%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{157}
%\documentclass[finalversion]{FPSAC2021}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

% Forbidden:  fullpage, authblk, babel
%\usepackage[top=30truemm,bottom=30truemm,left=25truemm,right=25truemm]{geometry}
%\usepackage{amssymb, amsthm}
%\usepackage{hyperref}


%--- packages ---
\usepackage{tikz}
		\usetikzlibrary{calc, patterns, intersections}
	\usepackage{tikz-cd}
	\usepackage{comment}
%%%\usepackage{graphicx}
%%%\usepackage{amsmath}
\usepackage{amscd}
\usepackage{enumerate}	
\usepackage{url}
%%%\usepackage{otf}
\usepackage{empheq}
%%%\usepackage[all]{xy}
%-----

\theoremstyle{plain}
%\theoremstyle{theorem}
\newtheorem*{theorem*}{Theorem}

\newtheorem{thm}{Theorem}[section]
\newtheorem{theorem}[thm]{Theorem}
%
\newtheorem{lemma}[thm]{Lemma}	
\newtheorem{lem}[thm]{Lemma}	
%
\newtheorem{proposition}[thm]{Proposition}
\newtheorem{prop}[thm]{Proposition}
%
\newtheorem{cor}[thm]{Corollary}
%
%----theoremstyle{definition}-----
\theoremstyle{definition}
\newtheorem{definition}[thm]{Definition}
\newtheorem{Def}[thm]{Definition}
%
\newtheorem{remark}[thm]{Remark}
\newtheorem{Rem}[thm]{Remark}
%
\newtheorem{example}[thm]{Example}
\newtheorem{Ex}[thm]{Example}


%	\usepackage{graphicx}
%	\usepackage{overpic}
%	%\usepackage{tikz-cd}
%	%\usetikzlibrary{decorations.pathreplacing}
%	%\usepackage[all]{xy}

%	\usepackage{lipsum}
%	\usepackage{comment}
%	\usepackage{multicol}



\newcommand{\Mapsto}{}
%\input{SET}
%\input{ALGEBRA}
%\input{PDIW}
%\input{INSET_input}


\numberwithin{equation}{section}
\allowdisplaybreaks

%\newcommand\alert{\textit}


\newcommand\Z{\mathbb{Z}}		% the set of integers
\newcommand\C{\mathbb{C}}		% the set of complex numbers
\newcommand\lm{\lambda}			% $\lambda$
\newcommand\ep{\epsilon}		% $\epsilon$ standard basis
\newcommand\al{\alpha}			% $\alpha$ root

% Lie algebra
\newcommand\g{\mathfrak{g}}
\newcommand\gl{\mathfrak{gl}}
\renewcommand\sl{\mathfrak{sl}}
\newcommand\h{\mathfrak{h}}		% Cartan subalgebra
\newcommand\fdwt{\varpi}		% fundamental domain
\newcommand\con{\mathbf{c}}		% content
\newcommand\predom{\zeta}	% predominant weight corresp to $\cdiag$
\newcommand\bra{\langle}		% the natural pairing
\newcommand\ket{\rangle}		%



\newcommand\cmin{b}	% the minimal element of c^{-1}(i)



\newcommand\pdiag{\Theta}		% periodic diagram
\newcommand\cdiag{\theta}		% cylindric diagram
\newcommand\peri{\omega}		% period


\newcommand\cyli{\mathcal{C}}	% cylinder
\newcommand\ptn{\mathcal{P}}	% the set of partitions
\newcommand\LE{{\mathrm{ST}}}	% the set of standard tableaux
\newcommand\lex{{\mathfrak{t}}}	% a standard tableau



\newcommand{\precdot}{\mathrel{\prec\!\!\!\cdot}}

\newcommand\rble{\prec}
\newcommand\rbleq{\preceq}


\newcommand\lble{\prec_L}		% weak left Bruhat order
\newcommand\lbge{\succ_L}		%
\newcommand\lbleq{\preceq_L}	%
\newcommand\lbgeq{\succeq_L}	%



\newcommand\sinf{\frac{\infty}{2}}		% semi-infinite

\newcommand\btm{\Gamma}			% the bottom set
\newcommand\btmmax{{\btm_{\mathrm{max}}}}
\newcommand\btmmin{{\btm_{\mathrm{min}}}}

\newcommand\hk{{\mathbf{h}}}
\newcommand\hkr{{{\gamma}_{\lex}}}

\newcommand\arm{{\mathrm{Arm}}}
\newcommand\leg{{\mathrm{Leg}}}

\newcommand\I{\mathcal{J}}
\newcommand\F{\mathcal{F}}		% the set of order filter
\newcommand\ideal{\xi}
\newcommand\ofil{\eta}			% order filter of $\cdiag$




%%%%%%%% Orders %%%%%%%%%%%
\newcommand{\oordneq}{{<^{\mathrm{or}}}}
\newcommand{\hpord}{\leq^{\mathrm{hp}}}
\newcommand{\hpordl}{{{}^{\mathrm{hp}\!\geq}}}

\newcommand\oord{{\leq^{\mathrm{or}}}}
\newcommand\oordl{{{}^{\mathrm{or}}\!\geq}}
\newcommand\tcord{{{\trianglelefteq^{\mathrm{tc}}}}}
\newcommand\tcordl{{{\trianglerighteq^{\mathrm{tc}}}}}
\newcommand\neword{\unlhd}%{{ \triangleleft}}
\newcommand\newordl{\unrhd}%{{\triangleright}}
\newcommand\cylord{\trianglelefteq}
\newcommand\cylordl{\trianglerighteq}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand\DS{{D}}

\newcommand{\CL}[3][white]{
	\put#2{%
	{\color{#1}%
	\put(0,-.1){\line(1,0){1}}%
	\put(0,-.2){\line(1,0){1}}%
	\put(0,-.3){\line(1,0){1}}%
	\put(0,-.4){\line(1,0){1}}%
	\put(0,-.5){\line(1,0){1}}%
	\put(0,-.6){\line(1,0){1}}%
	\put(0,-.7){\line(1,0){1}}%
	\put(0,-.8){\line(1,0){1}}%
	\put(0,-.9){\line(1,0){1}}%
	\put(.1,-1){\line(0,1){1}}%
	\put(.2,-1){\line(0,1){1}}%
	\put(.3,-1){\line(0,1){1}}%
	\put(.4,-1){\line(0,1){1}}%
	\put(.5,-1){\line(0,1){1}}%
	\put(.6,-1){\line(0,1){1}}%
	\put(.7,-1){\line(0,1){1}}%
	\put(.8,-1){\line(0,1){1}}%
	\put(.9,-1){\line(0,1){1}}}%
	\put(0,0){\line(1,0){1}}%
	\put(0,0){\line(0,-1){1}}%
	\put(0,-1){\line(1,0){1}}%
	\put(1,0){\line(0,-1){1}}%
	\put(.5,-.5){\makebox(0,0){#3}}%
	}
}
\newcommand{\NP}{}%{\newpage}


%% define your title in the usual way
\title{Colored hooks and poset structure of cylindric diagrams}


%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details




\author{\texorpdfstring{
  Kento Nakada
  \thanks{\href{mailto:nakada@okayama-u.ac.jp}{nakada@okayama-u.ac.jp}.}
  \addressmark{1},
%  \and
  Takeshi Suzuki
  \thanks{\href{mailto:suzuki@math.okayama-u.ac.jp}{suzuki@math.okayama-u.ac.jp}.}
  \addressmark{2},
  \and
  Yoshitaka Toyosawai
  \thanks{\href{mailto:prkr5rq9@s.okayama-u.ac.jp}{prkr5rq9@s.okayama-u.ac.jp}.}
  \addressmark{2}
}{
  Kento Nakada, Takeshi Suzuki, and Yoshitaka Toyosawai  
}}

%% then use \addressmark to match authors to institutions here
\address{
	\addressmark{1}
	Graduate School of Education, Okayama University, Japan.
	\\
	\addressmark{2}
	Graduate School of Natural Science and Technology, Okayama University, Japan.
%	\\
%	\addressmark{3}
%	Graduate School of Natural Science and Technology, Okayama University, Japan.
}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
	Cylindric diagrams admit structures of infinite $\mathrm{d}$-complete posets with natural ordering.
	The purpose of this paper is to provide a realization of a cylindric diagram as a subset of 
	an affine root system of type $A$ via colored hook lengths, 
	and to present several characterizations of its poset structure.
	Furthermore, the set of finite order ideals of a cylindric diagram is described
	as a weak right Bruhat interval of the affine Weyl group.
}

%% put your French abstract here, or comment this out if you don't have one
\resume{
	Les diagrammes cylindriques admettent des structures de posets infinis $\mathrm{d}$-complets avec un ordre naturel.
	Le but de cet article est de fournir une r\'ealisation d'un diagramme cylindrique en tant que sous-ensemble d'un syst\`eme affine de racines de type $A$ via des longueurs de crochets color\'es,
	et de pr\'esenter plusieurs caract\'erisations de sa structure de poset.
	De plus, l'ensemble des id\'eaux d'ordre fini d'un diagramme cylindrique est d\'ecrit
	comme un intervalle de Bruhat droit faible du groupe de Weyl affine.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{d-complete poset, cylindric diagrams, hooks.}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want

%\usepackage[backend=biber]{biblatex}
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{ref.bib}

%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above


\input{NSTfpsac_sec1.tex}	% introduction
\input{NSTfpsac_sec2.tex}	% cylindric diagram
\input{NSTfpsac_sec3.tex}	% main theorem


%\acknowledgements{
%}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
%\printbibliography

%\bibliographystyle{amsplain-url}
%\bibliography{ref}

\printbibliography

\end{document}
