\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


\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


Question: Let $X_0$ be a smooth affine surface over an algebraically closed field $K$. Assume that $\operatorname{Aut}(X_0)$ is non-elementary, meaning it contains two loxodromic automorphisms with no common iterate, where loxodromic means dynamical degree $>1$. Also assume that $X_0$ admits a smooth projective completion whose boundary divisor is a cycle of rational curves. Which description gives the complete classification of such surfaces?

Exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.

Choices:
B. Exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is the complement of a cycle of rational curves in an arbitrary smooth del Pezzo surface; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.
C. One always has the following dichotomy: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type.
D. If $X_0$ is completable by a cycle of rational curves and $\operatorname{Aut}(X_0)$ contains a loxodromic automorphism, then exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.
E. Exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is the complement of a triangle of lines in a cubic projective surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.

specific ambient cubic-surface conclusion

criterion via non-constant invertible regular functions

non-elementary hypothesis weakened to existence of one loxodromic automorphism


\label{thm: A}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\pi_2(V)} = 0$ and $\pi_1(\partial W) = 0$. There exists a $\Z$-graded algebra homomorphism
$$
\Phi: \SH^*(V; \Lda_V) \rightarrow \SH^*(W; \Lda_W)
$$
under a canonical identification $\Lda_V = \Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.
\[\begin{tikzcd}
        \SH^*(V) \arrow{r}{\Phi}  & \SH^*(W)  \\
        \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
    \end{tikzcd}
\]
where the vertical maps are the canonical $c^*$-map.

\begin{example}\label{ex: neglinebdl2}
Following Example \ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\pi: D(E) \rightarrow B$ now serves as a complement-exact subdomain. Assume that
\begin{itemize}
\item $[\ow] = c_1(B)$ and $[\ow]$ is primitive in $\Ho_2(B; \Z);$
\item $B$ is simply-connected.
\end{itemize}
Then $\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \ref{thm: A}, where $S(E)$ is the corresponding circle bundle.

For a more concrete example, one takes the blow-up $B : = \CP^2 \# k \overline{\CP}^2$ of $\CP^2$ at $k$ generic points, with $0 < k < 9$. This admits a symplectic form $\ow_k$ whose Poincar\'e dual is given by $3H - E_1 - \cdots -E_k$ where $H$ is the class of a line in $\CP^2$ and $E_i$ is the exceptional curve derived from $i$-th blow-up. In particular $[\ow_k] = c_1(B)$ is primitive in $H_2(B;\Z)$ and $B$ is simply-connected.
\end{example}

\begin{lemma}\label{lem: actionorbits}
For 1-orbits $x$ in the region \emph{(I)} and \emph{(II)}, we have $\mathcal{F}(x) \leq 0$. In the other regions, we have $\mathcal{F}(x) > 0$.
\end{lemma}

\begin{example}\label{ex: resolution}
    Let $X$ be a $\Q$-factorial variety with a unique singularity at the origin $O \in X$ and assume that the complement $X \setminus \{O\}$ admits an exact K\"ahler form $\ow_X$. Then by \cite[Lemma 3.2]{McRi23}, any resolution $\pi: Y \rightarrow X$ admits a K\"ahler form $\ow_Y$ such that $\pi^* \ow_X = \ow_Y$. In particular, $Y$ is a convex symplectic domain, and a neighborhood of $\pi^{-1}(\{O\}) \subset Y$ serves as a complement-exact subdomain.
\end{example}

\begin{remark}\label{rem: usualactionfctl}
For an admissible Hamiltonian $H$, a standard action functional $\mathcal{A}_H : \widetilde{\mathcal{L}_0 V} \rightarrow \R$ is given by
$$
\mathcal{A}_H(x, v) =  -\int_{D^2} v^*\ow + \int_{S^1} H(x(t)) dt.
$$
The class $(x, v)$ is a critical point of $\mathcal{A}_H$ if $x$ is a $1$-periodic Hamiltonian orbit of $H$. Here, we use the convention $\ow(\cdot, X_H)  = d H$ for the definition of the Hamiltonian vector field $X_H$.
\end{remark}

Transfer maps in Floer theory, introduced by Viterbo \cite{Vit}, have played a central role in fundamental questions of symplectic topology. For example, it serves as a quite nontrivial obstruction to the existence of certain Lagrangian submanifolds \cite{Vit}, and it is also an essential ingredient for constructing symplectic capacities from Floer theory, as e.g. in \cite{FHW, GH18}.

Roughly speaking, for a symplectic manifold $V$ and a codimension zero embedding $W \hookrightarrow V$, transfer maps appear as natural homomorphisms between various flavors of Floer (co)homology of $V$ and $W$.
To our knowledge, it is only constructed when the ambient symplectic manifold $V$ is globally exact in the literature, and
the purpose of this article is to give a construction of transfer maps with favorable functorial properties in symplectic cohomology for possibly non-exact symplectic manifolds. We do this when the complement of a subdomain is exact.

Let $V$ be a convex symplectic domain, that is, a compact symplectic manifold which is exact near the boundary $\partial V$ and the Liouville vector field points outward along $\partial V$. Under some additional assumptions depending on the context,
the symplectic cohomology $\SH^*(V)$ is defined as a $\Z$-graded algebra over the Novikov field $\Lda_{V}$. A standard construction of $\SH^*(V)$ is briefly reviewed in Section \ref{sec: symcohomology}; we refer the reader to \cite{BeRi20} for an intensive study of the symplectic cohomology of convex symplectic domains.

Let $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \setminus W$; we call it a \emph{complement-exact} subdomain. When the boundary $\partial W$ is simply-connected, the Novikov field $\Lda_W$ associated with $W$ can be canonically identified with the one $\Lda_V$ of the ambient domain $V$; see Theorem \ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\SH^*(V)$ to $\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \QH^*(V) \rightarrow \QH^*(W)$ on the quantum cohomology rings.

\begin{theorem}\label{thm: nonnegandW}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\pi_1(\p W) = 0$ and $c_1(V)|_{\pi_2(V)} = 0$. Then
\begin{enumerate}
\item The inclusion $i: W \rightarrow V$ induces an isomorphism between the Novikov fields $\Lda_W$ and $\Lda_V$.
\item Under the above identification $\Lda_V \cong \Lda_W$, we have a canonical vector space isomorphism
$$
\SH^*_{\leq 0}(V; \Lda_V) \cong \SH^*(W;\Lda_W).
$$
\end{enumerate}
\end{theorem}

Let $W$ be a codimension zero subdomain of $V$ such that the symplectic form is exact in the complement $V \setminus W$; we call it a \emph{complement-exact} subdomain. When the boundary $\p W$ is simply-connected, the Novikov field $\Lda_W$ associated with $W$ can be canonically identified with the one $\Lda_V$ of the ambient domain $V$; see Theorem \ref{thm: nonnegandW}. Under this identification, the main result of this article is to establish an algebra homomorphism from $\SH^*(V)$ to $\SH^*(W)$ that has a functorial property with respect to the restriction map $i^*: \QH^*(V) \rightarrow \QH^*(W)$ on the quantum cohomology rings.

The $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \cite[Section 4.1]{BeRi20} and \cite[Section 5]{Rit}.

\begin{example}\label{ex: neglinebdl2}
Following Example \ref{ex: neglinebdl}, the total space $E$ can be seen as a (completed) convex symplectic manifold, and the total space of a disk bundle $\pi: D(E) \rightarrow B$ now serves as a complement-exact subdomain. Assume that
\begin{itemize}
\item $[\ow] = c_1(B)$ and $[\ow]$ is primitive in $\Ho_2(B; \Z);$
\item $B$ is simply-connected.
\end{itemize}
Then $\pi_1(S(E)) = 0$ and $c_1(E) =0$ as in Theorem \ref{thm: A}, where $S(E)$ is the corresponding circle bundle.

\subsubsection{Admissible Hamiltonians} \label{sec: admHam} Let $(V, \ow, \lda)$ be a convex symplectic domain. We assume that the first Chern class $c_1(V)$ vanishes on the second homotopy group $\pi_2(V)$
A Hamiltonian $H: \widehat V \rightarrow \R$ is called \emph{admissible} if
\begin{itemize}
\item there exists $r_0 \in [1, \infty)$ such that $H$ depends only on the radial coordinate $r$ for $r \geq r_0$, say $H(z) = h(r)$, and $h'(r) \geq 0$;
\item $h(r) = \kappa r + b$ for sufficiently large $r$ where the \emph{slope} $\kappa$ is not the period of the Reeb orbit on the contact boundary $(\p V, \alpha)$.
\end{itemize}

\subsubsection{Ring structure} The symplectic cohomology $\SH^*(V)$ admits a ring structure by the pair-of-pants product.
Let $\mathcal{S}$ be the Riemann sphere with two positive punctures and one negative puncture; this means that $\mathcal{S}$ admits a parametrization $[0, \infty) \times S^1$ and $(-\infty, 0] \times S^1$, respectively, near the punctures. Given three admissible Hamiltonians $H_1, H_2, H_3$ on $\widehat V$, we take an $\mathcal{S}$-parametrized Hamiltonian $H_{\mathcal{S}}$ which coincides with $H_1, H_2$ near the positive punctures and with $H_3$ near the negative puncture. We likewise take an $\mathcal{S}$-parameterized admissible almost complex structure $J_{\mathcal{S}}$. We define a product
\[
\HF^k(H_1) \otimes \HF^{\ell}(H_2) \rightarrow \HF^{k+\ell}(H_3), \quad x_1 \otimes x_2 \mapsto x_3
\]
by counting the solutions $u: \mathcal{S} \rightarrow \widehat V$ of the Floer equation
\begin{equation} \label{eq: Floereq}
 (du - X_{H_{\mathcal{S}}} \otimes \beta)^{0, 1} = 0
\end{equation}
which converges to $x_1, x_2$ at the positive punctures and converges to $x_3$ at the negative puncture. Here, $\beta \in \Omega^1(\mathcal{S})$ is a one-form that agrees with $dt$ near the punctures (where $t$ denotes the $S^1$-coordinates of the parameterizations). As is well-known, the product is compatible with the continuation maps and hence induces a product on the symplectic cohomology as
\[
\SH^k(V) \otimes \SH^{\ell}(V) \rightarrow \SH^{k+\ell}(V).
\]
With this product, the symplectic cohomology $\SH^*(V)$ is now a unital $\Z$-graded algebra over the Novikov field $\Lda_V$. For more details on the construction of the ring structure, we refer the reader to \cite[Section 6]{Rit} and \cite[Section 4.7]{BeRi20}.

\begin{theorem}\label{thm: nonnegandW}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$ such that $\pi_1(\p W) = 0$ and $c_1(V)|_{\pi_2(V)} = 0$. Then
\begin{enumerate}
\item The inclusion $i: W \rightarrow V$ induces an isomorphism between the Novikov fields $\Lda_W$ and $\Lda_V$.
\item Under the above identification $\Lda_V \cong \Lda_W$, we have a canonical vector space isomorphism
$$
\SH^*_{\leq 0}(V; \Lda_V) \cong \SH^*(W;\Lda_W).
$$
\end{enumerate}
\end{theorem}

\subsection{Transfer maps} Transfer-admissible Hamiltonians $H: \widehat V \rightarrow \R$ form a cofinal family of admissible Hamiltonians. The symplectic cohomology $\SH^*(V; \Lda_V)$ is the direct limit of Hamiltonian Floer cohomology $\HF^*(H)$ along the slope $\kappa$ of transfer-admissible Hamiltonians $H$. Consequently, the action functional $\mathcal{F}$ defines a natural map into the quotient
\begin{equation}\label{eq: fulltononpositive}
 \SH^*(V; \Lda_V) \rightarrow \SH_{\leq 0}^*(V; \Lda_V),
\end{equation}
and by the discussion in Section \ref{sec: ring on nonneg}, this map preserves the ring structure. Now, the composition with the identification $\SH^*_{\leq 0}(V; \Lda_V) \cong \SH^*(W; \Lda_W)$ by Corollary \ref{cor: algebraisom} yields an algebra homomorphism
$$
\Phi: \SH^*(V; \Lda_V) \rightarrow \SH^*(W, \Lda_W).
$$
over $\Lda_V = \Lda_W$.
\begin{proof}[Proof of Theorem \ref{thm: A}]
It only remains to explain about the commutative diagram
\[
\begin{tikzcd}
 \SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
 \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
For a transfer-admissible Hamiltonian $H^0$ with the slope $\kappa$ (and hence $\frac{1}{2}\kappa$) sufficiently small, 1-periodic orbits of $H^0$ are precisely those of constant orbits corresponding to Morse critical points on $W$ and $V$ respectively. Moreover, the standard Floer theory, e.g. \cite{Fl88}, tells us that they recover the quantum cohomology $\QH^*(V)$ and $\QH^*(W)$ including the ring structure \cite[Lemma 13]{Ri14}. Now the restriction of the Hamiltonian $H^0: \widehat{V} \rightarrow \R$ to $H^0_W: \widehat{W}\rightarrow \R$, as in the proof of Theorem \ref{thm: nonnegandW}, corresponds to the natural restriction map $i^*: \QH^*(V) \rightarrow \QH^*(W)$. In other words, we have a commutative diagram:
\[
\begin{tikzcd}
 \SH^*(V) \arrow{r} & \SH_{\leq 0}^*(V) \\
 \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
Here, the upper horizontal map is the homomorphism in \eqref{eq: fulltononpositive}, and the canonical $c^*$-map is defined by the inclusion; see \cite[Section 5]{Rit}. Now, applying the identification $\SH^*_{\leq 0}(V) \cong \SH^*(W)$ in Corollary \ref{cor: algebraisom}, we obtain the desired diagram.
\end{proof}

\begin{remark}\label{rem: Weinattach}
 Given a convex symplectic domain $W$ with $\pi_1(\p W) = 0$ and $c_1(W)|_{\pi_2(W)} = 0$, attach the Weinstein handle $\mathcal{H}$ to $W$ along the boundary $\p W$. Denote the resulting convex domain by $V$. Note that $W$ is now a complement-exact subdomain of $V$ which meets the assumptions of Theorem \ref{thm: A}. As an analogue of the seminal result of Cieliebak \cite{Ci02}, it is likely that the transfer map $\Phi: \SH^*(V) \rightarrow \SH^*(W)$ gives rise to an algebra isomorphism when the handle $\mathcal{H}$ is subcritical.
\end{remark}

\begin{corollary}\label{cor: algebraisom}
    Under the assumptions of Theorem \ref{thm: nonnegandW}, we have a canonical $\Lda$-algebra isomorphism
    \[
    \SH^*_{ \leq 0}(V) \cong \SH^*(W).
    \]
\end{corollary}

\begin{theorem}\label{thm: A}
Let $W$ be a complement-exact subdomain in a convex symplectic domain $V$. Assume that $c_1(V)|_{\pi_2(V)} = 0$ and $\pi_1(\p W) = 0$. There exists a $\Z$-graded algebra homomorphism
$$
\Phi: \SH^*(V; \Lda_V) \rightarrow \SH^*(W; \Lda_W)
$$
under a canonical identification $\Lda_V = \Lda_W$. Moreover, it fits the following commutative diagram with quantum cohomology rings.
\[\begin{tikzcd}
        \SH^*(V) \arrow{r}{\Phi}  & \SH^*(W)  \\
        \QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*}& \QH^*(W) \arrow[swap]{u}{c^*}
    \end{tikzcd}
\]
where the vertical maps are the canonical $c^*$-map.
\end{theorem}

The $c^*$-map above is induced by a natural inclusion involved in the definition of symplectic cohomology; see \cite[Section 4.1]{BeRi20} and \cite[Section 5]{Rit}.

For construction, we basically follow McLean \cite[Section 10.2]{Mclean_extension} and Ritter \cite[Section 9]{Rit} where transfer maps are constructed in symplectic (co)homology for exact domains. An essential ingredient is to use a special type of admissible Hamiltonians which is adapted to the embedding $W \hookrightarrow V$, see Figure \ref{fig: tranfer_Ham}, and also properly interacts with an action filtration on the symplectic (co)homology. In particular, a careful estimate of the action values of the generators of the corresponding Hamiltonian Floer cochain complex shows that the part $\SH^*_{\leq 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\SH^*(W)$ of the subdomain $W$. Then the natural quotient map $\SH^*(V) \rightarrow \SH^*_{\leq 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\SH^*(V) \rightarrow \SH^*(W)$. This was shown to be a ring homomorphism \cite[Section 10.2]{Mclean_extension} and more generally compatible with TQFT operations \cite[Section 9]{Rit} for exact domains.

The main technical point of the current non-exact setup is that the standard action functional, as in Remark \ref{rem: usualactionfctl}, now depends not only on loops but on their capping disks, and this makes the action value estimates delicate. To handle this, we introduce a new action functional, inspired by the work of McLean--Ritter \cite{McRi23},
which is designed to deal with non-exact convex symplectic domains. The basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes to the action values, and moreover, they can be effectively understood by Reeb periods of the contact boundary.

In Section \ref{sec: newfiltration}, we define an action functional $\mathcal{F}$ and give the relevant action value estimates in Lemma \ref{lem: actionorbits}. In particular, we show that the non-positive part $\SH^*_{\leq 0}(V)$ with respect to the action filtration induced by $\mathcal{F}$ again corresponds to the symplectic cohomology $\SH^*(W)$ of the subdomain $W$; see Theorem \ref{thm: nonnegandW}. It is also shown in Corollary \ref{cor: algebraisom} that the correspondence is compatible with the respective ring structure. The commutative diagram in Theorem \ref{thm: A} then follows from a standard argument.

There are various interesting examples of convex symplectic domains with complement-exact subdomains for potential applications. We consider negative line bundles in Example \ref{ex: neglinebdl2} and resolutions of isolated singularities in Example \ref{ex: resolution}. For a given convex domain $W$, attaching an exact cobordism along the boundary produces a larger convex domain $V$, and $W$ is complement-exact in $V$; See Remark \ref{rem: Weinattach}.

For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\SH^*_{\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\SH^*(W)$.” Then “the natural quotient map $\SH^*(V)\to \SH^*_{\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\SH^*(V)\to \SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”

In the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\SH^*_{\le 0}(V)$ with respect to the action filtration induced by $\mathcal{F}$ again corresponds to $\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” “The commutative diagram in Theorem~\ref{thm: A} then follows from a standard argument.”

For construction, the authors “basically follow McLean … and Ritter … where transfer maps are constructed in symplectic (co)homology for exact domains.” The key ingredient is “a special type of admissible Hamiltonians which is adapted to the embedding $W\hookrightarrow V$ … and also properly interacts with an action filtration.” A “careful estimate of the action values of the generators” shows that “the part $\SH^*_{\le 0}(V)$ whose generators have non-positive action is canonically isomorphic to the symplectic cohomology $\SH^*(W)$.” Then “the natural quotient map $\SH^*(V)\to \SH^*_{\le 0}(V)$ modding out by the generators with positive action produces the desired homomorphism $\SH^*(V)\to \SH^*(W)$,” and in the exact case this “was shown to be a ring homomorphism … and more generally compatible with TQFT operations.”

In the “current non-exact setup,” the “main technical point” is that the “standard action functional … now depends not only on loops but on their capping disks,” making estimates delicate. To handle this they “introduce a new action functional, inspired by … McLean--Ritter,” whose “basic idea is to impose action values to be zero in the non-exact region so that only the exact part contributes,” and the values “can be effectively understood by Reeb periods.” They then “define an action functional $\mathcal{F}$ and give the relevant action value estimates,” proving that “the non-positive part $\SH^*_{\le 0}(V)$ with respect to the action filtration induced by $\mathcal{F}$ again corresponds to $\SH^*(W)$,” and that “the correspondence is compatible with the respective ring structure.” The commutative diagram in establishing the main theorem then follows from a standard argument.”

Question: Let $V$ be a convex symplectic domain, meaning a compact symplectic manifold $(V,\omega)$ that is exact near $\partial V$ and whose Liouville vector field points outward along $\partial V$. Let $W\subset V$ be a codimension-zero subdomain that is complement-exact, i.e. $\omega$ is exact on $V\setminus W$. Assume moreover that $c_1(V)|_{\pi_2(V)}=0$ and $\pi_1(\partial W)=0$. Write $\Lambda_V$ and $\Lambda_W$ for the Novikov fields associated to $V$ and $W$, $\SH^*$ for symplectic cohomology, $\QH^*$ for quantum cohomology, $i^*:\QH^*(V)\to\QH^*(W)$ for the restriction map induced by the inclusion $i:W\hookrightarrow V$, and $c^*$ for the canonical map from quantum cohomology to symplectic cohomology. Under these assumptions, which statement holds?

Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.

Choices:
B.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra isomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\xrightarrow{\cong}\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
C.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
D.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(W;\Lambda_W)\to\SH^*(V;\Lambda_V)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(W) \arrow{r}{\Phi} & \SH^*(V) \\
\QH^*(W) \arrow{r}{i_*} \arrow{u}{c^*} & \QH^*(V) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes.
E.
Under the canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
such that the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commutes, and moreover $\Phi$ is induced by restricting Hamiltonians from $V$ to $W$ on the full symplectic cohomology complex, without passing to any action filtration or quotient.

direction of transfer and induced map on quantum cohomology

need for action filtration and quotient by positive-action generators


Question: Let $V$ be a convex symplectic domain, meaning a compact symplectic manifold that is exact near its boundary and whose Liouville vector field points outward along the boundary. Let $W \subset V$ be a complement-exact subdomain, i.e. a codimension-zero subdomain such that the symplectic form is exact on $V \setminus W$. Assume that $c_1(V)|_{\pi_2(V)}=0$ and $\pi_1(\partial W)=0$. Let $\SH^*(V;\Lambda_V)$ and $\SH^*(W;\Lambda_W)$ denote the symplectic cohomologies over their Novikov fields, let $\QH^*(V)$ and $\QH^*(W)$ be the quantum cohomology rings, let $i^*:\QH^*(V)\to\QH^*(W)$ be the restriction map induced by the inclusion $i:W\hookrightarrow V$, and let the vertical maps $c^*$ be the canonical maps from quantum cohomology to symplectic cohomology. Under these assumptions, which existence statement holds?

There exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
under a canonical identification $\Lambda_V=\Lambda_W$, and it makes the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commute.

Choices:
B.
There exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
under a canonical identification $\Lambda_V=\Lambda_W$, and it induces an isomorphism from the positive-action summand $\SH^*_{>0}(V)$ onto $\SH^*(W)$; equivalently, it fits into a commutative diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
with the vertical maps the canonical $c^*$-maps.
C.
Under a canonical identification $\Lambda_V=\Lambda_W$, there exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W).
$$
D.
There exists a $\mathbb{Z}$-graded algebra homomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\to\SH^*(W;\Lambda_W)
$$
under a canonical identification $\Lambda_V=\Lambda_W$, and for every complement-exact embedding $W\hookrightarrow V$ the construction is independent of any action filtration or choice of transfer-admissible Hamiltonian, in the sense that it is obtained directly on the full complex rather than as a quotient by positive-action generators; moreover it makes the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commute.
E.
Under a canonical identification $\Lambda_V=\Lambda_W$, there exists a canonical $\mathbb{Z}$-graded algebra isomorphism
$$
\Phi:\SH^*(V;\Lambda_V)\xrightarrow{\cong}\SH^*(W;\Lambda_W)
$$
which makes the diagram
\[
\begin{tikzcd}
\SH^*(V) \arrow{r}{\Phi} & \SH^*(W) \\
\QH^*(V) \arrow{r}{i^*} \arrow{u}{c^*} & \QH^*(W) \arrow[swap]{u}{c^*}
\end{tikzcd}
\]
commute.