\begin{theorem}\label{thm:clique_classification}
For any clique $\mathcal K=\{P_1,\dots,P_m\}$ with $m\ge 1$ in $\B_k(G)$, exactly one of the following holds:
\begin{enumerate}
    \item $\mathcal K=\{P_1,P_2,P_3\}$ is a cyclic $T$-triangle;
    \item $\mathcal K=\{P_1,P_2,P_3\}$  is a radial $T$-triangle;
    \item $\mathcal K=\{P_1,P_2,P_3,P_4\}$  is a split $T$-tetrahedron;
    \item $\mathcal K=\{P_1,P_2,P_3,P_4\}$ is a fused $T$-tetrahedron; or
    \item $\mathcal K=\{P_1,\dots,P_m\}$ is an $S$-clique.
\end{enumerate}
\end{theorem}

\begin{tikzpicture}[
    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\small},
    edge label/.style={midway, fill=white, font=\footnotesize, inner sep=1.5pt},
    part label/.style={font=\scriptsize, color=black, align=center},
    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},
    scale=1.2
]

\coordinate (P1) at (0, -2, 0);
\coordinate (P2) at (2, 1, 0);
\coordinate (P3) at (-2, 1, 0);
\coordinate (P4) at (0, 0, 3);

\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};
\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$};
\draw (P2) -- (P3) node[edge label] {$w$};
\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};
\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};
\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};

\node[vertex] at (P1) {$P_1$};
\node[part label, below left=0.3cm of P1] {$\{\{w, v_1\}, \{v_2\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P2) {$P_2$};
\node[part label, below=0.4cm of P2] {$\{\{w, v_2\}, \{v_1\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P3) {$P_3$};
\node[part label, below right=0.3cm of P3] {$\{\{w, v_3\}, \{v_1\}, \{v_2\}, \emptyset \}$};

\node[vertex] at (P4) {$P_4$};
\node[part label, above=0.4cm of P4] {$\{\{w\}, \{v_1\}, \{v_2\}, \{v_3\}\}$};

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_4(K_{3}\sqcup K_1)\cong K_4$. The graph $K_3\sqcup K_1$ has vertex set $\{v_1, v_2, v_3, w\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K3-plus-K1}
\end{figure}

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

\begin{theoremA}[Theorem~\ref{thm:trees-cycles}]
All trees and all cycles are realizable as Bell coloring graphs.
\end{theoremA}

We prove Theorem~\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs.

The papers \cites{BFHRS16, ABFR17} examine forbidden subgraphs in standard coloring graphs. To treat the analogous question for Bell coloring graphs, we first obtain a structural description of their cliques (see Theorem~\ref{thm:clique_classification}), showing that every clique belongs to one of two explicit families. This classification allows us to construct an infinite family of forbidden induced subgraphs of Bell coloring graphs.

\begin{theoremA}[Theorem~\ref{thm:forbidden}]
The graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\ge 6$ is an infinite family of forbidden induced subgraphs.
\end{theoremA}

Another natural question is whether the structure of the reconfiguration graph uniquely determines the base graph. This is known for standard coloring graphs; in particular, if $T$ is a tree, $\mathcal{C}_3(T)$ uniquely determines $T$ (\cite{AKL25}*{Theorem 1.1} or \cite{BBHvdHHP25}*{Theorem 1.2}). Having lost the color labels, $\mathcal{B}_3(T)$ has less symmetry than the $3$-coloring graph $\mathcal{C}_3(T)$. Despite this compression, our next result guarantees that a tree $T$ can be reconstructed uniquely from its Bell $3$-coloring graph. Let $\mathsf{Trees}$ and $\mathsf{Graphs}$ denote the sets of isomorphism classes of all (finite) trees and graphs, respectively.

\begin{theoremA}[Theorem~\ref{thm:tree-reconstruction}]
The map $\mathsf{Trees} \to \mathsf{Graphs}$ given by $T\mapsto \mathcal{B}_3(T)$ is injective.
\end{theoremA}

We also find it useful to work with a multigraph variant, the \emph{Bell coloring multigraph}, denoted $\BellMulti{k}{G}$, which uses multiple edges to encode how many vertices are responsible for each adjacency. If $P$ and $P'$ are two stable partitions of $G$, then in the Bell coloring graph, $P$ and $P'$ are adjacent whenever there exists a vertex $v\in V(G)$ with $P-v = P'-v$. Thus, multiple vertices may witness the same adjacency, but $\B_k(G)$ still records only a single edge between $P$ and $P'$. In the multigraph version, we instead add a separate parallel edge between $P$ and $P'$ for each distinct vertex $v$ such that $P-v = P'-v$.

The additional information means that Bell coloring multigraphs serve as a more refined graph invariant. For example, Figures~\ref{fig:Bell3-K13}~and~\ref{fig:Bell3-K3-plus-K1} show that $\B_3(K_{1,3})\cong \B_4(K_3\sqcup K_1)$. However, each edge incident to the partition $P_4$ in the latter graph is witnessed by two vertices, so
$\BellMulti{3}{K_{1,3}}\not\cong\BellMulti{4}{K_3\sqcup K_{1}}$.

Yet even a Bell coloring multigraph cannot serve as a complete graph invariant. For instance, $\BellMulti{3}{K_{1,3}}\cong \BellMulti{2}{\overline{K}_3}$. This ambiguity arises because the center vertex in $K_{1,3}$ is adjacent to every other vertex. When a vertex $v$ is adjacent to every vertex in $G-v$, we call $v$ a \emph{universal vertex}.
If $G$ has a universal vertex $w$, then $\B_{k+1}(G) \cong \B_k(G-w)$ for every $k\in\mathbb{N}$ due to the natural bijection between stable $k$-partitions of $G-w$ and stable $(k+1)$-partitions of $G$ in which $\{w\}$ forms an additional singleton part.

To handle this, let $U(G)$ denote the set of universal vertices of $G$. We define the \emph{core} of $G$, denoted $G^{\circ}$, as the induced subgraph $G[V(G)\setminus U(G)]$. We define the equivalence relation $\sim_{\text{uni}}$ on $\mathsf{Graphs}$ as follows: $G\sim_{\text{uni}} G'$ if their cores are isomorphic, $G^{\circ} \cong (G')^{\circ}$.
The equivalence classes of graphs under this relation form the set $\mathsf{Graphs}^{\circ} = \mathsf{Graphs}/\!\sim_{\text{uni}}$.

Our final result shows that a base graph of order $n$ can be reconstructed from its Bell $n$-coloring multigraph up to universal vertices:

\begin{theoremA}[Theorem~\ref{thm:general-reconstruction-multigraph:intro}]
The map $\mathsf{Graphs}^{\circ} \to \mathsf{Multigraphs}$ given by $G\mapsto \BellMultiIntro{|V(G)|}{G}$  is injective.
\end{theoremA}

\textbf{Outline of the paper.} Section~\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\ge 6$. Section~\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell
$3$-coloring graphs. Section~\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.

\section{Preliminaries}\label{sec:preliminaries}

\subsection{Notation and conventions} Let $\mathbb{N}=\{1, 2, 3, \dots\}$ denote the set of positive integers. For sets $X$ and $Y$, we use the notation $X\setminus Y=\{x\in X \ | \ x\notin Y\}$ for the set difference and $X\cup Y$ for set union; in the special case when $Y=\{v\}$ is a singleton, we also denote $X\setminus\{v\}$ by $X-v$ and $X\cup\{v\}$ by $X+v$. Given a graph $G$, we let $\overline{G}$ denote its complement. We write $G\sqcup H$ for the disjoint union of graphs $G$ and $H$.

\begin{defn}\label{def:stable-partition}
A \emph{stable $k$-partition} of a graph $G$ is a multiset $P=\{V_1,\dots,V_k\}$ of independent sets that partition $V(G)$, where we allow some of the parts $V_i$ to be empty.
\end{defn}

Let $\cP_k(G)$ denote the set of all stable $k$-partitions of $G$. For $v\in V(G)$, we write
\[
P-v \colonequals \{V_i-v : 1\leq i\leq k\},
\]
viewed as a multiset of $k$ subsets.

A Bell $k$-coloring graph $\B_k(G)$ is the graph whose vertex set is $\cP_k(G)$. Two vertices $P_1, P_2 \in \cP_k(G)$ are adjacent if and only if $P_1-v=P_2-v$ for some $v\in V(G)$.

Throughout, partitions $P, Q, R \in \cP_k(G)$ are treated as \emph{unordered multisets} of parts. We write $P \sim Q$ to denote adjacency in $\B_k(G)$. To specify the vertex responsible for the adjacency, we use the following notation:
\[
P \sim_v Q \iff P \sim Q \text{ and } P-v = Q-v.
\]
We say that a vertex $v\in V(G)$ is \emph{responsible} for an edge $PQ$ in $\B_k(G)$ if $P\sim_v Q$. Equivalently, the edge $PQ$ is \emph{realized} by $v$. The condition $P-v=Q-v$ implies that $P$ and $Q$ agree on the structure of the partition when restricted to $V(G)\setminus\{v\}$.

We record the following useful fact, which we use several times in the paper: if $P_1\sim_v P_2$ and $P_2\sim_v P_3$, then $P_1$ and $P_3$ agree on $V(G)\setminus\{v\}$, so $P_1\sim_v P_3$.

\subsection{Example Bell coloring graphs} We begin with a few illustrative examples.

\begin{ex} Consider $\B_2(\overline{K}_2)$ where $V(\overline{K}_2)=\{1,2\}$. The stable 2-partitions are $R_1 = \{\{1,2\}, \emptyset\}$ and $R_2 = \{\{1\}, \{2\}\}$.
We have $R_1-1 = \{\{2\}, \emptyset\}$ and $R_2-1 = \{\emptyset, \{2\}\}$, so $R_1 \sim_1 R_2$. Similarly $R_1 \sim_2 R_2$. Thus $\B_2(\overline{K}_2) \cong K_2$.
\end{ex}

\begin{ex}\label{ex:K3}
Consider $G=K_3 \sqcup K_1$ on vertices $\{1,2,3,4\}$, where $\{1,2,3\}$ form the $K_3$ and the vertex $4$ is isolated. We examine $\B_3(G)$. Since $\{1,2,3\}$ is a clique, any stable 3-partition must place them in different parts. The vertex $4$ can join any part.
\begin{align*}
P_1 &= \{\{1,4\}, \{2\}, \{3\}\}, \\
P_2 &= \{\{1\}, \{2,4\}, \{3\}\}, \\
P_3 &= \{\{1\}, \{2\}, \{3,4\}\}.
\end{align*}
These are the only stable 3-partitions of $G$. We check adjacencies: $P_1-4 = P_2-4 = P_3-4 = \{\{1\}, \{2\}, \{3\}\}$. Thus, we have $P_1 \sim_4 P_2$, $P_2 \sim_4 P_3$, and $P_3 \sim_4 P_1$, so $\B_3(G) \cong K_3$.
\end{ex}

\begin{ex}\label{ex:P4}
Consider the path graph $G$ on vertices $\{1,2,3,4\}$ with edges 12, 23, 34. The stable $3$-partitions are:
\[
\begin{aligned}
R_1 &= \{\{1,3\}, \{2\}, \{4\}\}, & R_2 &= \{\{1,4\}, \{2\}, \{3\}\},\\
R_3 &= \{\{1\}, \{2,4\}, \{3\}\}, & R_4 &= \{\{1,3\}, \{2,4\}, \emptyset\}.
\end{aligned}
\]
A direct check shows that $R_1 \sim_1 R_2$, $R_2 \sim_4 R_3$, $R_3 \sim_1 R_4$, and $R_4 \sim_4 R_1$. The graph $\B_3(P_4)$ is the cycle $C_4$; see Figure~\ref{fig:B3_P4}.
\end{ex}

\begin{figure}[h]
    \centering
    \begin{tikzpicture}[
        partition/.style={rectangle, draw, rounded corners, thick, inner sep=5pt, align=center, fill=white},
        edge label/.style={midway, fill=white, font=\small, inner sep=1.5pt},
        scale=1.1
    ]

    \node[partition] (R1) at (0, 2) {$\{1,3\}, \{2\}, \{4\}$};

    \node[partition] (R2) at (5, 2) {$\{1,4\}, \{2\}, \{3\}$};

    \node[partition] (R3) at (5, 0) {$\{1\}, \{2,4\}, \{3\}$};

    \node[partition] (R4) at (0, 0) {$\{1,3\}, \{2,4\}, \emptyset$};

    \draw[thick] (R1) -- (R2) node[edge label] {$1$};

    \draw[thick] (R2) -- (R3) node[edge label] {$4$};

    \draw[thick] (R3) -- (R4) node[edge label] {$1, 3$};

    \draw[thick] (R4) -- (R1) node[edge label] {$2, 4$};

    \end{tikzpicture}

\begin{theorem}\label{thm:general-reconstruction-multigraph:intro}
Suppose $G_1$ and $G_2$ are two graphs of orders $n_1$ and $n_2$, respectively. If $\BellMulti{n_1}{G_1} \cong \BellMulti{n_2}{G_2}$, then $G_1$ and $G_2$ have isomorphic cores.
\end{theorem}

\begin{theorem}\label{thm:forbidden}
The graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\ge 6$ is an infinite family of forbidden induced subgraphs.
\end{theorem}

\label{sec:intro}

Graph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \cite{JT95}, the themed collection \cite{BW15}, and the recent text \cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \cites{HSTT24, BBHvdHHP25, AKL25}.

In this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\in\mathbb{N}$, we define the \emph{Bell $k$-coloring graph} $\mathcal{B}_k(G)$ as follows.
\begin{itemize}
    \item The vertices of $\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \emph{stable} $k$-partitions.
    \item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\in V(G)$, formalized as $P_1-v = P_2-v$.
\end{itemize}
The notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\ref{fig:Bell3-K13}~and~\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.

The name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \cite{Haa12}, \cite{FM14}) address Bell coloring graphs.

\begin{figure}[h]
\centering
\begin{tikzpicture}[
    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\small},
    edge label/.style={midway, fill=white, font=\footnotesize, inner sep=1.5pt},
    part label/.style={font=\scriptsize, color=black, align=center},
    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},
    scale=1.2
]

\coordinate (P1) at (0, -2, 0);
\coordinate (P2) at (2, 1, 0);
\coordinate (P3) at (-2, 1, 0);
\coordinate (P4) at (0, 0, 3);

\draw (P1) -- (P2) node[edge label, pos=0.53] {$v_3$};
\draw (P1) -- (P3) node[edge label, pos=0.55] {$v_2$};
\draw (P2) -- (P3) node[edge label] {$v_1$};
\draw (P4) -- (P1) node[edge label, pos=0.5] {$v_1$};
\draw (P4) -- (P2) node[edge label, pos=0.4] {$v_2$};
\draw (P4) -- (P3) node[edge label, pos=0.6] {$v_3$};

\node[vertex] at (P1) {$P_1$};
\node[part label, below left=0.3cm of P1] {$\{\{u\}, \{v_2, v_3\}, \{v_1\} \}$};

\node[vertex] at (P2) {$P_2$};
\node[part label, below=0.4cm of P2] {$\{\{u\}, \{v_1, v_3\}, \{v_2\} \}$};

\node[vertex] at (P3) {$P_3$};
\node[part label, below right=0.3cm of P3] {$\{\{u\}, \{v_1, v_2\}, \{v_3\} \}$};

\node[vertex] at (P4) {$P_4$};
\node[part label, above=0.4cm of P4] {$\{\{u\}, \{v_1, v_2, v_3\}, \emptyset\}$};

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_3(K_{1,3})\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\{u, v_1, v_2, v_3\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K13}
\end{figure}

\begin{figure}[h]
\centering
\begin{tikzpicture}[
    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\small},
    edge label/.style={midway, fill=white, font=\footnotesize, inner sep=1.5pt},
    part label/.style={font=\scriptsize, color=black, align=center},
    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},
    scale=1.2
]

\coordinate (P1) at (0, -2, 0);
\coordinate (P2) at (2, 1, 0);
\coordinate (P3) at (-2, 1, 0);
\coordinate (P4) at (0, 0, 3);

\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};
\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$};
\draw (P2) -- (P3) node[edge label] {$w$};
\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};
\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};
\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};

\node[vertex] at (P1) {$P_1$};
\node[part label, below left=0.3cm of P1] {$\{\{w, v_1\}, \{v_2\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P2) {$P_2$};
\node[part label, below=0.4cm of P2] {$\{\{w, v_2\}, \{v_1\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P3) {$P_3$};
\node[part label, below right=0.3cm of P3] {$\{\{w, v_3\}, \{v_1\}, \{v_2\}, \emptyset \}$};

\node[vertex] at (P4) {$P_4$};
\node[part label, above=0.4cm of P4] {$\{\{w\}, \{v_1\}, \{v_2\}, \{v_3\}\}$};

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_4(K_{3}\sqcup K_1)\cong K_4$. The graph $K_3\sqcup K_1$ has vertex set $\{v_1, v_2, v_3, w\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K3-plus-K1}
\end{figure}

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

Graph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \cite{JT95}, the themed collection \cite{BW15}, and the recent text \cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \cites{HSTT24, BBHvdHHP25, AKL25}.

In this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\in\mathbb{N}$, we define the \emph{Bell $k$-coloring graph} $\mathcal{B}_k(G)$ as follows.
\begin{itemize}
 \item The vertices of $\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \emph{stable} $k$-partitions.
 \item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\in V(G)$, formalized as $P_1-v = P_2-v$.
\end{itemize}
The notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\ref{fig:Bell3-K13}~and~\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.

The name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \cite{Haa12}, \cite{FM14}) address Bell coloring graphs.

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_3(K_{1,3})\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\{u, v_1, v_2, v_3\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K13}
\end{figure}

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_4(K_{3}\sqcup K_1)\cong K_4$. The graph $K_3\sqcup K_1$ has vertex set $\{v_1, v_2, v_3, w\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K3-plus-K1}
\end{figure}

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

We prove Theorem~\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs.

\textbf{Outline of the paper.} Section~\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\ge 6$. Section~\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell
$3$-coloring graphs. Section~\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.

In this section, we introduce the matching reconfiguration graph and establish its connection to Bell coloring graphs. As a consequence, we obtain Theorem~\ref{thm:trees-cycles}, stating that all trees and cycle graphs are realizable as Bell coloring graphs.

The isomorphism in Proposition~\ref{prop:matchings} enables us to realize all cycles and trees as Bell coloring graphs. We focus on the case where $n=|V(G)|=2k+1$. More formally, if $n=2k+1$ then we call $\mathcal{M}_k(G)$ the \emph{near-perfect matching graph} of $G$. The following lemma gives three equivalent conditions for two near-perfect matchings to be adjacent in a matching graph, which helps us reason about matching graphs that are cycles. In particular, it shows that near-perfect matchings are adjacent if and only if the unmatched vertices $u$ and $v$ have a common neighbor $w$ such that the symmetric difference of the matching is $\{uw,uv\}$.

Having shown that all cycles are realizable as near-perfect matching graphs and hence Bell coloring graphs, we turn to the task of realizing all trees as Bell coloring graphs. To do this, we first describe a procedure for joining near-perfect matching graphs to create larger near-perfect matching graphs. This joining construction relies on a particular property of odd-order graphs. For a graph $G$ on $n=2k+1$ vertices, a vertex $v$ is \emph{uniquely unmatched} if the graph $G\setminus v$ has a unique perfect matching. For two such graphs $H_1$ and $H_2$, Figure~\ref{fig:npm} depicts a near-perfect matching of these graphs joined on their uniquely unmatched vertices. We show that the near-perfect matching graph of this joined graph is exactly the joined near-perfect matching graphs of $H_1$ and $H_2$.

\begin{theorem}\label{thm:trees-cycles}
 All trees and all cycle graphs are realizable as Bell coloring graphs.
\end{theorem}

This section proves Theorem~\ref{thm:tree-reconstruction} through a sequence of lemmas that yield an algorithmic procedure for reconstructing a tree given its Bell 3-coloring graph. First, we prove that three categories of trees can be differentiated based on the degree sequence of their Bell coloring graphs, and then we show that within each category, other Bell coloring graph features determine the specific tree. As we can manually verify the result for all trees of order at most $5$, we assume $n\geq 6$ throughout this section.

\begin{theorem}\label{thm:tree-reconstruction}
For trees $T_1$ and $T_2$, we have $\B_3(T_1)\cong \B_3(T_2)$ if and only if $T_1\cong T_2$.
\end{theorem}

\begin{theorem}\label{thm:trees-cycles}
    All trees and all cycle graphs are realizable as Bell coloring graphs.
\end{theorem}

We prove Theorem~\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs. In particular, Section~\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs.

\begin{theorem}\label{thm:trees-cycles}
    All trees and all cycle graphs are realizable as Bell coloring graphs.
\end{theorem}
All trees and all cycles are realizable as Bell coloring graphs.

Question: Let \(H\) be a graph that is either a tree or a cycle graph. A graph is said to be realizable as a Bell coloring graph if there exist a graph \(G\) and an integer \(k\ge 1\) such that \(H\cong \mathcal{B}_k(G)\), where \(\mathcal{B}_k(G)\) is the graph whose vertices are the stable \(k\)-partitions of \(V(G)\) (that is, partitions of \(V(G)\) into \(k\) independent sets, some possibly empty), and where two such partitions \(P_1\) and \(P_2\) are adjacent when they differ only in the placement of a single vertex, equivalently \(P_1-v=P_2-v\) for some vertex \(v\in V(G)\). Which statement holds for every such \(H\)?

Every tree and every cycle graph is realizable as a Bell coloring graph; equivalently, for each such graph \(H\), there exist a graph \(G\) and an integer \(k\ge 1\) with \(H\cong \mathcal{B}_k(G)\).

Choices:
B. Every tree is realizable as a Bell coloring graph, but among cycle graphs only the even cycles are realizable; equivalently, for each tree or even cycle graph \(H\), there exist a graph \(G\) and an integer \(k\ge 1\) with \(H\cong \mathcal{B}_k(G)\).
C. Every tree is realizable as a Bell coloring graph; equivalently, for each tree \(H\), there exist a graph \(G\) and an integer \(k\ge 1\) with \(H\cong \mathcal{B}_k(G)\).
D. For every graph \(H\) that is either a tree or a cycle graph, there exists an integer \(k\ge 1\) such that for every such \(H\) one can find a graph \(G\) with \(H\cong \mathcal{B}_k(G)\).
E. Every tree and every cycle graph is realizable as a Bell coloring graph; moreover, for each such graph \(H\), one may always choose \(G=H\) and some integer \(k\ge 1\) so that \(H\cong \mathcal{B}_k(H)\).

existence of some realizing graph G replaced by G=H

The stem does not explicitly reveal the correct option. It defines the Bell coloring graph notion and asks which global statement is true, without embedding the theorem’s conclusion.

The correct choice is essentially a direct restatement of the target theorem: that every tree and every cycle graph is realizable as a Bell coloring graph. The item mainly tests recall of that exact statement.

There is some reasoning pressure because the options vary by strength, quantifiers, and special cases (e.g., weaker true statement, parity restriction, fixed-k trap, stronger G=H claim). However, if the theorem is known, the correct answer is immediate rather than generated through substantial reasoning.

The distractors are strong and mathematically meaningful: B introduces a plausible even-cycle restriction, C is a weaker true statement, D tests quantifier dependence, and E is an overstrong realization claim. These reflect common failure modes.

Low leakage and high-quality distractors, but the item is close to a theorem-recall question and thus only weakly tests generative mathematical reasoning.


\begin{thm}\label{mainthm1} Let $\blam\in\P_{\ell,n}$ and $\s,\t\in\te{Std}(\blam).$ Then
$$\tau_R^{\te{MM}}(\fm_{\s\t})=
\begin{cases}
q^{\sum\limits_{k=1}^n(k-k(\t_{\downarrow\leq k}))}\bigg(\prod\limits_{j=1}^n\prod\limits_{i=l(\t,j)+1}^{\ell}(-Q_i)\bigg), & \mbox{if\ \ }\s=\t, \\
0, & \mbox{if\ \ }\s\neq \t.
\end{cases}$$
\end{thm}

\begin{dfn}\label{degenerate} The degenerate cyclotomic Hecke algebra $H_{\ell,n}=H_{\ell,n}^R(\bu)$ of type $G(\ell,1,n)$ is the unital associative $R$-algebra with generators $s_{1},s_2,\dots,s_{n-1},L_1,L_2,\dots,L_n$ and the following defining relations:
\begin{align}
	&(L_1-u_1)(L_1-u_2)\cdots(L_1-u_\ell)=0;\\
\label{degquadraticrela}	&s_{i}^{2}=1,\quad \forall\,1\leq i\leq n-1;\\
\label{degbraidrela1}	&s_{i}s_{j}=s_{j}s_{i},\ \ \forall\,1\leq i<j-1<n-1;\\
\label{degbraidrela2}	&s_is_{i+1}s_i=s_{i+1}s_is_{i+1},\ \ \forall\,1\leq i<n-1;\\
\label{deg-L-comm1} &L_iL_k=L_kL_i,\ \  s_iL_l=L_ls_i,\ \ 1\leq i<n,\ 1\leq k,l\leq n,\ l\neq i,i+1;\\
\label{deg-L-comm2} &L_{i+1}=s_iL_is_i+s_i,\ \ 1\leq i<n .
\end{align}
\end{dfn}

\begin{cor}\label{maincor12}
Let $\blam\in\P_{\ell,n}$ and $\s,\t\in\te{Std}(\blam).$
\begin{enumerate}
  \item If $n(\s)= n(\t),$ then
for each $(\fp,\fq)\in \Std^2(\P_{\ell,n-1})$ with $(\fp,\fq)\rhd (\s_{\downarrow\leq (n-1)},\t_{\downarrow\leq (n-1)}),$ we have
$$\begin{aligned}
&c_{\t,n}+\sum_{\substack{(\u,\v)\in\te{Std}^2(\P_{\ell,n})\\(\u,\v)\rhd (\s,\t)
\\\u_{\downarrow\leq (n-1)}=\s_{\downarrow\leq (n-1)}\\
\v_{\downarrow\leq (n-1)}=\t_{\downarrow\leq (n-1)}}}
b_{\u\v}^{\s\t}c_{\v,n}
=q^{n-n(\t)}\Biggr(\prod\limits_{i=l(\t,n)+1}^{\ell}(-Q_i)\Biggr),\\
&\sum_{\substack{(\u,\v)\in\te{Std}^2(\P_{\ell,n})\\(\u,\v)\rhd (\s,\t)\\
\u_{\downarrow\leq (n-1)}=\fp\\
\v_{\downarrow\leq (n-1)}=\fq}} b_{\u\v}^{\s\t} c_{\v,n}=
q^{n-n(\t)}\Biggr(\prod\limits_{i=l(\t,n)+1}^{\ell}(-Q_i)\Biggr) b_{\fp\fq}^{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}.
\end{aligned} $$
  \item If $n(\s)\neq n(\t),$ then for each $(\fp,\fq)\in \Std^2(\P_{\ell,n-1})$
  with $(\fp,\fq)\rhd (\s_{\downarrow\leq (n-1)},\t_{\downarrow\leq (n-1)}),$ we have
$$\sum_{\substack{(\u,\v)\in\te{Std}^2(\P_{\ell,n})\\(\u,\v)\rhd (\s,\t)\\
\u_{\downarrow\leq (n-1)}=\fp\\
\v_{\downarrow\leq (n-1)}=\fq}} b_{\u\v}^{\s\t} c_{\v,n}=0. $$
\end{enumerate}
\end{cor}

\begin{cor}\label{maincor13}
Let $\blam\in\P_{\ell,n}$ and $\s,\t\in\te{Std}(\blam).$ Then we have
\begin{align*}
\delta_{\s\t}d_\t+\sum_{\substack{\u\in\te{Std}(\P_{\ell,n})\\(\u,\u)\rhd (\s,\t)}}a_{\u\u}^{\s\t}d_\u=\delta_{\s\t} c_\t,\quad
\delta_{\s\t}c_\t+\sum_{\substack{\u\in\te{Std}(\P_{\ell,n})\\(\u,\u)\rhd (\s,\t)}}b_{\u\u}^{\s\t}c_\u=\delta_{\s\t} d_\t.
\end{align*}
\end{cor}

\begin{cor}\label{maincor11}
Let $\blam\in\P_{\ell,n}$ and $\s,\t\in\te{Std}(\blam).$
\begin{enumerate}
  \item If $n(\s)= n(\t),$ then
for each $(\fp,\fq)\in \Std^2(\P_{\ell,n-1})$ with $(\fp,\fq)\rhd (\s_{\downarrow\leq (n-1)},\t_{\downarrow\leq (n-1)}),$ we have
$$\begin{aligned}
&q^{n-n(\t)}\Biggr(\prod\limits_{i=l(\t,n)+1}^{\ell}(-Q_i)\Biggr)+
\sum_{\substack{(\u,\v)\in\te{Std}^2(\P_{\ell,n})\\(\u,\v)\rhd (\s,\t)
\\\u_{\downarrow\leq (n-1)}=\s_{\downarrow\leq (n-1)}\\
\v_{\downarrow\leq (n-1)}=\t_{\downarrow\leq (n-1)}}}
a_{\u\v}^{\s\t}q^{n-n(\u)}\Biggr(\prod\limits_{i=l(\u,n)+1}^{\ell}(-Q_i)\Biggr)=c_{\t,n},\\
&\sum_{\substack{(\u,\v)\in\te{Std}^2(\P_{\ell,n})\\(\u,\v)\rhd (\s,\t)\\
\u_{\downarrow\leq (n-1)}=\fp\\
\v_{\downarrow\leq (n-1)}=\fq}} a_{\u\v}^{\s\t}q^{n-n(\u)}
\Biggr(\prod\limits_{i=l(\u,n)+1}^{\ell}(-Q_i)\Biggr)=c_{\t,n} a_{\fp\fq}^{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}.
\end{aligned} $$
  \item If $n(\s)\neq n(\t),$ then for each $(\fp,\fq)\in \Std^2(\P_{\ell,n-1})$
  with $(\fp,\fq)\rhd (\s_{\downarrow\leq (n-1)},\t_{\downarrow\leq (n-1)}),$ we have
$$\sum_{\substack{(\u,\v)\in\te{Std}^2(\P_{\ell,n})\\(\u,\v)\rhd (\s,\t)\\
\u_{\downarrow\leq (n-1)}=\fp\\
\v_{\downarrow\leq (n-1)}=\fq}} a_{\u\v}^{\s\t}q^{n-n(\u)}
\Biggr(\prod\limits_{i=l(\u,n)+1}^{\ell}(-Q_i)\Biggr)=0. $$
\end{enumerate}
\end{cor}

Let $R$ be an integral domain, $1\neq q\in R^\times$ and ${\mathbf Q}:=(Q_{1},Q_2,\dots, Q_{\ell})\in R^\ell$. The non-degenerate cyclotomic Hecke algebras $\HH_{\ell,n}^R(q;\mathbf{Q})$ of type $G(\ell,1,n)$ with Hecke parameter $q$ and cyclotomic parameters $Q_{1},Q_2,\dots, Q_{\ell}$ were first introduced in \cite[Definition 3.1]{AK1}, \cite[Definition 4.1]{BM:cyc} and \cite[before Proposition 3.2]{C} as certain deformations of the group ring $R[W_{\ell,n}]$ of the complex reflection group $W_{\ell,n}$ of type $G(\ell,1,n)$. By definition, $\mathscr{H}_{\ell,n}=\HH_{\ell,n}^R(q;\mathbf{Q})$ is the unital associative $R$-algebra with generators $T_0,T_1,\dots,T_{n-1}$ and the following defining relations:
\begin{align}
&(T_0-Q_1)(T_0-Q_2)\cdots(T_0-Q_\ell)=0;\\
&T_0T_1T_0T_1=T_1T_0T_1T_0;\\
\label{nondegquadraticrela}&(T_i-q)(T_i+1)=0,\quad \forall\,1\leq i\leq n-1;\\
\label{nondegbraidrela1}&T_iT_j=T_jT_i,\ \ \forall\,1\leq i<j-1<n-1;\\
\label{nondegbraidrela2}&T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\ \ \forall\,1\leq i<n-1.
\end{align}
If $\ell=1$ or $\ell=2$, then the cyclotomic Hecke algebra $\HH_{\ell,n}^R(q;\mathbf{Q})$ becomes the Iwahori-Hecke algebra of type $A_{n-1}$ or type $B_n$ respectively.

The symmetric group $\Sym_n$ on $\{1,2,\dots,n\}$ is generated by $\{\sigma_i:=(i,i+1)\,|\,1\leq i<n\}$. A word $w=\sigma_{i_{1}}\sigma_{i_{2}}\cdots\sigma_{i_{k}}$, where $1\leq i_1,i_2,\dots,i_k\leq n-1$, is called a reduced expression of $w$ if $k$ is minimal; in this case we say that $w$ has length $k$ and we write $\ell(w)=k$. Note that $T_w:=T_{i_1}T_{i_2}\cdots T_{i_k}$ depends only on $w$ but not on the choice of the reduced expression of $w$ because of the braid relations holding in $\HH_{\ell,n}$.

The $R$-subalgebra of $\mathscr{H}_{\ell,n}$ generated by $T_1,\dots,T_{n-1}$ is isomorphic to the Iwahori-Hecke algebra of type $A_{n-1}$, which has a standard basis $\{T_w\,|\, w\in\Sym_n\}$. We set $$
\mL_{1}=T_{0},\quad \mL_{k+1}=q^{-1}T_{k}L_{k}T_{k},\ \ \forall\,1\leq k\leq n-1 .
$$
These elements commute with each other, and are called the {\bf Jucys-Murphy operators} of $\HH_{\ell,n}^R(q;\mathbf{Q})$.

\begin{lem}\label{AK basis}\te{(}Ariki--Koike \cite[(3.10)]{AK1}\te{)}
The algebra $\mathscr{H}_{\ell,n}$ is free as an $R$-module with basis
$$\{\mL_1^{c_1}\mL_2^{c_2}\cdots \mL_n^{c_n}T_w\mid
       w\in\Sym_n,\ 0\leq c_i\leq \ell-1,\ \forall\,1\leq i\leq n\}.$$
In particular, the $R$-rank of $\mathscr{H}_{\ell,n}$ is equal to $\ell^nn!$.
\end{lem}

\begin{dfn}\label{ssf} For any $w\in \Sym_n$ and integers $0\leq c_1,c_2,\dots,c_n<\ell,$ we define
$$
\tau_{R}^{\rm{MM}}(\mL_1^{c_1}\mL_2^{c_2}\cdots \mL_n^{c_n}T_w):=\begin{cases}
1, &\text{if\ \,$w=1$\ and\ $c_1=c_2=\cdots=c_n=0$},\\
0, &\text{otherwise.}
\end{cases}
$$
We extend $\tau_{R}^{\rm{MM}}$ linearly to an $R$-linear function on $\HH_{\ell,n}$.
\end{dfn}
One can show directly that the $R$-linear function $\tau_{R}^{\rm{MM}}$ is symmetric, i.e., $\tau_{R}^{\rm{MM}}(xy)=\tau_{R}^{\rm{MM}}(yx)$, $\forall\,x,y\in \HH_{\ell,n}$, see \cite{MM} and \cite{HHL}. We call $\tau_R^{\rm{MM}}$ the {\bf{standard symmetrizing form}} on $\HH_{\ell,n}$. If furthermore, $(Q_{1},Q_2,\dots, Q_{\ell})\in (R^\times)^\ell$, then by a result of Malle-Mathas \cite{MM}, the $R$-linear function $\tau_{R}^{\rm{MM}}$ on $\HH_{\ell,n}$ is non-degenerate, which makes $\HH_{\ell,n}$ into a symmetric algebra over $R$.

We use $\P_{\ell,n}$ to denote the set of $\ell$-partitions of $n$. For each $\blam\in\P_{\ell,n}$, we use $\Std(\blam)$ to denote the set of standard $\blam$-tableaux. Each $\mathfrak{t}\in\Std(\blam)$ can be viewed as a bijective map
$\mathfrak{t}: [\blam]\rightarrow\{1,2,\dots,n\}$ from the Young diagram $[\blam]:=\{(r,c,l)\mid 1\leq l\leq\ell,\ r\geq 1,\ 1\leq c\leq\lam_r^{(l)}\}$ to $\{1,2,\dots,n\}$.

In \cite{DJM}, Dipper-James-Mathas constructed a cellular basis $\{\fm_{\mathfrak{s}\mathfrak{t}}\,|\, \mathfrak{s},\mathfrak{t}\in\Std(\blam),\ \blam\in\P_{\ell,n}\}$ for $\HH_{\ell,n}$
in the sense of \cite{GL}. This basis generalizes the well-known Murphy basis for the Iwahori-Hecke algebra associated to $\Sym_n$ (i.e., the level one case). We call it {\bf the Murphy basis for the non-degenerate cyclotomic Hecke algebra $\HH_{\ell,n}$}. When $\HH_{\ell,n}$ is semisimple, Mathas also constructed a seminormal basis $\{\mf_{\mathfrak{s}\mathfrak{t}}\,|\,\mathfrak{s},\mathfrak{t}\in\Std(\blam),\ \blam\in\P_{\ell,n}\}$ for $\HH_{\ell,n}$. We refer the readers to Sections 2.1 and 3.2 for their precise definitions.

A critical challenge in the study of $\HH_{\ell,n}$ lies in characterizing the transition matrix between the Murphy basis and the seminormal basis. The explicit transformation between the Murphy basis and the seminormal basis—whose entries encode essential structural correlations—lacks a systematic description. This gap persists partly due to the absence of precise computations of trace forms on
 the Murphy basis, as the values of the trace form on seminormal basis elements can be easily expressed in terms of the gamma coefficients and the Schur functions. Without such exact values, the transition matrix remains elusive, obstructing efforts to unify the combinatorial (Murphy basis) and representation-theoretic (seminormal basis) perspectives of Hecke algebras.

Another pressing problem concerns the integral basis of the Hecke algebra's cocenter. In \cite{HS25}, the first author and Lei Shi have constructed an integral basis for the cocenter of $\HH_{\ell,n}$. However, it is very hard to apply that integral basis of the cocenter because it is given by a subset of Bremke-Malle's standard basis $\{T_w\}$ whose definition depends on the choice of the reduced expression of $w$ and is not directly related to the cellular structure of $\HH_{\ell,n}$. We really expect an integral basis for the cocenter of $\HH_{\ell,n}$ which is directly related to the cellular structure of $\HH_{\ell,n}$. The Murphy basis, with its inherent integral and recursive properties, offer a promising avenue for constructing such a basis—but this requires precise knowledge of trace forms on the Murphy basis to ensure integrality and linear independence.

\begin{dfn}\label{keydefn} Let $k\in\mathbb N,\ \blam=(\lambda^{(1)},\dots,\lambda^{(\ell)})\in\P_{\ell,k}$ and $\mathfrak{t}\in\Std(\blam)$.
Let $1\leq k(\mathfrak{t})\leq k$ be the unique integer such that
\begin{equation*}\label{ntDef}\mathfrak{t}^{-1}(k)=(\mathfrak{t}^{\blam})^{-1}(k(\mathfrak{t})).
\end{equation*}
For each integer $1\leq j\leq n$, we let $r(\mathfrak{t},j),\,c(\mathfrak{t},j)$ and $l(\mathfrak{t},j)$ be the integers such that
\begin{equation*}\label{coordinate}
\mathfrak{t}^{-1}(j)=(r(\mathfrak{t},j),c(\mathfrak{t},j),l(\mathfrak{t},j))\in [\blam].
\end{equation*}
\end{dfn}

The following theorem is the first main result of this paper.

Let $R$ be an integral domain, $1\neq q\in R^\times$ and ${\mathbf Q}:=(Q_{1},Q_2,\dots, Q_{\ell})\in R^\ell$. The non-degenerate cyclotomic Hecke algebras $\HH_{\ell,n}^R(q;\mathbf{Q})$ of type $G(\ell,1,n)$ with Hecke parameter $q$ and cyclotomic parameters $Q_{1},Q_2,\dots, Q_{\ell}$ were first introduced in \cite[Definition 3.1]{AK1}, \cite[Definition 4.1]{BM:cyc} and \cite[before Proposition 3.2]{C} as certain deformations of the group ring $R[W_{\ell,n}]$ of the complex reflection group $W_{\ell,n}$ of type $G(\ell,1,n)$. By definition, $\mathscr{H}_{\ell,n}=\HH_{\ell,n}^R(q;\mathbf{Q})$ is the unital associative $R$-algebra with generators $T_0,T_1,\dots,T_{n-1}$ and the following defining relations:
\begin{align}
&(T_0-Q_1)(T_0-Q_2)\cdots(T_0-Q_\ell)=0;\\
&T_0T_1T_0T_1=T_1T_0T_1T_0;\\
\label{nondegquadraticrela}&(T_i-q)(T_i+1)=0,\quad \forall\,1\leq i\leq n-1;\\
\label{nondegbraidrela1}&T_iT_j=T_jT_i,\ \ \forall\,1\leq i<j-1<n-1;\\
\label{nondegbraidrela2}&T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\ \ \forall\,1\leq i<n-1.
\end{align}
If $\ell=1$ or $\ell=2$, then the cyclotomic Hecke algebra $\HH_{\ell,n}^R(q;\mathbf{Q})$ becomes the Iwahori-Hecke algebra of type $A_{n-1}$ or type $B_n$ respectively.

\begin{dfn}\label{ssf} For any $w\in \Sym_n$ and integers $0\leq c_1,c_2,\dots,c_n<\ell,$ we define
$$
\tau_{R}^{\rm{MM}}(\mL_1^{c_1}\mL_2^{c_2}\cdots \mL_n^{c_n}T_w):=\begin{cases}
1, &\text{if\ \,$w=1$\ and\ $c_1=c_2=\cdots=c_n=0$},\\
0, &\text{otherwise.}
\end{cases}
$$
We extend $\tau_{R}^{\rm{MM}}$ linearly to an $R$-linear function on $\HH_{\ell,n}$.
\end{dfn}
One can show directly that the $R$-linear function $\tau_{R}^{\rm{MM}}$ is symmetric, i.e., $\tau_{R}^{\rm{MM}}(xy)=\tau_{R}^{\rm{MM}}(yx)$, $\forall\,x,y\in \HH_{\ell,n}$, see \cite{MM} and \cite{HHL}. We call $\tau_R^{\rm{MM}}$ the {\bf{standard symmetrizing form}} on $\HH_{\ell,n}$. If furthermore, $(Q_{1},Q_2,\dots, Q_{\ell})\in (R^\times)^\ell$, then by a result of Malle-Mathas \cite{MM}, the $R$-linear function $\tau_{R}^{\rm{MM}}$ on $\HH_{\ell,n}$ is non-degenerate, which makes $\HH_{\ell,n}$ into a symmetric algebra over $R$.

We use $\P_{\ell,n}$ to denote the set of $\ell$-partitions of $n$. For each $\blam\in\P_{\ell,n}$, we use $\Std(\blam)$ to denote the set of standard $\blam$-tableaux. Each $\mathfrak{t}\in\Std(\blam)$ can be viewed as a bijective map
$\mathfrak{t}: [\blam]\rightarrow\{1,2,\dots,n\}$ from the Young diagram $[\blam]:=\{(r,c,l)\mid 1\leq l\leq\ell,\ r\geq 1,\ 1\leq c\leq\lam_r^{(l)}\}$ to $\{1,2,\dots,n\}$.

In \cite{DJM}, Dipper-James-Mathas constructed a cellular basis $\{\fm_{\mathfrak{s}\mathfrak{t}}\,|\, \mathfrak{s},\mathfrak{t}\in\Std(\blam),\ \blam\in\P_{\ell,n}\}$ for $\HH_{\ell,n}$
in the sense of \cite{GL}. This basis generalizes the well-known Murphy basis for the Iwahori-Hecke algebra associated to $\Sym_n$ (i.e., the level one case). We call it {\bf the Murphy basis for the non-degenerate cyclotomic Hecke algebra $\HH_{\ell,n}$}. When $\HH_{\ell,n}$ is semisimple, Mathas also constructed a seminormal basis $\{\mf_{\mathfrak{s}\mathfrak{t}}\,|\,\mathfrak{s},\mathfrak{t}\in\Std(\blam),\ \blam\in\P_{\ell,n}\}$ for $\HH_{\ell,n}$. We refer the readers to Sections 2.1 and 3.2 for their precise definitions.

\begin{dfn}\label{keydefn} Let $k\in\mathbb N,\ \blam=(\lambda^{(1)},\dots,\lambda^{(\ell)})\in\P_{\ell,k}$ and $\mathfrak{t}\in\Std(\blam)$.
Let $1\leq k(\mathfrak{t})\leq k$ be the unique integer such that
\begin{equation*}\label{ntDef}\mathfrak{t}^{-1}(k)=(\mathfrak{t}^{\blam})^{-1}(k(\mathfrak{t})).
\end{equation*}
For each integer $1\leq j\leq n$, we let $r(\mathfrak{t},j),\,c(\mathfrak{t},j)$ and $l(\mathfrak{t},j)$ be the integers such that
\begin{equation*}\label{coordinate}
\mathfrak{t}^{-1}(j)=(r(\mathfrak{t},j),c(\mathfrak{t},j),l(\mathfrak{t},j))\in [\blam].
\end{equation*}
\end{dfn}

The following theorem is the first main result of this paper.

The following theorem is the first main result of this paper.

Let $\bu=(u_1,u_2,\dots,u_\ell)\in R^{\ell}$. The degenerate cyclotomic Hecke algebra $H_{\ell,n}=H_{\ell,n}^R(\bu)$ of type $G(\ell,1,n)$ is the unital associative $R$-algebra with generators $s_{1} ,\dots,s_{n-1},L_1, \dots,L_n$ which satisfy certain relations, see Definition \ref{degenerate}. In \cite{AMR}, Ariki, Mathas and Rui constructed a cellular basis $\{m_{\s\t}\,|\, \s,\t\in\Std(\blam),\ \blam\in\P_{\ell,n}\}$ for $H_{\ell,n}$. This basis generalizes the well-known Murphy basis for the symmetric group algebra $R[\Sym_n]$ (i.e., the level one case). We call it {\bf the Murphy basis for the degenerate cyclotomic Hecke algebra $H_{\ell,n}$}. Brundan-Kleshchev \cite{BK08} introduced a symmetrizing form $\tau_R^{\te{BK}}$ on $H_{\ell,n}$ which makes it into a symmetric algebra over $R$. We refer the readers to Section 2.2 for their precise definitions. The following theorem is the second main result of this paper.

\begin{prop}\label{nondegvemst}
Let $\blam\in\mathscr{P}_{\ell,n}$ and $\s,\t\in\te{Std}(\blam)$. Suppose that either $n(\s)=n$ or $n(\t)=n$. Then
$$\ve_n(\fm_{\s\t})=\begin{cases} \bigg(\prod\limits_{i=l(\t,n)+1}^{\ell}(-Q_i)\bigg)\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}, &\text{if\ \ $n(\s)=n(\t),$}\\
0, &\text{if\ \ $n(\s)\neq n(\t).$}\end{cases}$$
\end{prop}

\begin{prop}\label{nondegvemst2}
Let $\blam\in\mathscr{P}_{\ell,n}$ and $\s,\t\in\te{Std}(\blam)$. Suppose that $n(\s)\neq n\neq n(\t)$. Then
$$\ve_n(\fm_{\s\t})=\begin{cases} q^{n-n(\t)}\bigg(\prod\limits_{i=l(\t,n)+1}^{\ell}(-Q_i)\bigg)\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}, &\text{if\ \,$n(\s)=n(\t),$}\\
0, &\text{if\ \,$n(\s)\neq n(\t).$}\end{cases}$$
\end{prop}

\smallskip\noindent
{\it Case 2.} Suppose $n(\s)\neq n=n(\t)$. Then $d(\s)\neq d(\s_{\downarrow\leq (n-1)})$ and $d(\t)=d(\t_{\downarrow\leq (n-1)}).$
Then for each $w\in\Sym_\blam$ such that $\sigma_{n-1}\not\leq w,$
using (\ref{decompositionofdt}), (\ref{deg-L-comm1}) and Lemma \ref{deg-L-comm3} (1) we deduce that
\begin{align}\label{degcompu3}
d(\s)^*w u_\blam^+ d(\t)
&=d(\s_{\downarrow\leq (n-1)})^*\gamma_{n-1,n(\s)} w
\Biggr(\prod_{s=2}^{\ell}\prod_{k=1}^{\fa_s}(L_k-u_s)\Biggr)d(\t_{\downarrow\leq (n-1)})\\
\notag &=\varpi_{\s,\t,w}^{(2,1)} s_{n-1}\varpi^{(2,2)}_{\s,\t,w} + \rp^{(2)}_{\s,\t,w},
\end{align}
where
$$\begin{aligned}
\varpi_{\s,\t,w}^{(2,1)}&=
d(\s_{\downarrow\leq (n-1)})^*
\Biggr(\prod\limits_{s=\ell_0+1}^{\ell}(L_{n-1}-u_s)\Biggr)\in H_{\ell,n-1},\\
\varpi_{\s,\t,w}^{(2,2)}&=\gamma_{n-2,n(\s)}w
\Biggr(\prod\limits_{s=2}^{\ell}\prod\limits_{\substack{1\leq k\leq \fa_s\\k\neq n}}(L_k-u_s)\Biggr) d(\t_{\downarrow\leq (n-1)})\in H_{\ell,n-1},\end{aligned}$$ and $$\begin{aligned}
\rp^{(2)}_{\s,\t,w}&=
\begin{cases}\begin{matrix}
d(\s_{\downarrow\leq (n-1)})^* \bigg(\sum\limits_{i=\ell_0+1}^{\ell}
\bigg( \prod\limits_{\ell_0+1\leq s<i}(L_{n-1}-u_s)
\prod\limits_{i< t\leq \ell}(L_{n}-u_t)\bigg)\bigg) \\
\times \gamma_{n-2,n(\s)}w\bigg(\prod\limits_{s=2}^{\ell}\prod\limits_{\substack{1\leq k\leq \fa_s\\k\neq n}}(L_k-u_s)\bigg)d(\t_{\downarrow\leq (n-1)}),\end{matrix} & \mbox{if\ \ } \ell_0<\ell,\\
 0, & \mbox{if\ \ } \ell_0=\ell.
\end{cases}
\end{aligned}$$
As in Case 1, using (\ref{easyobservationdeg}) we see that in this case,
\begin{equation}\label{Case2Vendeg1}
\e_n\bigl({d(\s)}^*w u_\blam^+ {d(\t)}\bigr)=\rp_{\ell-1}^{(n)}(\rp^{(2)}_{\s,\t,w})=0.
\end{equation}
On the other hand, for each $w\in\Sym_\blam$ such that $\sigma_{n-1} \leq w,$
using (\ref{decompositionofdt})-(\ref{decompositionofwlamr}), (\ref{degbraidrela1}), (\ref{deg-L-comm1}), Lemma \ref{keytoolsdeg} (1) and Lemma \ref{deg-L-comm3} (1)
and noticing that $b_{\ell_0,r_{\ell_0}}\geq a_{\ell_0,r_{\ell_0}-1}+1>n(\s)$, we have
\begin{align*}
d(\s)^*w u_\blam^+ d(\t)
& = d(\s_{\downarrow\leq (n-1)})^*\gamma_{n-1,n(\s)}
\Biggr(\prod_{i=1}^{\ell_0-1} w_{i}\Biggr)\Biggr(\prod_{j=1}^{r_{\ell_0}-1}w_{\ell_0,j}\Biggr)
\beta_{b_{\ell_0,r_{\ell_0},n-1}}w_{\ell_0,r_{\ell_0}}^{(1)} \\
\notag&\quad\ \times \Biggr(\prod_{s=2}^{\ell}\prod_{k=1}^{\fa_s}(L_k-Q_s)\Biggr)d(\t_{\downarrow\leq (n-1)})\\
&= d(\s_{\downarrow\leq (n-1)})^*\gamma_{n-1,n(\s)}\beta_{b_{\ell_0,r_{\ell_0},n-1}}
\Biggr(\prod_{i=1}^{\ell_0-1} w_{i}\Biggr)\Biggr(\prod_{j=1}^{r_{\ell_0}-1}w_{\ell_0,j}\Biggr)w_{\ell_0,r_{\ell_0}}^{(1)}\\
\notag&\quad\ \times
\Biggr(\prod_{s=2}^{\ell}\prod_{k=1}^{\fa_s}(L_k-u_s)\Biggr)d(\t_{\downarrow\leq (n-1)})\\
&= d(\s_{\downarrow\leq (n-1)})^*\beta_{b_{\ell_0,r_{\ell_0}}-1,n-2}s_{n-1}\gamma_{n-2,n(\s)}
\Biggr(\prod_{i=1}^{\ell_0-1} w_{i}\Biggr)\Biggr(\prod_{j=1}^{r_{\ell_0}-1}w_{\ell_0,j}\Biggr) w_{\ell_0,r_{\ell_0}}^{(1)}\\
\notag&\quad\ \times
\Biggr(\prod_{s=2}^{\ell}\prod_{k=1}^{\fa_s}(L_k-u_s)\Biggr)d(\t_{\downarrow\leq (n-1)})\\
&= \varpi_{\s,\t,w}^{(3,1)} s_{n-1}\varpi^{(3,2)}_{\s,\t,w} + \rp^{(3)}_{\s,\t,w},
\end{align*}
where
$$\begin{aligned}
\varpi_{\s,\t,w}^{(3,1)}&=
d(\s_{\downarrow\leq (n-1)})^*\beta_{b_{\ell_0,r_{\ell_0}}-1,n-2}
\Biggr(\prod\limits_{s=l(\blam,n)+1}^{\ell}(L_{n-1}-u_s)\Biggr)\in H_{\ell,n-1},\\
\varpi_{\s,\t,w}^{(3,2)}&=\gamma_{n-2,n(\s)}
\Biggr(\prod_{i=1}^{\ell_0-1} w_{i}\Biggr)\Biggr(\prod_{j=1}^{r_{\ell_0}-1}w_{\ell_0,j}\Biggr) w_{\ell_0,r_{\ell_0}}^{(1)}
\Biggr(\prod\limits_{s=2}^{\ell}\prod\limits_{\substack{1\leq k\leq \fa_s\\k\neq n}}(L_k-u_s)\Biggr) d(\t_{\downarrow\leq (n-1)})\in H_{\ell,n-1},\end{aligned}$$ and $$\begin{aligned}
\rp^{(3)}_{\s,\t,w} =
\begin{cases}\begin{matrix}
d(\s_{\downarrow\leq (n-1)})^* \beta_{b_{\ell_0,r_{\ell_0}}-1,n-2}
 \bigg(\sum\limits_{i=\ell_0+1}^{\ell}\bigg( \prod\limits_{\ell_0+1\leq s<i}(L_{n-1}-u_s)\\
\times \prod\limits_{i< t\leq \ell}(L_{n}-u_t)\bigg)\bigg)
\gamma_{n-2,n(\s)}\bigg(\prod\limits_{i=1}^{\ell_0-1} w_{i}\bigg)\bigg(\prod\limits_{j=1}^{r_{\ell_0}-1}w_{\ell_0,j}\bigg) w_{\ell_0,r_{\ell_0}}^{(1)}\\
\times \bigg(\prod\limits_{s=2}^{\ell}\prod\limits_{\substack{1\leq k\leq \fa_s\\k\neq n}}(L_k-u_s)\bigg)d(\t_{\downarrow\leq (n-1)}),\end{matrix} & \mbox{if\ \ }\ell_0<\ell,\\
 0, & \mbox{if\ \ } \ell_0=\ell.
\end{cases}
\end{aligned}$$
Then still using (\ref{easyobservationdeg}) we see that in this case, \begin{equation}\label{Case2Vendeg2}
\e_n\bigr(d(\s)^*w u_\blam^+ d(\t)\bigr)=\rp_{\ell-1}^{(n)}(\rp^{(3)}_{\s,\t,w})=0.
\end{equation}
Therefore, combining (\ref{Case2Vendeg1}) and (\ref{Case2Vendeg2}), we get
$\e_n(m_{\s\t}) =0.$

Let $\mathbf{u}=(u_1,u_2,\dots,u_\ell)\in R^{\ell}$. The degenerate cyclotomic Hecke algebra $H_{\ell,n}=H_{\ell,n}^R(\mathbf{u})$ of type $G(\ell,1,n)$ is the unital associative $R$-algebra with generators $s_{1} ,\dots,s_{n-1},L_1, \dots,L_n$ which satisfy certain relations, see Definition \ref{degenerate}. In \cite{AMR}, Ariki, Mathas and Rui constructed a cellular basis $\{m_{\mathfrak{s}\mathfrak{t}}\,|\, \mathfrak{s},\mathfrak{t}\in\Std(\blam),\ \blam\in\P_{\ell,n}\}$ for $H_{\ell,n}$. This basis generalizes the well-known Murphy basis for the symmetric group algebra $R[\Sym_n]$ (i.e., the level one case). We call it {\bf the Murphy basis for the degenerate cyclotomic Hecke algebra $H_{\ell,n}$}. Brundan-Kleshchev \cite{BK08} introduced a symmetrizing form $\tau_R^{\te{BK}}$ on $H_{\ell,n}$ which makes it into a symmetric algebra over $R$. We refer the readers to Section 2.2 for their precise definitions. The following theorem is the second main result of this paper.

\begin{thm}\label{mainthm2} Let $\blam\in\P_{\ell,n}$ and $\mathfrak{s},\mathfrak{t}\in\te{Std}(\blam).$ Then
$$\tau_R^{\te{BK}}(m_{\mathfrak{s}\mathfrak{t}})=
\begin{cases}
1, & \mbox{if\ \,}\mathfrak{s}=\mathfrak{t}\te{\ \,and\ \,} l(\mathfrak{t},i)=1,\ \forall\,1\leq i\leq n, \\
0, & \mbox{\text{otherwise}}.
\end{cases}$$
\end{thm}

The main idea of the proof of Theorems \ref{mainthm1} and \ref{mainthm2} comes from \cite{HLL},
where we identify the standard symmetrizing form on $\HH_{\ell,n}$ (resp., on $H_{\ell,n}$) with the composition of a series of descending map $\ve_k: \HH_{\ell,k}\rightarrow\HH_{\ell,k-1}$  (resp., $\e_k: H_{\ell,k}\rightarrow H_{\ell,k-1}$),
where each descending map $\ve_k$ (resp., $\e_k$) is given by the cyclotomic Mackey decomposition of $\HH_{\ell,n}$ (resp., of $H_{\ell,n}$).
Instead of computing the value of the standard symmetrizing form via its original definition, we shall show that
each $\ve_n(\fm_{\mathfrak{s}\mathfrak{t}})$ (resp., $\e_n(m_{\mathfrak{s}\mathfrak{t}})$) is zero or a scalar multiple of
$\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}$ (resp., of $m_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}$),
and we shall precisely compute this scalar.

The content of the paper is organised as follows.
In Section 2, we recall some preliminary results on the non-degenerate cyclotomic Hecke algebra and the degenerate cyclotomic Hecke algebra of type $G(\ell,1,n)$.
In Section 3, we use the cyclotomic Mackey decomposition of the non-degenerate cyclotomic Hecke algebra of type $G(\ell,1,n)$
to show in Proposition \ref{MainProp1}  that each $\ve_n(\fm_{\mathfrak{s}\mathfrak{t}})$ is zero or a scalar multiple of $\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}$, and thus get a proof of the first main result Theorem \ref{mainthm1} of this paper.
Then we apply Proposition \ref{MainProp1} and Theorem \ref{mainthm1}
to derive some equations in Corollaries \ref{maincor11}, \ref{maincor12} and \ref{maincor13}  on the coefficients
which occur in the transition matrix between the Murphy basis elements and the seminormal basis elements. In Section 4, we generalize all the results obtained in Section 3 to the degenerate cyclotomic Hecke algebra of type $G(\ell,1,n)$.

\bigskip\bigskip
\centerline{Acknowledgements}
\bigskip

The research was support by the National Natural Science Foundation of China (No. 12431002).
\bigskip

\bigskip

The post-theorem introduction says that “The main idea of the proof of Theorems \ref{mainthm1} and \ref{mainthm2} comes from \cite{HLL},” where they “identify the standard symmetrizing form on $\HH_{\ell,n}$ … with the composition of a series of descending map $\ve_k: \HH_{\ell,k}\rightarrow\HH_{\ell,k-1}$ … where each descending map $\ve_k$ … is given by the cyclotomic Mackey decomposition.”

“For Theorem \ref{mainthm1}” they avoid the original definition of the symmetrizing form and instead “show that each $\ve_n(\fm_{\mathfrak{s}\mathfrak{t}})$ is zero or a scalar multiple of $\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}$, and … precisely compute this scalar.” Concretely, in Section 3 they “use the cyclotomic Mackey decomposition … to show in Proposition \ref{MainProp1} that each $\ve_n(\fm_{\mathfrak{s}\mathfrak{t}})$ is zero or a scalar multiple of $\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}$, and thus get a proof of … Theorem \ref{mainthm1}.”

The post-theorem introduction says that “The main idea of the proof of Theorems mainthm1 and \begin{thm}\label{mainthm2} Let $\blam\in\P_{\ell,n}$ and $\s,\t\in\te{Std}(\blam).$ Then
$$\tau_R^{\te{BK}}(m_{\s\t})=
\begin{cases}
1, & \mbox{if\ \,}\s=\t\te{\ \,and\ \,} l(\t,i)=1,\ \forall\,1\leq i\leq n, \\
0, & \mbox{\text{otherwise}}.
\end{cases}$$
\end{thm} comes from \cite{HLL},” where they “identify the standard symmetrizing form on $\HH_{\ell,n}$ … with the composition of a series of descending map $\ve_k: \HH_{\ell,k}\rightarrow\HH_{\ell,k-1}$ … where each descending map $\ve_k$ … is given by the cyclotomic Mackey decomposition.”

“In establishing the main theorem,” they avoid the original definition of the symmetrizing form and instead “show that each $\ve_n(\fm_{\mathfrak{s}\mathfrak{t}})$ is zero or a scalar multiple of $\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}}$, and … precisely compute this scalar.” Concretely, they “use the cyclotomic Mackey decomposition … to show the following proposition.

\begin{prop}\label{MainProp1}
Let $\blam\in\mathscr{P}_{\ell,n}$ and $\s,\t\in\te{Std}(\blam).$ Then
$$\ve_n(\fm_{\s\t})=\begin{cases}
q^{n-n(\t)}\bigg(\prod\limits_{i=l(\t,n)+1}^{\ell}(-Q_i)\bigg)\fm_{\s_{\downarrow\leq (n-1)}\t_{\downarrow\leq (n-1)}},
& \text{if\ \ $n(\s)=n(\t),$}\\
0, & \text{if\ \ $n(\s)\neq n(\t)$.}
\end{cases} $$
\end{prop}

They then deduce from this proposition a proof of the main theorem.”

\label{mainthm1} Let $\blam\in\P_{\ell,n}$ and $\mathfrak{s},\mathfrak{t}\in\te{Std}(\blam).$ Then
$$\tau_R^{\te{MM}}(\fm_{\mathfrak{s}\mathfrak{t}})=
\begin{cases}
q^{\sum\limits_{k=1}^n(k-k(\t_{\downarrow\leq k}))}\bigg(\prod\limits_{j=1}^n\prod\limits_{i=l(\mathfrak{t},j)+1}^{\ell}(-Q_i)\bigg), & \mbox{if\ \ }\mathfrak{s}=\mathfrak{t}, \\
0, & \mbox{if\ \ }\mathfrak{s}\neq \mathfrak{t}.
\end{cases}$$,

Question: Let $R$ be an integral domain, let $1\neq q\in R^\times$, let $\mathbf Q=(Q_1,\dots,Q_\ell)\in R^\ell$, and let $\mathscr H_{\ell,n}=\mathscr H_{\ell,n}^R(q;\mathbf Q)$ be the non-degenerate cyclotomic Hecke algebra of type $G(\ell,1,n)$. Let $\tau_R^{\mathrm{MM}}$ be the $R$-linear map on $\mathscr H_{\ell,n}$ determined by $\tau_R^{\mathrm{MM}}(\mathcal L_1^{c_1}\cdots \mathcal L_n^{c_n}T_w)=1$ if $w=1$ and $c_1=\cdots=c_n=0$, and $0$ otherwise, extended linearly. For each $\lambda\in\mathscr P_{\ell,n}$, let $\mathrm{Std}(\lambda)$ be the set of standard $\lambda$-tableaux, and let $\{\fm_{\mathfrak u\mathfrak v}\mid \mathfrak u,\mathfrak v\in\mathrm{Std}(\lambda),\ \lambda\in\mathscr P_{\ell,n}\}$ be the Murphy basis of $\mathscr H_{\ell,n}$. For a standard tableau $\mathfrak t$, write $\mathfrak t^{-1}(j)=(r(\mathfrak t,j),c(\mathfrak t,j),l(\mathfrak t,j))$, so $l(\mathfrak t,j)$ is the component index containing $j$. For $1\le k\le n$, let $\mathfrak t_{\downarrow\le k}$ be the restriction of $\mathfrak t$ to the entries $1,\dots,k$, and let $k(\mathfrak t_{\downarrow\le k})$ be the unique integer such that the box containing $k$ in $\mathfrak t_{\downarrow\le k}$ is the same box that contains $k(\mathfrak t_{\downarrow\le k})$ in the initial standard tableau of the same shape. With the usual convention that an empty product equals $1$, which statement holds for every $\lambda\in\mathscr P_{\ell,n}$ and every $\mathfrak s,\mathfrak t\in\mathrm{Std}(\lambda)$?

$\tau_R^{\mathrm{MM}}(\fm_{\mathfrak s\mathfrak t})=\begin{cases}q^{\sum_{k=1}^n\bigl(k-k(\mathfrak t_{\downarrow\le k})\bigr)}\left(\prod_{j=1}^n\prod_{i=l(\mathfrak t,j)+1}^{\ell}(-Q_i)\right),&\text{if }\mathfrak s=\mathfrak t,\\0,&\text{if }\mathfrak s\ne\mathfrak t.\end{cases}$

Choices:
B. $\tau_R^{\mathrm{MM}}(\fm_{\mathfrak s\mathfrak t})=\begin{cases}q^{\sum_{k=1}^n\bigl(k-k(\mathfrak t_{\downarrow\le k})\bigr)}\left(\prod_{j=1}^n\prod_{i=l(\mathfrak t,j)}^{\ell}(-Q_i)\right),&\text{if }\mathfrak s=\mathfrak t,\\0,&\text{if }\mathfrak s\ne\mathfrak t.\end{cases}$
C. $\tau_R^{\mathrm{MM}}(\fm_{\mathfrak s\mathfrak t})=0\quad\text{if }\mathfrak s\ne\mathfrak t.$
D. $\tau_R^{\mathrm{MM}}(\fm_{\mathfrak s\mathfrak t})=\begin{cases}q^{\sum_{k=1}^n\bigl(k-k(\mathfrak t_{\downarrow\le k})\bigr)}\left(\prod_{j=1}^n\prod_{i=l(\mathfrak t,j)+1}^{\ell}(-Q_i)\right),&\text{if }n(\mathfrak s)=n(\mathfrak t),\\0,&\text{if }n(\mathfrak s)\ne n(\mathfrak t).\end{cases}$
E. $\tau_R^{\mathrm{MM}}(\fm_{\mathfrak s\mathfrak t})=\begin{cases}q^{\sum_{k=1}^n\bigl(k-k(\mathfrak s_{\downarrow\le k})\bigr)}\left(\prod_{j=1}^n\prod_{i=l(\mathfrak s,j)+1}^{\ell}(-Q_i)\right),&\text{if }\mathfrak s=\mathfrak t,\\0,&\text{if }\mathfrak s\ne\mathfrak t.\end{cases}$

component-product lower bound i=l(\mathfrak t,j)+1

dropped explicit diagonal scalar formula

equality criterion \mathfrak s=\mathfrak t replaced by n(\mathfrak s)=n(\mathfrak t)

dependence on terminal tableau switched from \mathfrak t to \mathfrak s in the scalar

The stem introduces notation and definitions but does not state or strongly hint at the full explicit formula. The correct answer is not leaked directly from the wording.

The item is essentially asking for the exact explicit formula of a stated result. This is a direct theorem-recall/restatement question rather than a problem requiring a new conclusion from premises.

Some reasoning is needed to distinguish the exact formula from close variants, especially the weaker true statement and the subtle indexing changes. However, the task mainly tests precise recall/recognition rather than genuinely generative mathematical reasoning.

The distractors are plausible and mathematically targeted: one changes an index bound, one gives only a weaker true statement, one alters the equality condition, and one swaps dependence from t to s. These reflect realistic failure modes.

A technically well-constructed recognition item with strong distractors and no answer leakage, but it is mostly theorem recall and thus weak on non-tautological, generative reasoning.


\label{thm:charac-Markov}
  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.
  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities.
  \begin{enumerate}
    \item $X_0 = \G_m^2$.
    \item $X_0$ is a cubic affine surface of Markov type.
  \end{enumerate}
  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.

\label{sec:intro}

Let $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \emph{completion} of $X_0$ is a
smooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the
\emph{dynamical degree} $\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,
then
\begin{equation}
 \lambda (f) = \lim_n \left( (f^n)^* H \cdot H \right)^{1/n}.
 \label{}
\end{equation}
The limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\lambda
(f) \geq 1$.
we say that an automorphism of $X_0$ is \emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's
work in \cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\mathbf{C}$ the topological entropy of $f$ is equal to $\log
\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\Aut (X_0)$ is
\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.

In \cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal
affine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal
affine surface with a loxodromic automorphism, then either $X_0$ is
completable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent
conditions. We denote by $\overline \kappa (X_0)$ the log Kodaira dimension of $X_0$.

\begin{prop}\label{prop:charac-cycle}
 Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either
 $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and
 the following are equivalent.
 \begin{enumerate}
 \item $X_0$ is completable by a cycle of rational curves.
 \item $\overline \kappa (X_0) = 0$.
 \item For every loxodromic automorphism $f$ of $X_0$, $\lambda (f)$ is a quadratic integer.
 \end{enumerate}
 Or
 \begin{enumerate}
 \item $X_0$ is completable by a tree of rational curves.
 \item $\overline \kappa (X_0) = - \infty$.
 \item For every loxodromic automorphism $f$ of $X_0$, $\lambda (f)$ is an integer.
 \end{enumerate}
\end{prop}
The case of a tree of rational curve is quite rich, the affine plane $\mathbf{A}^2$ is the main example of such an affine surface but there are
many other examples of non isomorphic affine surfaces. In \cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed
that there are affine surfaces completable by a zigzag with a huge automorphism group. In
\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational
complex affine surface with uncountably many non-isomorphic real forms.

For the cycle case, the main example is the algebraic torus $\G_m^2$ and cubic affine surfaces of Markov type, i.e the
complement of a triangle of lines $\Delta$ in a smooth cubic projective surface $S$ in $\mathbf{P}^3$, (see
\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in
$S$, it defines an involution $\sigma_p$ as follows: if $p \in S \setminus \Delta$, the line between $q$ and $p$
intersects $S \setminus \Delta$ in one other point which is $\sigma_q (p)$. Up to finite index the automorphism group of
such an affine surface is generated
by the three involutions $\sigma_{q_1}, \sigma_{q_2}, \sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection
points in $\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different
areas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or
the once punctured torus, see \cite{cantatHolomorphicDynamicsPainleve2007}.
The purpose of this paper is to prove the following theorem which states that these are the only smooth examples.

Let $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \emph{completion} of $X_0$ is a
smooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the
\emph{dynamical degree} $\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,
then
\begin{equation}
 \lambda (f) = \lim_n \left( (f^n)^* H \cdot H \right)^{1/n}.
 \label{}
\end{equation}
The limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\lambda
(f) \geq 1$.
we say that an automorphism of $X_0$ is \emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's
work in \cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\C$ the topological entropy of $f$ is equal to $\log
\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\Aut (X_0)$ is
\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.

\begin{rmq}\label{rmq:}
 If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$
 is a curve of degree 3 in $\P^2$ with a nodal singularity, then $X_0 := \P^2 \setminus C$ is a smooth affine surface
 completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of
 $X_0$ such that $\Delta_X = C_1 \cup C_2$ which are two smooth rational curves that meet at two points. Here
 $\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is
 elementary.
\end{rmq}

Let $X_0$ be a normal affine surface, we write $K[X_0]$ for its ring of regular functions. A $K$-valuation over
$K[X_0]$ is a map $v : K[X_0] \rightarrow \R \cup \left\{ \infty \right\}$ such that
\begin{enumerate}
 \item $v_{|K^\times} = 0$.
 \item $v(0) = \infty$.
 \item $\forall P,Q \in K[X_0], \quad v(PQ) = v(P) + v (Q)$.
 \item $\forall P,Q \in K[X_0], \quad v(P+Q) \geq \min \left( v(P), v(Q) \right)$.
\end{enumerate}
We give two examples that will be useful for this note. First we have \emph{divisorial} valuations. Let $X$ be a
completion of $X_0$ and $E$ a prime divisor at infinity, then $\ord_E$ the order of vanishing along $E$ is a valuations
over $K[X_0]$. The second example is as follows. Let $E,F$ be two divisors at infinity in $X$ intersecting at a point
$p$, then we can find local coordinates $(x,y)$ at $p$ such that $x = 0 $ is a local equation of $E$ and $y = 0$ is a
local equation of $F$. For $\alpha, \beta > 0$ we define $v_{\alpha,\beta}$ as follows. The completion of the local ring
at $p$ with respect to its maximal ideal is $K \left[ [ x,y ] \right]$ the ring of formal power series in $x,y$. Define
\begin{equation}
 v_{\alpha,\beta} \left( \sum_{i,j} a_{ij} x^i y^j \right) = \min \left( \alpha i + \beta j : a_{ij} \neq 0 \right).
 \label{<+label+>}
\end{equation}
This defines a valuation over $K[X_0]$ because every $P \in K[X_0]$ is in the fraction field of $\OO_{X,p}$ the local
ring at $p$ and this fraction field embeds into $K [ [ x ,y ] ]$. We have two possibilities, if $\alpha / \beta \in \Q$,
then $v_{\alpha,\beta} = \ord_G$ is also divisorial but we need to blow up $p$ and infinitely near points to make $G$
appear. Otherwise $v_{\alpha,\beta}$ is \emph{irrational}.

\section{Proof of Theorem \ref{thm:charac-Markov}}\label{sec:}
Let $X_0$ be a smooth affine surface completable by a cycle of rational curves and assume that $\Aut (X_0)$ is not elementary.
\begin{prop}\label{prop:vanishing-self-intersection}
 For any $[v] \in \cC_\infty$, we have $Z_v^2 = 0$.
\end{prop}
\begin{proof}
 Let $f,g $ be two loxodromic automorphisms generating a non-elementary subgroup. Write $v_f^\pm, v_g^\pm$ for the
 eigenvaluations of $f$ and $g$ respectively. Recall that they are all different by Proposition
 \ref{prop:different-eigenvaluations}. From Proposition \ref{prop:eigenvaluations-divisor}, we have that
 \begin{equation}
 Z_{v^\pm_f}^2 = Z_{v^\pm_g}^2 = 0.
 \label{<+label+>}
 \end{equation}
 Let $Y$ be a cyclic completion of $X_0$ such that the centers of these four eigenvaluations are distinct. Recall that
 they are irrational valuations so their centers is always a satellite point. Let
 $p = E \cap F = c_Y (v_f^+)$. We look at the monomial valuations centered at $p$. The function
 \begin{equation}
 \phi: s \in [0, +\infty) \mapsto Z_{v_{1,s}}^2
 \label{<+label+>}
 \end{equation}
 is a polynomial map of $s$ of degree $\leq 2$ by Lemma \ref{lemme:self-intersection-irrational}. We have that $v_f^+ = v_{1, s_0}$ for some $s_0 > 0$. Now, we know
 that $f^n_* [v_g^+] \rightarrow [v_f^+]$ by Theorem \ref{thm:dynamics-loxodromic-automorphisms}. This implies that after a finite number of steps we have up to
 renormalisation
 \begin{equation}
 f^n_* v_g^+ = v_{1, s_n}
 \label{<+label+>}
 \end{equation}
for some $s_n > 0$ with $s_n \neq s_0$ and $s_n \rightarrow s_0$. But since $Z_{v_g^+}^2 = 0$ and this is invariant by the action of $f$
we have that $Z_{v_{1,s_n}}^2 = 0$ so the function $\phi$ is zero. Since for any valuation $[v] \in \cC_\infty \setminus
\left\{ [v_f^-] \right\}, f^n_* [v] \rightarrow [v_f^+]$ we have that $Z_v^2 = 0$.
\end{proof}
This implies by Proposition \ref{prop:nef-valuation} that for every $v \in \cC_\infty, Z_v$ is nef.
\begin{cor}\label{cor:intersection-positive}
 If $E,F$ are different prime divisors at infinity of a cyclic completion $X$ of $X_0$, then
 \begin{equation}
 \hat E \cdot \hat F > 0.
 \label{<+label+>}
 \end{equation}
\end{cor}
\begin{proof}
 We have by Proposition \ref{prop:vanishing-self-intersection} that $(\hat E)^2 = (\hat F)^2 = 0$ so that they are nef
 and effective divisors by Proposition \ref{prop:nef-valuation}. This implies that $\hat E \cdot \hat F \geq 0$.
 If the intersection number is zero, then by the Hodge index theorem we must have that $\hat E = \hat F$ but this is a
 contradiction.
\end{proof}

\begin{prop}\label{prop:trivial-k-plus-delta}
 Let $X_0$ be a smooth affine surface completable by a cycle of rational curves with a non-elementary automorphism
 group, then the class $K+\Delta \in \cNS_{cyc} (X_0)$ is equal to $0$.
\end{prop}
\begin{proof}
Recall by Proposition \ref{prop:Cartier-canonical-class} that $(K+ \Delta) \in \Cartier_{cyc}(X_0)_\R$ and it is defined
 by $K_X + \Delta_X$ for any cyclic completion $X$ of $X_0$. Furthermore, it is fixed by $\Aut (X_0)$. We show that
 \begin{equation}
 \forall v \in \cC_\infty, \quad Z_v \cdot (K+\Delta) = 0.
 \end{equation}
 Let $f,g \in \Aut (X_0)$ be two loxodromic automorphisms not sharing a common iterate with their eigenvaluations
 $v_h^\pm$ for $h = f,g$. We have that
 \begin{equation}
 Z_{v_h^\pm} \cdot (K+\Delta) = (h^{\pm 1})^* \left( Z_{v_h^\pm} \cdot (K+\Delta) \right) = \lambda \left(h^{\pm 1}\right) Z_{v_h^\pm}
 \cdot (K+\Delta).
 \label{<+label+>}
 \end{equation}
 This implies that $Z_{v_h^\pm} \cdot (K+\Delta) = 0$. Now, let $X$ be a cyclic completion of $X_0$ and let $p = E \cap F = c_X (v_f^+)$. The function
 \begin{equation}
 \phi: s > 0 \mapsto Z_{v_{1,s}} \cdot (K+\Delta)
 \label{<+label+>}
 \end{equation}
 is a polynomial of degree at most 1 because we have
 \begin{equation}
 Z_{v_{1,s}} \cdot (K+\Delta) = (Z_{v_{1,s}})_X \cdot (K+\Delta)_X = (\hat E + s \hat F) \cdot (K_X + \Delta_X).
 \label{<+label+>}
 \end{equation}
 Now we have that $v_f^+ = v_{1,s_0}$ for some $s_0 > 0$. As in the proof of Proposition
 \ref{prop:vanishing-self-intersection}, for $n$ large enough, $f^n_* v_g^+ = v_{1,s_n}$ and we have $\phi(s_n) = 0$.
 So that the function $\phi$ is 0. Now, for any $[v] \in \cC_\infty \setminus \left\{ [v_{-,f}] \right\}$ we have that
 $f^n_* [v] \rightarrow [v_{+, f}]$ so it will belong to $[v_E, v_F[$ after finitely many iterations. So that we get
 $Z_v \cdot (K+\Delta) = (f^n)_* (Z_v \cdot (K+\Delta)) = Z_{f^n_* v} \cdot (K+\Delta) = 0 $.

\begin{rmq}\label{rmq:}
  If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$
  is a curve of degree 3 in $\mathbf{P}^2$ with a nodal singularity, then $X_0 := \mathbf{P}^2 \setminus C$ is a smooth affine surface
  completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of
  $X_0$ such that $\Delta_X = C_1 \cup C_2$ which are two smooth rational curves that meet at two points. Here
  $\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is
  elementary.
\end{rmq}

\subsection{Idea of the proof}\label{subsec:}
The proof goes as follows. We use the valuative techniques from \cite{abboudDynamicsEndomorphismsAffine2023}. In the
case of an affine surface completable by a cycle of rational curves, the space of valuations centered at infinity of
$X_0$ contains an $\Aut(X_0)$-equivariant set which is homeomorphic to a circle. This circle is homeomorphic to the
completion of the inverse limits of the dual graphs of cyclic completions. If $\Aut(X_0)$ is non-elementary, then its
action on this circle has very large orbits. This imposes strong constraints on the intersection form on the space of
divisors supported at infinity. We then conclude that $X_0$ has to be the complement of a triangle of lines in a smooth
cubic surface unless $X_0 = \G_m^2$.

\subsection{Study of endomorphisms of affine surfaces of Markov type}\label{subsec:endomorphisms-intro}
We fully classify the dynamics of cubic affine surfaces of Markov type in characteristic zero by showing that they do not have
non-invertible dominant endomorphisms.

\begin{bigthm}\label{thm:endomorphisms-affine-surface}
  Let $K$ be a field of characteristic zero. If $X_0$ is a smooth cubic affine surface of Markov type over $K$ and $f$ is a
  dominant endomorphism of $X_0$, then $f$ is an automorphism.
\end{bigthm}
To show this result we can assume that $K = \mathbf{C}$, then using valuative techniques and the geometry of the space of
valuations centered at infinity we show that $f$ must be proper and unramified. Thus, it is a covering of the complex
manifold $X_0 (\mathbf{C})$ which is simply connected (see eg \cite[Lemma 3.10]{cantatHolomorphicDynamicsPainleve2007}). Hence
$f$ must be a homeomorphism and therefore an automorphism.

Notice that this is not true for singular cubic affine surfaces. Indeed, let $\sC \subset \mathbf{A}^3$ be the Cayley cubic defined by
\begin{equation}
  x^2 + y^2 + z^2 = xyz + 4.
  \label{<+label+>}
\end{equation}
It is the quotient of $\G_m^2$ by the involution $(u,v) \mapsto (u^{-1}, v^{-1})$. The Cayley cubic has four singular
points which are orbifold singularities of order 2. It has many endomorphisms, namely every monomial endomorphism of
$\G_m^2$ descends to an endomorphism of $\sC$. The proof fails for this surface because it is not simply connected as an
orbifold. Indeed, its orbifold fundamental group is not trivial.

\subsection*{Acknowledgements} I thank Serge Cantat and Matteo Ruggiero for related discussions on this problem.

The proof of Theorem~\ref{thm:charac-Markov} is described in the subsection \emph{Idea of the proof}. One "use[s] the valuative techniques from \cite{abboudDynamicsEndomorphismsAffine2023}" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which "contains an $\Aut(X_0)$-equivariant set which is homeomorphic to a circle." This circle is identified as "homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions." If $\Aut(X_0)$ is non-elementary, then "its action on this circle has very large orbits," which "imposes strong constraints on the intersection form on the space of divisors supported at infinity." From these constraints one concludes that "$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \G_m^2$."

The proof of the main theorem is described in the subsection \emph{Idea of the proof}. One "use[s] the valuative techniques from \cite{abboudDynamicsEndomorphismsAffine2023}" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which "contains an $\Aut(X_0)$-equivariant set which is homeomorphic to a circle." This circle is identified as "homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions." If $\Aut(X_0)$ is non-elementary, then "its action on this circle has very large orbits," which "imposes strong constraints on the intersection form on the space of divisors supported at infinity." From these constraints one concludes that "$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \G_m^2$."

Question:
Let $K$ be an algebraically closed field, and let $X_0$ be a smooth affine surface over $K$. Assume that $\operatorname{Aut}(X_0)$ is non-elementary, meaning that it contains two loxodromic automorphisms with no common iterates, where an automorphism $f$ is called loxodromic if its dynamical degree satisfies $\lambda(f)>1$, with
\[
\lambda(f)=\lim_{n\to\infty}\bigl(((f^n)^*H)\cdot H\bigr)^{1/n}
\]
for an ample divisor $H$ on any smooth projective completion of $X_0$. Also assume that $X_0$ is completable by a cycle of rational curves, i.e. there exists a smooth projective surface $X$ containing $X_0$ as a Zariski dense open subset such that the boundary $X\setminus X_0$ is a cycle of rational curves. A cubic affine surface of Markov type means the complement $S\setminus \Delta$, where $S\subset \mathbf P^3$ is a smooth cubic surface and $\Delta$ is a triangle of lines on $S$. Under these hypotheses, which conclusion about $X_0$ holds?

There are exactly two mutually exclusive possibilities: either $X_0\cong \mathbf G_m^2$, or $X_0$ is a cubic affine surface of Markov type (that is, $X_0\cong S\setminus \Delta$ for a smooth cubic surface $S\subset \mathbf P^3$ and a triangle of lines $\Delta\subset S$). The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions.

Choices:
B. There are exactly two mutually exclusive possibilities: either $X_0\cong \mathbf G_m^2$, or $X_0$ is the complement of a cycle of rational curves in some smooth del Pezzo surface. In particular, every smooth affine surface with non-elementary automorphism group that is completable by a cycle of rational curves arises from a del Pezzo surface, and the distinction between the two cases is determined by whether $X_0$ admits non-constant invertible regular functions.
C. Under these hypotheses, $X_0$ is either isomorphic to $\mathbf G_m^2$ or is a cubic affine surface of Markov type.
D. There are exactly two mutually exclusive possibilities: either $X_0\cong \mathbf G_m^2$, or $X_0$ is the complement $S\setminus \Delta$ of a triangle of lines $\Delta$ in a cubic surface $S\subset \mathbf P^3$; moreover, the cubic surface $S$ need not be smooth. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions.
E. There are exactly two mutually exclusive possibilities: either, after replacing $X_0$ by a finite étale cover, one has $X_0\cong \mathbf G_m^2$, or, after such a cover, $X_0$ becomes a cubic affine surface of Markov type. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions.

precise geometric classification as smooth cubic with triangle of lines

drops mutual exclusivity and the criterion via invertible regular functions

classification of X_0 itself versus only after finite étale cover

The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the classification, without giving away the cubic/torus dichotomy or the criterion via invertible functions.

This is essentially a direct theorem-recall item: the hypotheses in the stem closely match the classification theorem, and the correct choice is basically the theorem statement itself.

Some reasoning is needed to distinguish the exact theorem from nearby variants (weaker true statement, weakened hypothesis, overgeneralization), but the main task is recognition/recollection rather than genuine derivation.

The distractors are plausible and mathematically meaningful. They reflect common failure modes: overgeneralizing to arbitrary del Pezzo surfaces, selecting a weaker true dichotomy, weakening the hypothesis from non-elementary to one loxodromic automorphism, or confusing the geometric model.

A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is quite tautological and tests recall more than generative mathematical reasoning.


Question:
Let $K$ be an algebraically closed field, and let $X_0$ be a smooth affine surface over $K$. Assume that $\operatorname{Aut}(X_0)$ is non-elementary, meaning that it contains two loxodromic automorphisms with no common iterates, where an automorphism $f$ is called loxodromic if its dynamical degree satisfies $\lambda(f)>1$, with
\[
\lambda(f)=\lim_{n\to\infty}\bigl(((f^n)^*H)\cdot H\bigr)^{1/n}
\]
for an ample divisor $H$ on any smooth projective completion of $X_0$. Also assume that $X_0$ is completable by a cycle of rational curves, i.e. there exists a smooth projective surface $X$ containing $X_0$ as a Zariski dense open subset such that the boundary $X\setminus X_0$ is a cycle of rational curves. A cubic affine surface of Markov type means the complement $S\setminus \Delta$, where $S\subset \mathbf P^3$ is a smooth cubic surface and $\Delta$ is a triangle of lines on $S$. Under these hypotheses, which conclusion about $X_0$ holds?

Choices:
B. There are exactly two mutually exclusive possibilities: either $X_0\cong \mathbf G_m^2$, or $X_0$ is the complement of a cycle of rational curves in some smooth del Pezzo surface. In particular, every smooth affine surface with non-elementary automorphism group that is completable by a cycle of rational curves arises from a del Pezzo surface, and the distinction between the two cases is determined by whether $X_0$ admits non-constant invertible regular functions.
C. Under these hypotheses, $X_0$ is either isomorphic to $\mathbf G_m^2$ or is a cubic affine surface of Markov type.
D. There are exactly two mutually exclusive possibilities: either $X_0\cong \mathbf G_m^2$, or $X_0$ is the complement $S\setminus \Delta$ of a triangle of lines $\Delta$ in a cubic surface $S\subset \mathbf P^3$; moreover, the cubic surface $S$ need not be smooth. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions.
E. There are exactly two mutually exclusive possibilities: either, after replacing $X_0$ by a finite étale cover, one has $X_0\cong \mathbf G_m^2$, or, after such a cover, $X_0$ becomes a cubic affine surface of Markov type. The distinction between these two cases is determined by whether $X_0$ admits non-constant invertible regular functions.

The stem gives hypotheses and definitions but does not state or strongly hint at the exact classification conclusion. The correct bifurcation and the criterion via invertible regular functions are not leaked.

This is largely a theorem-recall/classification question: the stem lists the assumptions and asks for the conclusion. However, it is not a pure tautology because the options include weaker, overgeneralized, and subtly altered variants that must be distinguished.

Some reasoning is needed to identify the strongest correct conclusion and reject nearby variants such as a weaker true statement, a smoothness error, and a finite-étale-cover modification. Still, the item mainly tests precise recall of the theorem rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: overgeneralization to del Pezzo surfaces, dropping smoothness, weakening the conclusion, and changing the quantifier to 'after finite étale cover.' They are distinct and nontrivial.

A solid theorem-classification MCQ with strong distractors and little answer leakage, but it mainly tests accurate recall of a specific result rather than deep generative reasoning.