambda)$ be an irreducible spin representation of $\tilde S_n$ indexed by a partition $\lambda\vdash n$ with distinct parts.
 Let $g=w_n$ be an element of $\tilde S_n$ with cycle type $(n)$.
 Then the minimal polynomial $p_\lambda(g)$ of $\phi_{\lambda}(g)$ has $n$ distinct roots except in the following cases:
 \begin{enumerate}
 \item $\phi_{(3)}$ and $p_\lambda(\sigma_3^\pm)= x^2 \mp x+1$.
 \item $\phi_{(4)}^\pm$ and $p_\lambda(\sigma_4^\pm)= x^2 \pm \sqrt{2} x +1$.
 \item $\phi_{(5)}$ and $p_\lambda(\sigma_5^\pm)= \frac{x^5\pm 1}{x\pm 1}$.
 \item $\phi_{(6)}^\pm$ and $p_\lambda(\sigma_6^\epsilon)= \frac{x^6 -1}{(x-\eta_1)(x-\eta_2)}$, where $\eta_1= \frac{1\pm \epsilon \sqrt{3}i}{2}$ and $\eta_2= \frac{-1\pm \epsilon \sqrt{3}i}{2}$.
 \item $\phi_{(8)}^\pm$ and $p_\lambda(\sigma_8^\epsilon)= \frac{x^8 -1}{(x\pm \epsilon 1)}$.
 \item $\phi_{(2,1)}^\pm$ and $p_\lambda(\sigma_{(3)}^\pm)= x \pm 1$.
 \end{enumerate}
\end{theorem}

\begin{lemma}\label{lemma:basic_spin_representation_eigenvalues}
Suppose that $\rho$ ( or $\rho^\pm$) is the basic spin representation of $\tilde S_n$ and $\sigma\in C_\alpha^\pm$.
Denote the least common multiple of elements of $\alpha$ by $k$.
The following statements are true.
\begin{enumerate}
 \item If $3$ is the unique part of $\alpha$ that is divisible by $3$, and it is not true that $5$ is the unique part of $\alpha$ that is divisible by $5$, then the minimal polynomial of $\rho(\sigma)$ equals $\frac{x^{k}\pm1}{x^{k/3}\pm1}$.
\item If $5$ is the unique part of $\alpha$ that is divisible by $5$, and it is not true that $3$ is the unique part of $\alpha$ that is divisible by $3$, then the minimal polynomial of $\rho(\sigma)$ equals $\frac{x^{k}\pm1}{x^{k/5}\pm1}$.
\item If $\alpha$ contains $3$ and $5$, while other parts of $\alpha$ are not divisible by $3$ and $5$, then the minimal polynomial of $\rho(\sigma)$ equals $\frac{(x^{k}-\tau1)(x^{k/15}-\tau1)}{(x^{k/3}-\tau1)(x^{k/5}-\tau1)}$, where $\tau\in\{\pm1\}$.
\item In all other cases, the minimal polynomial of $\rho(\sigma)$ equals $x^{k}\pm1$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $\sigma$ is an $n$-cycle, then the assertion follows from Theorem~\ref{theorem:cycle_double_S_n}.
Therefore, we can assume that $l(\alpha)\geqslant 2$.
Note that we can assume that $n\geqslant12$. We prove the statement using induction on $n$.

Then we have
 \begin{align*}
 w_p w_q (v \otimes v_1 \otimes v_2)
 &= w_p ( v \otimes v_1 \otimes w_q v_2 )\\
 &= \xi_1 v \otimes w_p v_1 \otimes S w_q v_2\\
 &= \kappa_V v \otimes w_p v_1 \otimes S w_q v_2.
 \end{align*}
 Since $V_\beta$ is self-associate, by Theorem~\ref{theorem:cycle_double_S_n}, the set of eigenvalues of $w_q$ are $\{\zeta \mid \zeta^q=\varepsilon_q\}$.
 Clearly, the set of eigenvalues of $S w_q$ is $\{\pm I\}.\{\zeta \mid \zeta^q=\varepsilon_q\}$.
 Therefore, the set of eigenvalues of $w_\mu$ contains
 \begin{align*}
 \{\kappa_V\}.\{\eta \mid \eta^p=\varepsilon_p\}.\{\pm I\}.\{\zeta \mid \zeta^q=\varepsilon_q\}\\
 = \{\pm I\}.\{\eta \mid \eta^p=\varepsilon_p\}.\{\zeta \mid \zeta^q=\varepsilon_q\}\\
 = \{\eta \mid \eta^{\lcm(p,q)}=\varepsilon_{\lcm(p,q)}\}.
 \end{align*}
 This completes the proof.
\end{proof}
\begin{lemma}\label{lemma:all_even_cycles_at_least_two_parts}
 Let $\mu$ be a strict partition of $n$ with at least two parts such that all parts of $\mu$ are even.
 Then the minimal polynomial of $w_\mu$ in the spin representation $V_\lambda$ is $x^{\lcm(\mu)} - \varepsilon_\mu$.
\end{lemma}
\begin{proof}
 Let us verify the lemma for $n\leq 16$ by direct computation and hence we assume that $n\geq 17$.
 If $\mu$ has exactly two parts, then the result follows from Lemma~\ref{lemma:two_even_cycles}.
 Now we assume that $\mu$ has at least three parts.
 Write $w_\mu = w_{\tilde\mu} w_{\mu_l}$ where $\tilde\mu=(\mu_1,\mu_2,\dotsc,\mu_{l-1})$.
 Let $\lambda$ be a strict partition of $n$.
 As in the proof of Lemma~\ref{lemma:two_even_cycles}, there exist strict partitions $\alpha,\beta$ of $|\tilde\mu|$ and $\mu_l$ respectively such that the shifted LR coefficient $f_{\alpha\beta}^\lambda>0$ and $\beta\neq (3,2,1)$.
 Then by induction, we may assume that the set of eigenvalues of $\rho_{\alpha}(w_{\tilde\mu})$ is $\{\eta \mid \eta^{\lcm(\tilde\mu)}=\varepsilon_{\tilde\mu}\}$.
 If $w_{\tilde\mu}$ is odd, then the proof proceeds exactly as in the proof of Lemma~\ref{lemma:two_even_cycles}.

We have, in fact, one more very useful information similar what we we have in the alternating group case which is as follows.
For a partition $\lambda \in DP_n^+$, we define the subtraction of those characters as
\begin{displaymath}
 \Delta_\lambda := \phi_\lambda^+ - \phi_\lambda^-.
\end{displaymath}
\begin{theorem}\label{theorem:Delta_details_double_alternating}\cite[Theorem 7.4]{Stembridge_Shifted}
 Let $\lambda\in DP_n^+$. Then we have
 $\Delta_\lambda(\sigma)= \pm i^{(n-l(\lambda))/2} \sqrt{z_\lambda} $ if $|\sigma|=\lambda$ and $0$ in all other cases.
\end{theorem}
We have the following obvious lemma.
\begin{lemma}\label{lemma:double_A_n_not_DP_n^-}
 Let $g$ be an element of $\tilde A_n$ with cycle type $\mu$ and $\lambda$ be a strict partition of $n$.
 Then for any irreducible spin representation $V_\lambda$ of $\tilde A_n$, we have that the minimal polynomial of $g$ in $V_\lambda$ is same as in the irreducible representation $V$ of $\tilde S_n$ which contains $V_\lambda$ upon restriction to $\tilde A_n$ if $\lambda \notin DP_n^+$ or $\lambda \in DP_n^+$ with $\lambda \neq \mu$.
\end{lemma}
\begin{proof}
 Let $\phi_\lambda$ (resp. $\phi_\lambda^\pm$) be the character of the irreducible representation $V_\lambda$ (resp. $V_\lambda^\pm$) of $\tilde A_n$.
 Using Lemma~\ref{lemma:conjugacy_classes_A_n}, Lemma~\ref{lemma:restriction_to_A_n}, and Theorem~\ref{theorem:Delta_details_double_alternating}, we see that $\phi_\lambda(g^i)$ (resp. $\phi_\lambda^\pm(g^i)$) is either equal to $\phi_\lambda(g^i)$ or $\phi_\lambda(g^i)/2$ for all $i$.
 Computing the inner product of characters, we see that the minimal polynomial of $g$ in $V_\lambda$ (or $V_\lambda^\pm$) is same as the minimal polynomial of $g$ in $V_\lambda$.
 This proves the lemma.
\end{proof}

The following results essentially follow from our proof of Theorem~\ref{theorem:cycle_double_S_n} and Lemma~\ref{lemma:Prime_cycle_sym}.
\begin{theorem}\label{theorem:cycle_double_A_n_odd}
Let $n\geq 3$ be an odd integer.
Let $\lambda \in DP_n^+$.
Then the minimal polynomial of $w_{(n)}$ in the irreducible spin representation $\phi_\lambda^\pm$ has $n$ distinct roots except in the following cases:
\begin{itemize}
 \item $n=3$ and $\lambda = (3)$,
 \item $n=5$ and $\lambda = (5)$ or $(3,2)$,
 \item $n=7$ and $\lambda = (7)$,
 \item $n=9$ and $\lambda=(9)$.
\end{itemize}
\end{theorem}
\begin{proof}
The theorem uses induction on $n$.
 Let us verify the theorem for $n\leq 63$.
 Therefore, we may from now on assume that $n>63$.
 Write $n=mp$, where $p$ is the smallest prime dividing $n$.
 In particular, $m>9$.
 Let $\Res^{\tilde A_n}_{\tilde A_m^{\times p}} V_\lambda\geq V \times V_1 \otimes V_2 \otimes \dotsb \otimes V_p$, for some irreducible spin representation $V \otimes V_1 \otimes V_2 \otimes \dotsb \otimes V_p$ of $\tilde A_m^{\times p}$.

The irreducible spin representations of $\tilde A_n$ are indexed by $DP_n$.
More precisely, for each $\lambda\in DP_n^-$, we have a unique irreducible spin representation $V_\lambda$ and for each $\lambda\in DP_n^+$, we have two non-isomorphic irreducible spin representations $V_\lambda^\pm$ of $\tilde A_n$.
Then the analogous theorem for $\tilde A_n$ is the following.
\begin{theorem}\label{theorem:main_alt}
	Let $(\rho,V)$ be an irreducible spin representation of $\tilde A_n$ indexed by a partition $\lambda$.
	Let $g\in \tilde A_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and $k=\lcm(\mu)$.
	Then the minimal polynomial $p_V(g)$ of $\rho(g)$ has $k$ distinct roots except in the following cases:

	\noindent For $n\leq 10$ and $((\rho,V),g)$ is one of the pairs listed in Tables~\ref{table:exceptional_cases_1_8_alt} and \ref{table:exceptional_cases_9_10_alt}.

	For $n>10$, the following four exceptions occur:
\begin{itemize}
		\item $V=V_{(n)}$ and $\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/3}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/5}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $3$ and $5$ and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}$.
		\item $V=V_{(n-1,1)}$ and $\mu$ contains $3$ and $5$, and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)}{(x^{k/15}\pm1)}$.
	\end{itemize}
\end{theorem}

The article is organized as follows. In Sections~\ref{section:Preliminaries for the double cover of the symmetric group} and ~\ref{section:combinatprics_on_tableaux}, we provide the necessary background while developing some new results on the structure theory of  $\tilde S_n$.
In Section~\ref{section:minimal_polynomial_of_n_cycles}, we prove Theorem~\ref{theorem:main} for the elements of $\tilde S_n$ with cycle type $(n)$.
In Section~\ref{section:minimal_polynomial_some_family}, we prove Theorem~\ref{theorem:main} for some specific families of elements and irreducible spin representations.
Sections~\ref{section:minimal_polynomial_of_any_element} and~\ref{section:minimal_poly_of_any_element_in_A_n} are devoted to the proofs of Theorems~\ref{theorem:main} and~\ref{theorem:main_alt}, respectively.

The post-theorem introduction does not give a step-by-step proof argument for Theorem~\ref{theorem:main}, but it does outline where the proof is carried out in the paper: Section~\ref{section:minimal_polynomial_of_n_cycles} proves Theorem~\ref{theorem:main} for elements of $\tilde S_n$ with cycle type $(n)$; Section~\ref{section:minimal_polynomial_some_family} proves Theorem~\ref{theorem:main} for "some specific families of elements and irreducible spin representations"; and Sections~\ref{section:minimal_polynomial_of_any_element} and~\ref{section:minimal_poly_of_any_element_in_A_n} are "devoted to the proofs" of Theorems~\ref{theorem:main} and~\ref{theorem:main_alt}, respectively.

The post-theorem introduction does not give a step-by-step proof argument for the main theorem, but it does outline where the proof is carried out in the paper: next the paper proves the main theorem for elements of $\tilde S_n$ with cycle type $(n)$; later it proves the main theorem for "some specific families of elements and irreducible spin representations"; and finally it is devoted to the proofs of the main theorem and the following theorem, respectively. We first prove the following theorem.

\begin{theorem}\label{theorem:main_alt}
	Let $(\rho,V)$ be an irreducible spin representation of $\tilde A_n$ indexed by a partition $\lambda$.
	Let $g\in \tilde A_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and $k=\lcm(\mu)$.
	Then the minimal polynomial $p_V(g)$ of $\rho(g)$ has $k$ distinct roots except in the following cases:

	\noindent For $n\leq 10$ and $((\rho,V),g)$ is one of the pairs listed in Tables~\ref{table:exceptional_cases_1_8_alt} and \ref{table:exceptional_cases_9_10_alt}.

	For $n>10$, the following four exceptions occur:
\begin{itemize}
		\item $V=V_{(n)}$ and $\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/3}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/5}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $3$ and $5$ and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}$.
		\item $V=V_{(n-1,1)}$ and $\mu$ contains $3$ and $5$, and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)}{(x^{k/15}\pm1)}$.
	\end{itemize}
\end{theorem}

Question: Let \(\tilde S_n\) be the double cover of the symmetric group \(S_n\). For a strict partition \(\lambda\in DP_n\) of \(n\) (that is, a partition of \(n\) with distinct parts), let \((\rho_\lambda,V_\lambda)\) be the corresponding irreducible complex spin representation of \(\tilde S_n\); when \(\lambda\) has an odd number of even parts, the two associate spin representations are denoted by \(\lambda^\pm\). Let \(g\in \tilde S_n\) project to a permutation of cycle type \(\mu=(\mu_1,\mu_2,\dotsc,\mu_r)\), and set \(k=\operatorname{lcm}(\mu_1,\mu_2,\dotsc,\mu_r)\). In the exceptional list, notation such as \((\mu)^\pm\) refers to one of the two lifted classes in \(\tilde S_n\) lying over the cycle type \(\mu\). Which statement holds for every such \(\lambda\) and \(g\) about the minimal polynomial \(p_\lambda(g)\) of \(\rho_\lambda(g)\)?

For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 8\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial:
- \(n=2\):
  - \(\lambda=(2)^\pm\): \(((2)^+,\frac{x^2+1}{x\pm i})\), \(((2)^-,\frac{x^2+1}{x\mp i})\).
- \(n=3\):
  - \(\lambda=(3)\): \(((3)^+,\frac{x^3+1}{x+1})\), \(((3)^-,\frac{x^3-1}{x-1})\).
  - \(\lambda=(2,1)^\pm\): \(((2,1)^+,\frac{x^2+1}{x\pm i})\), \(((2,1)^-,\frac{x^2+1}{x\pm i})\), \(((3)^+,\frac{x^3+1}{x^2-x+1})\), \(((3)^-,\frac{x^3-1}{x^2+x+1})\).
- \(n=4\):
  - \(\lambda=(4)^\pm\): \(((3,1)^+,\frac{x^3+1}{x+1})\), \(((3,1)^-,\frac{x^3-1}{x-1})\), \(((4)^+,x^2\mp \sqrt{2}x+1)\), \(((4)^-,x^2\pm \sqrt{2}x+1)\).
- \(n=5\):
  - \(\lambda=(5)\): \(((3,1^2)^+,\frac{x^3+1}{x+1})\), \(((3,1^2)^-,\frac{x^3-1}{x-1})\), \(((3,2)^\pm,\frac{x^6+1}{x^2+1})\), \(((5)^\pm,\frac{x^5\pm1}{x\pm1})\).
  - \(\lambda=(3,2)^\pm\): \(((5)^+,\frac{x^5+1}{x+1})\), \(((5)^-,\frac{x^5-1}{x-1})\), \(((3,2)^+,x^4\pm\sqrt{3}x^3+2x^2\pm\sqrt{3}x+1)\), \(((3,2)^-,x^4\mp\sqrt{3}x^3+2x^2\mp\sqrt{3}x+1)\).
- \(n=6\):
  - \(\lambda=(6)^\pm\): \(((6)^+,x^4\pm\sqrt{-3}x^3-2x^2\pm\sqrt{-3}x+1)\), \(((6)^-,x^4\mp\sqrt{-3}x^3-2x^2\mp\sqrt{-3}x+1)\), \(((3,1^3)^+,\frac{x^3+1}{x+1})\), \(((3,1^3)^-,\frac{x^3-1}{x-1})\), \(((5,1)^+,\frac{x^5+1}{x+1})\), \(((5,1)^-,\frac{x^5-1}{x-1})\), \(((3,2,1)^\pm,\frac{x^6+1}{x^2+1})\).
  - \(\lambda=(3,2,1)^\pm\): \(((3,2,1)^+,x^4\mp\sqrt{3}x^3+2x^2\mp\sqrt{3}x+1)\), \(((3,2,1)^-,x^4\pm\sqrt{3}x^3+2x^2\pm\sqrt{3}x+1)\), \(((3,3)^+,\frac{x^3-1}{x-1})\), \(((3,3)^-,\frac{x^3+1}{x+1})\), \(((5,1)^+,\frac{x^5+1}{x+1})\), \(((5,1)^-,\frac{x^5-1}{x-1})\), \(((6)^\pm,\frac{x^6-1}{x^2-1})\).
- \(n=7\):
  - \(\lambda=(7)\): \(((3,1^4)^\pm,\frac{x^3\pm1}{x\pm1})\), \(((5,1^2)^\pm,\frac{x^5\pm1}{x\pm5})\), \(((3,2,1^2),\frac{x^6+1}{x^2+1})\), \(((3,2,2),\frac{x^6+1}{x^2+1})\), \(((4,3)^\pm,\frac{x^{12}+1}{x^4+1})\), \(((5,1^2)^\pm,\frac{x^5\pm1}{x\pm1})\), \(((5,2)^\pm,\frac{x^{10}+1}{x^2+1})\).
- \(n=8\):
  - \(\lambda=(8)^\pm\): \(((5,1^3)^+,\frac{x^5+1}{x+1})\), \(((5,1^3)^-,\frac{x^5-1}{x-1})\), \(((3,1^5)^+,\frac{x^3+1}{x+1})\), \(((3,1^5)^-,\frac{x^3-1}{x-1})\), \(((5,3)^+,\frac{(x^{15}-1)(x-1)}{(x^5-1)(x^3-1)})\), \(((5,3)^-,\frac{(x^{15}+1)(x+1)}{(x^5+1)(x^3+1)})\), \(((8)^+,\frac{x^8-1}{x\pm1})\), \(((8)^-,\frac{x^8-1}{x\mp1})\), \(((5,2,1)^\pm,\frac{x^{10}+1}{x^2+1})\), \(((3,2^2,1),\frac{x^6+1}{x^2+1})\), \(((3,2,1^3),\frac{x^6+1}{x^2+1})\), \(((4,3,1)^\pm,\frac{x^{12}+1}{x^4+1})\).
  - \(\lambda=(7,1)\): \(((5,3)^\pm,\frac{x^{15}\mp1}{x\mp1})\).

For \(n>8\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with the remaining parts not divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with the remaining parts not divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\pm1}.
  \]

Choices:
B.
For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 10\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial. For \(n>10\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with the remaining parts not divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with the remaining parts not divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\pm1}.
  \]
C. For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), if \((\lambda,\mu)\) is not one of the exceptional pairs for \(n\le 8\) and does not belong to one of the four infinite families for \(n>8\), then the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\).
D.
For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 8\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial. For \(n>8\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with exactly one part divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with exactly one part divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\pm1}.
  \]
E.
For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 8\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial. For \(n>8\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with the remaining parts not divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\mp 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with the remaining parts not divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\mp 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\mp 1)}{(x^{k/3}\mp1)(x^{k/5}\mp 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\mp1}.
  \]