\label{thm:PrefMain} Let $(W,S)$ be a Coxeter system.
\begin{enumerate}
\item The language $\Pref_T$ of reduced words for reflection-prefixes is regular.
\item The language of palindromic reduced words is:
$$
\operatorname{Pal}=\{s_1\cdots  s_k s_{k+1}  s_{k}\cdots  s_1 \in S^*\mid  s_1\cdots  s_k s_{k+1}\in\Pref_T\}.
$$
\item The following Poincar\'e series is rational:
$$
\Pref_T(q)=\sum_{\bold w \in \Pref_T} q^{\ell(w)}=\sum_{k\geq 0} \pal_kq^k,
$$
where $\pal_k$ is the number of palindromic reduced words of length $k$.

\item The Poincar\'e series $\operatorname{Pal}(q)=q\Pref_T(q^2)$ is rational.
\end{enumerate}

\begin{Corollary}\label{cor:Refregular}
The language $\Pref_T$ is regular and the generating function
\begin{align*}
\sum_{u\in \Pref_T} q^{|u|}& =\sum_{t\in T} |\{s_1 \cdots s_k s_{k+1} s_k\cdots s_1 \in \Pref_T \mid t=s_1 \cdots s_k s_{k+1}s_k\cdots s_1\}| \, q^{\ell(t)}\\
&=\sum_{k\in\mathbb{N}} a_k q^k
\end{align*}
is rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\RPal(q)=q\Pref_T(q^2)$ is rational.
\end{Corollary}

\section{Poincare series of affine Coxeter groups}\label{se:affine}
In this section we provide a direct solution to Stembridge’s problem by providing an explicit description of the Poincaré series for affine Coxeter groups. This formula is based on a detailed description of the Hasse diagram of the associated root poset.

\subsection{Crystallographic root systems}

When one works with an affine Coxeter system $(W,S)$, with underlying finite Weyl group $(W_0,S_0)$, it is customary to use its associated crystallographic root system in a positive semi-definite quadratic space $(V,B)$. In this case, the classical geometric representation (see \S\ref{ss:InversionSets}) is the unique geometric representation, and  the simple system $\Delta$ is a basis of $V$. The crystallographic condition allows us to consider root systems with non-necessarily unitary roots. We briefly recall this construction below.

Let $\Phi_0$ be a reduced, irreducible, crystallographic root system for the finite
Weyl group $W_0$, realized in a Euclidean space $(V_0,B_0)$ with corresponding set of simple roots $\Delta_0=\{\alpha_s\mid s\in S_0\}$ and positive roots $\Phi_0^+$ (see for instance \cite{bour,Hum}). Then for $s,t\in S_0$, we have:
\[
B_0(\alpha_s, \alpha_t) =
\begin{cases}
-\|\alpha_s\| \|\alpha_t\| \cos \left(\frac{\pi}{m_{st}}\right) & \text{if } m_{st} < \infty, \\
-1 & \text{if } m_{st} = \infty.
\end{cases}
\]
The quadratic space $(V,B)$ is obtained as follows:  $V = V_0 \oplus \mathbb{R}\delta$, and $B$ is the extension of  $B_0$ by setting the radical of $B$ to be equal to  $Q = \mathbb{R}\delta$. Let $\omega$ denote the highest root of $\Phi_0$; the crystallographic affine simple system is $\Delta = \Delta_0 \cup \{ \delta - \omega \}$ with root system
$\Phi = \{ \alpha + k\delta \mid \alpha \in \Phi_0,\; k \in \mathbb{Z} \}$.
The set of positive roots is:
\[\Phi^+=
\{\,  k\delta + \alpha,\;  (k+1)\delta -\alpha \mid \alpha \in \Phi_0^+,\; k \in \mathbb{N} \,\}=(\mathbb{N}\delta+\Phi^+_0)\sqcup (\mathbb{N}^*\delta-\Phi^+_0).
\]
It follows that $\Phi$ is a {\em crystallographic (infinite) root system} for the affine Weyl group $(W,S)$, where $S=S_0 \cup \{s_{\delta-\omega}\}$.

\subsection{The root poset and dominance order in the affine case} Structural results for unitary root systems, specifically those determined by the sign of the bilinear form $B$, generally extend to crystallographic systems. However, properties that depend on the specific numerical values of the form, such as dominance and long edges, must be precisely reformulated for crystallographic root systems; see~\cite[Example 3.9]{DH} for more details.

From the crystallographic root system, $\Phi$, one obtains a {\em unitary} root system in the sense of \S\ref{ss:InversionSets} by normalizing each root via the map $\alpha\mapsto \frac{\alpha}{\|\alpha\|}$. Let $\alpha,\beta\in\Phi$, then:
$$
B\left(\frac{\alpha}{\|\alpha\|},\frac{\beta}{\|\beta\|}\right) = \frac{B(\alpha,\beta)}{\|\alpha\|\|\beta\|}.
$$
Therefore, the condition $B(\alpha',\beta')\leq -1$ for unitary roots $\alpha',\beta'$, which appear in Definition~\ref{def:long} or Proposition~\ref{prop:inf-depth}, becomes:  $$B(\alpha,\beta)\leq -\|\alpha\|\|\beta\|.$$
For this reason Definition~\ref{def:long} in the crystallographic setting becomes the following.

\begin{Definition}[Long and short edges in crystallographic root system] A covering $\alpha\lhd s(\alpha)$ in the root poset $(\Phi^+,\leq)$ is \textit{short} if $-\|\alpha\| \|\alpha_s\| < B(\alpha, \alpha_s)<0$. Otherwise we say that the covering is \textit{long}.
\end{Definition}

The Hasse diagram of the root poset is studied in more details in the subsequent section.

\smallskip
In the affine case, the Hasse diagram of the dominance order is the union of $2|\Phi_0^+|$ total ordered chains. Indeed, the only dominance relations in  $\Phi$ are of the form
$$
 k\delta +\alpha \;\leq_d\;  l\delta+\alpha,
\quad
\text{or} \quad
 (k+1)\delta -\alpha\;\leq_d\;  (l+1)\delta - \alpha,
$$
where $k\leq l$ in $\mathbb N$ and $\alpha\in\Phi_0^+$; see~\cite[Example 3.9]{DH}.

An interesting consequence of these dominance relations is that for $\alpha\in\Phi_0^+$ and $k\in\mathbb N$ we have:
$$
\dep_\infty(k\delta+\alpha)=k\quad\textrm{and}\quad\dep_\infty((k+1)\delta-\alpha)=k,
$$
see~\cite[Example 3.9, Eq.($\diamond$)]{DH}. In particular, the set of $k$-small roots (see Section \ref{ss:mSmallRoots}) for $k\in\mathbb N$ is in the affine case:
\begin{equation}
\label{eq:mSmall}
\Sigma_k=\left(k\delta+\Phi^+_0\right)\sqcup \left((k+1)\delta-\Phi_0^+\right).
\end{equation}

\begin{Example}
Let $(W,S)$ be a Coxeter system of type $\widetilde{A}_2$,

\begin{center}
\begin{tikzpicture}
	[scale=2,
	 q/.style={teal,line join=round},
	 racine/.style={blue},
	 racinesimple/.style={blue},
	 racinedih/.style={blue},
	 sommet/.style={inner sep=2pt,circle,draw=black,fill=blue!40,thick,anchor=base},
	 rotate=0]
 \tikzstyle{every node}=[font=\small]
\def\grosseursimple{0.025}
\coordinate (ancre) at (0,3);

\node[sommet,label=above:$s_{\delta-\omega}$] (a2) at ($(ancre)+(0.25,0.4)$) {};
\node[sommet,label=below right : $s_2$] (a3) at ($(ancre)+(0.5,0)$) {} edge[thick] node[auto,swap,right] {}(a2) ;
\node[sommet,label=below left:$s_1$] (a4) at (ancre) {} edge[thick] node[auto,swap,below] {} (a3) edge[thick] node[auto,swap,left] {} (a2);
\end{tikzpicture}
\end{center}
\noindent
then the underlying finite Coxeter system $(W_0,S_0)$ is of type $A_2$. Consider $S=\{s_1, s_2, s_{\delta-\omega}\}$, where $s_1,s_2\in S_0$ and $\omega=\alpha_1+\alpha_2$ is the highest root of $W_0$. The finite positive root system is $\Phi^+_0=\{\alpha_1, \alpha_2, \alpha_1+\alpha_2\}$ and the affine simple system is $\Delta=\{\alpha_1, \alpha_2, \delta-\omega\}$. The set of $k$-small roots for $k\in\mathbb N$ is:
\begin{eqnarray*}
\Sigma_k &=& \{\alpha_1 + k\delta, \alpha_2 + k\delta, \alpha_1+\alpha_2+k\delta\}\\
    &&\sqcup\  \{-\alpha_1+(k+1)\delta, -\alpha_2+(k+1)\delta, -(\alpha_1+\alpha_2)+(k+1)\delta\}.
\end{eqnarray*}
The affine positive root system is:
$$
\Phi^+=\bigsqcup_{m\in\mathbb N} \Sigma_m.
$$
The Hasse diagram of $(\Phi^+,\leq)$ is depicted in Figure \ref{fig:rootposetA2}, up to depth equal to 3, where each color of the edges corresponds to the simple reflection that is applied: red for $s_1$, yellow for $s_2$ and green for $s_3:=s_{\delta -\omega}$. Moreover, the dashed edges correspond to the long coverings.
\end{Example}

\begin{figure}[ht]
    \centering    \includegraphics[width=0.9\linewidth]{rootposetA22.pdf}
    \caption{Hasse diagram of the root poset of type $\widetilde{A}_2$ up to depth~3, with $\Sigma=\Sigma_0$ and $\Sigma_1$ that correspond to the blocks connected with non-dotted edges.}
    \label{fig:rootposetA2}
\end{figure}

\begin{Example}
    Let $(W,S)$ be a Coxeter system of type $\widetilde{B}_3$ with the following Coxeter graph
    \begin{center}
\begin{tikzpicture}
	[scale=2,
	 q/.style={teal,line join=round},
	 racine/.style={blue},
	 racinesimple/.style={blue},
	 racinedih/.style={blue},
	 sommet/.style={inner sep=2pt,circle,draw=black,fill=blue!40,thick,anchor=base},
	 rotate=0]
 \tikzstyle{every node}=[font=\small]
\def\grosseursimple{0.025}
\coordinate (ancre) at (0,3);

\node[sommet,label=right:$s_3$] (a2) at ($(ancre)+(0.8,0.25)$) {};
\node[sommet,label=above :$s_2$] (a3) at ($(ancre)+(0.5,0)$) {} edge[thick] node[auto,swap,right] {}(a2) ;
\node[sommet,label=above:$s_1$] (a4) at (ancre) {} edge[thick] node[auto,swap,below] {$4$} (a3);
\node[sommet,label=right :$s_{\delta-\omega}$] (a5) at ($(ancre)+(0.8,-0.25)$) {} edge[thick] node[auto,swap,right] {}(a3) ;
\end{tikzpicture}
\end{center}

\noindent
    In this case we have $\|\alpha_1\|=1$ and $\|\alpha_2\|=\|\alpha_3\|=\|\omega\|=\sqrt{2}$, where $\omega=2\alpha_1+2\alpha_2+\alpha_3$. For instance, $B(\alpha_2,\delta-\omega)=-B(\alpha_2,\omega)=-1>-\|\alpha_2\|\|\omega\|=-2$, and $B(\delta-\alpha_3, \alpha_3)=-\|\alpha_3\|^2=-2$. Hence the covering $\alpha_2\lhd s_{\delta-\omega}(\alpha_2)$ is short, while $\delta-\alpha_3\lhd s_3(\delta-\alpha_3)=\delta+\alpha_3$ is long. The Hasse diagram of the root poset  restricted to $\Sigma=\Sigma_0$ of type $\widetilde{B}_3$ is in Figure \ref{fig:root-poset-b3}.
\end{Example}

\subsection{Poset isomorphisms within the root poset}

Let $(W,S)$ be an affine Coxeter system with underlying finite system $(W_0,S_0)$ and crystallographic root system $\Phi_0$. Since $\Phi_0$ is irreducible, we have two cases for the decomposition of $\Phi_0$ into $W_0$-orbits:
\begin{itemize}
    \item $\Phi_0=\mo_1$ is a single orbit if $(W,S)$ is simply laced;
    \item $\Phi_0=\mo_1\sqcup\mo_2$ if $(W,S)$ is of type $\tilde B_n$, $\tilde C_n$, $\tilde G_2$ or $\tilde F_4$.
\end{itemize}
For simplification, we write:
$$
\Phi_0^+=\bigsqcup_{i=1}^h \mo_i^+,
$$
where $\mo_i^+=\mo_i \cap \Phi_0^+$ and $1\leq i\leq h$ ($h=1,2$). We also assume that $\mo_1^+$ is the $W_0$-orbit containing the highest root $\omega$.

\smallskip

Let $\mo$ be a $W_0$-orbit. For $k\in\mathbb N$, the set of $k$-small roots in $\mo$ is:
\begin{equation}
\Sigma_{\mo,k} =(k\delta+\mo^+)\sqcup \left((k+1)\delta-\mo^+\right) =\{k\delta + \alpha, (k+1)\delta - \alpha \mid \alpha \in \mo^+\}.
\end{equation}
Moreover, by Eq~(\ref{eq:mSmall}), we have $\Sigma_k=\sqcup_{i=1}^h\Sigma_{\mo_i,k}$, where $h=1,2$. Set
$$
\Phi_{\mo}=\bigsqcup_{k\in \mathbb N} \Sigma_{\mo,k}.
$$
So $\Phi^+=\sqcup_{i=1}^h\Phi_{\mo_i}$, where $h=1,2$.

The Hasse diagrams of the subposet $(\Phi_{\mo},\leq)$  exhibit strong symmetry properties that are fundamental to prove the main result of this section.

\subsection*{Symmetry at the level of small roots} We denote for simplification, the set of small root in $\mo$ by:
$$
\Sigma_{\mo}=\Sigma_{\mo,0}=\mo^+\sqcup (\delta-\mo^+).
$$
Recall that $\Sigma$ is an order ideal in the root poset (see~\cite[Corollary~4.7.7]{BB}) and any cover relation within $\Sigma$ is short (see~\cite[Lemma~4.7.5 and Theorem~4.7.6]{BB}).

\begin{Theorem}\label{th:iso1} Let $\mo$ be a $W_0$-orbit.
The map $\phi: \mo^+ \to \delta - \mo^+$ defined by $\phi(\alpha) = \delta - \alpha$ is an isomorphism of posets between $(\mo^+, \leq)$ and the opposite poset $(\delta - \mo^+, \leq)^{op}$: for any $\alpha, \beta \in \mo^+$, we have $\alpha \leq \beta$ if and only of  $\delta - \beta \leq \delta - \alpha$. Moreover all covering relations in $(\mo^+,\leq)$ and in $(\delta - \mo^+, \leq)$ are short.
\end{Theorem}

\begin{proof}
Since a poset isomorphism is uniquely determined by the preservation (or reversal) of covering relations, it suffices to show that $\alpha \lhd \beta$ in $\mo^+$ if and only if $\delta - \beta \lhd \delta - \alpha$ in $\delta - \mo^+$. Recall that for any $\alpha, \beta \in \Phi^+$, a covering $\alpha \lhd \beta$ exists if and only if there is a simple reflection $s \in S$ such that $\beta = s(\alpha)$ and $B(\alpha, \alpha_s) < 0$. We examine the two types of simple reflections in the affine Weyl group $W$.

\begin{itemize}
    \item[(i)] Let $s \in S_0$ be a finite reflection. For a finite simple root $\alpha_s$, we have:
    \[ B(\delta - \alpha, \alpha_s) = B(\delta, \alpha_s) - B(\alpha, \alpha_s) = -B(\alpha, \alpha_s). \]
    It follows that $B(\alpha, \alpha_s) < 0$ if and only if $B(\delta - \alpha, \alpha_s) > 0$. Thus, $\alpha \lhd s(\alpha)$ if and only if  $s(\delta - \alpha) = \delta - s(\alpha) \lhd \delta - \alpha$.

    \item[(ii)] Let $s_{\delta-\omega}$ be the affine reflection. Similarly, for the affine simple root $\delta - \omega$, the property of the radical implies:
    \[ B(\delta - \alpha, \delta - \omega) = B(\delta, \delta - \omega) - B(\alpha, \delta - \omega) = -B(\alpha, \delta - \omega). \]
    Again, the sign of the bilinear form is reversed: $B(\alpha, \delta - \omega) < 0$ if and only if $B(\delta - \alpha, \delta - \omega) > 0$. This ensures that $\alpha \lhd s_{\delta-\omega}(\alpha)$ if and only if $s_{\delta-\omega}(\delta - \alpha) = \delta - s_{\delta-\omega}(\alpha) \lhd \delta - \alpha$.
\end{itemize}

Now, let $\alpha \lhd s(\alpha)$ be a covering in $\mo^+$ for some $s \in S$. The type of this cover depends on the value $B(\alpha, \alpha_s)$. Under the map $\phi$, this corresponds to the cover $\delta-s(\alpha) \lhd \delta -\alpha$. By the properties of the reflection $s$ and of the radical, we have:
\[
B(\delta - s(\alpha), \alpha_s) = -B(s(\alpha), \alpha_s) = B(\alpha, \alpha_s).
\]
Since  $B(\cdot, \alpha_s)$ is identical in both case, the condition for being short or long in Definition \ref{def:long} is preserved. Since $\mo^+$ and $\delta-\mo^+$ decomposes the set of small roots~$\Sigma$, so they are all short.
\end{proof}

\begin{Example}
    If $(W,S)$ is of type $\widetilde{B}_3$, then $\Phi_0^+=\mo_1^+\sqcup\mo_2^+$, where $\mo_1^+=\{\alpha_1, s_2(\alpha_1), s_3s_2(\alpha_1)\}$ and $\mo_2^+=\{\alpha_2, \alpha_3, s_1(\alpha_2), s_3(\alpha_2), s_1s_3(\alpha_2), s_2s_1s_3(\alpha_2)\}$. The root poset restricted to $\Sigma_{\mo_1}\sqcup\Sigma_{\mo_2}$ is depicted in Figure \ref{fig:root-poset-b3}.
\end{Example}

\begin{figure}[hbtp]
\centering
\includegraphics[scale=.75]{rootposetB3.pdf}
\caption{Root poset of type $\widetilde{B}_3$ restricted to $\Sigma_{\mo_1}\sqcup \Sigma_{\mo_2}$. The portion of poset in the red lines is isomorphic to opposite in the black lines, as stated in Theorem \ref{th:iso1}.}
\label{fig:root-poset-b3}
\end{figure}

\subsection*{Symmetry of the root poset restricted to a $W_0$- orbit}
Following \cite[Ch. VI, \S 1]{bour}, for $\alpha, \beta\in \Phi^+$, we set
$$n(\alpha,\beta):=2\frac{B(\alpha,\beta)}{B(\beta,\beta)},$$
so that $s_\beta(\alpha)=\alpha - n(\alpha,\beta)\beta.$ By \cite[Ch. VI, \S 1, Proposition~8]{bour}, if $\alpha\neq \beta$ and $\|\alpha\|\leq \|\beta\|$, then $n(\alpha,\beta) \in \{0,\pm 1\}$. It follows that $n(\alpha, \omega) \in \{0,\pm 1\}$ when $\alpha\neq \omega$, since $\|\alpha\|\leq \|\omega\|$ for all positive roots $\alpha$. Finally, note that $n(k\delta+\alpha, \beta)=n(\alpha, \beta)=n(\alpha, m\delta+\beta)$ and $n(k\delta-\alpha, \beta)=-n(\alpha, \beta)=n(\alpha, m\delta-\beta)$ for $k,m\in\mathbb{N}^*$.
\smallskip

 We describe now the covering relations inside the sets $\Sigma_{\mo,k}$. Let $\alpha \in \Phi_0^+$ and $k \in \mathbb{N}$.

\begin{enumerate}
\item[(1)]
There are no covering relations of the form
\[
(k+1)\delta - \alpha \lhd s_{\delta-\omega}\big((k+1)\delta - \alpha\big).
\]
Indeed, if such a covering existed, then
$0 > B\big((k+1)\delta - \alpha,\, \delta - \omega\big) = B(\alpha,\omega)$,
so that $n(\alpha,\omega) = -1$. It follows that
\[
s_{\delta-\omega}(\alpha)
= \alpha - n(\alpha,\delta-\omega)(\delta-\omega)
= \alpha + n(\alpha,\omega)(\delta-\omega)
= \alpha - \delta + \omega.
\]
Thus, $s_{\delta-\omega}\big((k+1)\delta - \alpha\big)
= (k+2)\delta - (\alpha + \omega)
\notin \Sigma_{\mo,k},$ a contradiction.
\smallskip

\item[(2)]
All covering relations in $\Sigma_{\mo,k}$ are of one of the following three types:
\begin{itemize}
\item[(i)]
$k\delta + \alpha \lhd s(k\delta + \alpha) = k\delta + s(\alpha),
\qquad \text{when } B(\alpha,\alpha_s) < 0;
$

\item[(ii)]
$
(k+1)\delta - \alpha \lhd s\big((k+1)\delta - \alpha\big)
= (k+1)\delta - s(\alpha),
$
when $B(\alpha,\alpha_s) > 0$ and $\alpha \neq \alpha_s$.
Note that if $\alpha = \alpha_s$, then
$(k+1)\delta - s(\alpha) = (k+1)\delta + \alpha_s \in \Sigma_{\mo,k+1}$.

\item[(iii)]
$
k\delta + \alpha \lhd s_{\delta-\omega}(k\delta + \alpha)
= k\delta + s_{\delta-\omega}(\alpha),
$
when $B(\alpha,\delta-\omega) < 0$ and $\alpha \neq \omega$.
Otherwise, $s_{\delta-\omega}(k\delta + \omega) = (k+2)\delta - \omega
\in \Sigma_{\mo,k+1}.$
\end{itemize}
\end{enumerate}

We state now the second symmetry of the root poset.

\begin{Theorem}\label{th:iso2} Let $\mo$ be a $W_0$-orbit. There is a poset isomorphism $$(\Sigma_{\mo,k}, \leq) \cong (\Sigma_{\mo}, \leq)$$ for all $k\in\mathbb{N}$. Moreover, all covering in $\Sigma_{\mo,k}$ are short.
\end{Theorem}

\begin{proof}
The map $\pi_k : \Sigma_{\mo,k} \longrightarrow \Sigma_{\mo}$ defined by
\[
\pi_k(k\delta+\alpha)=\alpha,
\qquad
\pi_k((k+1)\delta-\alpha)=\delta-\alpha,
\]
for $\alpha\in\Phi_0^+$ is a bijection. By the discussion preceding the theorem, we have three types of coverings in $\Sigma_{\mo,k}$.
\smallskip

A covering $k\delta+\alpha \lhd k\delta+s(\alpha)$ of type (i) occurs if and only if
$0>B(k\delta+\alpha,\alpha_s)=B(\alpha,\alpha_s)$,
which is equivalent to $\alpha \lhd s(\alpha)$ being a covering in $\Sigma_{\mo}$.
\smallskip

A covering $(k+1)\delta-\alpha \lhd (k+1)\delta-s(\alpha)$ of type (ii), occurs if and only if
$ 0>B((k+1)\delta-\alpha,\alpha_s)=-B(\alpha,\alpha_s)$,
with $\alpha\neq\alpha_s$, which is equivalent to
$\delta-\alpha \lhd \delta-s(\alpha)$
being a covering in $\Sigma_{\mo}$.
\smallskip

A covering $k\delta+\alpha \lhd s_{\delta-\omega}(k\delta+\alpha)$ of type (iii)
occurs if and only if $0>B(k\delta+\alpha,\delta-\omega)=B(\alpha,\delta-\omega)$, with $\alpha\neq\omega$, which is equivalent to $\alpha \lhd s_{\delta-\omega}(\alpha)
$ being a covering in $\Sigma_{\mo}$.
\smallskip

Thus $\pi_k$ preserves and reflects covering relations and is therefore a poset isomorphism. The final statement follows from Theorem~\ref{th:iso1} since all covering are short in $\Sigma_\mo$.
\end{proof}

\begin{Example} In the Figure~\ref{fig:root-poset-b3-iso2} below, the the first three slices of the Hasse diagram of the root poset of type $\widetilde{B}_3$ are represented. The first slice coincides with the poset shown in Figure \ref{fig:root-poset-b3}, although the order relations are drawn differently. This alternative depiction makes the isomorphisms described in Theorem \ref{th:iso2} more apparent.
\end{Example}

\begin{figure}[hbtp]
\centering
\includegraphics[scale=.7]{periodicB3str.pdf}
\caption{}
\label{fig:root-poset-b3-iso2}
\end{figure}

We end this discussion by showing that a covering $\alpha\lhd \beta$ with $\alpha\in \Sigma_{\mo,k}$ and $\beta\in \Sigma_{\mo,k+1}$ is long.

\begin{Proposition}[Covering relations]\label{prop:long-cover}\
\begin{itemize}
    \item[(i)]  For all $k\in\mathbb{N}, s\in S_0$, $(k+1)\delta-\alpha_s\lhd s((k+1)\delta - \alpha_s)$ is a long covering, where $(k+1)\delta-\alpha_s\in \Sigma_{\mo,k}$ and $s((k+1)\delta - \alpha_s)\in \Sigma_{\mo, k+1}$.
    \item[(ii)] For all $k\in\mathbb{N}$, $k\delta +\omega\lhd s_{\delta-\omega}(k\delta + \omega)$ is a long covering, where $k\delta +\omega\in \Sigma_{\mo,k}$ and $s_{\delta-\omega}(k\delta + \omega)\in \Sigma_{\mo,k+1}$.
\end{itemize}

\end{Proposition}

\begin{proof}
    Consider the  $(k+1)\delta-\alpha_s\lhd s((k+1)\delta - \alpha_s)$, then $B((k+1)\delta-\alpha_s, \alpha_s)=-\|\alpha_s\|^2$, hence the covering is long, and $s((k+1)\delta - \alpha_s)=(k+1)\delta+\alpha_s \in \Sigma_{\mo,k+1}$. Analogously, if we consider the covering $k\delta +\omega\lhd s_{\delta-\omega}(k\delta + \omega)$, $B(k\delta +\omega, \delta-\omega)=-\|\omega\|^2$, so the covering is long. In particular, $s_{\delta -\omega}(k\delta +\omega)=(k+2)\delta -\omega \in \Sigma_{\mo,k+1}$.
    \end{proof}

\subsection{An explicit formula for the generating function of depth} We are ready now to provide a closed formula for the generating function of the depth of positive roots in $\Phi$, using the restriction of the root poset to the sets $\Sigma_\mo$ and $\Phi_\mo$ introduced in the previous section.
\medskip

Let $\mo \subseteq \Phi_0$ be $W_0$-orbit. Define
$$
M_{\mo}=\max\{\dep(\beta) \mid \beta \in \Sigma_{\mo}\}.
$$
We show that the depth of a positive root depends only on $M_{\mo}$ and on
the depth of a root in $\mo^+$.

\begin{Corollary}\label{cor:maxdep}
    Let $\alpha\in \mo^+$ and $k\in \mathbb{N}$, then :
    \begin{enumerate}
        \item $\dep(k\delta + \alpha)=\dep(\alpha)+k(M_\mo +1)$;
        \item $\dep((k+1)\delta -\alpha) = \dep(\delta-\alpha) +k(M_\mo +1)$.
    \end{enumerate}
\end{Corollary}

\begin{Proposition}\label{prop:PrefDom} Let $\beta \in \Phi^+$ and let $t=s_\beta$ be the corresponding reflection. For any $t$-prefix $p_t$ we have:
\begin{eqnarray*}
\dom(\beta) & = & \{ \alpha\in \Phi(p_t) \ \mid\  B(\alpha,\beta)\geq 1\}\\
&=&\{\beta\}\sqcup \{ \alpha_s\in \Phi(p_t) \ \mid\  |\langle s,t\rangle| =\infty\}.
\end{eqnarray*}
In particular, $\dep_\infty (\beta)=|\{ \alpha_s\in \Phi(p_t) \ \mid\ |\langle s,t\rangle| =\infty\}|$.
\end{Proposition}

\begin{proof} Since $\pi_m(w)\leq_R w$, we have $\Sigma_m(\pi_m(w))\subseteq \Sigma_m(w)$. The converse follows from  \cite[Theorem 6.4]{DFHM24} in which the authors prove that the set
$$
\{g\in W\mid \Sigma_m(g)=\Sigma_m(w)\}
$$
has a unique element of minimal length $x\in L_m$ and that $x\leq_R w$. So $\Sigma_m(w)=\Sigma_m(x)$. But $x\leq_R\pi_m(x)$ by definition. Therefore  $ \Sigma_m(w)=\Sigma_m(x)\subseteq \Sigma_m(\pi_m(w))$.
\end{proof}

\section{Reflection-prefixes and palindromic reduced words of reflections}\label{se:reflectionprefix}

In this section, we  first define and state properties of {\em reflection-prefixes} in relation to the root poset, to the dominance order and to the canonical generators of maximal dihedral subgroups.  Then we produce a family of automata built from the finite Garside shadows $L_m$ ($m\in\mathbb N)$  that recognize the language of reflection-prefixes.

\subsection{Reflection-prefixes}\label{ss:reflectionprefixes}

 Let $t\in T$ be a reflection of $(W,S)$. It is well known that $\ell(t)=2k+1$ for some $k\in\mathbb N$ and that  if $t=s_1\dots s_k s_{k+1} s_{k+2} \dots s_{2k+1}$ is a reduced word for $t$, the word $s_1 \dots s_k s_{k+1} s_k\dots s_1$ is a {\em palindromic reduced word} for $t$; see for instance \cite[Proposition~2.3]{DFHM24}.

 The following proposition shows in particular that $t$ is uniquely determined by the prefix $s_1\cdots s_ks_{k+1}$ of any of its palindromic reduced words.

 \begin{Proposition}\label{prop:Pref1} Let $t\in T$ with corresponding positive root $\alpha_t$. Let  $w\in W$ and assume there is $r\in D_R(w)$ such that $t=wrw^{-1}$ and $\ell(t)=2\ell(w)-1$. Then $D_R(w)=\{r\}$. Moreover:
 \begin{enumerate}

\item if $w=s_1\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\cdots s_k r s_k\cdots s_1$ is a palindromic reduced word for $t$;
\item $\alpha_t=w(\alpha_r)= s_1\cdots s_k(\alpha_r)$ and  $\dep(\alpha_t)=k=\ell(w)-1$.
\end{enumerate}
\end{Proposition}
\begin{proof}   Let $w=s_1\cdots s_k r$ be a reduced word.  We show that $D_R(w)=\{r\}$. By contradiction, assume that $|D_R(w)|\geq 2$. So there exists  a $s\in D_R(w)$ with $s\not = r$; set $I=\{s,r\}\subseteq D_R(w)$.  By Lemma~\ref{lem:Descents},  we know that $W_I$ is finite and  there is $u\in X_I$ such that $w=u w_{\circ,I}$ is a reduced product. So $\ell(w)=\ell(u)+m$, where $m=m_{sr}\geq 2$. Now, observe that  $w_{\circ,I}rw_{\circ,I}\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore
$$
t= s_1\cdots s_k s_{k+1} s_k\cdots s_1 =s_1\cdots s_k r s_k\cdots s_1=w r w^{-1}=uw_{\circ,I}rw_{\circ,I}u^{-1},
$$
implying that $\ell(t)\leq 2\ell(u)+1=2\ell(w)-2m+1<2\ell(w)-1=\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\ell(t)=2\dep(\alpha_t)+1=2\ell(w)-1$.
\end{proof}

\begin{Proposition}\label{prop:PrefRoot} Let $t\in T$ and $\alpha_t\in \Phi^+$ be the corresponding positive root.
\begin{enumerate}
\item Let  $\alpha_{r}\lhd s_k(\alpha_{r})\lhd\dots\lhd s_1\cdots s_{k-1} s_k(\alpha_r)=\alpha_t$ be a saturated chain in $(\Phi^+,\leq)$, where $r\in S$. Then $p_t=s_1\dots s_kr$ is a reduced word for some $t$-prefix $p_t$.

\item Let $p_t$ be a  $t$-prefix and fix a reduced word $p_t=s_1\cdots s_k r $, with $r$ the unique right-descent. Then $\alpha_{r}\lhd s_k(\alpha_{r})\lhd\cdots\lhd s_1\cdots s_{k-1} s_k(\alpha_r)=\alpha_t$ is a saturated chain in $(\Phi^+,\leq)$.
\end{enumerate}
\end{Proposition}

\begin{Theorem}\label{th:main}
    The generating function of the depth of positive roots is of the form
    $$\Phi^+(q)=\frac{P(q)}{1-q^M}$$
    where $M=\lcm(M_\mo+1\mid \mo\ \mbox{ $W_0$-orbit})$ and $P$ is palindromic.
\end{Theorem}

\label{se:openproblems}

We propose here some natural problems arising from this article.

\begin{Problem}
    Is the language of palindromic reduced word (or lexicographically ordered palindromic red

\begin{Theorem}\label{prop:algo} Let $r,t\in T$ with $r\not = t$ and let $W'=\langle r,t\rangle$ be the dihedral reflection subgroup generated by $r$ and $t$. Let $p_r$ be a $r$-prefix and $p_t$ be a $t$-prefix.
\begin{enumerate}
\item If $|\Phi_{W'}(p_r)|=|\Phi_{W'}(p_t)|=1$, then:
$$
\Delta_{W'}=\{\alpha_r,\alpha_t\}\quad\textrm{and}\quad\chi(W')=\{r,t\}.
$$
\item If   $|\Phi_{W'}(p_r)|>1$ (resp. $|\Phi_{W'}(p_t)|>1$), then:
 $$
 \Delta_{W'}=\{\alpha_1,\alpha_2\}\quad\textrm{and}\quad\chi(W')=\{s_1,s_2\},
 $$
 where $\alpha_1$ is the unique root with the smallest depth in $\Phi(p_r)\cap \Phi_{W'}$ (resp. $\Phi(p_t)\cap \Phi_{W'}$) and $s_1(\alpha_2)$ is the unique root with smallest depth in $\Phi(p_r)\cap \Phi_{W'}\setminus\{\alpha_1\}$ (resp. $\Phi(p_t)\cap \Phi_{W'}\setminus\{\alpha_1\}$). \qed
\end{enumerate}
 \end{Theorem}

Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\bold w =  s_1 \cdots  s_k$, while the resulting product in $W$ is denoted by $w=s_1\cdots s_k$. Let $w\in W$, a word $\bold w= s_1 \cdots  s_k\in S^*$ is a {\em reduced word for $w$} if  $w=s_1\cdots s_k$  and $k$ is minimal for this property; in this case the {\em length} of $w$ is $\ell(w)=k$. The identity $e\in W$ is represented by the empty word, and $\ell(e)=0$. We denote the set of all reduced words of $w\in W$ by $\Red(w)$, and the set of all reduced words in $W$ by $\Red=\bigsqcup_{w\in W} \Red(w)$, where $\sqcup$ denotes the disjoint union.
Finally, for any subset $A\subseteq W$, the {\em Poincar\'e series of $A$} is the formal power series
$A(q):=\sum_{w\in A} q^{\ell(w)}.$
\smallskip

In this article, we study two combinatorial problems regarding the {\em set of reflections} $T:=\bigcup_{w\in W} wSw^{-1}$ of a Coxeter group $W$:
\begin{enumerate}
    \item Is the language of {\em palindromic reduced words (for the reflections)} regular?
    \item Are there explicit and elegant formulas for the Poincar\'e series of the set $T$ of reflections ?
\end{enumerate}
Our main contributions to those questions are the following:
\begin{itemize}
\item We introduce the notion of  {\em reflection-prefixes}, a class of elements in $W$ arising naturally from  palindromic reduced words of reflections, and study their properties in relation to the {\em root poset}, the {\em dominance order on roots}  and {\em dihedral reflection subgroups}.
\item For any Coxeter system, we show that the {\em language of reduced words for reflection-prefixes}, $\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\cite[\S3.4]{HNW} and \cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\Pref_T$. As a consequence, we show that the generating function  of palindromic reduced words is rational.
\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.
\end{itemize}

We discuss now some history and motivations that lead to this article.
If $W$ is finite, $A(q)$ is clearly a polynomial for any $A\subseteq W$. A natural question in combinatorics of infinite Coxeter systems is to classify for which subsets $A$ the Poincar\'e series $A(q)$ is {\em rational}, that is, can be written as a ratio of two polynomials in $q$. It is well-known that $W(q)$ is rational, see for instance~\cite[Corollary~7.1.8]{BB}. Furthermore, an explicit recursive formula for $W(q)$ in term of standard parabolic subgroups is provided in~\cite[Proposition~7.1.7]{BB}.

A further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$.  More precisely, given the canonical projection $\pi:S^* \rightarrow W$ sending a word $\bold w= s_1 \cdots  s_k$ to the element $w=s_1\cdots s_k$,
it is natural to  consider the {\em ‘‘lifted'' Poincar\'e series} of a subset $B\subseteq S^*$ relative to $(W,S)$:
$$
B(q)=\sum_{\bold w\in B} q^{\ell(w)},
$$
where the notation ${\ell(w)}$ is understood to be $\ell(\pi({\bf w}))$.

A longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\Red(w)|$; for a detailed discussion see~\cite[p.123]{BB}, and for various partial results, see~\cite{Eriksson,Hart}.  However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\in \mathbb N$ is enumerated by  the following Poincar\'e series
$$
\Red(q)=\sum_{\bold w\in \Red} q^{\ell(w)}=\sum_{k\in\mathbb N} r_kq^k,
$$
which is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\Red$; see for instance~\cite[Theorem~4.9.1]{BB} for more details.

\smallskip

As reported by Brenti~\cite{B1}, Stembridge proposed the following problem during an open problem session at the Mathematical Sciences Research Institute at Berkeley in 1997: Is it true that the Poincar\'e series
$$
T(q)=\sum_{t\in T} q^{\ell(t)}\quad \textrm{is rational?}
$$

\textbf{}Since reflections are known to admit palindromic reduced words, a natural approach to this question is to construct a finite automaton that recognizes exactly one palindromic reduced word for each reflection. The existence of such a regular language would imply that the associated generating function is rational. However, the property of being regular is not generally preserved under the palindromic constraint. If a language is regular, it is well-known that the sublanguage of its palindromic words is context-free but not necessarily regular; for instance if $S=\{a,b\}$ then $S^*$ is regular but a standard application of the {\em pumping lemma} shows that the sublanguage of its palindroms is not.  Indeed, the regularity of $\operatorname{Pal}$, {\em the language of all palindromic reduced words}, which are necessarily reduced words for reflections, remains an open question. In a recent article, Mili\'cevi\'c ~\cite{Mili} provides palindromic reduced words for all reflections in finite Weyl groups, but highlights the challenge of finding a general algorithm applicable to any Coxeter group. To overcome these challenges, we shift to the language of reflection-prefixes which we define below.

\subsection*{The language of reflection-prefixes}

We recall that the {\em (right) weak order} $(W,\leq_R)$ is the poset defined by $u\leq_R w$ if there exists a reduced word for $u\in W$ that is a prefix for a reduced word for $w\in W$. Equivalently, $u\leq_R w$ if and only if $\ell(w)=\ell(u^{-1}w)+\ell(u)$.

It is well-known that every reflection has an odd length. Let $t\in T$ be a reflection of length $\ell(t)=2k+1$, we define a {\em $t$-prefix} to be any element $p_t\in W$ such that $p_t\leq_R t$ and $\ell(p_t)=k+1$.
  It turns out that $ s_1\cdots s_k s_{k+1}  s_{k}\cdots  s_1$ is a palindromic reduced word for $t$  if and only if $s_1\cdots  s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \S\ref{ss:reflectionprefixes}.

Reflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\ref{prop:PrefRoot}), the dominance order (Proposition~\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\ref{prop:algo}).  Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.

Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\bold w = s_1 \cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\cdots s_k$. Let $w\in W$, a word $\bold w= s_1 \cdots s_k\in S^*$ is a {\em reduced word for $w$} if $w=s_1\cdots s_k$ and $k$ is minimal for this property; in this case the {\em length} of $w$ is $\ell(w)=k$. The identity $e\in W$ is represented by the empty word, and $\ell(e)=0$. We denote the set of all reduced words of $w\in W$ by $\Red(w)$, and the set of all reduced words in $W$ by $\Red=\bigsqcup_{w\in W} \Red(w)$, where $\sqcup$ denotes the disjoint union.
Finally, for any subset $A\subseteq W$, the {\em Poincar\'e series of $A$} is the formal power series
$A(q):=\sum_{w\in A} q^{\ell(w)}.$
\smallskip

In this article, we study two combinatorial problems regarding the {\em set of reflections} $T:=\bigcup_{w\in W} wSw^{-1}$ of a Coxeter group $W$:
\begin{enumerate}
 \item Is the language of {\em palindromic reduced words (for the reflections)} regular?
 \item Are there explicit and elegant formulas for the Poincar\'e series of the set $T$ of reflections ?
\end{enumerate}
Our main contributions to those questions are the following:
\begin{itemize}
\item We introduce the notion of {\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\em root poset}, the {\em dominance order on roots} and {\em dihedral reflection subgroups}.
\item For any Coxeter system, we show that the {\em language of reduced words for reflection-prefixes}, $\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\cite[\S3.4]{HNW} and \cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.
\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.
\end{itemize}

A further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$. More precisely, given the canonical projection $\pi:S^* \rightarrow W$ sending a word $\bold w= s_1 \cdots s_k$ to the element $w=s_1\cdots s_k$,
it is natural to consider the {\em ‘‘lifted'' Poincar\'e series} of a subset $B\subseteq S^*$ relative to $(W,S)$:
$$
B(q)=\sum_{\bold w\in B} q^{\ell(w)},
$$
where the notation ${\ell(w)}$ is understood to be $\ell(\pi({\bf w}))$.

A longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\Red(w)|$; for a detailed discussion see~\cite[p.123]{BB}, and for various partial results, see~\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\in \mathbb N$ is enumerated by the following Poincar\'e series
$$
\Red(q)=\sum_{\bold w\in \Red} q^{\ell(w)}=\sum_{k\in\mathbb N} r_kq^k,
$$
which is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\Red$; see for instance~\cite[Theorem~4.9.1]{BB} for more details.

It is well-known that every reflection has an odd length. Let $t\in T$ be a reflection of length $\ell(t)=2k+1$, we define a {\em $t$-prefix} to be any element $p_t\in W$ such that $p_t\leq_R t$ and $\ell(p_t)=k+1$.
 It turns out that $ s_1\cdots s_k s_{k+1} s_{k}\cdots s_1$ is a palindromic reduced word for $t$ if and only if $s_1\cdots s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \S\ref{ss:reflectionprefixes}.

Reflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\ref{prop:PrefRoot}), the dominance order (Proposition~\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.

Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\bold w = s_1 \cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\cdots s_k$. Let $w\in W$, a word $\bold w= s_1 \cdots s_k\in S^*$ is a {\em reduced word for $w$} if $w=s_1\cdots s_k$ and $k$ is minimal for this property; in this case the {\em length} of $w$ is $\ell(w)=k$. The identity $e\in W$ is represented by the empty word, and $\ell(e)=0$. We denote the set of all reduced words of $w\in W$ by $\Red(w)$, and the set of all reduced words in $W$ by $\Red=\bigsqcup_{w\in W} \Red(w)$, where $\sqcup$ denotes the disjoint union.
Finally, for any subset $A\subseteq W$, the {\em Poincar\'e series of $A$} is the formal power series
$A(q):=\sum_{w\in A} q^{\ell(w)}.$
\smallskip

In this article, we study two combinatorial problems regarding the {\em set of reflections} $T:=\bigcup_{w\in W} wSw^{-1}$ of a Coxeter group $W$:
\begin{enumerate}
 \item Is the language of {\em palindromic reduced words (for the reflections)} regular?
 \item Are there explicit and elegant formulas for the Poincar\'e series of the set $T$ of reflections ?
\end{enumerate}
Our main contributions to those questions are the following:
\begin{itemize}
\item We introduce the notion of {\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\em root poset}, the {\em dominance order on roots} and {\em dihedral reflection subgroups}.
\item For any Coxeter system, we show that the {\em language of reduced words for reflection-prefixes}, $\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\cite[\S3.4]{HNW} and \cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.
\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.
\end{itemize}

A longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\Red(w)|$; for a detailed discussion see~\cite[p.123]{BB}, and for various partial results, see~\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\in \mathbb N$ is enumerated by the following Poincar\'e series
$$
\Red(q)=\sum_{\bold w\in \Red} q^{\ell(w)}=\sum_{k\in\mathbb N} r_kq^k,
$$
which is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\Red$; see for instance~\cite[Theorem~4.9.1]{BB} for more details.

Reflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\ref{prop:PrefRoot}), the dominance order (Proposition~\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.

The first assertion, the regularity of $\Pref_T$, is established in Corollary~\ref{cor:Refregular}. The proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m \in \mathbb{N})$; for further details, see~\cite{HNW, DFHM24}. The second assertion follows from Proposition~\ref{prop:Pref1}, while the third and fourth are consequence of the first two.

\item if $w=s_1\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\cdots s_k r s_k\cdots s_1$ is a palindromic reduced word for $t$;
\item $\alpha_t=w(\alpha_r)= s_1\cdots s_k(\alpha_r)$ and $\dep(\alpha_t)=k=\ell(w)-1$.
\end{enumerate}
\end{Proposition}
\begin{proof} Let $w=s_1\cdots s_k r$ be a reduced word. We show that $D_R(w)=\{r\}$. By contradiction, assume that $|D_R(w)|\geq 2$. So there exists a $s\in D_R(w)$ with $s\not = r$; set $I=\{s,r\}\subseteq D_R(w)$. By Lemma~\ref{lem:Descents}, we know that $W_I$ is finite and there is $u\in X_I$ such that $w=u w_{\circ,I}$ is a reduced product. So $\ell(w)=\ell(u)+m$, where $m=m_{sr}\geq 2$. Now, observe that $w_{\circ,I}rw_{\circ,I}\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore
$$
t= s_1\cdots s_k s_{k+1} s_k\cdots s_1 =s_1\cdots s_k r s_k\cdots s_1=w r w^{-1}=uw_{\circ,I}rw_{\circ,I}u^{-1},
$$
implying that $\ell(t)\leq 2\ell(u)+1=2\ell(w)-2m+1<2\ell(w)-1=\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\ell(t)=2\dep(\alpha_t)+1=2\ell(w)-1$.
\end{proof}

\begin{Example}[Universal Coxeter system]
 Let $(U_n, S)$ be the Universal Coxeter system of rank $n$, that is $U_n=\langle S\mid s^2=e \ \mbox{for all $s \in S$} \rangle$. Every element has a unique reduced word, so the language of palindromic reduced words for $(U_n,S)$ is given by $\RPal= \{s_1\cdots s_{i-1}s_is_{i-1}\cdots s_1 \mid s_j\in S \mbox{ and } s_j\neq s_{j+1}\}$. Furthermore, the language of reduced words for reflection-prefixes coincides with the set of all reduced words in $W$, that is, $\Pref_T=\Red$.
\end{Example}

\begin{Corollary}\label{cor:Refregular}
The language $\Pref_T$ is regular and the generating function
\begin{align*}
\sum_{u\in \Pref_T} q^{|u|}& =\sum_{t\in T} |\{s_1 \cdots s_k s_{k+1} s_k\cdots s_1 \in \Pref_T \mid t=s_1 \cdots s_k s_{k+1}s_k\cdots s_1\}| \, q^{\ell(t)}\\
&=\sum_{k\in\mathbb{N}} a_k q^k
\end{align*}
is rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\RPal(q)=q\Pref_T(q^2)$ is rational.
\end{Corollary}

The first assertion, the regularity of $\Pref_T$, is established in Corollary~\ref{cor:Refregular}. The proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m \in \mathbb{N})$; for further details, see~\cite{HNW, DFHM24}. The second assertion follows from Proposition~\ref{prop:Pref1}, while the third and fourth are consequence of the first two.

\subsection*{Closed formulas for $T(q)$ in affine Coxeter systems} Remarkably, a solution to Stembridge's problem was provided in 1999 by de~Man~\cite{DM}. However, the author seemed unaware of Stembridge's question and the combinatorics community was not aware of the solution. This oversight may be partly due to the title of de~Man's article {\em ``The generating function for the number of roots of a Coxeter group''}, which only address counting reflections by length at the end of the article in \cite[\S5.2]{DM}. His solution is based on the Brink--Howlett automaton~\cite{BH}  that recognizes the language of {\em lexicographically ordered reduced word}, that is, each element $w$ has a unique lexicographically reduced word in $\Red(w)$.

More precisely, in \cite[Theorem 4.1]{DM}, de Man shows that the generating function  $\Phi^+(q)$ of the depth of positive roots is a sum of rational functions that are obtained from a subset of states of the Brink-Howlett automaton and from the enumeration of the vertices of a family of graphs parameterized by $T$. So the generating function $\Phi^+(q)$ is rational. Since $T(q)=q\, \Phi^+(q^2)$, the Poincar\'e series $T(q)$  is rational as well. However, the Brink--Howlett automaton contains a lot of states, most of them ``dead states'', which make this automaton difficult to use in practice. For instance, in affine type $\widetilde{A}_2$, the number of states of the Brink--Howlett automaton is bounded by $2^6$; de Man provides some example of $T(q)$ using a computer to handle his formula.

 In this article, we give, in the case of affine Coxeter systems, an alternative solution to Stembridge's problem by providing a simple formula for  $T(q)$.

\begin{Theorem}\label{thm:Main1} Let $(W,S)$ be an affine Coxeter system. Then
$$
T(q):=\sum_{t\in T}q^{\ell(t)}=q\frac{P(q^2)}{1-q^{2M}},
$$
where $M\in \mathbb{N}$ and $P(q)$ is a palindromic polynomial.
\end{Theorem}

The proof of the above theorem relies on two fundamental symmetries of the root poset on $\Phi^+$ (Theorems \ref{th:iso1} and \ref{th:iso2}). These properties yield an explicit formula for $M$ and $P(q)$ (Theorem \ref{th:main})), from which the above result follows.

\subsection*{Plan of the article} The article is organized as follows. In~\S\ref{sec:root-poset}, we recall some needed facts about the combinatorics of words and roots and, in particular, about two partial orders on the root system of a Coxeter system: the root poset and the dominance order, which were introduced by Brink and Howlett~\cite{BH} in their work to show that Coxeter systems are automatic. In~\S\ref{se:reflectionprefix}, we discuss reflection-prefix, palindromic reduced words for reflections and automata and, in \S\ref{se:affine},  we give the enumeration of reflections by length in affine types. Finally, we propose some open problems in \S\ref{se:openproblems}.

\subsection*{Acknowledgements}  CH warmly thanks Francesco Brenti, James Parkinson and Christophe Reutenauer for very instructive discussions.
The authors are particularly indebted to Matthew Dyer for bringing de Man’s article~\cite{DM} to their attention during the early stages of this project.

For Theorem~\ref{thm:PrefMain}: the first assertion (regularity of $\Pref_T$) is proved in Corollary~\ref{cor:Refregular}, and the proof "relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m\in\mathbb{N})$" (with details in~\cite{HNW, DFHM24}). The second assertion "follows from Proposition~\ref{prop:Pref1}\." The third and fourth assertions are stated to be "consequence of the first two."

For Theorem~\ref{thm:PrefMain}: the first assertion (regularity of $\Pref_T$) is proved in
\begin{Corollary}\label{cor:Refregular}
The language $\Pref_T$ is regular and the generating function
\begin{align*}
\sum_{u\in \Pref_T} q^{|u|}& =\sum_{t\in T} |\{s_1 \cdots s_k s_{k+1} s_k\cdots s_1 \in \Pref_T \mid t=s_1 \cdots s_k s_{k+1}s_k\cdots s_1\}| \, q^{\ell(t)}\\
&=\sum_{k\in\mathbb{N}} a_k q^k
\end{align*}
is rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\RPal(q)=q\Pref_T(q^2)$ is rational.
\end{Corollary}
and the proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m\in\mathbb{N})$ (with details in~\cite{HNW, DFHM24}). The second assertion follows from
\begin{Proposition}\label{prop:Pref1} Let $t\in T$ with corresponding positive root $\alpha_t$. Let  $w\in W$ and assume there is $r\in D_R(w)$ such that $t=wrw^{-1}$ and $\ell(t)=2\ell(w)-1$. Then $D_R(w)=\{r\}$. Moreover:
 \begin{enumerate}

\item if $w=s_1\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\cdots s_k r s_k\cdots s_1$ is a palindromic reduced word for $t$;
\item $\alpha_t=w(\alpha_r)= s_1\cdots s_k(\alpha_r)$ and  $\dep(\alpha_t)=k=\ell(w)-1$.
\end{enumerate}
\end{Proposition}
The third and fourth assertions are stated to be consequence of the first two.

Question: Let (W,S) be a Coxeter system with S finite, let S* be the free monoid on S, and let π: S* → W be the canonical projection. A word s1⋯sk ∈ S* is reduced if k is minimal among words representing π(s1⋯sk), and then ℓ(π(s1⋯sk)) = k. Let T = ⋃_{w∈W} wSw^{-1} be the set of reflections. For u,v ∈ W, write u ≤_R v for the right weak order, i.e. ℓ(v) = ℓ(u) + ℓ(u^{-1}v). Since every reflection t ∈ T has odd length, if ℓ(t)=2k+1 then a t-prefix is an element p_t ∈ W such that p_t ≤_R t and ℓ(p_t)=k+1. Let Pref_T ⊆ S* be the set of reduced words of all reflection-prefixes, and let Pal be the language of palindromic reduced words for reflections. For B ⊆ S*, define B(q)=∑_{w∈B} q^{ℓ(π(w))}. Which statement holds for every such Coxeter system?

The language Pref_T is regular; moreover, the palindromic reduced words are exactly
Pal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk s_{k+1} ∈ Pref_T}.
In addition, the Poincaré series
Pref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k
is rational, where pal_k is the number of palindromic reduced words of length k; consequently,
Pal(q) = q Pref_T(q^2)
is rational.

Choices:
B.
The language Pref_T is regular; moreover, the palindromic reduced words are exactly
Pal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk ∈ Pref_T}.
In addition, the Poincaré series
Pref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k
is rational, where pal_k is the number of palindromic reduced words of length k; consequently,
Pal(q) = q Pref_T(q^2)
is rational.
C.
The language Pref_T is regular. Moreover, the Poincaré series
Pal(q)=\sum_{w\in \mathrm{Pal}} q^{\ell(\pi(w))}
is rational.
D.
The language Pref_T is regular; moreover, the palindromic reduced words are exactly
Pal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk s_{k+1} ∈ Pref_T}.
In addition, the Poincaré series
Pref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k
is rational, where pal_k is the number of palindromic reduced words of length k; consequently,
Pal(q) = Pref_T(q^2)
is rational.
E.
The language Pref_T is regular; moreover, every palindromic reduced word has the form
s1⋯sk s_{k+1} s_k⋯s1 with s1⋯sk s_{k+1} ∈ Pref_T, and every such word lies in Pal.
In addition, there is a single finite deterministic automaton, independent of any auxiliary parameter, that recognizes Pref_T for every Coxeter system; consequently the series Pref_T(q) and Pal(q) are rational.

full-prefix length k+1 in the characterization of palindromes

dropped the exact characterization of Pal and the explicit identity Pal(q)=q Pref_T(q^2)

odd-length shift encoded by Pal(q)=q Pref_T(q^2)

dependence on the family of m-Shi/Garside shadow automata rather than one parameter-free universal automaton


Question: Let $(W,S)$ be a Coxeter system, let $S^*$ be the free monoid on $S$, and let $T=\bigcup_{w\in W} wSw^{-1}$ be the set of reflections. A word in $S^*$ is reduced if it has minimal length among words representing its element of $W$. If $t\in T$ has odd length $\ell(t)=2k+1$, a $t$-prefix is an element $p\in W$ such that $p\le_R t$ and $\ell(p)=k+1$, and $\Pref_T$ is the language of reduced words for all reflection-prefixes. Let $\operatorname{Pal}$ be the language of palindromic reduced words, i.e. reduced words of the form $s_1\cdots s_k s_{k+1}s_k\cdots s_1$. For a language $B$ of reduced words, write $B(q)=\sum_{\boldsymbol w\in B} q^{\ell(\pi(\boldsymbol w))}$, where $\pi:S^*\to W$ is the canonical projection. Which statement correctly gives the complete description of palindromic reduced words in terms of reflection-prefixes, together with the resulting regularity and generating-series properties?

$\Pref_T$ is a regular language; moreover
$$
\operatorname{Pal}=\{s_1\cdots s_k s_{k+1}s_k\cdots s_1\in S^*\mid s_1\cdots s_k s_{k+1}\in \Pref_T\}.
$$
In addition,
$$
\Pref_T(q)=\sum_{\boldsymbol w\in \Pref_T} q^{\ell(\pi(\boldsymbol w))}=\sum_{k\ge 0} \operatorname{pal}_k q^k,
$$
where $\operatorname{pal}_k$ is the number of palindromic reduced words of length $k$, so $\Pref_T(q)$ is rational; and
$$
\operatorname{Pal}(q)=q\,\Pref_T(q^2)
$$
is also rational.

Choices:
B.
$\Pref_T$ is a regular language; moreover
$$
\operatorname{Pal}=\{s_1\cdots s_k s_{k+1}s_k\cdots s_1\in S^*\mid s_1\cdots s_k\in \Pref_T\}.
$$
In addition,
$$
\Pref_T(q)=\sum_{\boldsymbol w\in \Pref_T} q^{\ell(\pi(\boldsymbol w))}=\sum_{k\ge 0} \operatorname{pal}_k q^k,
$$
where $\operatorname{pal}_k$ is the number of palindromic reduced words of length $k$, so $\Pref_T(q)$ is rational; and
$$
\operatorname{Pal}(q)=q\,\Pref_T(q^2)
$$
is also rational.
C.
$\Pref_T$ is a regular language, and every word of the form
$$
s_1\cdots s_k s_{k+1}s_k\cdots s_1\in S^*
$$
with $s_1\cdots s_k s_{k+1}\in \Pref_T$ belongs to $\operatorname{Pal}$. Consequently, $\operatorname{Pal}(q)=q\,\Pref_T(q^2)$ is rational.
D.
$\Pref_T$ is a regular language; moreover
$$
\operatorname{Pal}=\{s_1\cdots s_k s_{k+1}s_k\cdots s_1\in S^*\mid s_1\cdots s_k s_{k+1}\in \Pref_T\}.
$$
In addition,
$$
\Pref_T(q)=\sum_{\boldsymbol w\in \Pref_T} q^{\ell(\pi(\boldsymbol w))}=\sum_{k\ge 0} \operatorname{pal}_k q^k,
$$
where $\operatorname{pal}_k$ is the number of palindromic reduced words of length $k$, so $\Pref_T(q)$ is rational; and
$$
\operatorname{Pal}(q)=\Pref_T(q^2)
$$
is also rational.
E.
$\Pref_T$ is a regular language; moreover
$$
\operatorname{Pal}=\{s_1\cdots s_k s_{k+1}s_k\cdots s_1\in S^*\mid s_1\cdots s_k s_{k+1}\in \Pref_T\}.
$$
In addition,
$$
\Pref_T(q)=\sum_{\boldsymbol w\in \Pref_T} q^{\ell(\pi(\boldsymbol w))}=\sum_{k\ge 0} \operatorname{pal}_k q^k,
$$
where $\operatorname{pal}_k$ is the number of palindromic reduced words of length $k$, so $\Pref_T(q)$ is rational; and for every Coxeter system there is a single integer $m\in\mathbb N$ such that the $m$-Shi Garside shadow automaton recognizes $\Pref_T$, hence
$$
\operatorname{Pal}(q)=q\,\Pref_T(q^2)
$$
is also rational.

prefix-length k+1 condition in the palindromic characterization

dropped the converse/equality in the description of $\operatorname{Pal}$ and omitted the full statement about $\Pref_T(q)$

missing initial factor $q$ in the substitution $\operatorname{Pal}(q)=q\Pref_T(q^2)$

uniform existence of one fixed $m$-Shi automaton for all Coxeter systems


\begin{Thm}\label{thm1.1}
	Assume $N\ge1$, $m\ge2$, and \textbf{\textup{$(A_1)$--$(A_3)$}}. Then for $\varepsilon>0$,
	\eqref{eq1.1} admits a sequence of sign-changing solutions
	\[
	\mathbf{u}^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H, ~ k\in\mathbb{N}^*,
	\]
	satisfying
	\[
	0<J_\varepsilon\bigl(\mathbf{u}^{(1)}_\varepsilon\bigr)
	< \cdots < J_\varepsilon\bigl(\mathbf{u}^{(k)}_\varepsilon\bigr)
    < J_\varepsilon\bigl(\mathbf{u}^{(k+1)}_\varepsilon\bigr)
	< \cdots, ~\text{and}~ \lim\limits_{k\to\infty}
	J_\varepsilon\bigl(\mathbf{u}^{(k)}_\varepsilon\bigr)=+\infty.
	\]
\end{Thm}

\label{introduction}
\par
In this paper we consider the following competitive Schr\"odinger system
\begin{equation}\label{eq1.1}
	\begin{cases}
		-\Delta u_i + u_i
		= \mu_i Q_\varepsilon(x-y_i) |u_i|^{2p-2}u_i
		+ \sum_{j\neq i} \lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\[6pt]
		u_i \in H^1(\mathbb{R}^N),\ i=1,\dots,m.
	\end{cases}
\end{equation}
where $N\ge1$, $m\ge2$. Nonlinear Schr\"odinger systems with competing interactions have been extensively studied in recent years, motivated both by physical models such as Bose--Einstein
condensates and nonlinear optics, and by the rich mathematical structure of coupled
elliptic equations.
Among the various phenomena that may occur in this context, a
particularly interesting situation arises when the nonlinear potentials are localized in
small regions that shrink to isolated points as a small parameter $\varepsilon>0$ tends to zero. In this case,
solutions may concentrate around finitely many prescribed points and exhibit a delicate
interaction between the different components.
\par
 We now state the structural assumptions that will be used throughout the paper. They are standard in this context and reflect the subcritical regime and the competitive nature of the system.
\begin{itemize}
  \item [$(A_1)$] $1<p<\frac{2^*}{2}$ if $N\ge3$, and $p>1$ if $N=1,2$,
where $2^*=\frac{2N}{N-2}$ if $N\ge3$.
  \item [$(A_2)$] $\mu_i>0$, $i=1,\dots,m$, $\lambda_{ij}=\lambda_{ji}<0$, for all $i\neq j$.
\item[$(A_3)$]
Let $Q_\varepsilon(x-y_i):=Q(\varepsilon x-y_i)$, where
$Q\in C(\mathbb{R}^N)$ is nonnegative, $\operatorname{supp}(Q)$ is bounded, and there exist exactly pairwise distinct points $y_1,\dots,y_m\in\mathbb{R}^N$ such that
\[
Q(-y_i)=\|Q\|_\infty=\max\limits_{x\in\mathbb{R}^N}Q(x)~\text{for all }i=1,\dots,m.
\]
\end{itemize}
\par
In this setting, the solutions $u_i$ may be interpreted as standing wave profiles of
different species which are attracted to regions where $Q_\varepsilon$ is
positive and repelled from their complement. The assumption $\lambda_{ij}<0$ means that
distinct components repel each other, which in turn favors spatial segregation, so the system \eqref{eq1.1} is competitive.
\begin{Exam}\label{exam1.1}
There are functions satisfying $(A_3)$.
Assume $N\ge2$ and fix $m$ pairwise distinct points $y_1,\dots,y_m\in\mathbb R^N$.
Set
\[
d:=\min_{i\neq j}|y_i-y_j|>0,
~
r:=\frac{d}{4}.
\]
Let $\rho\in C_0^\infty(\mathbb R^N)$ be the standard compactly supported bump
\[
\rho(x):=
\begin{cases}
\exp\!\bigl(-\frac{1}{1-|x|^2}\bigr), & |x|<1,\\[2mm]
0, & |x|\ge 1,
\end{cases}
~
\text{and define}~
\phi(x):=\frac{\rho(x)}{\rho(0)} .
\]
Then $\phi\in C_0^\infty(\mathbb R^N)$, $\phi\ge0$, $\operatorname{supp}(\phi)\subset \overline{B_1(0)}$,
$\phi(0)=1$, and $\phi(x)<1$ for all $x\neq0$.

For each $i=1,\dots,m$ define
\[
\phi_i(x):=\phi\!\left(\frac{x+y_i}{r}\right),
~ x\in\mathbb R^N,
\]
and set
\[
Q(x):=\max_{1\le i\le m}\phi_i(x),~ x\in\mathbb R^N.
\]
Then $Q$ satisfies $(A_3)$.
\end{Exam}
\par
In the scalar case, a fundamental contribution is due to Ackermann and Szulkin~\cite{AckermannSzulkin2013},
who showed that when the positive region of the nonlinear coefficient collapses to isolated
points, every nontrivial solution concentrates at one of these cores, and ground states
select a single core without splitting their mass.  In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or
symmetry assumption on the linear part. Since then, their approach has been extended
in various directions, including problems on the whole space and coupled systems;
see, e.g.,~\cite{ClappHernandezSantamariaSaldana2025,ClappSaldanaSzulkin2025,ZhongZou2014}
and the references therein. We point out that these works focus primarily on positive solutions or
the least energy solutions, for which the Mountain Pass Theorem (\cite{Willem1996}) and the
maximum principles are available. Very recently, in \cite{ClappSaldanaSzulkin2025}, Clapp, Salda\~na and Szulkin studied the system \eqref{eq1.1} with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as $\varepsilon\to0$. Furthermore, the
authors characterized the limit profile and described how the components either decouple
or remain coupled depending on the geometry of the attraction centers. Related results
for scalar equations, including concentration of semiclassical states and the description
of their limit profiles, were obtained in earlier works such as
\cite{AckermannSzulkin2013,FangWang2020}, and see
\cite{TavaresTerracini2012} and the references therein for phase separation phenomena of competitive systems. We refer \cite{Clapp-Pistoia-Saldana-2026-JMPA} to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.
\par
However, much less is known about sign-changing solutions for systems of the form
\eqref{eq1.1}. Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers \cite{ClappHernandezSantamariaSaldana2025,Clapp-Pistoia-Saldana-2026-JMPA} concerned with this topic. In \cite{ClappHernandezSantamariaSaldana2025}, Clapp, Hern\'andez-Santamar\'ia and Salda\~na obtained the existence and concentration of  nodal solutions via the nodal Nehari manifold method,  and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In \cite{Clapp-Pistoia-Saldana-2026-JMPA}, Clapp, Pistoia and Salda\~na established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method.
But for systems, the situation is substantially more
delicate, since one has to keep track simultaneously of the sign structure of each
component and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system \eqref{eq1.1}.
\par
First, we state the meaning of sign-changing solutions of \eqref{eq1.1}.
\begin{Def}\label{def1.1}
A solution $\mathbf{u}=(u_1,u_2,\cdots,u_m)$ of \eqref{eq1.1} is called \emph{sign-changing} if for each $i=1,\dots,m$, both the positive part $u_i^+=\max\{u_i,0\}$ and the negative part $u_i^-=\min\{u_i,0\}$ are nonzero in $H^1(\mathbb{R}^N)$, i.e., $\|u_i^\pm\|_{H^1}>0$.
\end{Def}
\par
The purpose of
this paper is to develop a variational framework to seek for infinitely many sign-changing solutions of \eqref{eq1.1}, and to analyze their concentration behavior as $\varepsilon\to 0$.
The main technical difficulties arise from the sign-changing nature of the solutions.
First, the natural Nehari manifold associated with nonnegative solutions does
not capture directly sign changing solutions, so for elements in the Nehari manifold, one has to impose constraints separately on the
positive and negative parts of each component. This leads to a nonlinear and
nonconvex constraint set on which the direct minimization is not available.
Second, the competitive couplings $\lambda_{ij}<0$ mix the components in a nontrivial way, and one must show that on the nodal constraint the energy still controls the
$H^{1}$-norm uniformly, so that Palais-Smale sequences are bounded.
Third, compactness issues are more severe in the nodal setting, one has to
prevent vanishing of some sign components and to rule out loss of mass at
infinity while keeping track of the nodal structure. So it is interesting to seek for sign-changing solutions of \eqref{eq1.1} since the above difficulties prevent one to use techniques based on the maximum principles, the comparison arguments, or the monotonicity methods.
These obstacles make it necessary to develop a variational strategy that is tailored to
the nodal structure of the problem and that remains effective in the presence of
competitive couplings.
\par
We are now in a position to state the main results. Our first theorem
shows that, for small $\varepsilon>0$, system~\eqref{eq1.1} admits a sequence of sign-changing solutions with unbounded energies.

For each $i=1,\dots,m$ define
\[
\phi_i(x):=\phi\!\left(\frac{x+y_i}{r}\right),
~ x\in\mathbb R^N,
\]
and set
\[
Q(x):=\max_{1\le i\le m}\phi_i(x),~ x\in\mathbb R^N.
\]
Then $Q$ satisfies $(A_3)$.
\end{Exam}
\par
In the scalar case, a fundamental contribution is due to Ackermann and Szulkin~\cite{AckermannSzulkin2013},
who showed that when the positive region of the nonlinear coefficient collapses to isolated
points, every nontrivial solution concentrates at one of these cores, and ground states
select a single core without splitting their mass. In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or
symmetry assumption on the linear part. Since then, their approach has been extended
in various directions, including problems on the whole space and coupled systems;
see, e.g.,~\cite{ClappHernandezSantamariaSaldana2025,ClappSaldanaSzulkin2025,ZhongZou2014}
and the references therein. We point out that these works focus primarily on positive solutions or
the least energy solutions, for which the Mountain Pass Theorem (\cite{Willem1996}) and the
maximum principles are available. Very recently, in \cite{ClappSaldanaSzulkin2025}, Clapp, Salda\~na and Szulkin studied the system \eqref{eq1.1} with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as $\varepsilon\to0$. Furthermore, the
authors characterized the limit profile and described how the components either decouple
or remain coupled depending on the geometry of the attraction centers. Related results
for scalar equations, including concentration of semiclassical states and the description
of their limit profiles, were obtained in earlier works such as
\cite{AckermannSzulkin2013,FangWang2020}, and see
\cite{TavaresTerracini2012} and the references therein for phase separation phenomena of competitive systems. We refer \cite{Clapp-Pistoia-Saldana-2026-JMPA} to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.
\par
However, much less is known about sign-changing solutions for systems of the form
\eqref{eq1.1}. Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers \cite{ClappHernandezSantamariaSaldana2025,Clapp-Pistoia-Saldana-2026-JMPA} concerned with this topic. In \cite{ClappHernandezSantamariaSaldana2025}, Clapp, Hern\'andez-Santamar\'ia and Salda\~na obtained the existence and concentration of nodal solutions via the nodal Nehari manifold method, and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In \cite{Clapp-Pistoia-Saldana-2026-JMPA}, Clapp, Pistoia and Salda\~na established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method.
But for systems, the situation is substantially more
delicate, since one has to keep track simultaneously of the sign structure of each
component and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system \eqref{eq1.1}.
\par
First, we state the meaning of sign-changing solutions of \eqref{eq1.1}.
\begin{Def}\label{def1.1}
A solution $\mathbf{u}=(u_1,u_2,\cdots,u_m)$ of \eqref{eq1.1} is called \emph{sign-changing} if for each $i=1,\dots,m$, both the positive part $u_i^+=\max\{u_i,0\}$ and the negative part $u_i^-=\min\{u_i,0\}$ are nonzero in $H^1(\mathbb{R}^N)$, i.e., $\|u_i^\pm\|_{H^1}>0$.
\end{Def}
\par
The purpose of
this paper is to develop a variational framework to seek for infinitely many sign-changing solutions of \eqref{eq1.1}, and to analyze their concentration behavior as $\varepsilon\to 0$.
The main technical difficulties arise from the sign-changing nature of the solutions.
First, the natural Nehari manifold associated with nonnegative solutions does
not capture directly sign changing solutions, so for elements in the Nehari manifold, one has to impose constraints separately on the
positive and negative parts of each component. This leads to a nonlinear and
nonconvex constraint set on which the direct minimization is not available.
Second, the competitive couplings $\lambda_{ij}<0$ mix the components in a nontrivial way, and one must show that on the nodal constraint the energy still controls the
$H^{1}$-norm uniformly, so that Palais-Smale sequences are bounded.
Third, compactness issues are more severe in the nodal setting, one has to
prevent vanishing of some sign components and to rule out loss of mass at
infinity while keeping track of the nodal structure. So it is interesting to seek for sign-changing solutions of \eqref{eq1.1} since the above difficulties prevent one to use techniques based on the maximum principles, the comparison arguments, or the monotonicity methods.
These obstacles make it necessary to develop a variational strategy that is tailored to
the nodal structure of the problem and that remains effective in the presence of
competitive couplings.
\par
We are now in a position to state the main results. Our first theorem
shows that, for small $\varepsilon>0$, system~\eqref{eq1.1} admits a sequence of sign-changing solutions with unbounded energies.

\medskip

By \eqref{eq2.3} for $\mathbf{u}_{\varepsilon_n}$ and make a change of variables we obtain
\begin{equation}\label{eq5.20}
\begin{aligned}
\|\tilde{u}_{j,n}^\sigma\|_{H^1}^2 &= \mu_j \int_{\mathbb{R}^N} Q_{\varepsilon_n}(x + x_{i,n} - y_j) |\tilde{u}_{j,n}^\sigma|^{2p} \, dx
+ \sum_{k \neq j} \lambda_{jk} \int_{\mathbb{R}^N} |\tilde{u}_{k,n}|^p
|\tilde{u}_{j,n}^\sigma|^p \, dx\\
&\le \mu_j \int_{\mathbb{R}^N} Q_{\varepsilon_n}(x + x_{i,n} - y_j) |\tilde{u}_{j,n}^\sigma|^{2p} \, dx,
\end{aligned}
\end{equation}
since $\lambda_{jk} < 0$.
\par
For any $M>0$, it follows from $\tilde{u}_{j,n}^\sigma \to 0$ in $L^{2p}(B_M(0))$ and $|Q({\varepsilon_n}(x + x_{i,n} - y_j)| \leq \|Q\|_\infty$ that
\begin{equation}\label{eq5.21}
\int_{B_M(0)} Q_{\varepsilon_n}(x + x_{i,n} - y_j) |\tilde{u}_{j,n}^\sigma|^{2p} \, dx \to 0.
\end{equation}
On the other hand, by the standard Agmon-type estimates (see \cite{Agmon1982}), there exist $C,\alpha>0$ independent of $n$, such that
\[
|\tilde{u}_{j,n}(x)|\leq C e^{-\alpha|x|} \quad \text{for all } x\in\mathbb{R}^N.
\]
Consequently,
\[
\int_{|x|>M} Q_{\epsilon_n}(x+x_{i,n}-y_j)|\tilde{u}_{j,n}^\sigma|^{2p}dx
\leq \|Q\|_\infty \int_{|x|>M} |\tilde{u}_{j,n}^\sigma|^{2p}dx
\leq \|Q\|_\infty\, C^{2p} \int_{|x|>M} e^{-2p\alpha|x|}dx.
\]
For any $\eta>0$, choose $M=M(\eta)>0$ such that
\[
\int_{|x|>M} e^{-2p\alpha|x|}dx < \eta,
\]
jointly with \eqref{eq5.20} and \eqref{eq5.21}, implies that $\|\tilde{u}_{j,n}^\sigma\|_{H^1}\to0$. This contradicts \eqref{eq5.19}.
 Therefore,
$U_j^\sigma \neq 0$ for every $j$ and $\sigma$, i.e., $\mathbf{U} \in \mathcal{A}$.
\par
We claim that the sequence $\{\varepsilon_n x_{i,n}\}$ is bounded.
 Assume by contradiction that, up to a subsequence, $|\varepsilon_n x_{i,n}|\to\infty$.
 Fix $\varphi\in C_c^\infty(\R^N)$, and set $K:=\operatorname{supp}\varphi$.
 Since $\mathbf u_{\varepsilon_n}$ is a weak solution of the system, changing variables gives
 \begin{align*}
 \int_{\R^N}\bigl(\nabla \tilde u_{\ell,n}\cdot\nabla\varphi+\tilde u_{\ell,n}\varphi\bigr)\,dx
 &=
 \mu_\ell\int_{\R^N} Q\bigl(\varepsilon_n(x+x_{i,n})-y_\ell\bigr)\,
 |\tilde u_{\ell,n}|^{2p-2}\tilde u_{\ell,n}\,\varphi\,dx \\
 &~
 +\sum_{j\neq \ell}\lambda_{\ell j}\int_{\R^N}
 |\tilde u_{j,n}|^{p}|\tilde u_{\ell,n}|^{p-2}\tilde u_{\ell,n}\,\varphi\,dx .
 \end{align*}
 By $(A_3)$, $\operatorname{supp}(Q)\subset B_{R_Q}(0)$ for some $R_Q>0$.
 Since $K$ is bounded and $|\varepsilon_n x_{i,n}|\to\infty$, for all large $n$ we have
 \[
 \bigl|\varepsilon_n(x+x_{i,n})-y_\ell\bigr|
 \ge |\varepsilon_n x_{i,n}|-|y_\ell|-\varepsilon_n|x|>R_Q
 ~\text{for all }x\in K,
 \]
 hence $Q(\varepsilon_n(x+x_{i,n})-y_\ell)\equiv0$ on $K$ for all large $n$.
 Letting $n\to\infty$ and using $\tilde{\mathbf u}_n\to\mathbf U$ in $L^{2p}_{\rm loc}(\R^N)^m$, we obtain
 \[
 \int_{\R^N}\bigl(\nabla U_\ell\cdot\nabla\varphi+U_\ell\varphi\bigr)\,dx
 =
 \sum_{j\neq \ell}\lambda_{\ell j}\int_{\R^N}|U_j|^{p}|U_\ell|^{p-2}U_\ell\,\varphi\,dx,
 ~\forall\,\varphi\in C_c^\infty(\R^N).
 \]
Choose $\chi_R\in C_c^\infty(\R^N)$ such that $0\le \chi_R\le 1$, $\chi_R\equiv 1$ on $B_R(0)$,
$\operatorname{supp}(\chi_R)\subset B_{2R}(0)$ and $|\nabla\chi_R|\le C/R$.
Testing the above identity with $\varphi=\chi_R^2U_\ell$ and using $\lambda_{\ell j}<0$, we obtain
\[
\int_{\R^N}\Bigl(\nabla U_\ell\cdot\nabla(\chi_R^2U_\ell)+\chi_R^2|U_\ell|^2\Bigr)\,dx
=\sum_{j\neq \ell}\lambda_{\ell j}\int_{\R^N}\chi_R^2|U_j|^p|U_\ell|^p\,dx\le 0.
\]
Moreover, by the product rule,
\[
\int_{\R^N}\nabla U_\ell\cdot\nabla(\chi_R^2U_\ell)\,dx
=\int_{\R^N}\Bigl(|\nabla(\chi_RU_\ell)|^2-|U_\ell|^2|\nabla\chi_R|^2\Bigr)\,dx.
\]
Consequently,
\[
\int_{\R^N}\Bigl(|\nabla(\chi_RU_\ell)|^2+\chi_R^2|U_\ell|^2\Bigr)\,dx
\le \int_{\R^N}|U_\ell|^2|\nabla\chi_R|^2\,dx.
\]
Since $|\nabla\chi_R|\le C/R$ and $\operatorname{supp}(\nabla\chi_R)\subset B_{2R}(0)\setminus B_R(0)$, we have
\[
\int_{\R^N}|U_\ell|^2|\nabla\chi_R|^2\,dx
\le \frac{C}{R^2}\int_{B_{2R}(0)\setminus B_R(0)}|U_\ell|^2\,dx\to 0
\]
as $R\to\infty$, which implies
\[
\int_{\R^N}\bigl(|\nabla U_\ell|^2+|U_\ell|^2\bigr)\,dx=0,
\]
i.e., $U_\ell= 0$, which contradicts $U_\ell\neq 0$. Consequently, passing to a subsequence, there exists $\xi_i\in\R^N$ such that
 \begin{equation}\label{eq5.22}
 \varepsilon_n x_{i,n}\to \xi_i~\text{as}~n\to\infty.
 \end{equation}
\par
 Next, we show that $\xi_i = 0$ by contradiction. Suppose that $\xi_i \neq 0$.
\par
We first claim that
\begin{equation}\label{eq5.23}
 Q(\xi_i - y_\ell)<Q(-y_\ell)~\text{for some}~\ell\in\{1,\cdots,m\}.
\end{equation}
Suppose to the contrary that $Q(\xi_i-y_\ell)=Q(-y_\ell)=\|Q\|_\infty$ for all $\ell$. Then, by $(A_3)$, for each $\ell$, there exists$j_\ell\in{1,\dots,m}$ such that $\xi_i-y_\ell=-y_{j_\ell}$. If $j_\ell=\ell$ for some $\ell$, then $\xi_i=0$, contradicting $\xi_i\neq0$. Hence $j_\ell\neq\ell$ for all $\ell$. Define a map $\sigma:\{1,\dots,m\}\to\{1,\dots,m\}$ by $\sigma(\ell)=j_\ell$. Since the points $y_1,\dots,y_m$ are pairwise distinct, $\sigma$ is a permutation of $\{1,\dots,m\}$ without fixed points. Moreover, we have
\begin{equation*}
y_{\sigma(\ell)}=y_\ell-\xi_i,~\text{for each}~\ell.
\end{equation*}
Iterating this relation gives $y_{\sigma^k(\ell)}=y_\ell-k\xi_i$ for any positive integer $k$. Because the set $\{y_1,\dots,y_m\}$ is finite, there exist $k_1\neq k_2$ such that $y_{\sigma^{k_1}(\ell)}=y_{\sigma^{k_2}(\ell)}$, which implies $(k_1-k_2)\xi_i=0$ and hence $\xi_i=0$, a contradiction. Therefore, \eqref{eq5.23} must hold.
\par
Observe that $\mathbf{U}$ is a weak solution of
\begin{equation}\label{eq5.24}
-\Delta U_\ell + U_\ell = \mu_\ell Q(\xi_i - y_\ell) |U_\ell|^{2p-2}U_\ell + \sum_{j\neq\ell}\lambda_{\ell j}|U_j|^p |U_\ell|^{p-2}U_\ell, ~ \ell=1,\dots,m .
\end{equation}
Denote by $J_\xi$ the energy functional associated with \eqref{eq5.24} and recall $J_0$ be the original limiting functional with coefficients $Q(-y_\ell)$
\begin{equation*}
-\Delta U_\ell + U_\ell = \mu_\ell Q(- y_\ell) |U_\ell|^{2p-2}U_\ell + \sum_{j\neq\ell}\lambda_{\ell j}|U_j|^p |U_\ell|^{p-2}U_\ell, ~ \ell=1,\dots,m .
\end{equation*}
\par
From Lemma \ref{Lem5.3} we have $c_{\varepsilon_n,k} \to c_{0,k}$. The concentration-compactness principle together with the Br\'ezis--Lieb lemma yield
\begin{equation}\label{eq5.25}
c_{0,k} = \lim_{n\to\infty} J_{\varepsilon_n}(\mathbf{u}_{\varepsilon_n}) \ge J_\xi(\mathbf{U}).
\end{equation}
\par
By \eqref{eq5.23} and $U_\ell \neq 0$, we obtain
\begin{equation}\label{eq5.26}
\begin{aligned}
J_\xi(\mathbf{U}) - J_0(\mathbf{U}) &= \frac{1}{2p}\sum_{\ell=1}^m \mu_\ell\bigl[Q(-y_\ell) - Q(\xi_i - y_\ell)\bigr] \int_{\mathbb{R}^N} |U_\ell|^{2p}\\
&\ge \frac{1}{2p}\mu_\ell\bigl[Q(-y_\ell) - Q(\xi_i-y_\ell)\bigr] \int |U_\ell|^{2p} > 0.
\end{aligned}
\end{equation}

\par
To treat sign-changing solutions of competitive Schr\"odinger systems with shrinking nonlinear potentials, we develop a variational framework tailored to the nodal setting of \eqref{eq1.1}, inspired by the ideas in
\cite{AckermannSzulkin2013,ClappSaldanaSzulkin2025} but adapted to the present
system. We work in the product space $H:=(H^1(\mathbb{R}^N))^m$ endowed with the natural norm, and we consider the energy functional associated with
\eqref{eq1.1} $J_\varepsilon:H\to\mathbb{R}$. Our first step is to introduce a fully sign-changing Nehari set
$\mathcal{N}^{\mathrm{sc}}_\varepsilon$ (see \S \ref{sec:framework}), by imposing Nehari type constraints
separately on the positive and negative parts of each component. On a suitable open
subset $\mathcal{A}$ of $H$, rather than on a
linear subspace, we define a nodal projection
\[
m_\varepsilon:\mathcal{A}\longrightarrow
\mathcal{N}^{\mathrm{sc}}_\varepsilon,
\]
and we use it to construct an even reduced functional
$\Psi_\varepsilon:=J_\varepsilon\circ m_\varepsilon$ on
$\mathcal{A}$. The symmetry of $\Psi_\varepsilon$ allows us to apply
Krasnosel'skii genus theory and to implement a symmetric minimax scheme, which provide infinitely many critical values and
	yield a sequence of nodal solutions with unbounded energies.
This feature is essential for applying genus theory in a genuinely nodal
setting and allows us to recover a symmetric minimax structure despite the strong
nonlinearity of the constraints. The proof of Theorem~\ref{thm1.1} will be carried out in Section~\ref{sec:existence}.

\medskip

The second result in this paper concerns the asymptotic behavior of the above sign-changing solutions as $\varepsilon\to 0^+$. After a suitable rescaling, the potentials $Q_\varepsilon$ converge to the constants $Q(-y_i)$, and one is naturally led to the limiting autonomous system
\begin{equation}\label{eq1.2}
	\begin{cases}
	-\Delta U_i + U_i
	= \mu_i Q(-y_i)\,|U_i|^{2p-2}U_i
	+ \sum_{j\neq i}\lambda_{ij}|U_j|^p|U_i|^{p-2}U_i,\\[6pt]
	 U_i\in H^1(\mathbb{R}^N),\ i=1,\dots,m.
\end{cases}
\end{equation}
The energy functional associated with
\eqref{eq1.2} is denoted by $J_0(\mathbf{u}): H\to\mathbb{R}$.
\begin{Thm}\label{thm1.2}
	Assume $N\ge1$, $m\ge2$, and \textup{(A$_1$)--(A$_3$)}.
	Let $\{\varepsilon_n\}\subset(0,+\infty)$ be a sequence with $\varepsilon_n\to 0$.
	For each fixed $k\in\mathbb{N}^*$, let $\{\mathbf{u}^{(k)}_{\varepsilon_n}\}$ be the sequence of
	sign-changing solutions given by Theorem~\ref{thm1.1}.
	Then there exist $m$ sequences $\{x^{(k)}_{\varepsilon_n,i}\}\subset\mathbb{R}^N$ and a sign-changing solution
	$\mathbf{U}^{(k)}=(U^{(k)}_1,\dots,U^{(k)}_m)\in H$ of the limiting system \eqref{eq1.2}
	such that, up to a subsequence,
	\begin{itemize}
		\item[(i)]
		$\varepsilon_n x_{\varepsilon_n,i}^{(k)}\to 0$ as $n\to\infty$ for $i=1,\dots,m$.
	In particular, for every $R>0$,
		\[
		Q\bigl(\varepsilon_n(\,\cdot+x_{\varepsilon_n,i}^{(k)})-y_i\bigr)\to Q(-y_i)
		~\text{uniformly in}~B_R(0),~i=1,\dots,m.
		\]

		\item[(ii)]
		$u^{(k)}_{\varepsilon_n,i}(\,\cdot + x^{(k)}_{\varepsilon_n,i})
		\longrightarrow U^{(k)}_i$ in $H^1(\mathbb{R}^N)$ as $n\to\infty$ for  $i=1,\dots,m$.

	\item[(iii)]
	for every $q\in[1,\infty)$,
	\[
	\lim_{R\to+\infty}\,\limsup_{n\to\infty}
	\int_{\mathbb{R}^N\setminus B_{R/{\varepsilon_n}}(0)}
	|u^{(k)}_{\varepsilon_n, i}(x)|^q\,dx = 0,
	~ i=1,\dots,m,
	\]
	and for $q=\infty$,
	\[
	\lim_{R\to+\infty}\,\limsup_{n\to\infty}
	\|u^{(k)}_{\varepsilon_n, i}\|_{L^\infty(\mathbb{R}^N\setminus B_{R/{\varepsilon_n}}(0))} = 0,
	~ i=1,\dots,m.
	\]

		\item[(iv)]
		$\{\mathbf{U}^{(k)}\}$ are pairwise distinct and satisfy
		\[
		J_0\big(\mathbf{U}^{(1)}\big) < J_0\big(\mathbf{U}^{(2)}\big) < \cdots \to +\infty.
		\]
	\end{itemize}
\end{Thm}

\begin{Rem}\label{rem1.2}
	Theorem~\ref{thm1.2} extends the concentration results for nonnegative solutions obtained in
	\cite{ClappSaldanaSzulkin2025} to the sign-changing case.
	Assertions \textup{(i)}--\textup{(iii)} show that, after suitable translations, each component concentrates
	around the origin, while assertion \textup{(iv)} reveals a new phenomenon: the limiting autonomous system
	inherits infinitely many sign-changing solutions with unbounded energies, arising from the genus-based
	construction for the original problem.
\end{Rem}

The proof of Theorem~\ref{thm1.2} is based on a concentration--compactness argument for suitable
translations of sign-changing solutions $\{\mathbf u_n\}$, which yields a nontrivial limiting profile solving
\eqref{eq1.2} and the strong convergence in (ii) via a Br\'ezis--Lieb type energy
decomposition.
The decay property in (iii) follows from localized estimates with cut--off functions, interpolation,
and Agmon-type exponential decay bounds; the case $q=\infty$ is obtained by a translation argument
combined with local elliptic regularity and the compact support of $Q$.
It is carried out in Section~\ref{sec:concentration}.

The paper is organized as follows.
In Section~\ref{sec:framework} we introduce the variational setting, the fully nodal Nehari set and the reduced functional, and state our assumptions on the nonlinear potentials.
In Section~\ref{sec:ps} we establish a compactness result for the reduced functional $\Psi_{\varepsilon}$, which will be crucial in the construction of sign-changing solutions.
Section~\ref{sec:existence} is devoted to a genus-based minimax scheme on $\mathcal N_{\varepsilon}^{\mathrm{sc}}$, and hence obtain infinitely many sign-changing critical points. In Section~\ref{sec:concentration} we prove the concentration result Theorem~\ref{thm1.2}, including the
description of the limiting profiles for the semiclassical states.
\medskip

\textbf{Notation.}
\begin{itemize}
	\item Let $w^+(x):=\max\{w(x),0\}$ and $w^-(x):=\min\{w(x),0\}$ be the positive and negative parts of $w$, respectively.
	\item For $r>0$ and $x\in\mathbb{R}^N$, $B_r(x)$ is the open ball centered at $x$ with radius $r$.
    \item Boldface $\mathbf{u}$ denote a $\mathbb{R}^m$ vector-value function, $u_i$ denotes the $i$-th component function of $\mathbf{u}$.
    \item "$\to$" denotes the strong convergence and "$\rightharpoonup$" denotes the weak convergence.
    \item For $k\in\mathbb{Z}^N$, we denote by $k*\mathbf u$ the integer translation of $\mathbf u$:
    \[
    (k*\mathbf u)(x):=\mathbf u(x-k),~ x\in\mathbb{R}^N .
    \]
    \item For a function space $E$, $E^m$ denotes the product space $\underbrace{E\times\cdots\times E}_{m}$.
  \end{itemize}

To prove Theorem~\ref{thm1.1}, the authors “develop a variational framework tailored to the nodal setting of \eqref{eq1.1}” in the product space $H:=(H^1(\mathbb{R}^N))^m$ and consider the associated energy functional $J_\varepsilon:H\to\mathbb{R}$. The “first step is to introduce a fully sign-changing Nehari set $\mathcal{N}^{\mathrm{sc}}_\varepsilon$ … by imposing Nehari type constraints separately on the positive and negative parts of each component.” On “a suitable open subset $\mathcal{A}$ of $H$, rather than on a linear subspace,” they define a nodal projection $m_\varepsilon:\mathcal{A}\to\mathcal{N}^{\mathrm{sc}}_\varepsilon$ and use it to construct “an even reduced functional $\Psi_\varepsilon:=J_\varepsilon\circ m_\varepsilon$ on $\mathcal{A}$.” Since “the symmetry of $\Psi_\varepsilon$ allows us to apply Krasnosel'skii genus theory,” they “implement a symmetric minimax scheme,” which “provide[s] infinitely many critical values” and “yield[s] a sequence of nodal solutions with unbounded energies,” i.e., the desired infinitely many sign-changing solutions with increasing energy levels. The proof is said to be “carried out in Section~\ref{sec:existence},” with compactness for $\Psi_\varepsilon$ established earlier as “crucial in the construction of sign-changing solutions.”

To prove the main theorem, the authors “develop a variational framework tailored to the nodal setting of
\begin{equation}\label{eq1.1}
	\begin{cases}
		-\Delta u_i + u_i
		= \mu_i Q_\varepsilon(x-y_i) |u_i|^{2p-2}u_i
		+ \sum_{j\neq i} \lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\[6pt]
		u_i \in H^1(\mathbb{R}^N),\ i=1,\dots,m.
	\end{cases}
\end{equation}
” in the product space $H:=(H^1(\mathbb{R}^N))^m$ and consider the associated energy functional $J_\varepsilon:H\to\mathbb{R}$. The “first step is to introduce a fully sign-changing Nehari set $\mathcal{N}^{\mathrm{sc}}_\varepsilon$ … by imposing Nehari type constraints separately on the positive and negative parts of each component.” On “a suitable open subset $\mathcal{A}$ of $H$, rather than on a linear subspace,” they define a nodal projection $m_\varepsilon:\mathcal{A}\to\mathcal{N}^{\mathrm{sc}}_\varepsilon$ and use it to construct “an even reduced functional $\Psi_\varepsilon:=J_\varepsilon\circ m_\varepsilon$ on $\mathcal{A}$.” Since “the symmetry of $\Psi_\varepsilon$ allows us to apply Krasnosel'skii genus theory,” they “implement a symmetric minimax scheme,” which “provide[s] infinitely many critical values” and “yield[s] a sequence of nodal solutions with unbounded energies,” i.e., the desired infinitely many sign-changing solutions with increasing energy levels. The proof is said to be “carried out later,” with compactness for $\Psi_\varepsilon$ established earlier as “crucial in the construction of sign-changing solutions.”

\label{thm1.1}
	Assume $N\ge1$, $m\ge2$, and \textbf{\textup{$(A_1)$--$(A_3)$}}. Then for $\varepsilon>0$,
	\begin{equation}\label{eq1.1}
	\begin{cases}
		-\Delta u_i + u_i
		= \mu_i Q_\varepsilon(x-y_i) |u_i|^{2p-2}u_i
		+ \sum_{j\neq i} \lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\[6pt]
		u_i \in H^1(\mathbb{R}^N),\ i=1,\dots,m.
	\end{cases}
\end{equation}
	admits a sequence of sign-changing solutions
	\[
	\mathbf{u}^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H, ~ k\in\mathbb{N}^*,
	\]
	satisfying
	\[
	0<J_\varepsilon\bigl(\mathbf{u}^{(1)}_\varepsilon\bigr)
	< \cdots < J_\varepsilon\bigl(\mathbf{u}^{(k)}_\varepsilon\bigr)
    < J_\varepsilon\bigl(\mathbf{u}^{(k+1)}_\varepsilon\bigr)
	< \cdots, ~\text{and}~ \lim\limits_{k\to\infty}
	J_\varepsilon\bigl(\mathbf{u}^{(k)}_\varepsilon\bigr)=+\infty.
	\]

Question:
Assume $N\ge 1$, $m\ge 2$, and hypotheses $(A_1)$--$(A_3)$. For a fixed $\varepsilon>0$, consider the system
\[
\begin{cases}
-\Delta u_i+u_i=\mu_i Q_\varepsilon(x-y_i)|u_i|^{2p-2}u_i+\displaystyle\sum_{j\ne i}\lambda_{ij}|u_j|^p|u_i|^{p-2}u_i,\\[4pt]
u_i\in H^1(\mathbb R^N),\quad i=1,\dots,m,
\end{cases}
\]
and write $H=(H^1(\mathbb R^N))^m$. A vector solution $\mathbf u=(u_1,\dots,u_m)$ is called sign-changing if for every $i=1,\dots,m$, both $u_i^+=\max\{u_i,0\}$ and $u_i^-=\min\{u_i,0\}$ are nonzero in $H^1(\mathbb R^N)$. Let $J_\varepsilon$ denote the energy functional associated with this system. Under these assumptions, which asymptotic statement in the index $k\to\infty$ holds for sign-changing solutions?

For every $\varepsilon>0$, the system admits a sequence of sign-changing solutions
\[
\mathbf u^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H,\qquad k\in\mathbb N^*,
\]
such that
\[
0<J_\varepsilon\bigl(\mathbf u^{(1)}_\varepsilon\bigr)<\cdots<J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)<J_\varepsilon\bigl(\mathbf u^{(k+1)}_\varepsilon\bigr)<\cdots,
\]
and
\[
\lim_{k\to\infty}J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)=+\infty.
\]

Choices:
B.
For every $\varepsilon>0$, the system admits a sequence of sign-changing solutions
\[
\mathbf u^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H,\qquad k\in\mathbb N^*,
\]
such that
\[
0<J_\varepsilon\bigl(\mathbf u^{(1)}_\varepsilon\bigr)<\cdots<J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)<J_\varepsilon\bigl(\mathbf u^{(k+1)}_\varepsilon\bigr)<\cdots,
\]
and
\[
\lim_{k\to\infty}J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)=:c_\varepsilon<+\infty.
\]
C.
For every $\varepsilon>0$, the system admits infinitely many sign-changing solutions
\[
\mathbf u^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H,\qquad k\in\mathbb N^*,
\]
with
\[
J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)>0\qquad\text{for all }k\in\mathbb N^*.
\]
D.
There exists a sequence of sign-changing solutions
\[
\mathbf u^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H,\qquad k\in\mathbb N^*,
\]
such that for every $\varepsilon>0$,
\[
0<J_\varepsilon\bigl(\mathbf u^{(1)}_\varepsilon\bigr)<\cdots<J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)<J_\varepsilon\bigl(\mathbf u^{(k+1)}_\varepsilon\bigr)<\cdots,
\]
and
\[
\lim_{k\to\infty}J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)=+\infty.
\]
E.
For every $\varepsilon>0$, the system admits a sequence of sign-changing solutions
\[
\mathbf u^{(k)}_\varepsilon=(u^{(k)}_{\varepsilon,1},\dots,u^{(k)}_{\varepsilon,m})\in H,\qquad k\in\mathbb N^*,
\]
such that
\[
0<J_\varepsilon\bigl(\mathbf u^{(1)}_\varepsilon\bigr)\leq\cdots\leq J_\varepsilon\bigl(\mathbf u^{(k)}_\varepsilon\bigr)\leq J_\varepsilon\bigl(\mathbf u^{(k+1)}_\varepsilon\bigr)\leq\cdots,
\]
and the set $\{J_\varepsilon(\mathbf u^{(k)}_\varepsilon):k\in\mathbb N^*\}$ is infinite.