%% 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[submission]{FPSAC2019}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

\articlenumber{9}
\addbibresource{9.bib}

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

\usepackage{amscd}
\usepackage{caption,subcaption}
%\usepackage{showkeys}

\theoremstyle{definition}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}

\newenvironment{demo}[1]{%
  \trivlist
  \item[\hskip\labelsep
        {\bf #1.}]
}{%
  \endtrivlist
}

\newcommand\vectz{\boldsymbol{z}}
\newcommand\vecta{\boldsymbol{a}}
\newcommand\Nat{\mathbb{N}}
\newcommand\Int{\mathbb{Z}}
\newcommand\Rat{\mathbb{Q}}
\newcommand\Comp{\mathbb{C}}
\renewcommand\tilde{\widetilde}
\renewcommand\hat{\widehat}
\newcommand\ep{\varepsilon}
\newcommand\Hom{\operatorname{Hom}}

\newcommand\AP{\mathcal{A}}
\newcommand\BB{\mathcal{B}}
\newcommand\EE{\mathcal{E}}
\newcommand\GG{\mathcal{G}}
\newcommand\LL{\mathcal{L}}
\newcommand\OO{\mathcal{O}}
\newcommand\PP{\mathcal{P}}
\newcommand\RR{\mathcal{R}}
\newcommand\TT{\mathcal{T}}
\newcommand\XX{\mathcal{X}}
\newcommand\point{\operatorname{pt}}

%\usepackage{lipsum}

%% define your title in the usual way
\title[Skew hook formula for $d$-complete posets via equivariant $K$-theory]%
{Skew hook formula for $d$-complete posets via equivariant $K$-theory}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{%
Hiroshi Naruse%
\thanks{\href{mailto:hnaruse@yamanashi.ac.jp}{hnaruse@yamanashi.ac.jp}.
Partially supported by JSPS No.~16H03921.
}%
\addressmark{1}
\and
Soichi Okada%
\thanks{\href{mailto:okada@math.nagoya-u.ac.jp}{okada@math.nagoya-u.ac.jp}.
Partially supported by JSPS No.~15K13425 
and ESI, Wien, Austria.
}%
\addressmark{2}
}

%% then use \addressmark to match authors to institutions here
\address{%
\addressmark{1}Graduate School of Education, University of Yamanashi, Japan
\\
\addressmark{2}Graduate School of Mathematics, Nagoya 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{%
Peterson and Proctor obtained a product formula for the multivariate generating function 
of $P$-partitions on a $d$-complete poset $P$ in terms of hooks in $P$.
In this article, we give a skew generalization of Peterson--Proctor's hook formula, 
i.e., a subtraction-free formula for the generating function of $(P \setminus F)$-partitions for 
a $d$-complete poset $P$ and its order filter $F$.
Our proof uses the equivariant $K$-theory of Kac--Moody partial flag varieties, 
and this generalization provides an alternate proof of Peterson--Proctor's hook formula.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{%
$d$-complete posets, hook formulas, $P$-partitions, equivariant $K$-theory
}

%% 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, ie. [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=bibtex]{biblatex}
%\addbibresource{fpsac19.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

%
% 1. Introduction
%
\section{%
Introduction
}

The origin of hook formulas is the Frame--Robinson--Thrall hook formula \cite{FRT}, 
which asserts that, for a partition $\lambda$, the number $f^\lambda$ of 
standard tableaux of shape $\lambda$ is given by
\begin{equation}
\label{eq:FRT-hook}
f^\lambda
 =
\frac{ |\lambda|! }
     { \prod_{v \in D(\lambda)} h_{D(\lambda)}(v) },
\end{equation}
where $h_{D(\lambda)}(v)$ denotes the hook length of the cell $v$ in the Young diagram $D(\lambda)$.
Later Stanley \cite{Stan2} obtained a hook formula for the univariate generating function 
of reverse plane partitions of shape $\lambda$ with respect to $|\sigma| = \sum_{v \in D(\lambda)} \sigma(v)$ :
\begin{equation}
\label{eq:Stanley-hook}
\sum_{\sigma \in \AP(D(\lambda))} q^{|\sigma|}
 =
\frac{ 1 }
     { \prod_{v \in D(\lambda)} (1 - q^{h_{D(\lambda)}(v)}) },
\end{equation}
and Gansner \cite{G} gave a multivariate generalization of (\ref{eq:Stanley-hook}).
Similar formulas hold for shifted Young diagrams and rooted trees.

Standard tableaux and reverse plane partitions of shape $\lambda$ can be regarded as 
linear extensions and $P$-partitions of the poset $P = D(\lambda)$ respectively.
Given an $n$-element poset $P$, 
a linear extension of $P$ is an order-preserving bijection from $P$ to $\{ 1, 2, \dots, n \}$, 
and a $P$-partition is an order-reversing map $\sigma$ from $P$ to $\Nat$, 
the set of nonnegative integers.
We denote by $\AP(P)$ the set of all $P$-partitions.
Using Stanley's theory of $P$-partitions, we can derive (\ref{eq:FRT-hook}) 
from (\ref{eq:Stanley-hook}).

Proctor \cite{Proc1,Proc2} introduced a wide class of posets, 
called $d$-complete posets, enjoying ``hook-length property'', 
as a generalization of Young diagrams, shifted Young diagrams and rooted trees.
Peterson and Proctor obtained the following theorem, which is a far-reaching generalization 
of the hook formulas (\ref{eq:FRT-hook}) and (\ref{eq:Stanley-hook}).

\begin{theorem}
\label{thm:PP}
(Peterson--Proctor, see \cite{Proc3})
Let $P$ be a $d$-complete poset.
The multivariate generating function of $P$-partitions is given by
\begin{equation}
\label{eq:PP-hook}
\sum_{\sigma \in \AP(P)} \vectz^{\sigma}
 =
\frac{ 1 }
     { \prod_{v \in P} (1 - \vectz[H_P(v)]) }.
\end{equation}
(Refer to Section~2 for undefined notations.)
\end{theorem}

However the original proof, based on representation theory, of this theorem is not yet published, 
though an outline of their proof is given in \cite{Proc3}.
Different proofs are sketched by Ishikawa--Tagawa \cite{IT2} (using Schur function identities) 
and Nakada \cite{N2} (using combinatorics of root systems).
Our skew generalization (\cref{thm:main} below) provides an alternate proof of \cref{thm:PP}, 
which is based on equivariant Schubert calculus.

Another direction of generalizing the Frame--Robinson--Thrall hook formula (\ref{eq:FRT-hook}) 
is to consider skew shapes.
However one cannot expect a nice product formula for the number $f^{\lambda/\mu}$ of 
standard tableaux of skew shape $\lambda/\mu$ in general.
Naruse \cite{Naruse} presented and sketched a proof of a subtraction-free formula for $f^{\lambda/\mu}$:
\begin{equation}
\label{eq:Naruse-hook}
f^{\lambda/\mu}
 =
|\lambda/\mu|!
\sum_{D \in \EE_{D(\lambda)}(D(\mu))} 
 \frac{ 1 }
      { \prod_{v \in D(\lambda) \setminus D} h_\lambda(v) },
\end{equation}
where and $D$ runs over all excited diagrams of $D(\mu)$ in $D(\lambda)$.
Morales--Pak--Panova \cite{MPP} gave a $q$-analogue of Naruse's skew hook formula 
for the univariate generating functions for $P$-partitions on $P = D(\lambda) \setminus D(\mu)$.

The main result of this article is the following skew generalization of Peterson--Proctor's 
hook formula (\cref{thm:PP}).
A subset $F$ of a poset $P$ is called an \emph{order filter} of $P$ 
if $x < y$ in $P$ and $x \in F$ imply $y \in F$.

\begin{theorem}
\label{thm:main}
Let $P$ be a connected $d$-complete poset and $F$ an order filter of $P$.
Then the multivariate generating function of $(P \setminus F)$-partitions, 
where $P \setminus F$ is viewed as an induced subposet of $P$, is given by
\begin{equation}
\label{eq:main}
\sum_{\sigma \in \AP(P \setminus F)} \vectz^\sigma
 =
\sum_{D \in \EE_P(F)}
 \frac{ \prod_{v \in B(D)} \vectz[H_P(v)] }
      { \prod_{v \in P \setminus D} ( 1 - \vectz[H_P(v)] ) },
\end{equation}
where $D$ runs over all excited diagrams of $F$ in $P$.
(See Section~2 for undefined notations.)
\end{theorem}

If $F = \emptyset$, then our main theorem (\cref{thm:main}) gives 
\cref{thm:PP}.
If $P = D(\lambda)$ and $F = D(\mu)$ are the Young diagrams of partitions $\lambda \supset \mu$, 
then (\ref{eq:main}) reduces to Morales--Pak--Panova's $q$-hook formula \cite[Corollary~6.17]{MPP} 
after specializing all variables $z_i$ to $q$, 
and then to Naruse's hook formula (\ref{eq:Naruse-hook}) by the theory of $P$-partitions.

This article is an extended abstract of \cite{NO} and is organized as follows.
In Section~2, we give basic definitions and notations for $d$-complete posets 
and excited diagrams/peaks.
In Section~3, we provide Lie theoretical interpretations of the combinatorial notions for $d$-complete posets.
In Section~4, we give a sketch of the proof of \cref{thm:main} 
by using the Billey-type formula and the Chevalley-type formula for 
the equivariant $K$-theory of the Kac--Moody partial flag variety.

%
% 2. Basic definitions and notations
%
\section{%
Basic definitions and notations
}

We give several definitions concerning $d$-complete posets 
and introduce the notion of excited diagrams and excited peaks.

\subsection{%
Definition of $d$-complete posets
}

For an integer $k \ge 3$, we denote by $d_k(1)$ the poset consisting of $2k-2$ elements 
$u_1, \cdots, u_{k-2}, x, y, v_{k-2}, \cdots, v_1$ with covering relations
$$
u_1 \gtrdot u_2 \gtrdot \cdots \gtrdot u_{k-2}, 
\quad
u_{k-2} \gtrdot x \gtrdot v_{k-2}, 
\quad
u_{k-2} \gtrdot y \gtrdot v_{k-2}, 
\quad
v_{k-2} \gtrdot \cdots \gtrdot v_2 \gtrdot v_1.
$$
Note that $x$ and $y$ are incomparable.
The poset $d_k(1)$ is called the \emph{double-tailed diamond}.
The Hasse diagram of $d_k(1)$ is shown in \cref{fig:Hasse} (a).
\begin{figure}[htb]
\centering
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\end{picture}
}
\subcaptionbox{$D(5,4,2,1)$}{
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% shape D(5,4,2,1)
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\end{picture}
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\caption{Double-tailed diamond, shape, shifted shape and swivel}
\label{fig:Hasse}
\end{figure}

Let $P$ be a poset.
An interval $[v,u] = \{ x \in P : v \le x \le u \}$ is called 
a \emph{$d_k$-interval} if it is isomorphic to $d_k(1)$.
Then $v$ and $u$ are called the \emph{bottom} and \emph{top} of $[v,u]$ respectively,
and the two incomparable elements of $[v,u]$ are called the \emph{sides}.
A subset $I$ of $P$ is called \emph{convex} if $x < y < z$ in $P$ and $x$, $z \in I$ imply $y \in I$.
A convex subset $I$ is called a \emph{$d_k^-$-convex set} 
if it is isomorphic to the poset obtained by removing the top element from $d_k(1)$.

\begin{definition}
\label{def:dc_poset}
A poset $P$ is \emph{$d$-complete} if it satisfies 
the following three conditions for every $k \ge 3$:
\begin{enumerate}
\item[(D1)]
If $I$ is a $d_k^-$-convex set, then there exists an element $u$ 
such that $u$ covers the maximal elements of $I$ and 
$I \cup \{ u \}$ is a $d_k$-interval.
\item[(D2)]
If $I = [v,u]$ is a $d_k$-interval and the top $u$ covers $u'$ in $P$, 
then $u' \in I$.
\item[(D3)]
There are no $d_k^-$-convex sets which differ only in the minimal elements.
\end{enumerate}
\end{definition}

It is clear that rooted trees, viewed as posets with their roots being the maximum elements, 
are $d$-complete posets.

\begin{example}
\label{ex:dc_poset}
We regard $\Int^2$ as a poset by defining $(i,j) \le (i',j')$ if and only if 
$i \ge i'$ and $j \ge j'$. 
The following induced subposets of $\Int^2$ are $d$-complete:
\begin{align*}
D(\lambda)
 &= 
\{ (i,j) \in \Int^2 : i \ge 1, \ 1 \le j \le \lambda_i \},
\\
S(\mu)
 &= 
\{ (i,j) \in \Int^2 : i \ge 1, \ i \le j \le \mu_i + i-1 \},
\\
e_6(1)
 & =
\left\{ 
 \begin{array}{l}
  (1,1), (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), \\
  (3,4), (3,5), (3,6), (4,4), (4,5), (4,6), (4,7), (4,8)
 \end{array}
\right\},
\end{align*}
where $\lambda$ is a partition and $\mu$ is a strict partition.
These posets are called a \emph{shape}, a \emph{shifted shape} and 
a \emph{swivel} respectively.
\cref{fig:Hasse} (b), (c), (d) illustrate the Hasse diagrams of 
$D(5,4,2,1)$, $S(5,4,2,1)$ and $e_6(1)$.
Sometimes we represent subposets of $\Int^2$ as collections of unit cells like Young diagrams.
\end{example}

A poset $P$ is called \emph{connected} if the Hasse diagram of $P$ is a connected graph.
It is easy to see that, if $P$ is a $d$-complete poset, 
then each connected component of $P$ is $d$-complete.
Hence, when considering the generating functions of $P$-partitions, 
we may assume that a $d$-complete poset is connected.

\begin{prop}
\label{prop:dc_connected}
(\cite[\S 3]{Proc1})
If a $d$-complete poset $P$ is connected, 
then $P$ has a unique maximal element.
\end{prop}

\subsection{%
Top tree and $d$-complete coloring
}

Let $P$ be a poset with a unique maximal element.
The \emph{top tree} $\Gamma$ of $P$ is an induced subgraph of the Hasse diagram of $P$, 
whose vertex set consists of all elements $x \in P$ 
such that the order filter generated by $x$ is a chain. 
% every $y \ge x$ is covered by at most one other element. 
We will regard the top tree as a simply-laced Dynkin diagram.
For example, the top trees of \cref{fig:Hasse} (a), (b), (c) and (d) are 
of type $D_k$, $A_8$, $D_6$ and $E_6$ respectively. 

\begin{prop}
\label{prop:dc_color}
(\cite[Proposition~8.6]{Proc2})
Let $P$ be a connected $d$-complete poset and $\Gamma$ its top tree.
Let $I$ be a set of colors whose cardinality is the same as $\Gamma$.
Then a bijective labeling $c : \Gamma \to I$ can be uniquely extended to a map, 
called a \emph{$d$-complete coloring}, 
$c : P \to I$ satisfying the following three conditions:
\begin{enumerate}
\item[(C1)]
If $x$ and $y$ are incomparable, then $c(x) \neq c(y)$.
\item[(C2)]
If an interval $[v,u]$ is a chain, then the colors $c(x)$ ($x \in [v,u]$) are 
distinct.
\item[(C3)]
If $[v,u]$ is a $d_k$-interval then $c(v) = c(u)$.
\end{enumerate}
\end{prop}

\begin{example}
\label{ex:dc_color}
The assignments given in \cref{fig:dc_color} are $d$-complete colorings.
In general, for a shape $D(\lambda)$ and a shifted shape $S(\mu)$ with $l(\mu) \ge 2$, 
the maps $c_{D(\lambda)} : D(\lambda) \to \{ -(\lambda'_1-1), \dots, -1, 0, 1, \dots, \lambda_1-1 \}$ 
and $c_{S(\mu)} : S(\mu) \to \{ 0, 0', 1, 2, \dots, \mu_1 - 1 \}$ given by
$$
c_{D(\lambda)}(i,j) = j-i,
\quad
c_{S(\mu)}(i,j)
 =
\begin{cases}
 j-i &\text{if $i < j$,} \\
 0 &\text{if $i=j$ and $i$ is odd,} \\
 0' &\text{if $i=j$ and $i$ is even.}
\end{cases}
$$
are $d$-complete colorings.
\begin{figure}[htb]
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\put(20,40){\line(0,-1){30}}
\put(30,40){\line(0,-1){40}}
\put(40,40){\line(0,-1){40}}
\put(50,40){\line(0,-1){20}}
\put(0,30){\makebox(10,10){\scriptsize$0$}}
\put(10,30){\makebox(10,10){\scriptsize$1$}}
\put(20,30){\makebox(10,10){\scriptsize$2$}}
\put(30,30){\makebox(10,10){\scriptsize$3$}}
\put(40,30){\makebox(10,10){\scriptsize$4$}}
\put(10,20){\makebox(10,10){\scriptsize$0'$}}
\put(20,20){\makebox(10,10){\scriptsize$1$}}
\put(30,20){\makebox(10,10){\scriptsize$2$}}
\put(40,20){\makebox(10,10){\scriptsize$3$}}
\put(20,10){\makebox(10,10){\scriptsize$0$}}
\put(30,10){\makebox(10,10){\scriptsize$1$}}
\put(30,0){\makebox(10,10){\scriptsize$0'$}}
\end{picture}
}
\subcaptionbox{$e_6(1)$}{
\begin{picture}(100,40)(-10,0)
\put(0,40){\line(1,0){50}}
\put(0,30){\line(1,0){50}}
\put(20,20){\line(1,0){40}}
\put(30,10){\line(1,0){50}}
\put(30,0){\line(1,0){50}}
\put(0,30){\line(0,1){10}}
\put(10,30){\line(0,1){10}}
\put(20,20){\line(0,1){20}}
\put(30,0){\line(0,1){40}}
\put(40,0){\line(0,1){40}}
\put(50,0){\line(0,1){40}}
\put(60,0){\line(0,1){20}}
\put(70,0){\line(0,1){10}}
\put(80,0){\line(0,1){10}}
\put(0,30){\makebox(10,10){\scriptsize$1$}}
\put(10,30){\makebox(10,10){\scriptsize$2$}}
\put(20,30){\makebox(10,10){\scriptsize$3$}}
\put(30,30){\makebox(10,10){\scriptsize$4$}}
\put(40,30){\makebox(10,10){\scriptsize$5$}}
\put(20,20){\makebox(10,10){\scriptsize$6$}}
\put(30,20){\makebox(10,10){\scriptsize$3$}}
\put(40,20){\makebox(10,10){\scriptsize$4$}}
\put(30,10){\makebox(10,10){\scriptsize$2$}}
\put(40,10){\makebox(10,10){\scriptsize$3$}}
\put(50,10){\makebox(10,10){\scriptsize$6$}}
\put(30,0){\makebox(10,10){\scriptsize$1$}}
\put(40,0){\makebox(10,10){\scriptsize$2$}}
\put(50,0){\makebox(10,10){\scriptsize$3$}}
\put(60,0){\makebox(10,10){\scriptsize$4$}}
\put(70,0){\makebox(10,10){\scriptsize$5$}}
\end{picture}
}
\caption{$d$-complete colorings}
\label{fig:dc_color}
\end{figure}
\end{example}

\subsection{%
Hook monomials
}

In the rest of this paper, we assume that $P$ is a connected $d$-complete poset 
with top tree $\Gamma$, 
and fix a $d$-complete coloring $c : P \to I$, where $I$ is identified with vertex set of $\Gamma$.
Take a set of indeterminates $\vectz = (z_i)_{i \in I}$ indexed by $I$.

Given an order filter $F$ of $P$, we regard $P \setminus F$ as the induced subposet.
For a $(P \setminus F)$-partition $\sigma \in \AP(P \setminus F)$, we put
$$
\vectz^\sigma = \prod_{v \in P \setminus F} z_{c(v)}^{\sigma(v)},
$$
and we are interested in the multivariate generating function 
$\sum_{\sigma \in \AP(P \setminus F)} \vectz^\sigma$ of $(P \setminus F)$-partitions.

\begin{definition}
\label{def:hook}
For each element $u \in P$, we define the monomial $\vectz[H_P(u)]$, 
called the \emph{hook monomial} of $u$, inductively as follows:
\begin{enumerate}
\item[(i)]
If $u$ is not the top of any $d_k$-interval, then we define 
$\vectz[H_P(u)] = \prod_{w \le u} z_{c(w)}$.
\item[(ii)]
If $u$ is the top of a $d_k$-interval $[v,u]$, then we define
$$
\vectz[H_P(u)]
 =
\frac{ \vectz[H_P(x)] \cdot \vectz[H_P(y)] }
     { \vectz[H_P(v)] },
$$
where $x$ and $y$ are the sides of $[v,u]$.
\end{enumerate}
\end{definition}

\begin{example}
\label{ex:hook}
If $P$ is a shape $D(\lambda)$ or a shifted shape $S(\mu)$, 
then there is the classical notion of hooks $H_{D(\lambda)}(u) \subset D(\lambda)$ 
and $H_{S(\mu)}(u) \subset S(\mu)$ given by
\begin{align*}
H_{D(\lambda)}(i,j)
 &=
\{ (i,l) \in D(\lambda) : l \ge j \}
\cup
\{ (k,j) \in D(\lambda) : k > i \},
\\
H_{S(\mu)}(i,j)
 &=
\{ (i,l) \in S(\mu) : l \ge j \}
\cup
\{ (k,j) \in S(\mu) : k > j \}
\cup
\{ (j+1,l) \in S(\mu) : l > j \}.
\end{align*}
Then the hook monomial $\vectz[H_P(u)]$ in \cref{def:hook} coincides with 
the product \linebreak $\prod_{v \in H_P(u)} z_{c(v)}$.
\end{example}

\subsection{%
Excited diagrams and excited peaks
}

In order to formulate a skew hook formula for $d$-complete posets, 
we need to generalize the notion of 
excited diagrams and excited peaks used in \cite{Naruse} and \cite{MPP} 
to general $d$-complete posets.

For $i \in I$, let $N_i$ be the subset of $P$ consisting of elements $x \in P$ whose color 
$c(x)$ is adjacent to $i$ in the Dynkin diagram $\Gamma$.
Note that, if $[v,u]$ is a $d_k$-interval, then $[v,u] \cap N_{c(u)}$ consists of elements $x \in [v,u]$ 
such that $x$ is covered by $u$ or covers $v$.

\begin{definition}
\label{def:excited}
Let $P$ be a connected $d$-complete poset and let $F$ be an order filter of $P$.
\begin{enumerate}
\item[(a)]
Let $D$ be a subset of $P$ and $u \in D$.
We say that $u$ is \emph{$D$-active} 
if there exists an element $v \in P \setminus D$ 
such that $v < u$, $[v,u]$ is a $d_k$-interval 
and $[v,u] \cap D \cap N_{c(u)} = \emptyset$.
\item[(b)]
Let $D$ be a subset of $P$ and $u \in D$.
If $u$ is $D$-active, then we define $\alpha_u(D)$ to be the subset of $P$ obtained from $D$ by 
replacing $u \in D$ by the bottom element $v$ of the $d_k$-interval $[v,u]$.
We call this replacement an \emph{elementary excitation}.
\item[(c)]
An \emph{excited diagram} of $F$ in $P$ is a subset of $P$ 
obtained from $F$ after a sequence of elementary excitations on active elements. 
Let $\EE_P(F)$ be the set of all excited diagrams of $F$ in $P$.
\item[(d)]
To an excited diagram $D \in \EE_P(F)$ we associate a subset $B(D) \subset P$ as follows:
If $D = F$, then $B(F) = \emptyset$.
If $D$ is an excited diagram with an active element $u$, then we define
$$
B(\alpha_u(D))
 = 
\left( B(D) \setminus ([v,u] \cap N_{c(u)}) \right) \cup \{ u \},
$$
where $[v,u]$ is the $d_k$-interval with top element $u$.
We call $B(D)$ the set of \emph{excited peaks} of $D$.
(It can be shown that $B(D)$ is a well-defined subset of $P \setminus D$.)
\end{enumerate}
\end{definition}

\begin{example}
\label{ex:excited}
\begin{enumerate}
\item[(1)]
If $P = D(5,3,2)$ and $F = D(2,1)$, 
then there are five excited diagrams of $F$ in $P$.
$$
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\fboxsep=0mm
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\put(30,30){\line(0,-1){20}}
\put(40,30){\line(0,-1){10}}
\put(50,30){\line(0,-1){10}}
\end{picture}
}
@>{\alpha_{(1,2)}}>>
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\colorbox[gray]{0.7}{\makebox(10,10){}}}
\put(10,20){\makebox(10,10){$\times$}}
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\put(10,30){\line(0,-1){30}}
\put(20,30){\line(0,-1){30}}
\put(30,30){\line(0,-1){20}}
\put(40,30){\line(0,-1){10}}
\put(50,30){\line(0,-1){10}}
\end{picture}
}
\\[5pt]
@V{\alpha_{(2,1)}}VV
@V{\alpha_{(2,1)}}VV
\\[5pt]
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\colorbox[gray]{0.7}{\makebox(10,10){}}}
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\put(50,30){\line(0,-1){10}}
\end{picture}
}
@>{\alpha_{(1,2)}}>>
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\fboxsep=0mm
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\put(10,20){\makebox(10,10){$\times$}}
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\put(50,30){\line(0,-1){10}}
\end{picture}
}
@>{\alpha_{(1,1)}}>>
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\makebox(10,10){$\times$}}
\put(10,10){\colorbox[gray]{0.7}{\makebox(10,10){}}}
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\put(40,30){\line(0,-1){10}}
\put(50,30){\line(0,-1){10}}
\end{picture}
}
\end{CD}
$$
Here the shaded cells form an exited diagram and a cell with $\times$ is an excited peak.
\item[(2)]
If $P = S(5,3,2)$ and $F = S(2)$, 
then there are four excited diagrams of $F$ in $P$.
$$
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\begin{CD}
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\colorbox[gray]{0.7}{\makebox(10,10){}}}
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\put(0,30){\line(1,0){50}}
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\put(10,10){\line(1,0){30}}
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\put(30,30){\line(0,-1){30}}
\put(40,30){\line(0,-1){30}}
\put(50,30){\line(0,-1){10}}
\end{picture}
}
@>{\alpha_{(1,2)}}>>
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\colorbox[gray]{0.7}{\makebox(10,10){}}}
\put(10,20){\makebox(10,10){$\times$}}
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\put(30,30){\line(0,-1){30}}
\put(40,30){\line(0,-1){30}}
\put(50,30){\line(0,-1){10}}
\end{picture}
}
@>{\alpha_{(2,3)}}>>
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\colorbox[gray]{0.7}{\makebox(10,10){}}}
\put(10,20){\makebox(10,10){$\times$}}
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\end{picture}
}
@>{\alpha_{(1,1)}}>>
\raisebox{-12.5pt}{
\begin{picture}(50,30)
\fboxsep=0mm
\put(0,20){\makebox(10,10){$\times$}}
\put(20,0){\colorbox[gray]{0.7}{\makebox(10,10){}}}
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\put(30,30){\line(0,-1){30}}
\put(40,30){\line(0,-1){30}}
\put(50,30){\line(0,-1){10}}
\end{picture}
}
\end{CD}
$$
\item[(3)]
If $P = e_6(1)$ and $F$ is the order filter consisting of two elements, 
then there are four excited diagrams of $F$ in $P$.
$$
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\begin{CD}
\raisebox{-20pt}{
\begin{picture}(80,40)
\fboxsep=0mm
\put(0,30){\colorbox[gray]{0.7}{\makebox(10,10){}}}
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\put(0,40){\line(1,0){50}}
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}
@>{\alpha_{(1,2)}}>>
\raisebox{-20pt}{
\begin{picture}(80,40)
\fboxsep=0mm
\put(0,30){\colorbox[gray]{0.7}{\makebox(10,10){}}}
\put(10,30){\makebox(10,10){$\times$}}
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@>{\alpha_{(3,4)}}>>
\raisebox{-20pt}{
\begin{picture}(80,40)
\fboxsep=0mm
\put(0,30){\colorbox[gray]{0.7}{\makebox(10,10){}}}
\put(10,30){\makebox(10,10){$\times$}}
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}
@>{\alpha_{(1,1)}}>>
\raisebox{-20pt}{
\begin{picture}(80,40)
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\end{CD}
$$
\end{enumerate}
\end{example}

%
% 3. d-Complete posets, Weyl groups and root systems
%
\section{%
$d$-Complete posets, Weyl groups and root systems
}

In this section, we provide connections of combinatorics of $d$-complete posets 
with Lie theory involving Weyl groups and root systems.

\subsection{%
Lie theoretical interpretations
}

Let $P$ be a connected $d$-complete poset with top tree $\Gamma$, 
and regard $\Gamma$ as a (simply-laced) Dynkin diagram with node set $I$.
We fix an associated root datum $(\Lambda, \Lambda^*, \Pi, \Pi^\vee)$ consisting of 
a free $\Int$-module $\Lambda$ (the weight lattice), 
its dual lattice $\Lambda^*$ (the coweight lattice), 
a subset $\Pi = \{ \alpha_i : i \in I \} \subset \Lambda$ (the set of simple roots), 
and a subset $\Pi^\vee = \{ \alpha^\vee_i : i \in I \} \subset \Lambda^*$ (the set of simple coroots) 
subject to certain conditions.
Let $W$ be the corresponding Weyl group generated by the simple reflections $\{ s_i : i \in I \}$, 
$l : W \to \Nat$ the length function and $<$ the Bruhat order on W.
Let $\Phi = W \Pi$ and $\Phi^\vee = W \Pi^\vee$ be the sets of real roots and real coroots respectively.
The simple system $\Pi$ (resp. $\Pi^\vee$) determines 
the decomposition of $\Phi$ (resp. $\Phi^\vee$) into the positive system $\Phi_+$ (resp. $\Phi^\vee_+$) and 
the negative system $\Phi_-$ (resp. $\Phi^\vee_-$).
Then the standard partial ordering $>$ on $\Phi_+$ (resp. $\Phi^\vee_+$) is defined by setting 
$\alpha > \beta$ if $\alpha - \beta$ is a sum of simple roots (resp. simple coroots).

By using the $d$-complete coloring $c: P \to I$, we write 
$\alpha(p) = \alpha_{c(p)}$, $\alpha^\vee(p) = \alpha^\vee_{c(p)}$ and $s(p) = s_{c(p)}$ 
for each $p \in P$.
Take a linear extension and label the elements of $P$ with $p_1, \cdots, p_N$ ($N= \# P$) 
so that $p_i < p_j$ in $P$ implies $i<j$.
Then, for a subset $D = \{ p_{i_1}, \dots, p_{i_r} \}$ ($i_1 < \dots < i_r$) of $P$, 
we define
$$
w_D = s(p_{i_1}) \dots s(p_{i_r}) \in W.
$$
If $p = p_k \in P$, then we define 
$$
\beta(p_k) = s(p_1) \cdots s(p_{k-1}) \alpha(p_k) \in \Phi,
\quad
\gamma^\vee(p_k) = s(p_N) \cdots s(p_{k+1}) \alpha^\vee(p_k) \in \Phi^\vee.
$$
It turns out that these elements $w_D$, $\beta(p)$ and $\gamma^\vee(p)$ are independent 
of the choices of linear extensions of $P$.

Let $i_P$ be the color of the unique maximal element of $P$, 
and $\lambda_P \in \Lambda$ the corresponding fundamental weight.
Let $W_{\lambda_P}$ be the stabilizer of $\lambda_P$ in $W$, 
which is the maximal parabolic subgroup corresponding to $I \setminus \{ i_P \}$.
Let $W^{\lambda_P}$ be the set of minimum length coset representatives of $W/W_{\lambda_P}$, 
viewed as an induced subposet of $W$ with respect to the Bruhat order.

Then we summarize connections of combinatorics of $d$-complete posets 
with Weyl groups and root systems.
Proofs can be found in \cite{Proc1}, \cite{Proc3} and \cite{Stem2}.

\begin{prop}
\label{prop:Lie}
\begin{enumerate}
\item[(a)]
The element $w_P \in W$ is $\lambda_P$-minuscule, 
i.e., $\langle \gamma^\vee(p), \lambda_P \rangle = 1$ for all $p \in P$, 
where $\langle \ , \ \rangle : \Lambda^* \times \Lambda \to \Int$ is the canonical pairing.
\item[(b)]
$w_P$ is fully commutative, i.e.,
any reduced expression of $w$ can be obtained from any other by using only 
the Coxeter relations of the form $st = ts$.
\item[(c)]
The poset $P$ is isomorphic to the order dual of $\Phi^\vee_+ \cap w_P^{-1} \Phi^\vee_-$ 
via the correspondence $p \mapsto \gamma^\vee(p)$.
\item[(d)]
Under the identification $z_i = e^{\alpha_i}$ ($i \in I$), we have
$\vectz[H_P(p)] = e^{\beta(p)}$ for any $p \in P$.
\item[(e)]
The map $F \mapsto w_F$ gives a poset isomorphism 
from the set of all order filters of $P$ ordered by inclusion 
to the Bruhat interval $[e,w_P]$ in $W^{\lambda_P}$.
\item[(f)]
If $F$ is an order filter, then $w_F$ is $\lambda_P$-minuscule 
and $w_F \lambda_P = \lambda_P - \sum_{p \in F} \alpha(p)$.
\end{enumerate}
\end{prop}

\subsection{%
Excited diagrams and Weyl groups
}

Let $* : W \times W \to W$ be the associative product, called the Demazure product, defined by
$$
s_i * w
 =
\begin{cases}
 s_i w &\text{if $l(s_i w) = l(w) + 1$,} \\
 w &\text{if $l(s_i w) = l(w) - 1$.}
\end{cases}
$$
For a fixed linear extension of $P$ and 
a subset $D = \{ p_{i_1}, \cdots, p_{i_r} \}$ ($i_1 < \cdots < i_r$) of $P$, 
we define an element $w^*_D \in W$ by putting
$$
w^*_D = s(p_{i_1}) * s(p_{i_2}) * \cdots * s(p_{i_r}).
$$
It follows from \cref{prop:Lie}(b) that the element $w^*_D$ is 
independent of the choices of linear extensions of $P$
The following proposition is one of the key ingredients of the proof of \cref{thm:main}.
See \cite[Section~3]{NO} for the proof.

\begin{prop}
\label{prop:E=R}
Let $F$ be an order filter of a connected $d$-complete poset $P$ and $E \subset P$.
\begin{enumerate}
\item[(a)]
$E$ is an excited diagram of $F$ in $P$, i.e., $E \in \EE_P(F)$, if and only if 
$\# E = \# F$ and $w_E = w_F$.
\item[(b)]
$E$ is of the form $E = D \sqcup S$ for some $D \in \EE_P(F)$ and $S \subset B(D)$ 
if and only if $w^*_E = w_F$.
\end{enumerate}
\end{prop}

%
% 4. Equivariant K-theory and the proof of Theorem
%
\section{%
Equivariant $K$-theory and proof of \cref{thm:main}
}

In this section, we use the equivariant $K$-theory of a thick partial flag variety 
(see \cite{LSS} for example) to give a sketch of the proof 
of our main theorem (\cref{thm:main}). 

\subsection{
Equivariant $K$-theory of Kac--Moody partial flag varieties
}

Let $P$ be a $d$-complete poset with top tree $\Gamma$. 
From a fixed root datum $(\Lambda, \Lambda^*, \Pi, \Pi^\vee)$ associated to the Dynkin diagram $\Gamma$, 
we can construct the Kac--Moody group $\GG$ over $\Comp$, 
its Borel subgroup $\BB_-$ (corresponding to $\Phi_-$) 
and maximal torus $\TT \subset \BB_-$.
Let $i_P \in I$ be the color of the maximum element of $P$ and 
$\lambda_P$ the corresponding fundamental weight.
Let $\PP_- \supset \BB_-$ be the parabolic subgroup of $\GG$ corresponding 
to $J = I \setminus \{ i_P \}$.
Then we consider the Kashiwara thick partial flag variety $\XX = \GG/\PP_-$.

Let $K_\TT(\XX)$ be the $\TT$-equivariant $K$-theory of $\XX$.
Then $K_\TT(\XX)$ has a commutative associative $K_\TT(\point)$-algebra structure.
Here the $\TT$-equivalent $K$-theory $K_\TT(\point)$ of a point is isomorphic 
to the group algebra $\Int[\Lambda]$ with basis $\{ e^\lambda : \lambda \in \Lambda \}$.
In the following, we identify $e^{\alpha_i}$ with the indeterminates $z_i$ corresponding to the color $i \in I$.
Any elements of $K_\TT(\XX)$ is a (possibly infinite) $K_\TT(\point)$-linear combination 
of the equivariant Schubert classes $\{ [\OO_v] : v \in W^{\lambda_P} \}$, 
where $[\OO_v]$ is the class of the structure sheaf $\OO_v$ of the Schubert subvariety $\XX_v$ 
and $W^{\lambda_P}$ is the set of minimum length coset representatives in $W/W_{\lambda_P}$.

Each $w \in W^{\lambda_P}$ gives a $\TT$-fixed point $e_w = w \PP_-/\PP_- \in \XX$, 
and the inclusion map $\iota_w : \{ e_w \} \to \XX$ induces the pull-back ring homomorphism 
$\iota_w^* : K_{\TT}(\XX) \to K_{\TT}(e_w) \cong \Int[\Lambda]$, 
called the localization map at $w$.
For two elements $v$, $w \in W^{\lambda_P}$, we define 
$$
\xi^v|_w = \iota_w^* ( [\OO_v] ) \in \Int[\Lambda].
$$
Then, by using \cref{prop:Lie} (d), we can derive the following explicit expression 
from the Billey-type formula \cite[Proposition~2.10]{LSS}.

\begin{prop}
\label{prop:Billey_P}
For a connected $d$-complete poset $P$ and its order filter $F$, we have
\begin{equation}
\label{eq:Billey_P}
\xi^{w_F}|_{w_P}
 =
\sum_{E : w^*_E = w_F}
 (-1)^{\# E - \# F}
 \prod_{p \in E} \left( 1 - \vectz[H_P(p)] \right),
\end{equation}
where the summation is taken over all subsets $E \subset P$ satisfying $w^*_E = w_F$.
\end{prop}

\subsection{%
Equivariant $K$-theoretical Littlewood--Richardson coefficients
}

We consider the structure constants for the multiplication in $K_{\TT}(\XX)$ 
with respect to the equivariant Schubert classes.
For $u$, $v$, $w \in W^J$, we denote by $c^w_{u,v} \in K_{\TT}(\point)$ 
the structure constant determined by
\begin{equation}
\label{eq:LR}
[\OO_u] [\OO_v] = \sum_{w \in W^{\lambda_P}} c^w_{u,v} [\OO_w].
\end{equation}
Then we have $c^w_{u,v} = 0$ unless $u \le w$ and $v \le w$.
It is not difficult to prove the following proposition.

\begin{prop}
\label{prop:LR}
\begin{enumerate}
\item[(a)]
For $v$, $w \in W^{\lambda_P}$, we have $c^w_{v,w} = \xi^v|_w$.
\item[(b)]
Let $u$, $v$, $w \in W^{\lambda_P}$ and put $s = s_{i_P} \in W^{\lambda_P}$.
If $c^w_{s,w} \neq c^u_{s,u}$, then we have
\begin{equation}
\label{eq:recLR}
c^w_{u,w}
 =
\frac{ 1 }
     { c^w_{s,w} - c^u_{s,u} }
\sum_{u < x \le w}
 c^x_{s,u} c^w_{x,w}.
\end{equation}
\end{enumerate}
\end{prop}

%\begin{demo}{Proof}
%(a) is obtained by applying $\iota^*_w$ to (\ref{eq:LR}) with $u=w$.
%(b) can be shown by comparing the coefficients of $[\OO_w]$ in the associativity law 
%$\left( [\OO_s] [\OO_u] \right) [\OO_w]
% =
%[\OO_s] \left( [\OO_u] [\OO_w] \right).
%$
%\qed
%\end{demo}

By using \cref{prop:Lie} (c), (e) and (f), 
we can prove the following explicit formula 
from the Chevalley-type formula \cite[Theorem~4.8]{LS}.
(See \cite[Section~4]{NO} for the proof.)

\begin{prop}
\label{prop:Chevalley_P}
Let $P$ be a connected $d$-complete poset and put $s = s_{i_P}$.
For two order filters $F$ and $F'$ of $P$, we have
\begin{equation}
\label{eq:Chevalley_P}
c^{w_{F'}}_{s,w_F}
 =
\begin{cases}
 1 - \vectz[F]
 &\text{if $F' = F$,}
\\
 (-1)^{\# (F' \setminus F) -1} \vectz[F]
 &\text{if $F' \supsetneq F$ and $F' \setminus F$ is an antichain,}
\\
 0
 &\text{otherwise,}
\end{cases}
\end{equation}
where $\vectz[F] = \prod_{p \in F} z_{c(p)}$.
\end{prop}

\subsection{
Outline of the Proof of the Main Theorem
}

Now we are ready to prove our Main \cref{thm:main}.
\cref{thm:main} follows from the following two identities:
\begin{equation}
\label{eq:xi}
\sum_{\sigma \in \AP(P \setminus F)} \vectz^\sigma
 =
\frac{ \xi^{w_F}|_{w_P} }
     { \xi^{w_P}|_{w_P} },
\quad
\frac{ \xi^{w_F}|_{w_P} }
     { \xi^{w_P}|_{w_P} }
 =
\sum_{D \in \EE_P(F)}
 \frac{ \prod_{q \in B(D)} \vectz[H_P(q)] }
      { \prod_{p \in P \setminus D} (1 - \vectz[H_P(p)]) }.
\end{equation}

We use \cref{prop:Chevalley_P} to prove the first identity of (\ref{eq:xi}).
We proceed by induction on $\# (P \setminus F)$.
For an order filter $F$ of $P$, we put
$$
G_{P/F}(\vectz) = \sum_{\sigma \in \AP(P \setminus F)} \vectz^\sigma,
\quad
Z_{P/F}(\vectz)
 = 
\frac{ \xi^{w_F}|_{w_P} }
     { \xi^{w_P}|_{w_P} }.
$$
Since $Z_{P/P}(\vectz) = G_{P/P}(\vectz) = 1$, 
it is enough to show that $Z_{P/F}(\vectz)$ and $G_{P/F}(\vectz)$ satisfy the same recurrence of the form
\begin{equation}
\label{eq:rec}
X_{P/F}(\vectz)
 =
\frac{ 1 }
     { 1 - \vectz[P \setminus F] }
\sum_{F'}
 (-1)^{\# (F' \setminus F) -1 }
 X_{P/F'}(\vectz),
\end{equation}
where $F'$ runs over all order filters such that $F \subsetneq F' \subset P$ and 
$F' \setminus F$ is an antichain.

First we show that $G_{P/F}$ satisfies (\ref{eq:rec}).
Let $M$ be the set of all maximal elements of $P \setminus F$.
For a subset $I \subset M$, let
\begin{align*}
\AP(P \setminus F)_I 
 &=
\{ \sigma \in \AP(P \setminus F) : \sigma(x) = 0 \text{ for all $x \in I$} \},
\\
\AP'(P \setminus F)
 &=
\{ \sigma \in \AP(P \setminus F) :
 \text{$\sigma(x) = 0$ for some $x \in M$}
\}.
\end{align*}
Then $G_{P/(F \sqcup I)}(\vectz) = \sum_{\sigma \in \AP(P \setminus F)_I} \vectz^\sigma$ 
and by using the Inclusion-Exclusion Principle we have
$$
\sum_{F'}
 (-1)^{\# (F' \setminus F) -1 }
 G_{P/F'}(\vectz)
 =
\sum_{I \subset M, \ I \neq \emptyset}
 (-1)^{\# I - 1}
 \sum_{\sigma \in \AP(P \setminus F)_I} \vectz^\sigma
 =
\sum_{\sigma \in \AP'(P \setminus F)} \vectz^\sigma.
$$
Also it is not difficult to show
$$
\sum_{\sigma \in \AP(P \setminus F)} \vectz^\sigma
  =
\frac{ 1 }
     { 1 - \vectz[P \setminus F] }
\sum_{\sigma \in \AP'(P \setminus F)} \vectz^\sigma.
$$
Hence $G_{P/F}(\vectz)$ satisfies (\ref{eq:rec}).
On the other hand, by using \cref{prop:Lie} (e), \cref{prop:LR} (a) 
and \cref{prop:Chevalley_P}, we can rewrite (\ref{eq:recLR}) 
to show that $Z_{P/F}(\vectz)$ satisfies the recurrence (\ref{eq:rec}).
Hence we obtain the first identity of (\ref{eq:xi}).

The second identity of (\ref{eq:xi}) is derived by using \cref{prop:Billey_P} and 
\cref{prop:E=R} as follows:
\begin{align*}
\xi^{w_F}|_{w_P}
 &=
\sum_{D \in \EE_P(F)}
 \prod_{p \in D} \left( 1 - \vectz[H_P(p)] \right)
\sum_{S \subset B(D)}
 (-1)^{\# S}
 \prod_{p \in S} \left( 1 - \vectz[H_P(p)] \right)
\\
 &=
\sum_{D \in \EE_P(F)}
 \prod_{p \in D} \left( 1 - \vectz[H_P(p)] \right)
 \prod_{p \in B(D)} \vectz[H_P(p)].
\end{align*}
By dividing the both sides by $\xi^{w_P}|_{w_P} = \prod_{p \in P} ( 1 - \vectz[H_P(p)] )$, 
we obtain the second identity in (\ref{eq:xi}).
This completes the proof of \cref{thm:main}.


%\acknowledgements{\lipsum[20]}

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