\documentclass[submission]{FPSAC2017}

\articlenumber{59}
\addbibresource{59_Takigiku.bib}

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\crefname{theo}{Theorem}{Theorems}
\crefname{prop}{Proposition}{Propositions}

\title{On some factorization formulas of $K$-$k$-Schur functions}
\author{Motoki Takigiku\thanks{\href{mailto:takigiku@ms.u-tokyo.ac.jp}{takigiku@ms.u-tokyo.ac.jp}}}

\address{Graduate School of Mathematical Sciences, the University of Tokyo, Japan}
\received{\today}

\abstract{
We give some new formulas about factorizations of $K$-$k$-Schur functions $\kl$,
analogous to
the $k$-rectangle factorization formula
$s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t} s^{(k)}_{\lambda}$
of $k$-Schur functions,
where $\lambda$ is any $k$-bounded partition and $R_t$
denotes the partition $(t^{k+1-t})$ called a \textit{$k$-rectangle}.
Although a formula of the same form does not hold for $K$-$k$-Schur functions,
we can prove that $g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t\cup\lambda}$, and in fact more generally that
$g^{(k)}_{P}$ divides $g^{(k)}_{P\cup\lambda}$ for any multiple $k$-rectangles $P$ and any $k$-bounded partition $\lambda$.
We give the factorization formula of such $g^{(k)}_{P}$ and
the explicit formulas of $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}$ in some cases.

}

\keywords{$K$-$k$-Schur function, $k$-rectangle factorization, affine Schubert calculus}


\begin{document}
\maketitle

\section{Introduction}

Let $k$ be a positive integer.
\textit{$k$-Schur functions} $\ks{\la}$ and their $K$-theoretic analogues
$\kks{\la}$,
which are called {\it $K$-$k$-Schur functions},
are symmetric functions parametrized by
$k$-bounded partitions $\la$, 
namely by the weakly decreasing strictly positive integer sequences
$\la=(\la_1,\dots,\la_l)$, $l\in\Z_{\ge 0}$, whose terms are all bounded by $k$.

Historically, 
$k$-Schur functions were first introduced by Lascoux, Lapointe and Morse \cite{MR1950481},
and subsequent studies led to several (conjecturally equivalent) characterizations of $\ks{\la}$
such as 
the Pieri-like formula due to Lapointe and Morse \cite{MR2331242},
and Lam proved that $k$-Schur functions correspond to the Schubert basis of homology of the affine Grassmannian \cite{Lam08}.
%
Moreover it was shown by Lam and Shimozono that
$k$-Schur functions play a central role in the explicit description of the Peterson isomorphism
between quantum cohomology of the Grassmannian and homology of the affine Grassmannian
up to suitable localizations
\cite{MR2923177}.

These developments have analogues in $K$-theory.
%
%
Lam, Schilling and Shimozono \cite{MR2660675} characterized the $K$-theoretic 
$k$-Schur functions as the Schubert basis of the $K$-homology of the affine
Grassmannian, and Morse \cite{Morse12} investigated them from a combinatorial
viewpoint, giving various properties including the Pieri-like
formulas using affine set-valued strips (the form using cyclically
decreasing words are also given in \cite{MR2660675}).


In this paper 
we start from this combinatorial characterization (see \cref{KkSchur_def})
and show certain new factorization formulas of $K$-$k$-Schur functions.

Among the $k$-bounded partitions, 
those of the form $(t^{k+1-t})=(\underbrace{t,\dots,t}_{k+1-t}=:R_t)$, $1\le t \le k$,
called {\it $k$-rectangles}, play a special role.
In particular, 
if a $k$-bounded partition has the form $R_t\cup \la$,
where the symbol $\cup$ denotes the operation of concatenating the two sequences and reordering the terms in the weakly decreasing order,
then the corresponding $k$-Schur function has the following factorization property \cite[Theorem 40]{MR2331242}:
\begin{equation}\label{intro:zero}
	\ks{R_t\cup\la} = \ks{R_t} \ks{\la}.
\end{equation}
%

It is suggested in \cite[Remark 7.4]{MR2660675} that 
the $K$-$k$-Schur functions should also possess similar properties,
including the divisibility of $\kks{R_t\cup\la}$ by $\kks{R_t}$, 
and that it should be interesting to explore such properties.
The present work is an attempt to materialize his suggestion.

We do show 
that 
$\kks{R_t}$ divides $\kks{R_t\cup\la}$ 
in the ring $\Lk=\mathbb{Z}[h_1,\dots,h_k]$,
where $h_i$ denotes the complete homogeneous symmetric functions of degree $i$,
of which the $K$-$k$-Schur functions form a basis.
However, unlike the case of $k$-Schur functions,
the quotient $\kks{R_t\cup\la}/\kks{R_t}$ is not a single term $\kks{\la}$
but, in general, a linear combination of $K$-$k$-Schur functions with leading term $\kks{\la}$,
namely in which $\kks{\la}$ is the only highest degree term.
Even the simplest case where $\la$ consists of a single part $(r)$,
$1\le r \le k$, displays this phenomenon:
we show
that
\begin{equation}\label{intro:one}
\kks{R_t\cup(r)} = 
\begin{cases}
	\kks{R_t}\cdot \kks{(r)} & ({\rm if}\ t<r), \\
	\kks{R_t}\cdot \left(\kks{(r)} + \kks{(r-1)} + \cdots + \kks{\emptyset}\right) & ({\rm if}\ t\ge r)
\end{cases}
\end{equation}
(actually we have $\kks{(s)} = h_s$ for $1 \le s \le k$, and $\kks{\emptyset}=h_0=1$).
So we may ask:

\vspace{2mm}
\noindent{\bf Question 1.}
{\it
	Which $\kks{\mu}$, besides $\kl$, appear in the quotient $\krtl/\krt$?
	With what coefficients?
}

\vspace{2mm}
A $k$-bounded partition can always be written in the form
$R_{t_1}\cup\dots\cup R_{t_m} \cup \la$ with 
$\la$ not having so many repetitions of any part as to form a $k$-rectangle.
In such an expression we temporarily call $\la$ the remainder.
Proceeding in the direction of Question 1,
one ultimate goal may be to give a factorization formula in
terms of the $k$-rectangles and the remainder.
In the case of $k$-Schur functions, the straightforward factorization in \eqref{intro:zero} above leads
to the formula $\ks{\RRm\cup\la}=\ks{R_{t_1}}\dots\ks{R_{t_m}}\kl$.
On the contrary, with $K$-$k$-Schur functions, the simplest case having a multiple $k$-rectangle %,
gives
\begin{equation}\label{intro:two}
	\kks{R_t\cup R_t} = \krt \sum_{\la\subset R_t} \kl.
\end{equation}
Hence we cannot expect $\kks{R_t\cup R_t}$ to be divisible by $\krt$ twice.
Instead, upon organizing the part consisting of $k$-rectangles in the form 
$\RRma$ with $t_1 < \dots < t_m$ and $a_i \ge 1$ ($1\le i \le m$),
with $R_t^a = \underbrace{R_t\cup\dots\cup R_t}_{a}$, actually we show 
that

\centerline{$\kks{\RRma\cup\la}$ is divisible by $\kks{\RRma}$,}
which actually holds whether or not $\la$ is the remainder.
Then we can subdivide our goal as follows:

\vspace{2mm}
\noindent{\bf Question $1'$.}
{\it
	Which $\kks{\mu}$, besides $\kl$, appear in the quotient $\kks{P\cup \la}/\kks{P}$
	where $P=\RRma$, and with what coefficients?
}

\vspace{2mm}
\noindent{\bf Question $2$.}
{\it
	How can $\kks{\RRma}$ be factorized?
}

\vspace{2mm}
In this paper, we give a reasonably complete answer to Question 2 (\cref{theo:Rta}),
and partial answers to Question $1'$ (\cref{R_t_r,P_mu_r,samek_goal}).








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
He is also greatly indebted to the anonymous
referee and Joachim Cuntz for correcting a large number of errors in an earlier
version of the manuscript.

This work was supported by the Research Fellow of the
Japan Society for the Promotion of Science and the Program for Leading Graduate
Schools, MEXT, Japan
\end{comment}

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




\section{Preliminaries}\label{sec:prel}

In this section we review some
 requisite combinatorial backgrounds.
First recall that the Pieri rule characterizes Schur functions.
In the theory of ($K$-)$k$-Schur functions, 
the underlying combinatorial objects are the set of \textit{$k$-bounded partitions} (instead of partitions), 
which is isomorphic to the set of \textit{$(k+1)$-cores},
and we have to consider \textit{weak strips} instead of horizontal strips.
For detailed definitions,
see for instance \cite[Chapter 2]{MR3379711} or \cite[Chapter I]{MR1354144}.

\begin{comment}
Schur function is an important class of symmetric functions and the theory of symmetric functions involves the theory of partitions as underlying combinatorial objects.
Although Schur function has several equivalent characterization, we focus on the way using Pieri rules (or semistandard tableaux, equivalently).

$k$-Schur function is in a sense an ``affine analogue\footnote{The reason to use the term ``affine'' is that $k$-Schur function has relation to the geometry of affine Grassmannian, in contrast with Schur function relating to Grassmannian.}'' of Schur function, and it involves a certain class of partitions called {\it $k$-bounded partitions} as underlying objects,
which in fact bijectively corresponds to other good combinatorial objects such as {\it cores}, {\it affine Grassmannian permutation}, etc.

$K$-$k$-Schur function is a ``$K$-theoretic'' version of $k$-Schur function, which came from the replacement of cohomology by $K$-theory in geometry.
This modification can be seen in combinatorics as to consider ``set-valued version'' of tableaux.


In \cref{prel_partition,prel_sym} we review about definition of Schur function by using tableaux and Pieri rules, after preparing generalities about partitions and symmetric functions.
In \cref{prel_pca,prel_wo} we review the combinatorial backgrounds of $k$-Schur theory including bounded partitions, cores, the affine symmetric group, and their order.
Then in \cref{prel_kS} we give the definition of $k$-Schur function by using an analogue of Pieri rule for Schur function, and we devote \cref{prel_KkS} for that of $K$-$k$-Schur function.
Finally in \cref{prel_prop} we give some properties about $k$-Schur combinatorics being used later.

\end{comment}

\subsection{Partitions and Schur functions}\label{prel_partition}


%%% partition
Let $\mathcal{P}$ denote the set of partitions.
%%% diagram
A partition $\la=(\la_1 \ge \la_2 \ge \dots)\in\mathcal{P}$ is identified with
its {\it Young diagram} (or {\it shape}), for which we use the French notation here.
\begin{figure}[h]
\centering
\input{fig/prel-0.tex}
\caption{The Young diagram of $(4,2)$.}
\end{figure}
We denote the \textit{size} of a partition $\la$ by $|\la|$, the \textit{length} by $l(\la)$, and the \textit{conjugate} by $\la'$.
%
For partitions $\la$, $\mu$ we say $\la\subset\mu$ if $\la_i\le \mu_i$ for all $i$.
%
%%% dominance order
For a partition $\la$ and a cell $c=(i,j)$ in $\la$, 
we denote the \textit{hook length} of $c$ in $\la$ by $\hook{c}{\la}=\la_i+\la'_j-i-j+1$.

%%% removable/addable corner, extremal cell
For a partition $\la$, a {\it removable corner} of $\la$ (or {\it $\la$-removable corner}) is a cell $(i,j)\in\la$ with $(i,j+1),(i+1,j)\notin\la$.
$(i,j)\in(\Z_{>0})^2\sm\la$ is said to be
an {\it addable corner} of $\la$ (or {\it $\la$-addable corner})
if $(i,j-1),(i-1,j)\in\la$
with the understanding that $(0,j),(j,0)\in\la$.



Let $\La=\Z[h_1,h_2,\dots]$ be the ring of symmetric functions,
generated by the 
\textit{complete symmetric functions}
$ h_r = \sum_{i_1\le i_2\le\dots\le i_r} x_{i_1}\dots x_{i_r}$.

%%% Pieri
The \textit{Schur functions} $\{s_\la\}_{\la\in\mathcal{P}}$ are the family of symmetric functions satisfying the \textit{Pieri rule}:
$h_r s_\la = \sum s_\mu$, summed over $\mu$ such that $\mu/\la$ is a horizontal $r$-strip.


\subsection{Bounded partitions, cores and $k$-rectangles $R_t$}\label{prel_pca}

%%% Pk
A partition $\lambda$ is called {\it $k$-bounded} if $\lambda_1 \le k$.
Let $\Pk$ be the set of all $k$-bounded partitions.
An {\it $r$-core} (or simply a \textit{core} if no confusion can arise) is a partition none of whose cells have a hook length equal to $r$.
We denote by $\mathcal{C}_r$ the set of all $r$-core partitions.


\vspace{2mm}
{\it Hereafter we fix a positive integer $k$.}

%%% content, residue
For a cell $c=(i,j)$, the {\it content} of $c$ is $j-i$ and
the {\it residue} of $c$ is $\mathrm{res}(c)=j-i \mod (k+1) \in \mathbb{Z}/(k+1)$.
%
For a set $X$ of cells, we write
$\mathrm{Res}(X) = \{\,\res(c) \mid c\in X\,\}$.
%
We will write a $\la$-removable corner of residue $i$ simply a $\la$-removable $i$-corner.
%
For simplicity of notation, we may use an integer to represent a residue, omitting ``mod $(k+1)$''.

%%% R_t
We denote by $R_t$ the partition 
$(t^{k+1-t})=(t,t,\dots,t) \in \Pk$ for $1\le t \le k$,
which is called a {\it $k$-rectangle}.
Naturally a $k$-rectangle is a $(k+1)$-core.

\vspace{2mm}
Now we recall the
bijection between
the $k$-bounded partitions in $\Pk$ and the $(k+1)$-cores in $\Cn$:
The map
	$\bdd \colon \Cn \lra \Pk ; \ka \mapsto \la$
is defined by
$\la_i=\#\{j\mid (i,j)\in\ka,\ \hook{(i,j)}{\ka}\le k\}$.
%
Then in fact $\bdd$ is bijective and we put $\core=\bdd^{-1}$.
See \cite[Theorem 7]{MR2167475} for details.
Note that if $\la$ is contained in a $k$-rectangle then $\la\in\Pk$ and $\la\in\Cn$, and besides $\bdd(\la)=\la=\core(\la)$.

\vspace{2mm}

For $i=0,1,\dots,k$,
an action $s_i$ on $\Cn$ is defined as follows:
For $\ka\in\Cn$,
\begin{itemize}
	\item if there is a $\ka$-addable $i$-corner, 
		then let $s_i\cdot \ka$ be 
		$\ka$ with all $\ka$-addable $i$-corners added,
	\item if there is a $\ka$-removable $i$-corner, 
		then let $s_i\cdot \ka$ be 
		$\ka$ with all $\ka$-removable $i$-corners removed,
	\item otherwise, let $s_i\cdot \ka$ be $\ka$.
\end{itemize}

In fact the first and second case never occur simultaneously and
$s_i\cdot \ka\in\Cn$.





\subsection{Weak order and weak strips}\label{prel_wo}

We review the weak order on $\Cn$.


\begin{defi}\label{weakorder}
The {\it weak order} $\prec$ on $\Cn$ is defined by the following covering relation:
\[
	\tau \wcover \ka \iff \text{$\exists i$ such that $s_i \tau=\ka$, $\tau\subsetneq\ka$.}
\]
\end{defi}

\begin{defi}\label{weakstrip}
	For $(k+1)$-cores $\tau\subset\ka\in\Cn$,
	$\ka/\tau$ is called a {\it weak strip} of size $r$
	(or a {\it weak $r$-strip})
	when
\begin{enumerate}
\renewcommand{\labelenumi}{ }
\item $\ka/\tau$ is horizontal strip and 
	$\tau \wcover \exists \tau^{(1)} \wcover \dots \wcover \exists \tau^{(r)} = \ka$.
\end{enumerate}
\end{defi}

\subsection{$k$-Schur functions}\label{prel_kS}

We recall a characterization of $k$-Schur functions given in \cite{MR2331242},
since it is a model for and has a relationship with $K$-$k$-Schur functions.

\begin{defi}[$k$-Schur function via ``weak Pieri rule'']
$k$-Schur functions
$\{\ks{\la}\}_{\la\in\Pk}$ are the family of symmetric functions
such that 
$\ks{\emptyset}=1$ and 
\begin{align*}
	h_r \ks{\la} &= \sum_{\mu} \ks{\mu} \quad \text{for $r\le k$ and $\mu\in\Pk$,}
\end{align*}
summed over $\mu\in\Pk$ such that $\core(\mu)/\core(\la)$ is a weak strip of size $r$.
\end{defi}

In fact
$\{\ks{\la}\}_{\la\in\Pk}$ forms a basis of $\Lk=\Z[h_1,\dots,h_k]\subset\La$.
In addition $\ks{\la}$ is homogeneous of degree $|\la|$.
It is proved in \cite[Theorem 40]{MR2331242} that
\begin{prop}[$k$-rectangle property]\label{prop:kS_fac}
For $1\le t \le k$ and $\la\in\Pk$, we have
$
	\ks{R_t\cup\la} = \ks{R_t} \ks{\la} (=s_{R_t} \ks{\la}).
$
\end{prop}



\subsection{$K$-$k$-Schur functions $\kl$}\label{prel_KkS}


In \cite{Morse12} a combinatorial characterization of $K$-$k$-Schur functions is given
via an analogue of the Pieri rule, using some kind of strips called \textit{affine set-valued strips}.

\vspace{2mm}
For a partition $\la$, $(i,j)\in(\Z_{>0})^2$ is called {\it $\la$-blocked} if $(i+1,j)\in \la$.

\begin{defi}[affine set-valued strip]\label{asvs}
	For $r\le k$,
	$(\ga/\beta,\rho)$ is called an {\it affine set-valued strip} of size $r$
	(or an {\it affine set-valued $r$-strip})
	if
	$\rho$ is a partition and
	$\beta\subset\ga$ are cores both containing $\rho$ such that
	\begin{enumerate}
			\renewcommand{\labelenumi}{(\arabic{enumi})}
	\item $\ga/\beta$ is a weak $(r-m)$-strip where we put $m=\#\mathrm{Res}(\beta/\rho)$,
	\item $\beta/\rho$ is a subset of $\beta$-removable corners,
	\item $\ga/\rho$ is a horizontal strip,
	\item For all $i\in\Res(\beta/\rho)$, 
		all $\beta$-removable $i$-corners which are not $\ga$-blocked are in $\beta/\rho$.
	\end{enumerate}
\end{defi}

In this paper we employ the following characterization \cite[Theorem 48]{Morse12} of the $K$-$k$-Schur function as its definition.
\begin{defi}[$K$-$k$-Schur function via an ``affine set-valued'' Pieri rule]\label{KkSchur_def}
	$K$-$k$-Schur functions $\{\kks{\la}\}_{\la\in\Pk}$ are the family of symmetric functions such that 
	$\kks{\emptyset}=1$ and 
	for $\la \in \Pk$ and $0 \le r \le k$,
	\begin{equation}\label{eq:Pieri_moto}
		h_r \cdot \kks{\la} =
		\sum_{(\mu,\rho)} (-1)^{|\la|+r-|\mu|} \kks{\mu},
	\end{equation}
	summed over $(\mu,\rho)$ such that $(\core(\mu)/\core(\la),\rho)$ is an affine set-valued strip of size $r$.
\end{defi}






		In fact	$\{\kks{\la}\}_{\la\in\Pk}$ forms a basis of $\Lk$.
		Moreover,
		though $\kks{\la}$ is an inhomogeneous symmetric function in general,
		the degree of $\kks{\la}$ is $|\la|$
		and its homogeneous part of highest degree is equal to $\ks{\la}$.


\section{Results}\label{sec:results}
\subsection{Possibility of factoring out $\kks{\RRma}$ and some other general results}\label{sec:general}

As discussed above, it does not hold that
$\kks{R_t\cup\la} = \kks{R_t}\kks{\la}$ for any $\la\in\Pk$.
However, it holds that $\kks{R_t}$ divides $\kks{R_t \cup\la}$.
We prove it in a slightly more general form.

The following notation is often referred later:
\begin{itemize}
	\item[$\NP$]\label{nota:NP}
		Let $1\le t_1,\dots,t_m \le k$ be distinct integers
		and $a_i\in\Z_{>0}\ (1\le i\le m)$,
		where $m\in\Z_{>0}$.
		Then we put
		\begin{align*}
			P&=\RRma,	 \\
			\al{u} &= \#\{t_v\mid 1\le v \le m,\ t_v\ge u\} \quad\text{for each $u\in\Z_{>0}$}.
		\end{align*}
\end{itemize}

\begin{prop}\label{P_factor}
Let $P$ be as in the above $\NP$.
Then, for $\lambda=(\lambda_1,\cdots,\lambda_l) \in \Pk$,
we have $\kks{P} | \kks{\lambda \cup P}$ in the ring $\Lk$.
\end{prop}

\noindent\textit{Remark.} 
Note that $\la$ may still have the form $\la=R_t\cup\mu$.
Hereafter we will not repeat the same remark in similar statements.


Since the homogeneous part of highest degree of $\kks{\la}$ is equal to $\ks{\la}$ for any $\la$,
it follows from \cref{prop:kS_fac,P_factor} that
\begin{coro}
	Let $P$ be as in $\NP$.
	Then, for any $\la\in\Pk$,
	we can write
	$$
	\kks{P \cup \la} 
	= \kks{P} \left(\kks{\la} + \sum_{\mu} a_{\la\mu}\kks{\mu}\right),
	$$
	summing over $\mu\in\Pk$ such that $|\mu|<|\la|$,
	for some coefficients $a_{\la\mu}$ (depending on $P$).
\end{coro}

\vspace{2mm}
Now we are interested in finding a explicit description of $\kks{P\cup\la}\big/\kks{P}$.
Let us consider the case $P=R_t$ for simplicity.

As noted above, 
a linear map $\Theta$ extending $\kl \mapsto \kRtl$ ($\forall\la\in\Pk$) does not
coincide with the multiplication of $\kRt$
because it does not commute with the multiplication by $h_r$ in the first place.

However,
we can prove that the restriction of $\Theta$ to the subspace spanned by $\{\kks{R_t\cup\mu}\}_{\mu\in\Pk}$ 
(in fact this is the principal ideal generated by $\krt$)
commutes with the multiplication by $h_r$,
and thus it coincides with the multiplication of $\Theta(\krt)\big/\krt = \kks{R_t\cup R_t}\big/\krt$ on that ideal (\cref{Rt_twice}).
Thus it is of interest to describe the value of
$\kks{R_t\cup R_t}\big/\krt$,
which is shown to be $\sum_{\nu\subset R_t}\kks{\nu}$
later.


\begin{prop}\label{Rt_twice}
For $\lambda \in \Pk$ and $1 \le t \le k$, we have
$\kks{\lambda\cup R_t \cup R_t} = \kks{\lambda \cup R_t} \cdot \displaystyle\frac{\kks{R_t\cup R_t}}{\kks{R_t}}$.
\end{prop}

As a corollary,
it turns out that the value of $\kks{P\cup\la} / \kks{P}$ is independent of 
$a_1,\dots,a_m$, the ``multiplicities'' of $k$-rectangles.

\begin{theo}\label{Result_two}
Let $P=\RRma$ be as in $\NP$, and
put $Q=R_{t_1}\cup\cdots\cup R_{t_m}$.
Then, for $\la\in\Pk$ we have
$$ \frac{\kks{P\cup\la}}{\kks{P}}=\frac{\kks{Q\cup\la}}{\kks{Q}}.$$
\end{theo}


Thus we can reduce Question $1'$ to the case where the $k$-rectangles are of all different sizes.






\subsection{Answer to Question $2$}

For Question 2, we first show 
that multiple $k$-rectangles of different sizes entirely split,
namely, 

\begin{theo}\label{dist_krec_factor}
	For $1\le t_1 < \cdots < t_m \le k$
	and $a_1,\dots,a_m>0$, 
\[
	\kks{R_{t_1}^{a_1}\cup\cdots\cup R_{t_m}^{a_m}} 
	= \kks{R_{t_1}^{a_1}}\cdots\kks{R_{t_m}^{a_m}}.
\]
\end{theo}

Then
we show that for each $1\le t \le k$ and $a>1$,
we have a nice factorization 
generalizing the formula \eqref{intro:two}:

\begin{theo}\label{theo:Rta}
	For $1\le t \le k$ and $a>0$, we have
$$\kks{R_t^a}=\krt \left(\sum_{\la\subset R_t} \kl\right)^{a-1}.$$
Thus, substituting this into \cref{dist_krec_factor}, we have
\[
	\kks{\RRna} = \kks{R_{t_1}} \left(\sum_{\la^{(1)}\subset R_{t_1}} \kks{\la^{(1)}}\right)^{a_1-1}
	\dots
	\kks{R_{t_n}} \left(\sum_{\la^{(n)}\subset R_{t_n}} \kks{\la^{(n)}}\right)^{a_n-1}.
\]
\end{theo}




\subsection{(Partial) Answer to Question $1'$}

An easiest case is where $\la=(r)$ consists of a single part, 
which generalizes the case \eqref{intro:one} in Introduction.
Namely we show that

\begin{theo}\label{R_t_r}
Let $P, \al{u}$ be as in $\NP$ and $1\le r\le k$.
Then we have
\[\frac{\kks{P\cup(r)}}{\kks{P}} = \sum_{s=0}^{r}\binom{\al{r}+r-s-1}{r-s}h_s.\]

In particular, if $t_m < r$, which means $\al{r}=0$, we have
\[\frac{\kks{P\cup(r)}}{\kks{P}} = h_r = \kks{(r)}\]

On the other hand, when $m=1$,
\[
\frac{\kks{R_t\cup(r)}}{\kks{R_t}} = 
\begin{cases}
h_r & ({\rm if}\ r>t), \\
h_r+h_{r-1}+\cdots+h_0 & ({\rm if}\ r\le t).
\end{cases}
\]
\end{theo}


Then
generalizing this case, 
we derive explicit formulas in the cases where
$\la=(\la_1,\dots,\la_l)$ satisfies the following condition $\Nla$  and that
the parts of $\la$ except for $\la_l$ are all larger than the widths of the $k$-rectangles.



\begin{itemize}
	\item [$\Nla$]\label{nota:Nla}
		Let $(\emptyset\neq)\la\in\Pk$ with satisfying $\bla\subset\Rbl$,
		where we write
		$\bla=(\la_1,\la_2,\dots,\la_{l(\la)-1})$,
		$l=l(\la)$
		and
		$\bl = l(\bla) = l - 1$.
		(Here we consider $R_{t}$ to be $\emptyset$ unless $1\le t\le k$)
\end{itemize}
		({\it Note}:
		when $l(\la)=1$, we have $\bl = 0$ and $\bla = \emptyset = \Rbl$ thus $\la$ satisfies $\Nla$.
		When $l(\la) > k+1$, we have $\bl > k$ and $\bla \neq \emptyset = \Rbl$ thus $\la$ does not satisfy $\Nla$.)
		
\vspace{2mm}
Namely, we prove that		
	
\begin{theo}\label{P_mu_r} 
	Let $P$ and $\al{u}$ $($for $u\in\Z_{>0})$ be as in $\NP$ in \cref{sec:general}, before \cref{P_factor}.
	Let $\la, l, \bla,\bl$ be as in $\Nla$ above.
	Assume $\max_i\{t_i\} < \bla_\bl$.
	Then we have
	\begin{flalign*}
	(1) &&
		\kks{P}\kks{\la} &= \sum_{s=0}^{\la_{l}} (-1)^s \binom{\al{\la_{l}+1-s}}{s} \kks{P\cup\bar\la\cup(\la_l-s)}. & \\
	(2) &&
		\kks{P\cup\la} &= \kks{P} \sum_{s=0}^{\la_{l}} \binom{\al{\la_{l}}+s-1}{s} \kks{\bar\la\cup(\la_{l}-s)}. &
	\end{flalign*}
	In particular, if $t_n < \la_{l}$ then $\al{\la_{l}}=0$ and
	\[
		\kks{P\cup\la} = \kks{P} \kks{\la}.
	\]
	\end{theo}
	
\begin{figure}[h]
\centering
\input{fig/overview-1.tex}
\caption{In this figure $p=m-\al{\la_l}$ and $a_i=1$ for all $i$.}
\end{figure}






Moreover, we show a formula in a slightly different case where
$P$ is a single $k$-rectangle $R_t$ and
$\la=(\la_1,\dots,\la_l)$ satisfies $\Nla$ and that
the parts of $\la$ except for $\la_l$ are all larger than \textit{or equal to}
the widths of the $k$-rectangles.

\bigskip
\noindent\textit{Notation.}
For any partition $\la$,
let $\la\ci=(\la_1,\dots,\la_i)$ if $\la_i > t \ge \la_{i+1}$ (we set $\la\ci=\emptyset$ if $t\ge \la_1$).

\begin{theo}\label{samek_goal}
	Let $\la, l,\bla,\bl$ be as in $\Nla$.
	Assume
	$\bla_\bl \ge t \ge \la_{l}$.
	Then we have
	\begin{equation*}
		\kks{R_t\cup\la} = \kks{R_t}\sum_{\core(\la\ci)\subset\core(\nu)\subset\core(\la)} \kks{\nu}.
	\end{equation*}

\end{theo}


\subsection{Example}

Let us illustrate the sketch of the proof of \cref{P_mu_r} with a small example.

	Consider the case $P=R_t$: we shall show that $\kks{R_t\cup \la} = \kks{R_t}\kks{\la}$ if $\la_{l}>t$ and $\la_1+l\le k+2$.
	Let us assume \cref{R_t_r} (the case $l=1$)
	and consider the case $l=2$.
	Set $\la=(a,b)$ with $k \ge a \ge b > t$.

\bigskip

\noindent \underline{Step (A)}: Expand $\kks{(a,b)}$ into a linear combination of products of complete symmetric functions and $K$-$k$-Schur functions labeled by partitions with fewer rows:

By using the Pieri rule \eqref{eq:Pieri_moto} we have
{
\begin{align*}
	\kks{(a)} h_i &= \left(\kks{(a,i)} - \kks{(a,i-1)}\right) \\
				  &+ \left(\kks{(a+1,i-1)} - \kks{(a+1,i-2)}\right) \\
				  &+ \dots \\
				  &  \begin{cases}
						\begin{minipage}{7cm}
							$\dots + 
							\left(\kks{(a+i-1,1)} - \kks{(a+i-1,0)}\right)$ \\
							$\qquad +\kks{(a+i,0)} $
						\end{minipage}
						& \text{(if $a+i\le k$)} \\
						\begin{minipage}{7cm}
							$\dots + 
							\left(\kks{(k-1,a+i-k+1)} - \kks{(k-1,a+i-k)}\right)$ \\
							$\qquad +\left(\kks{(k,a+i-k)} - \kks{(k,a+i-k-1)}\right)$
						\end{minipage}
						& \text{(if $a+i >k$)}
					\end{cases}
\end{align*}
}
for $i\le a$, and summing this over $0 \le i \le b$, we have 
{
\begin{align}
\kks{(a)} \left( h_b+\dots+h_0 \right) 
&= \kks{(a,b)} + \kks{(a+1,b-1)} + \cdots 
 \begin{cases}
	 \kks{(a+b,0)} & \text{(if $a+b\le k$)} \\
	 \kks{(k,a+b-k)} & \text{(if $a+b\ge k$)}
 \end{cases} \label{eq:ab_sum}\\
&= \sum_{\substack{\mu/(a)\text{:horizontal strip} \\ |\mu|=a+b \\ \mu_1\le k}} \kks{\mu}. \notag
\end{align}
}
Similarly we have
{
\[\kks{(a+1)} \left( h_{b-1}+\dots+h_0 \right) = \kks{(a+1,b-1)} + \kks{(a+2,b-2)} + \dots = \sum_{\substack{\mu/(a+1)\text{:horizontal strip} \\ |\mu|=a+b \\ \mu_1\le k}} \kks{\mu},\]
}
hence 
{
\[\kks{(a,b)} = \kks{(a)}\left(h_b+\dots+h_0\right) - \kks{(a+1)}\left(h_{b-1}+\dots+h_0\right).\]
}

\noindent \underline{Step (B)}:
	Multiply $\kks{(a,b)}$ by $\kks{R_t}$. Then we have
	{
	\begin{align*}
		\kks{R_t}\kks{(a,b)}
		&= \kks{R_t} \kks{(a)}\left(h_b+\dots+h_0\right)
		- \kks{R_t} \kks{(a+1)}\left(h_{b-1}+\dots+h_0\right) \\
		&= \kks{R_t\cup(a)} \left(h_b+\dots+h_0\right)
		- \kks{R_t\cup(a+1)} \left(h_{b-1}+\dots+h_0\right)
	\end{align*}
	}
	because $\kks{R_t}\kks{(a)}=\kks{R_t\cup(a)}$ since $t<a$. Then carry out calculations similar to Step (A).
	
	\bigskip

\noindent{\it Notation.} For a proposition $P$, we shall write $\de\left[P\right]=1$ if $P$ is true and $\de\left[P\right]=0$ if $P$ is false.

Since the number of residues of $\core(R_t\cup(a,j))$-nonblocked $\core(R_t\cup(a))$-removable corners is $1+\de\left[t>j\right]$,
{\small
\begin{align*}
	\kks{R_t\cup(a)} h_i &= \left(\kks{R_t\cup(a,i)}-\binom{1+\de\left[t>i-1\right]}{1}\kks{R_t\cup(a,i-1)} + \binom{1+\de\left[t>i-2\right]}{2}\kks{R_t\cup(a,i-2)}\right) \\
						 &+ \left(\kks{R_t\cup(a+1,i-1)}-\binom{1+\de\left[t>i-2\right]}{1}\kks{R_t\cup(a+1,i-2)}+\binom{1+\de\left[t>i-3\right]}{2}\kks{R_t\cup(a+1,i-3)}\right) \\
				   &+ \dots\ .
\end{align*}
}
Summing this over $0\le i \le b$, we have
{
\begin{multline*}
\kks{R_t\cup(a)}\left(h_b+\dots+h_0\right) = \left(\kks{R_t\cup(a,b)}-\de\left[t>b-1\right] \kks{R_t\cup(a,b-1)} \right) \\
+ \left(\kks{R_t\cup(a+1,b-1)}-\de\left[t>b-2\right] \kks{R_t\cup(a+1,b-2)}\right) +\dots
\end{multline*}
}
Similarly we have
{
\begin{multline*}
\kks{R_t\cup(a+1)}\left(h_{b-1}+\dots+h_0\right) = \left(\kks{R_t\cup(a+1,b-1)}-\de\left[t>b-2\right] \kks{R_t\cup(a+1,b-2)}\right) \\
+\left(\kks{R_t\cup(a+2,b-2)}-\de\left[t>b-3\right] \kks{R_t\cup(a+2,b-3)}\right) +\dots,
\end{multline*}
}
hence we have 
{
\[\kks{R_t}\kks{(a,b)} = \kks{R_t\cup(a,b)} - \de\left[t>b-1\right] \kks{R_t\cup(a,b-1)} = \kks{R_t\cup(a,b)} \]
}
since we have assumed $b>t$.

\section{Discussions}\label{sec:discuss}


It is worth noting that,
in all cases we have seen, 
$\kks{P\cup\la}\big/\kks{P}$ is a linear combination 
of $K$-$k$-Schur functions
with {\it positive coefficients}:

\begin{conj}
	For all $\la \in \Pk$ and $P=\RRma$, write
	$$\kks{P\cup\la} = \kks{P} \sum_{\mu} a_{P,\la,\mu} \kks{\mu}.$$
	Then $a_{P,\la,\mu}\ge 0$ for any $\mu$.
\end{conj}

In the case $P=R_t$, it is observed that $a_{R_t,\la,\mu}=0$ or $1$.
Moreover, the set of $\mu$ such that $a_{R_t,\la,\mu}=1$ is expected to be an ``interval'',
but we have to consider the \textit{strong order} on $\Pk\simeq\Cn$, 
which can be seen as just inclusion as shapes in the poset of cores.
Namely, the strong order $\la\le\mu$ on $\Pk$ is defined by $\core(\la)\subset\core(\mu)$.
Notice that $\la\preceq\mu \Lra \la\subset\mu \Lra \la\le\mu$ for $\la,\mu\in\Pk$.
Then,

\begin{conj}\label{conj:interval}
	For all $\la \in \Pk$ and $1\le t \le k$, 
	there exists $\mu \in \Pk$ such that
	\[
		\kks{R_t\cup \la} = \kks{R_t} \sum_{\mu\le\nu\le\la} \kks{\nu}.
	\]
\end{conj}

It will be interesting to study the geometric meaning of these results and conjectures.

	





\begin{acknowledgements}
The author would like to express his gratitude to
T.\ Ikeda
for suggesting the problem to the
author and helping him with many fruitful discussions and advice.
%
He is grateful to
H.\ Hosaka and I.\ Terada
%
for many valuable comments.
This work was supported by
the Program for Leading Graduate
Schools, MEXT, Japan.
%
The contents of this abstract, including their detailed proofs,
are in the author's master-thesis.
The full paper will be published elsewhere.

\end{acknowledgements}


\printbibliography

\end{document}

