\begin{theorem}\label{thm1}
There exists a constant $C>1$ such that
\begin{equation}\label{equnthm1}
C^{-1}\lambda^{d_S/2}\leq{N}(u)\leq C\lambda^{d_S/2}
\end{equation}
holds for any global Dirichlet or Neumann eigenfunction $u$  on $\mathcal {SG}$ with  eigenvalue $\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.
\end{theorem}

\label{sec1}
The study of Laplacians on fractals and their spectral properties constitutes a cornerstone of analysis on fractals, a field largely pioneered by Kigami \cite{K2,K4}. Unlike their Euclidean counterparts, Laplacians on fractals exhibit a wealth of novel and unexpected phenomena, leading to a rich and distinct spectral theory \cite{FS, Ka1,Ka2,K3,KL, Sh, Sh2, T}. Among these phenomena, the intricate oscillatory behavior of eigenfunctions — such as the distribution and growth of their local extrema — remains a topic of profound interest.

In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \cite{Ho} show that the $L^\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.

The landscape on fractals is quite different. The existence of pre-localized eigenfunctions — eigenfunctions that vanish identically on the boundary along with their normal derivatives — is a hallmark of fractals \cite{K3,K4}. Such eigenfunctions can be highly localized and give rise to  a cascade of new eigenfunctions through a localization process. This structure fundamentally alters the asymptotic distribution of eigenvalues and the qualitative behavior of eigenfunctions. Consequently, classical tools and intuition from elliptic PDEs often fail, necessitating new frameworks for understanding the fine properties of eigenfunctions on fractals.

\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{SG2.pdf}
\caption{The Sierpinski gasket $\mathcal{SG}$.}
\label{figure1}
\end{center}
\end{figure}

This paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\mathcal{SG}$, see Figure \ref{figure1}) serves as a typical  example. We introduce and investigate the extremum counting function
$N(x)$ for the Laplacian $-\Delta_\mu$ where $\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\lambda$, the count
$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:

$$N(x)=\sup\{N(u): u \text{ is a }\lambda \text{-eigenfunction with } 0\leq \lambda\leq x\}.$$

Our primary goal is to establish the asymptotic growth rate of $N(x)$ as $x\to \infty$, linking it directly to the spectral exponent $d_S$ of the fractal, which characterizes a sharp phase transition in the behavior of $N(x)$.

For the upper bound, we prove in Proposition \ref{prothm1} that under a natural condition (A) — namely, that eigenfunctions with sufficiently small eigenvalues possess at most one extreme set — the growth of $N(x)$ is at most of the order $x^{d_S/2}$:
$$\limsup_{x\to\infty} \frac{N(x)}{x^{d_S/2}}<\infty.$$
Condition (A) can be interpreted as a form of ``low-energy simplicity''. It is not universal for all fractals, as evidenced by the modified Koch curve \cite{M,Sh2}, where even low-energy eigenfunctions can have infinitely many extrema. Verifying its validity is therefore a key step in the analysis for a given fractal.

Conversely, for the lower bound, Proposition \ref{prothm2} shows that the existence of  a  pre-localized eigenfunction forces \( N(x) \) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:
\[
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\]
The proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.

Our main result  is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\mathcal{SG}$.

In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \cite{Ho} show that the $L^\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.

\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{SG2.pdf}
\caption{The Sierpinski gasket $\mathcal{SG}$.}
\label{figure1}
\end{center}
\end{figure}

This paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\mathcal{SG}$, see Figure \ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function
$N(x)$ for the Laplacian $-\Delta_\mu$ where $\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\lambda$, the count
$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:

$$N(x)=\sup\{N(u): u \text{ is a }\lambda \text{-eigenfunction with } 0\leq \lambda\leq x\}.$$

Conversely, for the lower bound, Proposition \ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \( N(x) \) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:
\[
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\]
The proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.

Our main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\mathcal{SG}$.

\begin{definition}\label{def1}

(a).
Let $u\in\mathscr{D}_\mu$, $\emptyset\neq A\subset K$ and $c\in\mathbb{R}$, we say that $u$ has a \textit{local maximum ({\rm resp.} minimum)} at $A$ with value $c$ if

(a-1). $A$ is a connected component of $u^{-1}(c)$;

(a-2). $A\cap V_0=\emptyset$;

(a-3). there exists $\delta>0$ such that $u(p)\leq c$ (resp. $u(p)\geq c$) for any $ p\in A_\delta$, where
$A_\delta$ is the $\delta$-neighborhood of $A$.

Call such a set $A$ a \textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\it extreme point} of $u$.

(b). For $u\in\mathscr{D}_\mu$, denote
\[{N}(u)=\#\{A\subset K: A \text{ is an extreme set of } u\}.\]
For $x>0$, define
\[N(x)=\sup\{{N}(u): u \text { is a $\lambda$-eigenfunction with}\ 0\leq \lambda\leq x\},\]
and call it \textit{the extremum counting function} of $-\Delta_\mu$.
\end{definition}

\begin{proposition}\label{prothm2}
If there exists a pre-localized eigenfunction, then
\begin{equation*}
\liminf_{x\to\infty}\frac{\log N(x)}{\log x}\geq\kappa
\end{equation*}
for some $0<\kappa\leq d_S/2$.
In particular, for the lattice case (see the exact meaning in Section \ref{sec2}),
\begin{equation*}
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\end{equation*}
\end{proposition}

\noindent{\bf Remark.}
For a $\lambda$-eigenfunction with $\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).
Indeed, suppose that $u$ has a local maximum at $A$ with value $c\leq 0$. Since $u$ is a $\lambda$-eigenfunction, there exists $\delta>0$ such that $u\leq c\leq 0$ and $\Delta_\mu u=-\lambda u\geq 0$ on $A_\delta$. Moreover, $u$ is non-constant on $A_\delta$. Let $h$ be the harmonic function satisfying $h|_{\partial A_\delta}=u|_{\partial A_\delta}$, and denote by $g(\cdot,\cdot)$ the Green function for $-\Delta_\mu$ on $A_\delta$. It is known that $g(p,q)>0$ for all $p,q\in A_\delta$. Then, for $p\in A$, we have
\begin{equation*}
u(p)=\int_{A_\delta}g(p,q)\lambda u(q)\mathrm{d}\mu(q)+h(p)<h(p)\leq \max u|_{\partial A_{\delta}},
\end{equation*}
which yields a contradiction.

\noindent{\bf Remark.} By Lemma \ref{pro1}-(a), $\mathbf{A}_\lambda$ and $\mathbf{B}_\lambda^1$ are cones, i.e. if $\mathbf{a}\in \mathbf{A}_\lambda\ (\text{or }\mathbf{B}_\lambda^1)$, then $k\mathbf{a}\in \mathbf{A}_\lambda \ (\text{or }\mathbf{B}_\lambda^1)$ for any $k\neq 0$. Let $\pi:\mathbb{R}^L\setminus\{\mathbf{0}\}\to\mathbb{R}\mathrm{P}^{L-1}$ be the canonical projection that sends each point to the line through the origin it generates.
For $\mathbf{a},\mathbf{b}\in\mathbb{R}^L$, define $\mathbf{a}\sim\mathbf{b}$ if there exists $k\in\mathbb{R}\setminus\{0\}$ such that $\mathbf{a}=k\mathbf{b}$, then $\pi$ can be regarded as a bijection from $(\mathbb{R}^L\setminus\{\mathbf{0}\})/\sim$ to $\mathbb{R}\mathrm{P}^{L-1}$.
Write \[\mathcal{A}_\lambda=\pi(\mathbf{A}_\lambda),\quad \mathcal{B}_\lambda^1=\pi(\mathbf{B}_\lambda^1),\] and define
\begin{equation}\label{defabT}
\mathcal{T}_\lambda^i:\mathbb{R}\mathrm{P}^{L-1}\to\mathbb{R}\mathrm{P}^{L-1} \quad \text{by}\quad \mathcal{T}_\lambda^i=\pi\circ T_\lambda^i\circ\pi^{-1}.
\end{equation}
Then the condition (\ref{eq1}) is equivalent to
\begin{equation}\tag{\ref{eq1}*}\label{eq2}
\begin{aligned}
&\mathcal{A}_\lambda=\big(\bigcup_{i\in S}(\mathcal{T}_\lambda^i)^{-1}(\mathcal{A}_{\gamma^2_i\lambda})\big)\cup\mathcal{B}^1_\lambda,\quad\text{with}\\
(\mathcal{T}_\lambda^i&)^{-1}(\mathcal{A}_{\gamma^2_i\lambda}),\ i\in S\text{ and }\mathcal{B}^1_\lambda\text{ pairwise disjoint.}
\end{aligned}
\end{equation}
Consequently, condition (A) is equivalent to \eqref{eq2}.

First we prove \eqref{exC}, i.e. $\mathcal{C}_{\lambda}=\mathcal{D}_{\lambda_0}$. Noticing that for any $\mathbf{b}\in\mathbb{R}^3\setminus\{{\bf 0}\}$, by \eqref{eq4.4}, we have
\begin{equation}\label{pixxi}
\pi(\{\mathbf{x}\in\mathbb{R}^3\setminus\{{\bf 0}\}:\mathbf{b}^\mathrm{t}\mathbf{x}=0\})=\Big\{\boldsymbol{\xi}\in\mathbb{R}\mathrm{P}^2:\ \mathbf{b}^\mathrm{t}(\mathbf{1},Q)^{-\mathrm{t}}\begin{pmatrix}1\\ \boldsymbol{\xi}\end{pmatrix}=0\Big\},
\end{equation}
and a direct calculation gives
\begin{equation*}
(4-\lambda_0,-2,-2)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,0,2(6-\lambda_0)\big).
\end{equation*}
So by Lemma \ref{pro43}, we have
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(1)}=0\}\Big)\\
=&\Big\{\boldsymbol{\xi}\in\mathbb{R}\mathrm{P}^2:\ \big(-\lambda_0,0,2(6-\lambda_0)\big)\begin{pmatrix}1\\ \boldsymbol{\xi}\end{pmatrix}=0\Big\}\\
=&\Big\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ \xi^{(2)}-\frac{\lambda_0}{2(6-\lambda_0)}=0\Big\}\cup\{[(1,0)^\mathrm{t}]_\infty\}.
\end{aligned}
\end{equation*}
Similarly, $(-2,4-\lambda_0,-2)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,-\sqrt{3}(6-\lambda_0),-(6-\lambda_0)\big)$ gives
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(2)}=0\}\Big)\\
=&\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ \sqrt{3}\xi^{(1)}+\xi^{(2)}+\frac{\lambda_0}{6-\lambda_0}=0\}\cup\{[(1,-\sqrt{3})^\mathrm{t}]_\infty\},
\end{aligned}
\end{equation*}
and $(-2,-2,4-\lambda_0)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,\sqrt{3}(6-\lambda_0),-(6-\lambda_0)\big)$ gives
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(3)}=0\}\Big)\\
=&\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ -\sqrt{3}\xi^{(1)}+\xi^{(2)}+\frac{\lambda_0}{6-\lambda_0}=0\}\cup\{[(1,\sqrt{3})^\mathrm{t}]_\infty\}.
\end{aligned}
\end{equation*}
Recall that $\mathbf{C}=\{(a_1,a_2,a_3)\in\mathbb{R}^3:\text{either }a_i>0\text{ for all }i, \text{ or }a_i<0\text{ for all }i\}$ and $\mathcal{C}_\lambda=\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}(\mathbf{C})$. By the symmetry of $\mathcal{SG}$, we have $\boldsymbol{\theta}\in\mathcal{C}_\lambda$. So the above calculation gives
$\mathcal{C}_{\lambda}=\mathcal{D}_{\lambda_0}$.

(b). For $5^{-m}\lambda<\lambda_1^D$ with $m\geq 1$, it is easy to see that
$\lambda_{m}=\psi^{-1}(5^{-m}\lambda)\in(0,6)$ by Remark \ref{re33}. Considering the function $v=u\circ F_2F_3^{m-1}$, by (\ref{gamma-delta01pre}), (\ref{gamma-delta02pre}) and $p_{23}=F_2F_3^{m-1}p_3\in A$, we see
\begin{equation}\label{gamma-delta0102r}
\begin{aligned}
&2v(p_1)+2v(p_2)-(4-\lambda_m)v(p_3)=0,\\
&\big(v(p_{3})-v(p_2)\big)\big(v(p_{3})-v(p_1)\big)\geq 0.
\end{aligned}
\end{equation}
Using (\ref{pixxi}) we find that
\begin{equation*}
\begin{aligned}
&\tau(v)\in\pi\circ\tau_\lambda^D\Big(\big\{v\in E(5^{-m}\lambda)\setminus\{0\}: v\text{ satisfies (\ref{gamma-delta0102r})}\big\}\Big)\\
=&\ \big\{(\xi^{(1)},\xi^{(2)})\in\mathbb{R}^2:\xi^{(2)}=\sqrt{3}\xi^{(1)}-\frac{\lambda_m}{6-\lambda_m},\ 0\leq\xi^{(1)}\leq\frac{\sqrt{3}\lambda_m}{2(6-\lambda_m)}\big\}\\
=&\ \mathcal{I}_{\lambda_m,3}\cup\{\boldsymbol{\zeta}_{\lambda_m,23}\}\cup\{\boldsymbol{\zeta}_{\lambda_m,31}\},
\end{aligned}
\end{equation*}
since
\begin{equation*}
\begin{aligned}
&(2,2,-(4-\lambda_m))(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(\lambda_m,-\sqrt{3}(6-\lambda_m),6-\lambda_m\big),\\
&(0,-1,1)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(0,2\sqrt{3},0\big),\\
&(-1,0,1)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(0,\sqrt{3},-3\big),
\end{aligned}
\end{equation*}
and $\pi\circ \tau_\lambda^D(v')=\frac{\lambda_m}{4(6-\lambda_m)}(\sqrt{3},-1)^\mathrm{t}\in\mathcal{I}_{\lambda_m,3}$ for a function $v'\in E(5^{-m}\lambda)$ satisfying (\ref{gamma-delta0102r}) with $v'(p_1)=v'(p_2)=4-\lambda_m$ and $v'(p_3)=4$.

(b). for $u$ in the (D5) or (N5) case, there exists a constant $c_5>1$ such that
\begin{equation}\label{esti5}
c_5^{-1}\lambda^{d_S/2}\leq N(u)\leq c_5\lambda^{d_S/2};
\end{equation}

\begin{lemma}\label{lm62}
Let $u\in E_D(\lambda)\setminus\{0\}$ or $E_N(\lambda)\setminus\{0\}$ with $\mathrm{supp}\,u=\mathcal{SG}$. For $u$ in the (D6) or (N6) case and $\eps=\varnothing$, we have
\begin{equation}\label{esti6}
c_6^{-1}\lambda^{d_S/2}\leq N(u)\leq c_6\lambda^{d_S/2}
\end{equation}
for some constant $c_6>1$.
\end{lemma}
\begin{proof}
For $u$ in the (D6) or (N6) case, we have $\lambda_m=3$, $\lambda_{m-1}=6$ and $\lambda=5^m\psi(3)$. Now we consider $u\circ F_{w'}$ for each $w'\in W_{m-2}$. Since $-\Delta_{m-1}u|_{V_{m-1}}=\lambda_{m-1}u|_{V_{m-1}}$, for distinct $i,j,k\in S$ we have
\begin{equation*}
\begin{aligned}
-2u(p_{ij}^{w'})=u(p_{i}^{w'})+u(p_{j}^{w'})+u(p_{ki}^{w'})+u(p_{jk}^{w'}),
\end{aligned}
\end{equation*}
and solving the above equations gives $u(p_i^{w'})=-u(p_{ki}^{w'})-u(p_{ij}^{w'})$. Without loss of generality, we assume $u(p_{23}^{w'})\leq u(p_{31}^{w'})\leq u(p_{12}^{w'})$, then

The result demonstrates the highly regular behavior of  $\mathcal{SG}$, where eigenfunction complexity — quantified by the count of local extrema — grows  precisely as the  power law  $d_S/2$. The proof relies on a detail analysis on the eigenfunction decimation, originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.

In the classical setting, the number of critical points of an eigenfunction on an $n$-dimensional manifold is generally expected to grow linearly with $\lambda^{(n-1)/2}$, a rate related to the wave propagation and the Bohr-Sommerfeld quantization. However, in highly  symmetric setting such as $n$-dimensional rectangles or balls — where eigenfunctions can oscillate at high frequencies independently along different directions — this order improves to $\lambda^{n/2}$.

On fractals, as shown  in Theorem \ref{thm1} and Propositions \ref{prothm1} and \ref{prothm2}, the growth is governed by the spectral exponent $d_S$, which takes the role played by the geometric Hausdorff dimension $n$ in the Euclidean setting. The observed order $d_S/2$ for $\mathcal{SG}$ suggests that its high symmetry is a key factor in realizing this maximal growth rate. We therefore conjecture that for a broader class of p.c.f. self-similar sets, the growth order of local extrema is at most $d_S/2$, with this upper bound being attained only in highly symmetric cases  — such as nested fractals, or even beyond the p.c.f. setting, as exemplified by the  Sierpinski carpet — and strictly smaller in the presence of lower symmetry.

\subsection{Notation and Propositions \ref{prothm1} and \ref{prothm2}}Before ending this section, let us introduce the exact definition of   $N(u)$ and $N(x)$  on a p.c.f. self-similar set $K$, which is always assumed to be  connected.

Let $V_0$ denote the boundary of $K$, and $\mathscr{D}_\mu$ denote the domain of $\Delta_\mu$. For a function $u\in \mathscr{D}_\mu$ and $p\in V_0$, denote by $(du)_p$ the normal derivative of $u$ at $p$.
For $\lambda\geq 0$, call a non-trivial  function $u\in\mathscr{D}_\mu$ satisfying $-\Delta_\mu u=\lambda u$ on $K\setminus V_0$ a {\it $\lambda$-eigenfunction} of $-\Delta_\mu$.  Say an eigenfunction $u$ a \textit{Dirichlet (Neumann) eigenfunction} if  $u|_{V_0}=0$ ($du|_{V_0}=0$). In particular, say $u$ a \textit{pre-localized eigenfunction}  if  both $u|_{V_0}=0$ and $du|_{V_0}=0$ hold; and a \textit{global eigenfunction} if $\mathrm{supp}\,u=K$.

\begin{definition}\label{def1}

(a).
Let $u\in\mathscr{D}_\mu$, $\emptyset\neq A\subset K$ and $c\in\mathbb{R}$, we say that $u$ has a \textit{local maximum ({\rm resp.} minimum)} at $A$ with value $c$ if

(a-1). $A$ is a connected component of $u^{-1}(c)$;

(a-2). $A\cap V_0=\emptyset$;

(a-3). there exists $\delta>0$ such that $u(p)\leq c$ (resp. $u(p)\geq c$) for any $ p\in A_\delta$, where
$A_\delta$ is the $\delta$-neighborhood of $A$.

Call such a set $A$ a \textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\it extreme point} of $u$.

(b). For $u\in\mathscr{D}_\mu$, denote
\[{N}(u)=\#\{A\subset K: A \text{ is an extreme set of } u\}.\]
For $x>0$, define
\[N(x)=\sup\{{N}(u): u \text { is a $\lambda$-eigenfunction with}\ 0\leq \lambda\leq x\},\]
and call it \textit{the extremum counting function} of $-\Delta_\mu$.
\end{definition}

\noindent{\bf Remark.}
For a $\lambda$-eigenfunction with $\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).
Indeed, suppose that $u$ has a local maximum at $A$ with value $c\leq 0$. Since $u$ is a $\lambda$-eigenfunction, there exists $\delta>0$ such that $u\leq c\leq 0$ and $\Delta_\mu u=-\lambda u\geq 0$ on $A_\delta$. Moreover, $u$ is non-constant on $A_\delta$. Let $h$ be the harmonic function satisfying $h|_{\partial A_\delta}=u|_{\partial A_\delta}$, and denote by $g(\cdot,\cdot)$ the Green function for $-\Delta_\mu$ on $A_\delta$. It is known that $g(p,q)>0$ for all $p,q\in A_\delta$. Then, for $p\in A$, we have
\begin{equation*}
u(p)=\int_{A_\delta}g(p,q)\lambda u(q)\mathrm{d}\mu(q)+h(p)<h(p)\leq \max u|_{\partial A_{\delta}},
\end{equation*}
which yields a contradiction.

We have certain asymptotic estimates of $N(x)$ as $x\to \infty$.

\noindent{\bf(A)}: there exists $\lambda_0>0$ such that ${N}(u)\leq 1$ for any $\lambda$-eigenfunction $u$ with $0\leq\lambda<\lambda_0$.

\begin{proposition}\label{prothm1}
If condition (A) holds, then
\begin{equation*}
\limsup_{x\to\infty}\frac{N(x)}{x^{d_S/2}}< \infty.
\end{equation*}
\end{proposition}

\begin{proposition}\label{prothm2}
If there exists a pre-localized eigenfunction, then
\begin{equation*}
\liminf_{x\to\infty}\frac{\log N(x)}{\log x}\geq\kappa
\end{equation*}
for some $0<\kappa\leq d_S/2$.
In particular, for the lattice case (see the exact meaning in Section \ref{sec2}),
\begin{equation*}
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\end{equation*}
\end{proposition}

We structure the paper as follows.

In Section \ref{sec2}, we present the proofs of  Propositions \ref{prothm1} and \ref{prothm2}, and provide an equivalent characterization of condition (A).

Beginning in Section \ref{sec3}, we focus on the canonic Laplacian on the Sierpinski gasket $\mathcal{SG}$. There, we recall the spectral decimation method and state two key preparatory theorems — Theorems \ref{thm4} and \ref{thm5}.  The former confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions.

The proof of  Theorem \ref{thm4} is given in Section \ref{sec4},  followed by the proof of Theorem \ref{thm5} in Section \ref{sec5}. We conclude in Section \ref{sec6} with the proof of Theorem \ref{equnthm1}.

The post-theorem introduction says that the proof of Theorem~\ref{thm1} “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.” It further indicates the overall organization supporting Theorem~\ref{thm1}: in Section~\ref{sec3} the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems — Theorems~\ref{thm4} and \ref{thm5},” where “\ref{thm4} confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions,” and then “conclude[s] in Section~\ref{sec6} with the proof” of the main estimate \eqref{equnthm1} (i.e. Theorem~\ref{thm1}).

The post-theorem introduction says that the proof of the main theorem “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.” It further indicates the overall organization supporting the main theorem: next the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems.” We first record the following theorem.

\begin{theorem}\label{thm4}
For $0<\lambda<\lambda_1^D$, we have
\begin{equation}\label{eq3}
\begin{aligned}
\mathcal{A}_\lambda=\ & \mathcal{C}_\lambda=\mathcal{D}_{\psi^{-1}(\lambda)},\quad \mathcal{B}_\lambda^1=\big(\bigcup_{ij\in S_1}\mathcal{L}_{\psi^{-1}(\lambda),ij}\big)\cup\{\boldsymbol{\theta}\},\quad\text{and}\\
&(\mathcal{T}_\lambda^i)^{-1}(\mathcal{A}_{5^{-1}\lambda})=
\mathcal{G}_{\psi^{-1}(\lambda),i}\quad\text{for }i\in S.
\end{aligned}
\end{equation}
In particular, condition (A) holds.
\end{theorem}

The introduction then points to the later preparatory result
\begin{theorem}\label{thm5}
For  $\eps\in\{-1,1\}^n$ of length $n\geq 1$ with $\eps_n=1$, we have
\begin{equation}\label{equn}
c^{-1}(\lambda^{\eps})^{d_S/2}\leq N(u^{\eps})\leq c(\lambda^{\eps})^{d_S/2}
\end{equation}
with some constant $c>1$ independent of $\lambda_0$, $\mathbf a$ and $\eps$.
\end{theorem}

and finally says it “conclude[s] later with the proof” of the main estimate
\begin{equation}\label{equnthm1}
C^{-1}\lambda^{d_S/2}\leq{N}(u)\leq C\lambda^{d_S/2}
\end{equation}
(i.e. the main theorem).

Question: On the Sierpiński gasket \(\mathcal{SG}\), let \(N(u)\) denote the number of extreme sets of a function \(u\), where an extreme set is a connected component \(A\) of a level set \(u^{-1}(c)\), disjoint from the boundary vertex set \(V_0\), such that for some \(\delta>0\) the function is everywhere \(\le c\) on the \(\delta\)-neighborhood of \(A\) (local maximum case) or everywhere \(\ge c\) there (local minimum case). Let \(d_S\) be the spectral dimension of \(\mathcal{SG}\). A global Dirichlet or Neumann eigenfunction means an eigenfunction of the Laplacian on all of \(\mathcal{SG}\) satisfying Dirichlet or Neumann boundary conditions on \(V_0\). Which uniform estimate holds for such eigenfunctions?

There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2}\), except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).

Choices:
B. There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2}\), with no exceptional Neumann eigenfunction.
C. There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(N(u)\le C\lambda^{d_S/2}\).
D. For every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), there exists a constant \(C=C(u)>1\) such that \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2}\), except for the first non-constant Neumann eigenfunction where \(N(u)=0\).
E. There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda\), except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).

The stem gives definitions and setup but does not state or strongly hint at the exact estimate, exception, or exponent. The correct answer is not leaked directly.

The item is largely theorem-recognition: it asks which estimate holds, and the correct option is essentially the theorem statement. However, it is not a pure tautology because the options vary in meaningful ways (exception, quantifier uniformity, exponent, one-sided vs two-sided bound).

Some reasoning is needed to distinguish the strongest correct uniform estimate from nearby variants, especially against the weaker true statement and the quantifier/exponent perturbations. Still, the task is closer to precise recall/recognition than to genuine derivation.

The distractors are plausible and mathematically targeted: omission of the exceptional Neumann case, weakening to a one-sided bound, loss of uniformity via dependence on u, and substitution of an incorrect growth exponent. These reflect realistic failure modes.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it tests precise recall more than fully generative mathematical reasoning.


Let $\Delta$ be a spherical building.
There are order-preserving maps
\[ \Gamma: \mathrm{sd} \PD(\Delta) \to {\rm CB}(\Delta),\]
\[ \phi :\OPD(\Delta) \to \Delta * \Delta,\]
such that for every group $H$ acting on $\Delta$ by simplicial automorphisms, $\Gamma$ and $\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets:
\[ \Gamma_H : \PD(\Delta)^H \to {\rm CB}(\Delta)^H,\]
\[ \phi_H :\OPD(\Delta)^H \to \Delta^H * \Delta^H.\]
In particular, $\Gamma$ and $\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.

\begin{definition}
\label{def:decompBuilding}
Let $\Delta$ be a spherical building.
We denote by $\D(\Delta)$ the poset of all non-empty Levi spheres, with order induced by reverse inclusion.
\end{definition}

\begin{proposition}
\label{prop:lowLeviIntervalPD}
Let $L\in \D(\GG,k)$.
Then $\PD(\GG,k)_{\prec L} \simeq \Delta(\GG,k)^L * \PD(L,k)$.
Moreover, in homology, we have an isomorphism of $L(k)$-modules
\begin{equation}
\label{eq:isoHomologyIntervalsPDLevi}
\widetilde{H}_*\big(\PD(\GG,k)_{\prec L},R \big) \cong \widetilde{H}_*\big(\, \Delta(\GG,k)^L *\PD(L,k), R \, \big) \cong \widetilde{H}_*\big(\,\PD(L,k), R \, \big),
\end{equation}
provided that $RL(k)$ is semisimple.
\end{proposition}

\begin{theorem}
\label{thm:homologyWedge}
Suppose that $G$ is a finite group, and $f:X\to Y$ is a $G$-equivariant order-preserving map between finite-dimensional $G$-posets.
Let $R$ be a ring such that $RG$ is semisimple.
Then, under the conditions of Theorem \ref{thm:wedgeDecomposition}, for all $m\geq 0$ we have an isomorphism of $RG$-modules:
\[ \widetilde{H}_{m}(X, R) \cong_G \widetilde{Y}_m(Y, R) \oplus \bigoplus_{\overline{y} \in Y/G} \bigoplus_{i+j=m-1} \Ind_{\Stab_G(y)}^G \big( \widetilde{H}_i(f^{-1}(Y_{\leq y})) \otimes \widetilde{H}_j(Y_{> y})   \big). \]
\end{theorem}

Here we are denoting by $\overline{y}$ the image of an element $y\in Y$ in the orbit poset $Y/G$.

\subsection{Buildings}
We work with spherical buildings in the sense of \cite{AB}, so we do not assume that our buildings are thick.
From now on, $\Delta$ will denote a (spherical) building.
By the Solomon-Tits theorem, $\Delta$ is spherical in the sense that it has the homotopy type of a wedge of spheres of dimension $\dim \Delta$ \cite[Theorem 4.73]{AB}.
On the other hand, for every simplex $\sigma\in \Delta$, its link $\Lk_{\Delta}(\sigma)$ is a building of dimension $\codim_{\Delta}(\sigma)$ \cite[Proposition 4.9]{AB}.
In particular, buildings are Cohen-Macaulay.

We write $\A(\Delta)$ for the complete system of apartments of $\Delta$.
From now on, when we speak of an apartment of $\Delta$ we mean an
element of $\A(\Delta)$.
If $\sigma_1,\ldots,\sigma_r\in \Delta$ are simplices, we denote by $\A(\Delta,\sigma_1,\ldots,\sigma_r)$ the set of apartments $\Sigma\in \A(\Delta)$ such that $\sigma_i\in \Sigma$ for all $i$.
Recall that, since we work with spherical buildings, apartments are finite Coxeter complexes.

\subsubsection{Opposition}
\label{sub:opposition}
Let $\Sigma$ be a (finite) Coxeter complex.
Two chambers of $\Sigma$ are called opposite if their distance coincides with the diameter of $\Sigma$.
It is well known that every chamber of $\Sigma$ has a unique opposite, giving rise to an involutory bijection $\cham \Sigma\to \cham \Sigma$ that extends uniquely to an involutory automorphism $\op_{\Sigma}:\Sigma\to\Sigma$ \cite[2.39]{Tits1}.
Thus, two simplices $\sigma,\sigma' \in \Sigma$ are opposite if $\op_{\Sigma}(\sigma) = \sigma'$.
By convention, the opposite of the empty simplex is the empty simplex.

Two simplices of $\Delta$ are called opposite if they are opposite in some apartment (and hence in every apartment that contains both of them).
It follows that two opposite chambers lie in a unique apartment (see \cite[Lemma 4.69]{AB} and \cite[3.25]{Tits1}).
Note that a simplex may have multiple opposites in $\Delta$.

There is a bijection between apartments containing two given opposite simplices and apartments in the link of one of these simplices:

\begin{lemma}
\label{lm:bijectionApartmentsLinkOpposite}
Let $\Delta$ be a spherical building, and let $\sigma,\sigma'$ be two opposite simplices.
Then we have a bijection:
\[ \Sigma\in \A(\Delta, \sigma,\sigma') \longmapsto \Lk_{\Sigma}(\sigma)\in \A(\Lk_{\Delta}(\sigma)).\]
The inverse of this map is given as follows.
If $\widetilde{\Sigma}\in \A(\Lk_{\Delta}(\sigma))$ and $c,c'\in \widetilde{\Sigma}$ are two opposite chambers there, then $\widetilde{\Sigma} = \Lk_{\Sigma}(\sigma)$ where $\Sigma$ is the convex hull in $\Delta$ of the opposite chambers $c$ and $\proj_{\sigma'}(c')$. Thus we map $\widetilde{\Sigma}$ to $\Sigma$.
\end{lemma}

\begin{proof}
This is part of Proposition 2.1 in \cite{vH}.
\end{proof}

Here, $\proj_{\sigma}(\tau)$ denotes the projection of a simplex $\tau$ to $\sigma$, in the sense of Tits \cite[2.30]{Tits1} (see next subsection for the definition).

\subsubsection{Convexity}
\label{sub:convexity}
Now we look at convex subcomplexes of $\Delta$, in the sense we defined above.
Our notion of convexity coincides with Tits' definition \cite[1.5]{Tits1}, and it is stronger than the notion given in \cite[Definition 4.120]{AB}.
See also \cite[Remark 4.122]{AB}.
Moreover, we will mostly work with convex subcomplexes of $\Delta$ that are contained in some apartment (from the complete system of apartments).

Suppose first that $\Sigma$ is a Coxeter complex.
A root of $\Sigma$ is the image of a folding \cite[1.8]{Tits1}.
If $\alpha$ is a root of $\Sigma$, we write $-\alpha$ for its opposite root, and $\partial \alpha =\alpha \cap (-\alpha)$ is (by definition) a wall (see discussion at the top of page 11 in \cite{Tits1}).
Note that $-\alpha = \op_{\Sigma}(\alpha)$.
Write $\roots(\Sigma)$ for the set of roots of $\Sigma$.
Recall that a panel is a codimension-one simplex.
Given a chamber $\sigma\in \Sigma$ and a panel $\tau\subseteq \sigma$, there exists a unique root $\alpha\in \roots(\Sigma)$ such that $\tau\in \partial \alpha$ and $\sigma\in \alpha$.
Every root (and hence every wall) is a convex subcomplex of $\Sigma$.
Indeed, by \cite[2.19]{Tits1}, convex subcomplexes of $\Sigma$ are exactly those obtained as intersections of roots.
If $K$ is a subcomplex of $\Sigma$, we write $\roots_{\Sigma}(K)$ for the set of roots $\alpha\in \roots(\Sigma)$ that contain $K$.
Thus, $K$ is convex if and only if
\[ K = \bigcap_{\alpha\in \roots_{\Sigma}(K)} \alpha.\]

The convex hull of two opposite simplices $\sigma,\sigma'\in \Sigma$ is termed a Levi sphere of the Coxeter complex $\Sigma$.
Indeed, Levi spheres are exactly the subcomplexes of $\Sigma$ obtained as intersections of walls.
This terminology is
borrowed from the work in \cite{Serre} of Serre, who defines a Levi sphere as an intersection of $|\Sigma|$, viewed as the classical Euclidean sphere, with sets of reflecting hyperplanes of the underlying Coxeter group.
This concept generalizes the notion of Levi subgroups in algebraic groups to the context of buildings, as we will explain later in Lemma \ref{lm:LeviSubgroupsAndLeviSpheres}.

Now we go back to $\Delta$.
Let $S$ be a set of simplices of $\Delta$.
If $S \subseteq \Sigma$ for some apartment $\Sigma$, then the convex hull of $S$ is the same whether it is taken in $\Sigma$ or in $\Delta$, that is, $\Conv_{\Sigma}(S) = \Conv_{\Delta}(S)$.
If $\sigma,\tau\in \Delta$ are two simplices, then they lie in some apartment $\Sigma$.
The projection of $\tau$ onto $\sigma$, denoted by $\proj_{\sigma}(\tau)$ is the unique maximal simplex containing $\sigma$ in the convex hull $\Conv_{\Delta}(\sigma,\tau)=\Conv_{\Sigma}(\sigma,\tau)$ (see \cite[2.30, 3.19]{Tits1}).

A Levi sphere of $\Delta$ is the convex hull of two opposite simplices, which is then a Levi sphere of any apartment containing these simplices.
If $\sigma,\sigma'$ are opposite chambers, then $\Conv_{\Delta}(\sigma,\sigma')$ is the unique apartment that contains them and hence a $\dim \Delta$-sphere.
If $\sigma,\sigma'$ are opposite vertices, then $\Conv_{\Delta}(\sigma,\sigma') = \{\sigma,\sigma'\}$ is a $0$-sphere.

More generally, Levi spheres are always spheres (see also Remarques 2
on the bottom of page 200 in \cite{Serre}).

\begin{lemma}
\label{lm:convexIsleviSphere}
Let $\Sigma$ be a finite Coxeter complex, and let $K\subseteq \Sigma$ be a convex subcomplex.
Then $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\sigma_1,\sigma_2$ with dimension $\dim \sigma_i = \dim K$.
In such a case, $K$ is the convex hull of $\sigma_1,\sigma_2$ and $|K|$ is a sphere of dimension $\dim K$.
\end{lemma}

\begin{proof}
Suppose that $K$ contains a pair of opposite simplices $\sigma_1,\sigma_2$ of dimension $\dim K$.
Then, in the terminology of \cite[Théorème 2.1]{Serre}, $K$ must be completely reducible. This means that every point of $K$ has an opposite.
But every point of $K$ has at most one opposite as $K$ lies in the sphere $\Sigma$, so $K$ must be exactly the Levi sphere spanned by $\sigma_1$ and $\sigma_2$.
In particular, $K$ is the convex hull of $\sigma_1,\sigma_2$, and it triangulates a sphere of dimension $\dim K$.

The converse of this statement is clear (see \cite{Serre}).
\end{proof}

A root (resp. a wall) of $\Delta$ is a root (resp. a wall) of some apartment.
We write $\roots(\Delta)$ for the set of roots of $\Delta$, so
\[ \roots(\Delta) = \bigcup_{\Sigma \in \A(\Delta)} \roots(\Sigma).\]
Similarly, if $K\subseteq \Delta$, $\roots_{\Delta}(K)$ denotes the set of roots $\alpha\in \roots(\Delta)$ that contain $K$.
If $K$ is a convex subcomplex of $\Delta$ that is contained in some apartment, then $K$ is a convex subcomplex of any apartment containing it.
In particular, if $K\subseteq \Sigma\in \A(\Delta)$, then
$K = \bigcap_{\alpha \in \roots_{\Sigma}(K)} \alpha$, and more generally,
\[ K  =\bigcap_{\alpha\in \roots_{\Delta}(K)} \alpha.\]

\subsubsection{The CAT(1) metric}
\label{subsub:cat1metric}
The geometric realization of a spherical building $\Delta$ admits a canonical metric $\dcat$ that makes it a complete CAT(1) space.
See \cite[II.10 Theorem 10A.4]{BH} and \cite[Example 12.39]{AB}.
With this metric, an apartment becomes isometric to the unit sphere $\SS^d$ of the vector space on which the underlying reflection group acts, where  $d = \dim \Delta$.
In particular, the opposition map $\op_{\Sigma}:\Sigma\to\Sigma$ of an apartment $\Sigma$ gives rise to the involution $-\id_{\SS^d}:\SS^d\to \SS^d$ of the unit sphere.
Hence, two simplices $\sigma,\sigma'\in \Sigma$ are opposite if and only if their barycenters are opposite points when regarded in $\SS^d$ via this identification.

The diameter of $|\Delta|$, which is $\pi$ with this metric, is also the diameter of any apartment, which equals the distance between two opposite points.
In particular, for two points $x,y\in |\Delta|$ at distance $\dcat(x,y) < \pi$, there exists a unique geodesic from $x$ to $y$.
Recall that a subspace $X$ of a CAT(1) space is convex if for every two points $x,y\in X$ at distance $<\pi$, the unique geodesic segment joining $x,y$ is completely contained in $X$.
It follows that a convex subcomplex $K$ of $\Delta$ gives rise to a convex subspace of $|\Delta|$.
Notice that not every convex subspace of $|\Delta|$ arises in this way.
Now, if $\sigma,\sigma'$ are two non-opposite simplices in $\Delta$, then there is a unique geodesic (in the geometric realization) that joins their barycenters.

\subsubsection{Automorphism group}
\label{subsub:autDelta}
We denote by $\Aut(\Delta)$ the group of simplicial automorphisms of $\Delta$.
These automorphisms might not be type-preserving.
By a group acting on $\Delta$, we mean a group inducing simplicial automorphisms on $\Delta$.

Any simplicial automorphism on $\Delta$ gives rise to an isometry of $|\Delta|$ with the metric $\dcat$.
Therefore, if $H$ is a group acting simplicially on $\Delta$, and $x,y\in |\Delta|^H$ are two points at distance $<\pi$, then $H$ must fix the unique geodesic joining $x$ and $y$.
In particular, if $\sigma,\sigma'$ are non-opposite simplices of $\Delta$ that are invariant under the action of $H$, then $H$ fixes the unique geodesic in $|\Delta|$ that joins their barycenters.

From these observations, we get the following lemma.

\begin{lemma}
\label{lm:contractibleIntersectionApartments}
Let $\Delta$ be a spherical building, and let $H$ be a group acting on $\Delta$ by simplicial automorphisms.
Let $\tau\in \Delta^H$ be a non-empty simplex fixed by $H$, and let $\S \subseteq \A(\Delta,\tau)$ be a set of apartments containing $\tau$.

If $\bigcap_{\Sigma\in \S} (\Sigma^H)$ does not contain an opposite of $\tau$, then it is contractible.
\end{lemma}

\begin{proof}
Let $X := \bigcap_{\Sigma\in \S} \Sigma^H$, and note that $|X| =  \bigcap_{\Sigma\in \S} |\Sigma'|^H$.
Assume that there is no opposite of $\tau$ in $X$.
Then every point $x\in X$ is at distance $ < \pi$ from the barycenter of $\tau$ (say, $x_0$).
Hence, by the discussion in Subsection \ref{subsub:autDelta}, $H$ fixes the unique geodesic joining $x$ with $x_0$.
Thus, we can contract $X$ to $x_0$ using these geodesics.
\end{proof}

\section{Posets and simplicial complexes in the case of vector spaces}
\label{sec:vectorspaces}

In this section we take a closer look at the poset of partial decompositions, the poset of ordered partial decompositions
and the common bases complex for vector spaces.
We demonstrate how building-related constructions show up.
The results then motivate and guide the more general definitions for buildings in Sections \ref{sec:CBBuildings}, \ref{sec:leviSpheres} and
\ref{sec:opd}.

Let $V$ be a finite-dimensional vector space defined over a field $k$.
Write $T(V)$ for the poset of proper non-zero subspaces of $V$, ordered by inclusion.
We denote by $\Delta(V)$ the order complex of $T(V)$, which is also the building associated with the group $\GL(V)$.

\begin{definition}
\label{def:DandPDVectorSpaces}
Let $V$ be a finite-dimensional vector space defined over a field $k$.
A partial decomposition of $V$ is a subset $\{S_1,\ldots,S_r\} \subseteq T(V)$ such that
\[ \gen{S_1,\ldots,S_r} \cong S_1\oplus \cdots \oplus S_r.\]

We denote by $\PD(V)$ the poset of partial decompositions of $V$ other than $\emptyset$ and $\{V\}$, with order given by refinement; that is
for $d_1,d_2\in \PD(V)$
\[ d_1\leq d_2 \text{ if for all } S\in d_1 \text{ there is } T\in d_2 \text{ such that } S\leq T.\]

A full decomposition $V$ is a partial
decomposition $\{S_1,\ldots, S_r\}$ such that $V \cong S_1 \oplus\cdots \oplus S_r$. The poset of full decompositions of $V$ is the subposet
$\D(V)$ of $\PD(V)$ on the set of full decompositions of $V$.
\end{definition}

Note that we are not including $\{V\}$ in the poset $\D(V)$
and that
\[ d \text{ is a partial decomposition } \Leftrightarrow
\ \dim \gen{S\tq S\in d} = \sum_{S\in d} \dim S.\]

We will also work with the ordered versions of $\PD(V)$ and $\D(V)$:

\begin{definition}
\label{def:ODandOPDVectorSpaces}
Let $V$ be a finite-dimensional vector space defined over a field $k$.
The poset of ordered partial decompositions of $V$, denoted by $\OPD(V)$ consists of tuples of distinct subspaces $(S_1,\ldots,S_r)$ such that $\{S_1,\ldots,S_r\}\in \PD(V)$.
The ordering in $\OPD(V)$ is given by refinement that preserves the order of the elements of the tuples.
That is, if $d_1 = (S_1,\ldots,S_r)$ and $d_2 = (W_1,\ldots,W_t)$, then $d_1\leq d_2$ if for all $1\leq i\leq j\leq r$ there are $1\leq k\leq l\leq t$ such that $S_i\leq W_k$ and $S_j\leq W_l$.

The poset $\OD(V)$ of ordered full decompositions of $V$ is the subposet of
$\OPD(V)$ on the set of $(S_1,\ldots,S_r)$ such that $\{S_1,\ldots,S_r\}\in \D(V)$.
\end{definition}

As observed first in \cite{LR}, the poset $\OD(V)$ is naturally isomorphic to the poset of opposite pairs of the building $\Delta(V)$.

\begin{remark}
[{Opposite simplices}]
\label{rk:ODlinearCase}
An ordered full decomposition $d = (S_1,\ldots,S_r)\in \OD(V)$ determines the pair $\big(\,P(d),Q(d)\,\big)$ of flags given by
\[ P(d) = (\,S_1 < S_1 \oplus S_2 < \cdots < S_1\oplus\cdots \oplus S_{r-1}\,),\]
and
\[ Q(d) = (\, S_r < S_r\oplus S_{r-1} < \cdots < S_2\oplus\cdots \oplus S_r\,).\]
In the language of buildings, this means that $P(d),Q(d)$ are opposite simplices of $\Delta(V)$.
For $\Delta(V)$ the notion of opposition
of simplices from Subsection \ref{sub:opposition} translates into
the following. Two simplices $\sigma$ and $\tau$ from $\Delta(V)$ are opposite if they have the same dimension and for all $S\in \sigma$ there exists a unique $T\in \tau$ such that $V = \gen{S,T} \cong S\oplus T$.

Therefore, the poset $\OD(V)$ can be alternatively described as the poset of pairs of opposite simplices of the building $\Delta(V)$,
where the ordering is given by coordinate-wise reverse inclusion.
The isomorphism is given by $d\mapsto \big(\,P(d), Q(d)\,\big)$.

Another description of $\OD(V)$ is in terms of the Charney poset.
Recall that the Charney poset $\Ch(V)$ consists of pairs $(S,T)$ of proper non-zero subspaces of $V$ such that $S\oplus T = V$.
The ordering in $\Ch(V)$ is given by zig-zag containment:
\[ (S_1,T_1) \leq (S_2,T_2) \  \Leftrightarrow \ S_1\leq S_2 \text{ and } T_1\geq T_2.\]
Then it is not hard to see that
\[ \OD(V) = \X\big(\,\K(\,\Ch(V)\,)\,\big)^{\op}.\]
The Charney poset was introduced by R. Charney \cite{Charney} in the context of free modules over Dedekind domains $R$, where it was used to establish homological stability results for the linear groups $\GL_n(R)$.
\end{remark}

We have seen that $\OD(V)$ has an intrinsic description in terms of building properties.
Our next theorem describes $\D(V)$ in terms of Levi subgroups, hinting at a possible definition of the full decomposition poset for arbitrary buildings that arise from the BN-pair of the $k$-points of a connected reductive algebraic group.

In what follows, suppose that $\overline{k}$ is the algebraic closure of a field $k$.
We say that $L$ is a $k$-Levi subgroup of $\GL_n(k)$ if $L$ is an algebraic group defined over $k$ and it is the Levi complement in a parabolic subgroup of $\GL_n$ that is also defined over $k$.

\begin{theorem}
\label{thm:decompAndLeviVectorSpaces}
Let $\L(\GL_n,k)$ denote the poset of (proper) $k$-Levi subgroups of $\GL_n$ ordered by inclusion.
Then we have a $\GL_n(k)$-equivariant poset isomorphism $\L(\GL_n,k)\groupiso_{\GL_n(k)} \D(k^n)$.
\end{theorem}

\begin{definition}
\label{def:ODandOPDbuildings}
Let $\Delta$ be a spherical building.
We define the posets:
\begin{itemize}
\item $\OD(\Delta) := \Opp(\Delta) = \{ (\sigma,\sigma')\tq \sigma,\sigma'$ are opposite simplices$\}$, with order relation
$(\sigma,\sigma') \leq (\tau,\tau')$ if $\tau\subseteq \sigma$ and $\tau'\subseteq \sigma'$.
\item $\OPD(\Delta) = \X(\Delta) \cup \Opp(\Delta)$ with order relation $\preceq$ defined as follows. Among elements of $\X(\Delta)$ and
among elements of $\Opp(\Delta)$ the order $\preceq$ is inherited from the order on the respective posets. If $\sigma \in \X(\Delta)$ and
$(\tau,\tau') \in \Opp(\Delta)$ then we set
$\sigma \prec (\tau,\tau')$ if there is an apartment containing $\sigma,\tau$, and $\tau'$.
\end{itemize}
We call $\OD(\Delta)$ the ordered decomposition poset associated with $\Delta$, and $\OPD(\Delta)$ the ordered partial decomposition poset.
\end{definition}

\begin{definition}
\label{def:PDbuildings}
Let $\Delta$ be a spherical building.
We define $\PD(\Delta)$ as the poset on the disjoint union of
$\X(\Delta)$ and ${\D(\Delta)}$ with order relation $\preceq$
defined as follows.
Among elements of $\X(\Delta)$ and among elements of $\D(\Delta)$ the
order $\preceq$ is the one inherited from the order on the
respective posets.
If $\sigma \in \Delta$ and $S\in {\D(\Delta)}$, then we set $\sigma \prec S$ if and only if there exists an apartment $\Sigma$ of $\Delta$ such that $\sigma\in \Sigma$ and $S$ is a Levi sphere of $\Sigma$.
\end{definition}

\begin{definition}
\label{def:commonBasis}
Let $\Delta$ be a spherical building.
We write $\CB(\Delta)$ for the simplicial complex on the same
vertex set as $\Delta$ whose maximal simplices are
$\displaystyle{\bigcup_{\sigma \in \Sigma}} \sigma$ for
$\Sigma \in \A(\Delta)$.

\end{definition}

\label{sec:intro}

The order complex of the lattice of non-trivial subspaces of a finite-dimensional vector space is well known to be the spherical building of type A (see e.g., \cite{Tits1}).
Recently, simplicial complexes defined
on the set of non-trivial subspaces
via constraints on common bases (see e.g., \cite{Rognes,MPW}) or the relative position of the subspaces were studied (see e.g., \cite{LR,BPW24}). Such complexes are often order complexes of partially ordered sets.
The goals of this paper are to provide definitions of such
simplicial complexes and posets that are independent of Lie type, and to provide results on the associated homology groups
and homotopy types. In some cases, the complexes we construct in type A are the same as the previously studied ones.  In other cases, our type A complexes are different than those previously studied but have the same equivariant homotopy type. We start by defining the motivating posets and simplicial complexes and reviewing the relevant literature.

Let $V$ be an $n$-dimensional vector space over a field $k$.
A partial decomposition of $V$ is a set $\{V_1,\ldots,V_r\}$ of non-zero and proper subspaces of $V$ such that $\langle V_1,\ldots, V_r \rangle \cong V_1\oplus\cdots\oplus V_r$;
that is, the $V_1,\ldots, V_r$ are in internal direct sum.
The set $\PD(V)$ of all non-empty partial decompositions of $V$ is a poset with order given by refinement; i.e., $\{V_1,\ldots, V_r\}\leq \{W_1,\ldots, W_s\}$ if for all $1\leq i \leq r$ there is $1\leq j\leq s$ such that $V_i\leq W_j$.

In \cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.
In particular, $\PD(V)$ is spherical (see Section \ref{sec:preliminaries} for definitions).
More recently, in \cite{BPW24}, it was proved that for any field $k$,\,$\PD(V)$ has the homotopy type of the common bases complex ${\rm CB}(V)$ of $V$, as defined by J. Rognes \cite{Rognes}.
Concretely, ${\rm CB}(V)$ is the simplicial complex whose simplices are sets $\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\sigma$ is generated by some subset of $B$.
In this case, we say that the elements of $\sigma$ have a common basis.
Rognes showed that, although ${\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\leq 2n-3$, and he conjectured that ${\rm CB}(V)$ must indeed be $(2n-4)$-connected.
In \cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.
In particular, via \cite{BPW24} this implies that $\PD(V)$ is spherical for any field.

Considering the ordered version $\OPD(V)$ of $\PD(V)$
appears to be equally natural.
By definition, $\OPD(V)$ is the poset whose elements are tuples
$(V_1,\ldots,V_r)$, $r\geq 1$, of distinct subspaces of $V$ such that $\{V_1,\ldots,V_r\} \in \PD(V)$.
The ordering in $\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \ref{def:ODandOPDVectorSpaces}).
We call the elements of $\OPD(V)$ ordered partial decompositions.
It follows from \cite[Theorem 6.8]{PW25} that $\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.
If $V$ is a finite vector space then \cite[Theorem 6.10]{PW25} implies
that $\OPD(V)$ has the homotopy type of the two-fold join $\Delta * \Delta$ of the Tits building $\Delta$ of $V$.
Using GAP we also verified in some small examples that the top homology group of
$\OPD(V)$, as a $\GL(V)$-module, is the tensor square of the Steinberg module.

Later, we found an alternative approach to determining the
homotopy type of $\OPD(V)$ using closely related constructions, which inspired the work in this paper.
Let $\OD(V)$ denote the subposet of $\OPD(V)$ consisting of full ordered decompositions $(V_1,\ldots, V_r)$; i.e., those decompositions satisfying
$V \cong V_1\oplus \cdots \oplus V_r$.
Let $X = \Delta \cup \OD(V)$ be the poset on the disjoint union of
$\Delta$ (regarded as a poset without the empty simplex) and $\OD(V)$ with the following
ordering.
We keep the inclusion ordering among elements of $\Delta$, and the refinement ordering in $\OD(V)$.  No element of $\OD(V)$ lies below any element of $\Delta$. If $\sigma\in \Delta$ and $d\in \OD(V)$, we set $\sigma \prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\sigma$.
In Theorem \ref{thm:PDandPDBuildingVectorSpace} we show that
$\OPD(V) \simeq X$.
The poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \cite{MPW}, which is shown therein to have
the homotopy type of $\Delta * \Delta$.
This again allows us to conclude that $\OPD(V) \simeq \Delta * \Delta$ for any finite-dimensional vector space $V$.
Since all homotopy equivalences can be seen to be $\GL(V)$-equivariant, it now follows that the top homology group of
$\OPD(V)$ carries the tensor square of the Steinberg representation.

The poset $\OD(V)$ already appears in work {\cite{LR} of  G.I. Lehrer and L.J. Rylands, who observe that
$\OD(V)$ can be identified with poset of pairs of opposite parabolics of $\GL(V)$.
\footnote{This observation is motivated by previous work by R. Charney \cite{Charney}, and $\OD(V)$ is commonly known as the Charney complex. See also \cite[Proposition 4.15]{PW25}.}
Recall that two parabolic subgroups are called opposite if their intersection is a Levi complement in both of them.
Indeed, two parabolic subgroups are opposite if and only if they are opposite as simplices of the building $\Delta$ of $V$, which is a notion intrinsically defined for every building (see Subsection \ref{sub:opposition}).
Thus, $\OD(V)$ can be described in terms of simplices of $\Delta$, and we define
\[\, \OD(\Delta) = \big\{\,(\sigma,\sigma') \tq \sigma,\sigma'\in \Delta \text{ are (non-empty) opposite simplices}\,\big\},\]
where the ordering of this poset is inclusion-reversing in each coordinate (so a pair of opposite maximal simplices is an ordered frame of $V$, which is a minimal element of the poset).
Using this identification the ordering between crossed terms in $X = \Delta \cup \OD(\Delta)$ becomes $\sigma\prec (\sigma_1,\sigma_2)$ if there is a basis of $V$ spanning the subspaces from $\sigma\cup \sigma_1\cup \sigma_2$.
This condition is equivalent to  saying that $\sigma,\sigma_1,\sigma_2$ lie in a common apartment of $\Delta$. This shows
that
the poset $\Delta\cup \OD(\Delta)$ can be defined in terms of intrinsic combinatorial properties of the building.

These observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.
In this paper, we study the following questions and answer all of them positively.
Let $\Delta$ be a (spherical) building.
\begin{enumerate}
    \item Can we define a poset $\OPD(\Delta)$ that, up to homotopy, coincides with $\OPD(V)$ when $\Delta$ is the building of the vector space $V$?
    \item Can we similarly define $\PD(\Delta)$? Is it highly connected?
    \item Do we have a common basis complex ${\rm CB}(\Delta)$ for $\Delta$ that coincides with ${\rm CB}(V)$ when $\Delta$ is the building of the vector space $V$?
    \item If questions (2) and (3) have a positive answer, do we have ${\rm CB}(\Delta) \simeq \PD(\Delta)$?
    \item Is $\OPD(\Delta)\simeq \Delta * \Delta$?
\end{enumerate}

While the definitions of $\OPD(\Delta)$ and ${\rm CB}(\Delta)$ arise naturally, a suitable definition of $\PD(\Delta)$ is less obvious.
Indeed, as in the case of $\OPD$, Theorem \ref{thm:PDandPDBuildingVectorSpace} shows that $\PD(V)\simeq \Delta \cup \D(V)$, where $\D(V)$ is the poset of full decompositions of $V$, and the ordering between crossed terms in $\Delta \cup \D(V)$ is defined as in the ordered case: if $\sigma\in \Delta$ and $d\in \D(V)$, then we set $\sigma\prec d$ if $\sigma\cup d\in {\rm CB}(V)$ ($\sigma$ and $d$ have a common basis).
Our next observation is, that $\D(V)$ can be identified with the poset of (split) Levi subgroups of $\GL(V)$.
In the language of buildings, split Levi subgroups correspond to Levi spheres, as introduced by J.P. Serre in \cite{Serre} (see Subsection \ref{sub:convexity} and Section \ref{sec:leviSpheres}).
If we identify a full decomposition $d\in \D(V)$ with a Levi sphere, then the condition $\sigma \prec d$ becomes ``$\sigma$ and the Levi sphere $d$ lie in a common apartment".
Therefore, for a spherical building $\Delta$, we propose:

\begin{itemize}
    \item ${\rm CB}(\Delta) = \{\sigma \tq \sigma$ is a subset of the vertex set of some apartment of $\Delta\}$ (Definition \ref{def:commonBasis})
    \item $\D(\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \ref{def:decompBuilding}).
    \item $\PD(\Delta) = \Delta\cup \D(\Delta)$, with crossed-term ordering $\sigma\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\sigma$ (Definition \ref{def:PDbuildings}).
    \item $\OD(\Delta) = \{(\sigma_1,\sigma_2)\tq \sigma_1,\sigma_2$ are non-empty opposite simplices of $\Delta\}$, with coordinate-wise reverse-inclusion ordering (Definition \ref{def:ODandOPDbuildings}).
    \item $\OPD(\Delta) = \Delta \cup \OD(\Delta)$, with crossed-term ordering $\sigma\prec (\sigma_1,\sigma_2)$ if there is an apartment containing $\sigma,\sigma_1,\sigma_2$ (Definition \ref{def:ODandOPDbuildings}).
\end{itemize}

The main theorems of the paper are the following:

In \cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.
In particular, $\PD(V)$ is spherical (see Section \ref{sec:preliminaries} for definitions).
More recently, in \cite{BPW24}, it was proved that for any field $k$,\,$\PD(V)$ has the homotopy type of the common bases complex ${\rm CB}(V)$ of $V$, as defined by J. Rognes \cite{Rognes}.
Concretely, ${\rm CB}(V)$ is the simplicial complex whose simplices are sets $\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\sigma$ is generated by some subset of $B$.
In this case, we say that the elements of $\sigma$ have a common basis.
Rognes showed that, although ${\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\leq 2n-3$, and he conjectured that ${\rm CB}(V)$ must indeed be $(2n-4)$-connected.
In \cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.
In particular, via \cite{BPW24} this implies that $\PD(V)$ is spherical for any field.

Considering the ordered version $\OPD(V)$ of $\PD(V)$
appears to be equally natural.
By definition, $\OPD(V)$ is the poset whose elements are tuples
$(V_1,\ldots,V_r)$, $r\geq 1$, of distinct subspaces of $V$ such that $\{V_1,\ldots,V_r\} \in \PD(V)$.
The ordering in $\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \ref{def:ODandOPDVectorSpaces}).
We call the elements of $\OPD(V)$ ordered partial decompositions.
It follows from \cite[Theorem 6.8]{PW25} that $\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.
If $V$ is a finite vector space then \cite[Theorem 6.10]{PW25} implies
that $\OPD(V)$ has the homotopy type of the two-fold join $\Delta * \Delta$ of the Tits building $\Delta$ of $V$.
Using GAP we also verified in some small examples that the top homology group of
$\OPD(V)$, as a $\GL(V)$-module, is the tensor square of the Steinberg module.

Later, we found an alternative approach to determining the
homotopy type of $\OPD(V)$ using closely related constructions, which inspired the work in this paper.
Let $\OD(V)$ denote the subposet of $\OPD(V)$ consisting of full ordered decompositions $(V_1,\ldots, V_r)$; i.e., those decompositions satisfying
$V \cong V_1\oplus \cdots \oplus V_r$.
Let $X = \Delta \cup \OD(V)$ be the poset on the disjoint union of
$\Delta$ (regarded as a poset without the empty simplex) and $\OD(V)$ with the following
ordering.
We keep the inclusion ordering among elements of $\Delta$, and the refinement ordering in $\OD(V)$. No element of $\OD(V)$ lies below any element of $\Delta$. If $\sigma\in \Delta$ and $d\in \OD(V)$, we set $\sigma \prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\sigma$.
In Theorem \ref{thm:PDandPDBuildingVectorSpace} we show that
$\OPD(V) \simeq X$.
The poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \cite{MPW}, which is shown therein to have
the homotopy type of $\Delta * \Delta$.
This again allows us to conclude that $\OPD(V) \simeq \Delta * \Delta$ for any finite-dimensional vector space $V$.
Since all homotopy equivalences can be seen to be $\GL(V)$-equivariant, it now follows that the top homology group of
$\OPD(V)$ carries the tensor square of the Steinberg representation.

These observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.
In this paper, we study the following questions and answer all of them positively.
Let $\Delta$ be a (spherical) building.
\begin{enumerate}
 \item Can we define a poset $\OPD(\Delta)$ that, up to homotopy, coincides with $\OPD(V)$ when $\Delta$ is the building of the vector space $V$?
 \item Can we similarly define $\PD(\Delta)$? Is it highly connected?
 \item Do we have a common basis complex ${\rm CB}(\Delta)$ for $\Delta$ that coincides with ${\rm CB}(V)$ when $\Delta$ is the building of the vector space $V$?
 \item If questions (2) and (3) have a positive answer, do we have ${\rm CB}(\Delta) \simeq \PD(\Delta)$?
 \item Is $\OPD(\Delta)\simeq \Delta * \Delta$?
\end{enumerate}

\begin{itemize}
 \item ${\rm CB}(\Delta) = \{\sigma \tq \sigma$ is a subset of the vertex set of some apartment of $\Delta\}$ (Definition \ref{def:commonBasis})
 \item $\D(\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \ref{def:decompBuilding}).
 \item $\PD(\Delta) = \Delta\cup \D(\Delta)$, with crossed-term ordering $\sigma\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\sigma$ (Definition \ref{def:PDbuildings}).
 \item $\OD(\Delta) = \{(\sigma_1,\sigma_2)\tq \sigma_1,\sigma_2$ are non-empty opposite simplices of $\Delta\}$, with coordinate-wise reverse-inclusion ordering (Definition \ref{def:ODandOPDbuildings}).
 \item $\OPD(\Delta) = \Delta \cup \OD(\Delta)$, with crossed-term ordering $\sigma\prec (\sigma_1,\sigma_2)$ if there is an apartment containing $\sigma,\sigma_1,\sigma_2$ (Definition \ref{def:ODandOPDbuildings}).
\end{itemize}

The main theorems of the paper are the following:

Later, we found an alternative approach to determining the
homotopy type of $\OPD(V)$ using closely related constructions, which inspired the work in this paper.
Let $\OD(V)$ denote the subposet of $\OPD(V)$ consisting of full ordered decompositions $(V_1,\ldots, V_r)$; i.e., those decompositions satisfying
$V \cong V_1\oplus \cdots \oplus V_r$.
Let $X = \Delta \cup \OD(V)$ be the poset on the disjoint union of
$\Delta$ (regarded as a poset without the empty simplex) and $\OD(V)$ with the following
ordering.
We keep the inclusion ordering among elements of $\Delta$, and the refinement ordering in $\OD(V)$. No element of $\OD(V)$ lies below any element of $\Delta$. If $\sigma\in \Delta$ and $d\in \OD(V)$, we set $\sigma \prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\sigma$.
In Theorem \ref{thm:PDandPDBuildingVectorSpace} we show that
$\OPD(V) \simeq X$.
The poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \cite{MPW}, which is shown therein to have
the homotopy type of $\Delta * \Delta$.
This again allows us to conclude that $\OPD(V) \simeq \Delta * \Delta$ for any finite-dimensional vector space $V$.
Since all homotopy equivalences can be seen to be $\GL(V)$-equivariant, it now follows that the top homology group of
$\OPD(V)$ carries the tensor square of the Steinberg representation.

The main theorems of the paper are the following:

Here, $\CB(\Delta)$ and $\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.

If $\Delta$ is the building of a finite-dimensional vector space $V$, then $\CB(V) = \CB(\Delta)$, $\PD(\Delta)\simeq \PD(V)$ and $\OPD(\Delta)\simeq \OPD(V)$.
In particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \cite{MPW}.
Our proof is independent of that of \cite{MPW}, and it is based on geometric properties of buildings.
To prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.
Key ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible." For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.
This implies that $K$ is contractible by \cite{Serre}.
We extend this idea to prove that certain fixed-point subposets are contractible.

\begin{proof}
First, we show that $\Gamma$ is well-defined.
Note that if $c\in \PD(\Delta)'$, then there is an apartment $\Sigma$ containing all the vertices involved in $c$ by definition of $\PD(\Delta)$.
In particular, the convex hull of $c$ is a convex subcomplex of $\Sigma$, so it lies in $Y(\Delta)$.
If in addition the maximal element of $c$, say $x$, satisfies $x\prec S$, we can take $\Sigma$ containing $S$ by definition of the ordering in $\PD(\Delta)$, and hence $\Gamma(c) = \Conv_{\Delta}(c,S)\in Y(\Delta)_{\supseteq S}$.
Thus $\Gamma$ is an order-preserving map between posets.
Also, as pointed out at the end of Subsection \ref{subsub:chambercomplexes}, $\Gamma$ is $H$-equivariant.

Next, we prove that $\OPD(\Delta)$ has the homotopy type of the simplicial join $\Delta*\Delta$, and that it carries the tensor-square Steinberg representation in homology.
For a poset $X$, recall that $X^{(i)}$ denotes the subposet of elements of height at most $i$.
If $X = \X(K)$ where $K$ is a Cohen-Macaulay simplicial complex, then $X$ is Cohen-Macaulay as a poset.
Also, every rank selection of a Cohen-Macaulay poset is spherical (see Theorem 6.2 of \cite{Baclawski}).
In particular, rank selections of the face poset of a (spherical) building are Cohen-Macaulay.

\begin{theorem}
\label{thm:OPDandDeltaDelta}
Let $\Delta$ be a spherical building of dimension $m$ and let $H$ be a group acting on $\Delta$ by simplicial automorphisms.
Write $D_1 = \X(\Delta)$ and $D_2 = \X(\Delta)^{\op}$.
Note that $\OPD(\Delta) = D_1 \cup \OD(\Delta)$.

Let $\phi:\OPD(\Delta)\to D_1 \ojoin D_2$ be the following map:
\begin{align*}
 \phi(\sigma,\sigma') & = \sigma \in D_2 \quad \text{ if } (\sigma,\sigma')\in \OD(\Delta) \subset \OPD(\Delta),\\
 \phi(\sigma) & = \sigma \in D_1 \quad \text{ if } \sigma\in D_1 \subset \OPD(\Delta).
\end{align*}
Then $\phi$ is an $H$-equivariant order-preserving map.
Moreover, the following hold:
\begin{enumerate}
\item The restriction
\[ \phi_H: (\OPD(\Delta)^{(i+m+1)})^H \to (D_1 \ojoin (D_2^{(i)}))^H\]
is a homotopy equivalence for all $0 \leq i \leq m$.
Thus, $\OPD(\Delta)^{(i+m+1)} \simeq D_1 \ojoin D_2^{(i)}$ is spherical of dimension $m + i + 1$.
\item In particular, $\phi$ gives rise to a homotopy equivalence
\[\OPD(\Delta) \simeq \Delta * \Delta,\]
which is $H$-equivariant if $H$ is a compact Lie group, and to an $H$-equivariant isomorphism
\[ \widetilde{H}_{*}(\OPD(\Delta)) \cong_H \widetilde{H}_*(\Delta*\Delta).\]
\end{enumerate}
\end{theorem}

\begin{theorem}
\label{thm:PDandPDBuildingVectorSpace}
Let $V$ be an $n$-dimensional vector space over a field $k$.
Let $\sigma \in \OPD(V)'$, and write $\sigma = \sigma_0\cup \sigma_1$ where $\sigma_0$ is the set of ordered partial decompositions in $\sigma$ that span a proper subspace of $V$, and $\sigma_1$ is the set of full ordered decompositions in $\sigma$.
Then, for $\sigma \in \OPD(V)'$, we set
\[ \phi(\sigma) = \begin{cases}
\{ \gen{s} \tq s\in \sigma_0\} & \sigma_1 = \emptyset,\\
\max \sigma_1 & \sigma_1\neq\emptyset.
\end{cases}\]
Then $\phi$ is a $\GL(V)$-equivariant poset map $\phi: \OPD(V)' \to \OKtwo{V}$ such that for all $H\leq \GL(V)$, the induced map $\phi_H: (\OPD(V)')^H \to \OKtwo{V}^H$ is a homotopy equivalence.

We get analogous conclusions for the corresponding map $\PD(V)'\to \Ktwo{V}$.
\end{theorem}

Here, ${\rm CB}(\Delta)$ and $\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.

There is a canonical map $F:\OD(\Delta)\to \D(\Delta)$ that maps a pair of opposite simplices $(\sigma_1,\sigma_2)\in\OD(\Delta)$ to the Levi sphere spanned by $\sigma_1,\sigma_2$ (which is actually their convex hull, see Subsection \ref{sub:convexity}).

In \cite{vH}, A. von Heydebreck showed that $\OD(\Delta)$ is Cohen-Macaulay of dimension $\dim\Delta$.
Using the map $F$, we conclude:

\begin{corollary}
\label{coro:DDeltaCM}
Let $\Delta$ be a spherical building.
Then the poset of Levi spheres $\D(\Delta)$ is Cohen-Macaulay of dimension $\dim \Delta$.
\end{corollary}

We mention that Corollary \ref{coro:DDeltaCM} can potentially be applied to compute the homotopy type of Quillen's $p$-subgroup posets \cite{Qui78} of connected reductive algebraic groups $G$ over finite fields $\GF{q}$, where $p$ is a ``good" prime for which $q$ has order $1$ modulo $p$.
This is based on the work by D. Rossi \cite{Rossi}, where it is proved that for such a good prime $p$, the $p$-subgroup poset of the finite group $G(\GF{q})$ has the homotopy type of the poset of $\GF{q}$-Levi subgroups, which coincides with the poset $\D(\Delta(G,\GF{q}))$.
Here $\Delta(G,\GF{q})$ is the building of $G(\GF{q})$.

We will write $|S|$ to denote the geometric realization of the Levi sphere $S$, which is a sphere of dimension $\dim S$ (see Lemma \ref{lm:convexIsleviSphere}).

The map $F$ naturally extends to an order-preserving map $F:\OPD(\Delta)\to \PD(\Delta)$ as the identity on the $\Delta$-part.
Studying fibers of this map, we prove:

\begin{theorem}
Let $\Delta$ be a spherical building.
Then we have a homotopy equivalence
\begin{equation}
\OPD(\Delta) \simeq \PD(\Delta) \bigvee_{ T \in {\D(\Delta)}} |T| * |T|*\PD(\Lk_{\Delta}(\sigma_T)),
\end{equation}
where $\sigma_T\in T$ is some maximal simplex.
In particular, $\PD(\Delta)$ is spherical of dimension $2\dim\Delta + 1$, and ${\rm CB}(\Delta)$ has the homotopy type of a wedge of spheres of dimension $2\dim\Delta + 1$.

Moreover, for a Levi sphere $S$ and a pair of opposite maximal simplices $\sigma_1,\sigma_2\in S$, we have
\begin{align*}
\PD(\Delta)_{\prec S} & \simeq |S| * \PD(\Lk_{\Delta}(\sigma_1))\\
\OPD(\Delta)_{\prec (\sigma_1,\sigma_2)}  & \simeq |S| * \OPD(\Lk_{\Delta}(\sigma_1)).
\end{align*}
\end{theorem}

Here we are using the bar notation for Levi spheres specifically to distinguish between $S$ as a sphere (via $|S|$) and $S$ as an element of the poset $\D(\Delta)$.

If $\Delta$ is the building of a finite-dimensional vector space $V$, then ${\rm CB}(V) = {\rm CB}(\Delta)$, $\PD(\Delta)\simeq \PD(V)$ and $\OPD(\Delta)\simeq \OPD(V)$.
In particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \cite{MPW}.
Our proof is independent of that of \cite{MPW}, and it is based on geometric properties of buildings.
To prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.
Key ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible."  For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.
This implies that $K$ is contractible by \cite{Serre}.
We extend this idea to prove that certain fixed-point subposets are contractible.

\subsection*{Organization of the article}

The paper is organized as follows.
In Section \ref{sec:preliminaries} we introduce the notation from algebra, combinatorics, geometry, and topology that we will use throughout the paper. We also recall results that will play an important role in the rest of the paper. In particular,  we outline the results of the theory of Tits buildings, mostly following \cite{AB} and sometimes \cite{Tits1}.
In Section \ref{sec:vectorspaces}, we study the motivating type A situation.
More precisely, we give exact definitions of the ordered and unordered (partial) decomposition posets and the common basis complex for vector spaces. We also provide the proof that the ordered partial vector space decompositions are equivariantly homotopy equivalent to the poset of ordered partial decompositions of the corresponding building.

In Sections \ref{sec:CBBuildings}, \ref{sec:leviSpheres} and \ref{sec:opd}, we define common basis complexes, ordered and unordered decomposition, and partial decomposition posets for arbitrary buildings.
We provide results about homotopy types and homotopy equivalences, including equivariant versions and fixed points. Some of them extend
facts from the linear case discussed in
Section \ref{sec:vectorspaces}.

Finally, in Section \ref{sec:algebraicgroups} we specialize these results to the case of rational points of connected reductive algebraic groups, obtaining explicit descriptions of the posets and complexes in terms of rational parabolics and Levi subgroups.
In particular, as an application of the equivariance of our maps and the connectivity results, in Proposition \ref{prop:lowLeviIntervalPD} we provide a long exact sequence in terms of Steinberg-square modules of rational Levi subgroups for the rational points of a connected reductive algebraic group, ending in the Steinberg module of such a group.

Computer calculations were performed with GAP \cite{GAP4}
and software package \cite{posets}.

\subsection*{Acknowledgments}
We thank Bernhard Mühlherr and Richard Weiss for helpful and motivating conversations.
The first author was supported by the FWO grant 12K1223N.

To prove the stated homotopy equivalences, the authors say they “use the notion of convexity and complete reducibility introduced by Serre.” The “key ingredients” are statements that “a certain (convex) subcomplex $K$ of the building is contractible.” Their method is: “we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$,” and then “this implies that $K$ is contractible by \cite{Serre}.” They also note that they “extend this idea to prove that certain fixed-point subposets are contractible.”

They further indicate a structural approach via the canonical map $F:\OD(\Delta)\to \D(\Delta)$ (sending an opposite pair $(\sigma_1,\sigma_2)$ to the Levi sphere they span), which “naturally extends to an order-preserving map $F:\OPD(\Delta)\to \PD(\Delta)$ as the identity on the $\Delta$-part,” and that “studying fibers of this map” yields the wedge-type homotopy decomposition stated in the later theorem.

Question: Let \(\Delta\) be a spherical building. Define the common-basis complex \({\rm CB}(\Delta)\) to be the simplicial complex whose simplices are subsets of the vertex set of some apartment of \(\Delta\). Let \(\mathcal D(\Delta)\) be the poset of nonempty Levi spheres ordered by reverse inclusion, and let \(\mathrm{PD}(\Delta)=\Delta\cup \mathcal D(\Delta)\), with cross relation \(\sigma\prec S\) when there is an apartment containing both the simplex \(\sigma\) and the Levi sphere \(S\). Let \(\mathrm{OPD}(\Delta)\) be the associated ordered partial-decomposition poset, let \(\mathrm{sd}\,\mathrm{PD}(\Delta)\) denote the barycentric subdivision, and let \(\Delta*\Delta\) be the simplicial join. For a group \(H\) acting on \(\Delta\) by simplicial automorphisms, write \(X^H\) for the fixed-point subposet or subcomplex. Which statement is valid for these objects?

There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences on fixed points, namely \(\Gamma_H:\mathrm{PD}(\Delta)^H\to {\rm CB}(\Delta)^H\) and \(\phi_H:\mathrm{OPD}(\Delta)^H\to \Delta^H*\Delta^H\). In particular, \(\Gamma\) and \(\phi\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology.

Choices:
B. There exist order-preserving maps \(\Gamma:\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences on fixed points, namely \(\Gamma_H:\mathrm{PD}(\Delta)^H\to {\rm CB}(\Delta)^H\) and \(\phi_H:\mathrm{OPD}(\Delta)^H\to \Delta^H*\Delta^H\). In particular, \(\Gamma\) and \(\phi\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology.
C. There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that \(\Gamma\) and \(\phi\) are homotopy equivalences. In particular, they induce isomorphisms in homology and cohomology.
D. There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences on fixed points, namely \(\Gamma_H:\mathrm{PD}(\Delta)^H\to {\rm CB}(\Delta)^H\) and \(\phi_H:\mathrm{OPD}(\Delta)^H\to (\Delta*\Delta)^H\). In particular, \(\Gamma\) and \(\phi\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology.
E. There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences between the underlying spaces. Moreover, for every such \(H\), the fixed-point subposets \(\mathrm{PD}(\Delta)^H\) and \(\mathrm{OPD}(\Delta)^H\) are contractible.

barycentric-subdivision requirement for the PD-to-CB map

fixed-point equivariance and fixed-point-level homotopy equivalences for every H

identification of the fixed-point target as \(\Delta^H*\Delta^H\), not \((\Delta*\Delta)^H\)

contractibility applies only to certain convex fibers/fixed-point subposets used in the proof, not to all fixed-point subposets themselves

The stem gives only definitions and notation; it does not state the theorem’s conclusion or uniquely signal choice A. The correct answer is not leaked explicitly or by an obvious cue.

This is not a direct restatement of information already given in the stem. The respondent must distinguish between several closely related technical claims, so the item is non-tautological.

The item requires some reasoning or precise theorem-level recognition to separate subtle variants (e.g., barycentric subdivision, fixed points of joins, strength of equivariant claims). However, it mainly tests recall/discrimination of a known result rather than substantial generative mathematical construction.

The distractors are strong: they are plausible, mathematically nearby, and reflect realistic failure modes such as omitting subdivision, weakening the theorem, confusing \((\Delta*\Delta)^H\) with \(\Delta^H*\Delta^H\), or overclaiming contractibility.

A high-quality MCQ with no meaningful answer leakage and very good distractors. Its main limitation is that it tests precise theorem recognition more than deep generative reasoning.