\section{Introduction}

%%Let $\peri\in \Z_{\geqq 1}\times\Z_{\leqq -1}$.
%A { periodic $($Young$)$ diagram} %of period $\peri\in \Z_{\geqq 1}\times\Z_{\leqq -1}$ 
%%skew diagram of period $\peri$ 
%is a Young diagram consisting of infinitely 
%many cells in $\Z^2$ which is invariant under parallel 
%translations generated by a certain vector $\peri\in \Z^2$ called the period
%(see Fig. \ref{periodic diagram}).
%
An order filter of a cylinder \( \Z^2/\Z\peri \,\,\,(\peri\in\Z_{\geqq 1}\times\Z_{\leqq -1}) \)
%The image of a periodic diagram under the natural projection 
%%$\pi:\Z^2\to \Z^2/\Z\peri$ 
%onto the cylinder $\Z^2/\Z\peri$
is called a \emph{cylindric (Young) diagram}.
A diagram given as set-difference of two cylindric diagrams is called a \emph{cylindric skew diagram} (see Fig. \ref{fig:skew diag}).

%The set of standard tableaux on a periodic diagram can be identified 
%with the set of  standard tableaux on the corresponding cylindric 
%skew diagram.
We note that cylindric skew diagrams have been known to 
parameterize a
certain class of irreducible modules over 
the Cherednik algebras (double affine Hecke algebras)
(\cite{Suz2005,SuzVaz2005})
and the (degenerate) affine Hecke algebras (\cite{Kle1996, Ruff2006}) of type $A$,
where standard tableaux on those diagrams also appear.
%We also note that %the authors presented in \cite{SuzToy2022} the conjecture of the hook formula giving the number of cylindric standard tableaux on cylindric skew diagrams.
%the conjecture of hook formulas for cylindric skew diagrams was given in \cite{SuzToy2022}.



Let $\peri=(m,-\ell)$ and
let $\cdiag$ be a cylindric diagram in $\Z^2/\Z\peri$.
%of period $\peri$.
The lattice $\Z^2$ admits a partial order $\leq$ defined by
$$(a,b)\leq (c,d) \iff a\geqq c\text{ and }\ b\geqq d,$$
which induces %and it induces 
a poset structure on $\Z^2/\Z\peri$ and also on $\cdiag$.
%
Together with the content map $\con:\cdiag\to \Z/\kappa\Z$,
 where $\con(a,b)=b-a\mod \kappa$ and  $\kappa=\ell+m$,
the cylindric digram $\cdiag$ is a locally finite $\Z/\kappa\Z$-colored
$d$-complete poset in the sense of \cite{Str2020,Str2021}.

Our first goal of the present paper is to describe the poset structure $(\cdiag,\leq)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We briefly review a description in  the classical case.
Let $\lm$ be a finite Young diagram. 
The associated  Grassmannian permutation $w_\lm$
 is an element of the Weyl group of a finite root system $R$ of type $A$.
% where  $n=\sharp\{\con(x)\mid x\in \lm\}$.
%
In \cite{Stem2001}, J.~R.~Stembridge has shown
%It is known %has known been  shown by Stembridge \cite{Stem2001}
that
%It holds that
 the poset  $(\lm,\leq)$ is dually isomorphic to the poset 
$(R(w_\lm^{-1}),\oord)$, where 
$R(w_\lm^{-1}):=R_+\cap w_\lm^{-1} R_-$
%is the inversion set of $w$
% where   $R_+$ (resp. $R_-$) is the set of positive (resp. negative) roots
%Moreover the map $\hk$ gives an isomorphism $(\lm,\leq)\cong (R(w_\lm),\oord)$ as posets,
and $\oord$ is the ordinary order (or the standard order) 
defined by
%as the transitive closure of the relation
$$
  \al\oord \beta \iff \beta-\al \text{ is a sum of simple roots}.
 %\sum_{i \in [1,n]}\Z_{\geqq 0}\al_i
$$
%for $\al,\beta\in R(w_\lm)$
%where $\{\al_1,\al_2,\dots,\al_{n}\}$
%is the set of simple roots.


Let $\cdiag$ be a cylindric diagram
in $\Z^2/\Z\peri$. %of period $\peri=(m,-\ell)$.
We would like to describe the poset $(\cdiag,\leq)$
in terms of the root system of type $A_{\kappa-1}^{(1)}$ with $\kappa=\ell+m$.
A key ingredient in our approach is 
the {\it colored hook length} (\cite{Nak2008, Proc2014}),
 %which have appeared in \cite{Proc2014, Nak2008}, 
which is given by
$$
  \hk(x)=\sum_{y\in H(x)}\al_{\con(y)}\ \ (x\in\cdiag),
$$
where $H(x)$ denotes the hook at $x$ and $\al_i$  are  simple roots.
(See \Cref{ss:hook} for precise definitions.)
%
We will show
 that the map $\hk$ embeds the cylindric diagram $\cdiag$ into the set $R_+$ of positive (real) roots,
 and that the image $\hk(\cdiag)$ is also characterized as
the subset $\DS(\predom_\cdiag)$ 
of $R_+$ consisting of those elements satisfying
$$\bra\predom_\cdiag,\al^\vee\ket=-1,$$
where $\predom_\cdiag$ is 
 a predominant integral weight determined by $\cdiag$ (see \Cref{sec:predom} for details).
The set $\DS(\predom_\cdiag)$ can be thought as
 the inversion set $R(w_\cdiag)$ 
associated with a ``semi-infinite Grassmannian permutation'' $w_\cdiag$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%We will give  a natural generalization of the colored hook length $\hk$ 
%and the inversion set $R(w_\lm)$ and then 
%obtain a bijection
%$$\hk:\cdiag\to R(w_\cdiag).$$




Unlike the classical case, 
the ordinary order in $R(w_\cdiag)$ does not lead a poset isomorphism,
and we need to introduce a modified order $\neword$ in $R(w_\cdiag)$ by
\begin{equation*}
  \al\neword \beta \iff
  \beta-\al
   \text{ is a sum of elements of $\Pi_\cdiag$}
%  \in \sum_{\gamma\in \Pi_\cdiag}\Z_{\geqq 0}\gamma,
\end{equation*}
to obtain a poset isomorphism $(\cdiag,\leq)\cong
(R(w_\cdiag),\neword)$, where $\Pi_\cdiag$ is a certain subset of the affine root system (see \Cref{sec:inversion set}).


Another description of the poset $\cdiag$ is given by 
a linear extension or (reverse) standard tableau $\lex$ on $\cdiag$,
 namely a bijective order preserving map $\cdiag\to\Z_{\geqq1}$.
%A standard tableau 
A linear extension $\lex:\cdiag\to\Z_{\geqq1}$ brings a partial order \( \hpord_\lex \) to $\Z_{\geqq1}$
 and the resulting poset is an infinite analogue of a heap \cite{Stem2001}.
%
%
\begin{comment}
%For a cylindric diagram $\cdiag$ and a standard tableau $\lex$,
%the following posets are associated:
\begin{itemize}
\item $(\cdiag,\le)$, 
\item $(R(w_\cdiag),\neword)=(D(\predom_\cdiag),\neword)$,
\item $(\Z_{\geqq1},\hpord_\lex)$,
\end{itemize}
where the partial order $\neword$ is a new order to be introduced in \Cref{sec:1stresult}
and the partial order $\hpord_\lex$ is an analogy to cylindric diagrams of the heap ordering.
The set $D(\predom_\cdiag)$ is a subset of positive roots determined by some integral weight $\predom_\cdiag$ (see \Cref{sec:hooks}).
 %and a standard tableau $\lex$ on $\cdiag$,
%some integral weight $\predom_\cdiag$ and the inversion set $R(w_\cdiag)$ determine.
%The integral weight $\predom_\cdiag$ is pre-dominant, namely,
%$\bra \predom_\cdiag, \al^\vee \ket \geqq -1$ for all $\al\in R_+$.
The colored hook length $\hk$ can extend to cylindric diagrams.
To describe of $\im\hk$,
we will show that the map $\hk$ gives a bijection
from $\cdiag$ to $D(\predom_\cdiag)$ (\Cref{thm:hkinR+}).
The set $R(w_\cdiag)$ is a subset of positive roots associated a ``semi-infinite word'' $w_\cdiag$.
Moreover, the set $D(\predom_\cdiag)$ coincides with the inversion set $R(w_\cdiag)$ (see \Cref{sec:hooks}).
\end{comment}
In summary,  we  have the following:
%
%%%%%%%%%%%%%%%%%%%
%%%%%% inthm >> theorem
\begin{thm}[\Cref{th:cylord}, \Cref{prop:cdiagandheap}]
%\begin{theorem*}[\cref{th:cylord}, \cref{prop:tcord}, \ref{prop:cdiagandheap}, and \ref{prop:cdiagandheap2}]
  The followings are poset isomorphisms:
  \begin{equation*}
    (\Z_{\geqq1},\hpord_\lex) \stackrel{\lex}{\leftarrow} 
    (\cdiag,\le) \stackrel{\hk}{\rightarrow}
    (R(w_\cdiag),\neword).
%    = (R(w_\cdiag),\tcord) = (R(w_\cdiag),\hpord)
  \end{equation*}
\end{thm}
We will also give several other description of the order in \( R(w_\cdiag) \) (\Cref{prop:tcord,prop:cdiagandheap2}).
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%

Another goal of this paper is to describe the poset structure %of the set 
$\I(\cdiag)$.
%of cylindric skew diagrams included in $\cdiag$
%or equivalently, the set of 
%(or proper order ideals of $\cdiag$)
%with the partial order given by the inclusion relations.
%
%
For a finite Young diagram $\lm$,
%it is determined some dominant integral weight $\zeta_\lm$.
%Let $W^\lm$ denote the set of the minimal length coset representatives of $W/W_\lm$,
%where $W_\lm=\{w \in W \mid w \zeta_\lm=\lm\}$.
it is known that the set $\I(\lm)$ of order ideals
 of $\lm$  is isomorphic to
the interval $[e,w_\lm]=\{u \in W \mid e \rbleq u \rbleq w_\lm\}$ 
with weak right Bruhat order $\rbleq$ (\cite[Proposition I]{Proc2014}).
For a cylindric diagram $\cdiag$,
we define a ``semi-infinite weak right Bruhat interval'' $[e,w_{\cdiag})$,
and we have the following:
%For a cylindric diagram $\cdiag$,
%some integral weight $\predom_\cdiag$ is determined.
%In particular, it is predominant, namely, $\bra \predom_\cdiag, \al^\vee \ket \geqq -1$ for all $\al \in R_+$.
%As the second result,
%we will see the following:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% inthm >> theorem
\begin{thm}[\Cref{thm:ideal}]
  The map 
  $ \Psi: ([e,w_{\cdiag}),\rbleq) \to (\I(R(w_\cdiag)),\subset) \simeq(\I(\cdiag),\subset) $
  given by $\Psi(w) = R(w) $ is a poset isomorphism.
%  The poset \( (\I(\theta), \subset) \) is order-isomorphic to
%  \( ([e,w_\cdiag),\rbleq) \).
%  The map 
%  $$\Phi: (\I(\cdiag),\subset) \to ([e,w_{\cdiag}),\rbleq)$$
%  given by $\Phi(\ideal) = w_{\ideal}$ is a poset isomorphism.
\end{thm}

The detailed proofs of the statements in this exteded abstract
can be read in \cite{NST2024}.
