3$. \end{prop} \begin{proof} Let $k=n+2\ge 5$, consider the Coxeter system of type $(W',S')$ of type~$\tilde C_{k-1}$, and let $J=\{1,2,\dots,n\}\se S'$. The group $B_n$ is naturally isomorphic to parabolic subgroup $W'_J$ of $W'$ via the isomorphism $\phi: W\ra W'_J$ with $\phi(i)=i$ for each generator $i\in S$. Thus, Lemma~\ref{lemm:parabolic} implies that if $n>3$ then $I(24,24)=I_{W}(24,24)\se \{\phi\inverse(w'):w'\in I_{W'}(24,24),\supp(w')\se J\}$, where the last set equals $\{\phi\inverse(24),\phi\inverse(2124)\}=\{24,2124\}$. It follows that $I(24,24)\se \{24,2124\}$. Similarly, if $n=3$, then using the methods from Remark~\ref{rmk:relating0cells} we can see that $I_{W'}(13,13)=\{13,134543,132413, 13245413\}$, whence Lemma~\ref{lemm:parabolic} implies that $I(13,13)\se \{13\}$ because~$4\notin J$. \end{proof} \subsubsection{Type $A_n$} We complete the proof of Theorem~\ref{thm:oneI}.(1) by deducing the following proposition from Proposition~\ref{prop:B}. \begin{prop} \label{prop:A} If $X=A_n$ $(n\ge 3)$, then $I(13,13)\se \{13\}$. \end{prop} \begin{proof} We use parabolic restriction as in the proof of Proposition~\ref{prop:B}. More precisely, let $(W,S)$ and $(W',S')$ be the Coxeter systems of type $A_n$ and $B_{n+1}$, respectively, let $J=\{2,3,\dots, n+1\}$, and consider the isomorphism $\phi:W\ra W'_{J}$ with $\phi(i)=i+1$ for all $i\in S$. We have $I_{W'}(24,24)\se\{24,2124\}$ by Proposition~\ref{prop:B} and $I(13,13)=I_{W}(13,13)\se \{\phi\inverse(z):z\in I_{W'}(24,24), \supp(z)\in J\}$ by Lemma~\ref{lemm:parabolic}; therefore $I(13,13)\se\{\phi\inverse(24)\}=\{13\}$. \end{proof} \subsubsection{Type $E_{q,r}$} We complete the proof of Theorem~\ref{thm:oneI}.(5) by deducing the following result from Theorem~\ref{thm:oneI}.(1): \begin{prop} \label{prop:Eqr} If $X=E_{q,r}$ where $r\ge q\ge 1$, then $I(x,x)\se \{x\}$ for all $x\in \{(-1)v,1v,(-1)1\}$. \end{prop} \begin{proof} Let $x\in \{(-1)v, 1v,(-1)1\}$. To prove the proposition we may assume that $r\ge q> 1$ in the Coxeter system $W$ of type $E_{q,r}$: any system $(W',S')$ of type $E_{1,r}$ lies in $W$ as a parabolic subgroup, and Lemma~\ref{lemm:parabolic} implies that if $I_{W}(x,x)\se\{x\}$ then $I_{W'}(x,x)\se \{x\}$. Under the assumption that $r\ge q> 1$, it further suffices to show that $I((-1)1,(-1)1)\se\{(-1)1\}$: since $\sw$ contains a single $\sim$-class by Proposition~\ref{prop:desClasses}.(1), if $I((-1)1,(-1)1)\se\{(-1)1\}$ then all 0-cells have size 1 in $W_2$ by Proposition~\ref{prop:invariance}, which in turn implies that the 0-cell $I(y,y)$ has to contain only the element $y$ that is obviously in it for all short stubs $y\in \sw$. Let $x=(-1)1$ and let $w\in I(x,x)$. Suppose $v\in \supp(w)$, so that some element~$i$ in the heap $H(w)$ has label $v$. Since $\calr(w)=\call(w)=\{-1,1\}$, the element $i$ is neither minimal nor maximal in $H(w)$, so it must cover some element $j$ and be covered by some element $j'$ in $H(w)$ by Lemma~\ref{lemm:vertical}. The labels of $j$ and $j'$ must both be $0$, the only neighbor of $v$ in $S$. Since $m(v,0)=3$, the chain $j\prec i\prec j'$ cannot be convex by Proposition~\ref{prop:FCCriterion}, so Lemma~\ref{lemm:horizontal} implies that $j$ and $j'$ are connected by a chain in~$H(w)$ that contains an element $k$ labeled by either $-1$ or $1$. Lemma~\ref{lemm:3chain} then implies that the set~$\{i,k\}$ forms a trapped antichain relative to either the special quadruple~$(v,0,-1,-2)$ or the special quadruple $(v,0,1,2)$. This contradicts Proposition~\ref{prop:trapped}, so~$v\notin\supp(w)$. But then $w$ lies in the parabolic subgroup of $W$ of type $A_{q+r+1}$ generated by the set~$\{-q,\cdots,-1,0,1,\cdots,r\}$. It follows from Theorem~\ref{thm:oneI}.(1), Lemma~\ref{lemm:parabolic} and Proposition~\ref{prop:invariance} that $I(x,x)$ is the singleton $\{x\}$. The proof is complete. \end{proof} \subsubsection{Type $F_n$} We complete the proof of Theorem~\ref{thm:oneI}.(6)--(7) by deducing the following proposition from Proposition~\ref{prop:B}. \begin{prop} \label{prop:F} If $X=F_n$ $(n\ge 3)$, then $I(24,24)\se \{24\}$ if $n=4$ and $I(24,24)\se \{24,243524\}$ if $n>4$. \end{prop} \begin{proof} Consider the parabolic group $W_J$ of $W$ generated by the set $J=\{2,3,\cdots, n\}$. It is a Coxeter group of type $B_{n-1}$, so $I_{W_J}(24,24)=\{24\}$ if $n-1=3$ and $I_{W_J}(35,35)=\{35, 3235\}$ if $n-1>3$ by Theorem~\ref{thm:oneI}.(2)--(3). In the latter case, computing the 0-cell $I_{W_J}(24,24)$ from $I_{W_J}(35,35)$ using the methods of Remark~\ref{rmk:relating0cells} gives $I_{W_J}(24,24)=\{24, 243524\}$. Let $w\in I(24,24)$. We claim that $1\notin \supp(w)$, so that $w$ lies in the parabolic subgroup of type $B_{n-1}$ generated by the set $J=\{2,3,\dots,n\}$. It follows that $I(24,24)=I_{W}(24,24)\se I_{W_J}(24,24)$, so $I(24,24)\se \{24\}$ if $n=4$ and $I(24,24)\se \{24,243524\}$ if $n>4$ by the last paragraph. To prove the claim, suppose that $1\in \supp(w)$, so that some element $i$ in the heap $H(w)$ has label $1$. This leads to a contradiction to Proposition~\ref{prop:trapped} in the same way the element $i$ does in the proof of Proposition~\ref{prop:Eqr}: Lemmas ~\ref{lemm:vertical},~\ref{lemm:horizontal} and~\ref{lemm:3chain} force $i$ to be part of an antichain $\{i,k\}$ contained in an $2$-interval $[j,j']\se H(w)$ where $j\prec i\prec j'$ and $e(k)=3$, and this antichain is a trapped antichain relative to the special quadruple $(1,2,3,4)$. It follows that $1\notin \supp(w)$, as claimed. \end{proof} \subsubsection{Type $H_n$} We complete the proof of Theorem~\ref{thm:oneI}.(8)--(9) by deducing the following proposition from Proposition~\ref{prop:B}. \begin{prop} \label{prop:H} If $X=H_n$ $(n\ge 3)$, then $I(13,13)\se \{13\}$ if $n=3$ and $I(24,24)\se \{24,2124\}$ if $n>3$. \end{prop} \begin{proof} If we can show $I(24,24)=\{24,2124\}$ whenever $n>3$, then we can use Remark~\ref{rmk:relating0cells} to obtain $I(13,13)=\{13,132413\}$ when $n=4$. Viewing $H_3$ as the parabolic subgroup of $H_4$ generated by $\{1,2,3\}$ naturally, we can then use Lemma~\ref{lemm:parabolic} to deduce that $I(13,13)=\{13\}$ in $H_3$. Thus, it suffices to prove that $I(24,24)\se \{24,2124\}$ in~$H_n$ whenever $n>3$. Assume $n>3$ and let $w\in I(24,24)$. Let $\ul w$ be a reduced word of $w$. We claim that the heap $H(w)=H(\ul w)$ does not contain a convex chain of the form $i\prec j\prec i'\prec j'$ where $e(i)=e(i')=1,e(j)=e(j')=2$ or where $e(i)=e(i')=2, e(j)=e(j')=1$. By Proposition~\ref{prop:FCCriterion}, the claim implies that $\ul w$ is the reduced word of an element $w'$ in the Coxeter group of type $B_n$. Moreover, we have $\la(w')=n(w')=n(w)=2$ by Proposition~\ref{prop:a.vs.n}.(3), whence Theorem~\ref{thm:CellViaDecomp}.(3) implies that $w'\in I_{B_n}(24,24)=\{24,2124\}$ by reduced word considerations. It follows that $w\in \{24,2124\}$ in $H_n$, so that $I(24,24)\se \{24,2124\}$ in $H_n$, as desired. It remains to prove the claim, which we do by contradiction. Suppose that $e(i)=e(i')=1$ and $e(j)=e(j')=2$ in the chain $C$. Since $1\notin\call(w)$, the element $i$ must cover an element $h$ with label $2$ in $H(w)$ by Lemma~\ref{lemm:vertical}. Let $C=\{h,i,j,i',j'\}$ and let $k\in [h,j']$. Applying Lemma~\ref{lemm:horizontal} to the chains $h\prec i\prec j$ and $j\prec i'\prec j'$, we note that if~$e(k)\in \{1,2\}$ then $k\in C$. Also observe that $e(k)\neq 3$, for otherwise $\{i,k\}$ is a trapped antichain in $H(w)$ with relative to the special quadruple $(1,2,3,4)$, contradicting Proposition~\ref{prop:trapped}. Finally, we cannot have $e(k)>3$ either, because otherwise $e(k)$ and~$2$ lie on different sides of the vertex 3 in $G$, so $h$ must be connected to $k$ by a chain in~$[h,j']$ passing through an element labeled by 3, contradicting the last observation. It follows that $[h,j']=C$ as sets. But then the chain $h\prec i\prec j\prec i'\prec j'$ is convex, which contradicts the fact that $w\in \fc(H_n)$ by Proposition~\ref{prop:FCCriterion}. A similar contradiction can be derived if $e(i)=e(i')=2$ and $e(j)=e(j')=1$, and the proof is complete. \end{proof} \longthanks{ We thank Dana Ernst for useful discussions. We also thank the anonymous referee for reading the paper carefully and suggesting many improvements.} \bibliographystyle{mersenne-plain} \bibliography{ALCO_Xu_708} \end{document}