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\documentclass[submission]{FPSAC2019}
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%% define your title in the usual way
\title{Skew key polynomials and the key poset}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Sami Assaf\thanks{\href{mailto:shassaf@usc.edu}{shassaf@usc.edu}. S.A. supported in part by NSF DMS-1763336.}\addressmark{1} \and Stephanie van Willigenburg\thanks{\href{mailto:steph@math.ubc.ca}{steph@math.ubc.ca}. S.v.W. supported in part by NSERC.}\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Southern California, Los Angeles CA 90089, USA \\ \addressmark{2}Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada}

%% put the date of submission here
\received{15 November 2018}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We generalize Young's lattice on integer partitions to a new partial order on weak compositions called the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomial basis for the polynomial ring, a generalization of the Schur basis for symmetric functions. Generalizing skew Schur functions, we define skew key polynomials in terms of the poset, and, using weak dual equivalence, we give a nonnegative composition Littlewood--Richardson rule for the key expansion of skew key polynomials.}

%% 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{Young's lattice, composition poset, key polynomials, skew Schur functions}

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%\usepackage[backend=bibtex]{biblatex}
%\addbibresource{FPSAC_keyposet.bib}
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\begin{document}

\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Preliminaries}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:intro}

Schur polynomials are central to the study of the representation theories of the general linear group and of the symmetric group, as well as to the geometry of the Grassmannian. The combinatorics of Young tableaux associated with Schur polynomials often sheds light on important representation theoretic or geometric properties such as tensor products, induction and restriction of modules, and intersection numbers.

The celebrated Littlewood--Richardson rule \cite{LR34} gives a combinatorial description for the Schur expansion of a product of two Schur polynomials or, equivalently, for the Schur expansion of a \emph{skew Schur polynomial} as
\[ s_{\lambda} s_{\mu} = \sum_{\nu} c_{\lambda,\mu}^{\nu} s_{\nu},
\hspace{1em} \text{or} \hspace{1em}
s_{\nu\setminus\lambda} = \sum_{\mu} c_{\lambda,\mu}^{\nu} s_{\mu}, \]
where $c_{\lambda,\mu}^{\nu}$ is the number of saturated chains in \emph{Young's lattice} from $\lambda$ to $\nu$ satisfying certain conditions depending on $\mu$. Here Young's lattice is the partial order on integer partitions given by containment of Young diagrams. These coefficients have deep interpretations in representation theory as the irreducible multiplicities for the tensor product of two irreducible representations for the general linear group and as the irreducible multiplicities for the induced tensor product of two irreducible representations for the symmetric group. They also appear geometrically, giving the number of points lying in a suitable intersection of three Grassmannian Schubert varieties.

One of myriad generalizations of the Littlewood--Richardson rule comes in \cite{BLvW11} where Bessenrodt, Luoto and van Willigenburg define a partial order on strong compositions that gives rise to a nonnegative Littlewood--Richardson rule for a skew analog of the quasisymmetric Schur functions of Haglund, Luoto, Mason and van Willigenburg \cite{HLMvW11-2}. Another example comes in \cite{HLMvW11}, where the latter authors use the quasisymmetric Schur functions to derive a nonnegative Littlewood--Richardson rule for the product of a \emph{key polynomial} and a Schur polynomial with sufficiently many variables.

The key polynomials are polynomial generalizations of Schur polynomials first studied by Demazure \cite{Dem74a} in connection with Schubert varieties. Key polynomials are irreducible characters of Demazure modules for the general linear group \cite{Dem74} and represent Schubert classes for vexillary permutations \cite{LS90}. Their structure constants are not, in general, nonnegative, though Assaf and Quijada \cite{AQ18} have made progress on understanding the signs in the Pieri case. Assaf \cite{Ass-W} considered a diagram containment-based skew analog of key polynomials, but obtained nonnegativity results only in very special cases.

In this abstract, we generalize Young's lattice to a partial order on \emph{weak} compositions that we call the \emph{key poset}. In contrast with the generalization to strong compositions in \cite{BLvW11}, we give explicit cover relations as well as explicit criteria for comparability in the poset, though as with the strong composition poset, the key poset is not a lattice. Using this paradigm, we re-define skew key polynomials with respect to the poset and give a general nonnegative Littlewood--Richardson rule for skew key polynomials, vastly generalizing \cite{Ass-W}. Moreover, we show that under the more general containment definition for skew key polynomials considered in \cite{Ass-W}, all key polynomial coefficients are nonnegative if and only if the shapes are comparable in the key poset. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Posets}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:poset}

An integer \newword{partition} $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_{\ell})$ is a weakly decreasing sequence of positive integers. The \newword{Young diagram} of a partition $\lambda$ is the collection of $\lambda_i$ unit cells left-justified in row $i$ indexed from the bottom. Integer partitions become a \newword{poset}, or partially ordered set, under containment of diagrams, i.e. $\lambda \subseteq \mu$, if $\lambda_i \leq \mu_i$ for all $i$ or, equivalently, if the diagram for $\lambda$ is a subset of the diagram for $\mu$. We call this poset \newword{Young's lattice}.

The \newword{cover relations} for Young's lattice may be described by $\lambda \cover \mu$ if and only if $\mu$ is obtained from $\lambda$ by incrementing a single part $\lambda_i$ for which $i=1$ or $\lambda_{i-1} > \lambda_{i}$ by $1$ or, equivalently, by adding a single box to the end of a row for which the higher row is strictly shorter. The cover relations show that Young's lattice is ranked by the number of cells of the diagrams. \cref{fig:Young} depicts Young's lattice up to rank $4$.

\begin{figure}[ht]
  \begin{center}
    \begin{tikzpicture}[xscale=1.6,yscale=1.5]
      \node at (2,0) (Y0)      {$\varnothing$}; 
      \node at (2,0.6) (Y1)    {$\smtab{\ }$}; 
      \node at (1.5,1.25) (Y2)    {$\smtab{\ & \ }$}; 
      \node at (2.5,1.25) (Y11)   {$\smtab{\ \\ \ }$}; 
      \node at (1,2) (Y3)   {$\smtab{\ & \ & \ }$}; 
      \node at (2,2) (Y21)  {$\smtab{\ \\ \ & \ }$}; 
      \node at (3,2) (Y111) {$\smtab{\ \\ \ \\ \ }$};     
      \node at (0,3) (Y4)   {$\smtab{\ & \ & \ & \ }$}; 
      \node at (1,3) (Y31)  {$\smtab{\ \\ \ & \ & \ }$}; 
      \node at (2,3) (Y22)  {$\smtab{\ & \ \\ \ & \ }$}; 
      \node at (3,3) (Y211) {$\smtab{\ \\ \ \\ \ & \ }$}; 
      \node at (4,3) (Y1111){$\smtab{\ \\ \ \\ \ \\ \ }$}; 
      \draw[thin] (Y0.north) -- (Y1.south) ;
      \draw[thin] (Y1.north) -- (Y2.south) ;
      \draw[thin] (Y1.north) -- (Y11.south) ;
      \draw[thin] (Y2.north) -- (Y3.south) ;
      \draw[thin] (Y2.north) -- (Y21.south) ;
      \draw[thin] (Y11.north) -- (Y21.south) ;
      \draw[thin] (Y11.north) -- (Y111.south) ;
      \draw[thin] (Y3.north) -- (Y4.south) ;
      \draw[thin] (Y3.north) -- (Y31.south) ;
      \draw[thin] (Y21.north) -- (Y31.south) ;
      \draw[thin] (Y21.north) -- (Y22.south) ;
      \draw[thin] (Y21.north) -- (Y211.south) ;
      \draw[thin] (Y111.north) -- (Y211.south) ;
      \draw[thin] (Y111.north) -- (Y1111.south) ;
    \end{tikzpicture}
  \end{center}
  \caption{\label{fig:Young}The Hasse diagram of Young's lattice up to rank $4$.}
\end{figure}      

Young's lattice is a \newword{lattice} with least upper bound and greatest lower bound given by the set-theoretic union and the set-theoretic intersection of the diagrams, respectively. 

Young's lattice is a prominent tool in algebraic combinatorics, used to study symmetric functions, representations of finite and affine Lie groups, and intersection numbers for finite and affine Grassmannians. We generalize the construction from integer partitions to weak compositions in such a way that maintains these connections.

A \newword{weak composition} $a = (a_1,a_2,\ldots,a_{n})$ is a sequence of nonnegative integers. The \newword{key diagram} of a weak composition $a$ is the collection of $a_i$ unit cells left-justified in row $i$ indexed from the bottom. 

\begin{definition}
  The \newword{key poset} is the partial order $\prec$ on weak compositions of length $n$ defined by the relation $a \preceq b$ if and only if $a_i \leq b_i$ for $i=1,2,\ldots,n$ and for any indices $1 \leq i<j \leq n$ for which $b_j > a_j$ and $a_i > a_j$, we have $b_i > b_j$.
  \label{def:above}
\end{definition}

In terms of key diagrams, we have $a \prec b$ if and only if $a \subset b$ and whenever a cell of $b\setminus a$ lies above a cell of $a$, the lower row is strictly longer in $b$; see \cref{fig:above}. 

\begin{figure}[ht]
  \begin{displaymath}
    \vline\smtab{ %
      \cbox{white} & \cbox{white} & \cbox{blue} & \cbox{blue} \\ \\
      \cbox{white} & \cbox{white} & \cbox{white} & \cbox{purple} & \cbox{purple}}
  \end{displaymath}
  \caption{\label{fig:above}An illustration of the partial order on weak compositions in terms of key diagrams. Here, cells $\cbox{white}$ lie in $a \subseteq b$, cells $\cbox{blue}$ lie in $b \setminus a$, and cells $\cbox{purple}$ lie in $b = a \cup b$.}
\end{figure}

This partial order is not given simply by containment of diagrams. For example, $a=(2,1,1) \subset b=(2,1,2)$, but $a \not\prec b$ since $b_3 > a_3$ and $a_1 > a_3$ but $b_1 = b_3$. % See Fig.~\ref{fig:contain}.

%% \begin{figure}[ht]
%%   \begin{displaymath}
%%     \vline\smtab{ \ \\ \ \\ \ & \ }
%%     \hspace{3\cellsize} 
%%     \vline\smtab{ \ & \cbox{purple} \\ \ \\ \ & \ }
%%   \end{displaymath}
%%   \caption{\label{fig:contain}Two incomparable diagrams where the left is contained within the right but is not below in the poset as evidenced by the set-theoretic difference shown as $\cbox{purple}$.}
%% \end{figure}

As with Young's lattice, this partial order on weak compositions is ranked by the number of boxes, and we may describe the covering relations in terms of adding cells.

\begin{theorem}
  The key poset is ranked by number of cells with covering relation $a \cover b$ if and only if $b$ is obtained from $a$ by incrementing $a_j$ by $1$ where for any $i<j$ we have $a_i \neq a_j + 1$.
  \label{thm:cover}
\end{theorem}

On key diagrams, $a \cover b$ if and only if $a \subset b$, there is a single cell of $b\setminus a$, and this cell does not sit above any cell at the end of its row. \cref{fig:Key} shows the key poset up to rank $3$.

\begin{figure}[ht]
  \begin{center}
    \begin{tikzpicture}[xscale=1.25,yscale=2]
      \node at (4.5,0.25) (K000) {$\varnothing$};
      \node at (3,1) (K001) {$\vline\smtab{\ \\ \\ \\\hline }$};
      \node at (4.5,1) (K010) {$\vline\smtab{\\ \ \\ \\\hline }$};
      \node at (6,1) (K100) {$\vline\smtab{\\ \\ \ \\\hline }$};
      \node at (1.25,2) (K002) {$\vline\smtab{\ & \ \\ \\ \\\hline }$};
      \node at (2.5,2) (K011) {$\vline\smtab{\ \\ \ \\ & \\\hline}$};
      \node at (3.75,2) (K101) {$\vline\smtab{\ \\ \\ \ & \\\hline}$};
      \node at (5.25,2) (K020) {$\vline\smtab{\\ \ & \ \\ & \\\hline}$};
      \node at (6.5,2) (K110) {$\vline\smtab{\\ \ \\ \ & \\\hline}$};
      \node at (7.75,2) (K200) {$\vline\smtab{\\ \\ \ & \ \\\hline}$};
      \node at (0,3) (K003) {$\vline\smtab{\ & \ & \ \\ \\ & \\\hline}$};
      \node at (1,3) (K012) {$\vline\smtab{\ & \ \\ \ \\ & & \\\hline}$};
      \node at (2,3) (K102) {$\vline\smtab{\ & \ \\ \\ \ & & \\\hline}$};
      \node at (3,3) (K021) {$\vline\smtab{\ \\ \ & \ \\ & & \\\hline}$};
      \node at (4,3) (K111) {$\vline\smtab{\ \\ \ \\ \ & & \\\hline}$};
      \node at (5,3) (K201) {$\vline\smtab{\ \\ \\ \ & \ & \\\hline}$};
      \node at (6,3) (K030) {$\vline\smtab{\\ \ & \ & \ \\ & \\\hline}$};
      \node at (7,3) (K120) {$\vline\smtab{\\ \ & \ & \\ \ \\\hline}$};
      \node at (8,3) (K210) {$\vline\smtab{\\ \ \\ \ & \ & \\\hline}$};
      \node at (9,3) (K300) {$\vline\smtab{\\ \\ \ & \ & \ \\\hline}$};
      \draw[thin] (K000.north) -- (K100.south) ;
      \draw[thin] (K000.north) -- (K010.south) ;
      \draw[thin] (K000.north) -- (K001.south) ;
      \draw[thin] (K001.north) -- (K002.south) ;
      \draw[thin] (K001.north) -- (K011.south) ;
      \draw[thin] (K001.north) -- (K101.south) ;
      \draw[thin] (K010.north) -- (K110.south) ;
      \draw[thin] (K010.north) -- (K020.south) ;
      \draw[thin] (K100.north) -- (K200.south) ;
      \draw[thin] (K002.north) -- (K003.south) ;
      \draw[thin] (K002.north) -- (K012.south) ;
      \draw[thin] (K002.north) -- (K102.south) ;
      \draw[thin] (K011.north) -- (K012.south) ;
      \draw[thin] (K011.north) -- (K111.south) ;
      \draw[thin] (K011.north) -- (K021.south) ;
      \draw[thin] (K101.north) -- (K102.south) ;
      \draw[thin] (K101.north) -- (K201.south) ;
      \draw[thin] (K110.north) -- (K120.south) ;
      \draw[thin] (K110.north) -- (K210.south) ;
      \draw[thin] (K020.north) -- (K021.south) ;
      \draw[thin] (K020.north) -- (K120.south) ;
      \draw[thin] (K020.north) -- (K030.south) ;
      \draw[thin] (K200.north) -- (K201.south) ;
      \draw[thin] (K200.north) -- (K210.south) ;
      \draw[thin] (K200.north) -- (K300.south) ;
    \end{tikzpicture}
  \end{center}
  \caption{\label{fig:Key}The Hasse diagram of the key poset up to rank $3$.}
\end{figure}

While containment is not sufficient for covering in general, it is for the partition case.

\begin{proposition}
  Given $a,b$ with $a$ weakly increasing, we have $a \subseteq b$ if and only if $a \preceq b$. 
  \label{prop:keypart}
\end{proposition}

In particular, any finite subposet of Young's lattice is a subposet of the key poset.

If $a,b \preceq a \cup b$, then $a\cup b$ is the unique least upper bound, and if $a\cap b \preceq a,b$, then $a \cap b$ is the unique greatest lower bound. However, in general, this is not the case.

\begin{proposition}
  The key poset is not a lattice.
  \label{prop:no-lattice}
\end{proposition}

This is demonstrated by the example illustrated in \cref{fig:no-lattice}, where $a = (2,1,1)$, $b=(2,1,2)$, $c = (1,1,1)$, and $d=(1,0,1)$. Here both $c$ and $d$ are greatest lower bounds for $a,b$, and, conversely, both $a$ and $b$ are least upper bounds for $c,d$.

\begin{figure}[ht]
  \begin{displaymath}
    \arraycolsep=2\smcellsize
    \begin{array}{ccccc}
    \vline\smtab{ \ \\ \ \\ \ & \cbox{red}} &
    \vline\smtab{ \ & \cbox{blue} \\ \ \\ \ & \cbox{blue}} &
    \vline\smtab{ \ \\ \cbox{red} \\ \ & \cbox{red}} &
    \vline\smtab{ \ & \cbox{blue} \\ \cbox{blue} \\ \ & \cbox{blue}} &
    \vline\smtab{ \ \\ \cbox{green} \\ \ } \\
    c \prec a &
    c \prec b &
    d \prec a &
    d \prec b &
    d \not\prec c
    \end{array}
  \end{displaymath}
  \caption{\label{fig:no-lattice}An example showing that the key poset is not a lattice. Here ${\cbox{white}}\in c$ or $d$, ${\cbox{red}} \in a\setminus c$ or $a\setminus d$, ${\cbox{blue}} \in b\setminus c$ or $b\setminus d$, and ${\cbox{green}} \in c\setminus d$.}
\end{figure}

%% Unions and intersections are natural candidates for least upper bounds and greatest lower bounds, though as Proposition~\ref{prop:no-lattice} shows, they are not necessarily above or below their constituent parts in the key poset. While this problem cannot be overcome completely, there are derived diagrams which do always lie above or below their constituent parts and which are contained in or contain all other least or greatest bounds.
%% 
%% \begin{proposition}
%%   Given weak compositions $a$ and $b$, let $a \nabla b$ denote $a \cup b$ together with all cells $y$ such that the cell immediately left of $y$ sits under a cell of $(a \cup b) \setminus (a \cap b)$ and some cell left of $y$ either lies in $a$ with the cell above in $b \setminus (a \cap b)$ or lies in $b$ with the cell above in $a\setminus (a \cap b)$; see Fig.~\ref{fig:join}. Then $a,b \preceq a \nabla b$, and for any $c$ for which $a, b \preceq c$, we have $a \nabla b \subseteq c$.
%%   \label{prop:join}
%% \end{proposition}
%% 
%% \begin{figure}[ht]
%%   \centering
%%   \begin{minipage}{0.475\textwidth}
%%   \begin{displaymath}
%%     \vline\smtab{ \ & \ & \cbox{blue} & \cbox{blue} \\ \\ \ & \ & \ & \cbox{purple} & \cbox{purple}}\hspace{-5\smcellsize}\smtab{ \\ \\ & & & + & + }
%%   \end{displaymath}
%%   \captionof{figure}{\label{fig:join}An illustration of the rule for padding $a \cup b$ up to $a \nabla b$. Here cells $\cbox{white}$ lie in $a$, cells $\cbox{blue}$ lie in $b\setminus (a\cap b)$, and cells $\cbox{purple}$ marked $+$ must be added to make $a \nabla b$.}
%%   \end{minipage}%
%%   \begin{minipage}{0.5\textwidth}
%%   \begin{displaymath}
%%     \vline\smtab{ \ & \ & \cbox{blue} & \cbox{blue} \\ \\ \ & \ & \times}
%%   \end{displaymath}
%%   \captionof{figure}{\label{fig:meet}An illustration of the rule for culling $a \cap b$ down to $a \Delta b$. Here cells $\cbox{white}$ lie in $a \cap b$, cells $\cbox{blue}$ lie in $(a \cup b)\setminus (a\cap b)$, and the cell marked $\times$ is deleted to make $a \Delta b$.}
%%   \end{minipage}
%% \end{figure}
%% 
%% \begin{proposition}
%%   Given weak compositions $a$ and $b$, let $a \Delta b$ denote the subset of cells $x \in a \cap b$ such that for any $y \in a \setminus (a\cap b)$ and any $z\in b \setminus (a\cap b)$ lying above $x$, the row of $x$ is strictly longer than that of $y$ in $a$ and of $z$ in $b$; see Fig.~\ref{fig:meet}. Then $a \Delta b \preceq a, b$, and for any $c$ for which $c \preceq a, b$, we have $c \subseteq a \Delta b$.
%%   \label{prop:meet}
%% \end{proposition}

While this paper does not consider topological properties of the key poset, questions such as shellability are certainly worth exploring.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Polynomials}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:poly}

Given a partition $\lambda$ of rank $n$, the set $\SYT(\lambda)$ of \newword{standard Young tableaux of shape $\lambda$} consists of bijective fillings of the Young diagram of $\lambda$ with numbers $1,2,\ldots,n$ such that row entries increase left to right and column entries increase bottom to top.

We may identify standard Young tableaux of shape $\lambda$ with \newword{saturated chains} in Young's lattice from $\varnothing$ to $\lambda$, i.e. $\varnothing = \lambda^{(0)} \cover \lambda^{(1)} \cover \cdots \cover \lambda^{(n)} = \lambda$, by the correspondence that places $i$ in the unique cell of $\lambda^{(i)}\setminus\lambda^{(i-1)}$. For example, the two saturated chains from $\varnothing$ to $(2,1)$ are shown in \cref{fig:SYT}. The cover relations for Young's lattice are precisely equivalent to the increasing rows and columns conditions.

\begin{figure}[ht]
  \begin{displaymath}
    \arraycolsep=2pt
    \begin{array}{ccccccc}
    \raisebox{-0.75\smcellsize}{$\varnothing$} & \cover & \smtab{ \\ 1 } & \cover & \smtab{ \\ 1 & 2 } & \cover & \smtab{ 3 \\ 1 & 2 } 
    \end{array} \hspace{4\cellsize}
    \arraycolsep=2pt
    \begin{array}{ccccccc}
    \raisebox{-0.75\smcellsize}{$\varnothing$} & \cover & \smtab{ \\ 1 } & \cover & \smtab{ 2 \\ 1 } & \cover & \smtab{ 2 \\ 1 & 3 } 
    \end{array}
  \end{displaymath}
  \caption{\label{fig:SYT}Two saturated chains in Young's lattice from $\varnothing$ to $(2,1)$.}
\end{figure}

\newword{Schur functions} are ubiquitous throughout mathematics, arising as irreducible characters for polynomial representations of the general linear group, Frobenius characters for irreducible representations of the symmetric group, and polynomial representatives for the cohomology classes of Schubert cycles in Grassmannians. Combinatorially, Schur functions are the \newword{quasisymmetric generating functions} for standard Young tableaux.

Gessel introduced the \newword{fundamental quasisymmetric functions} \cite{Ges84}, indexed by strong compositions, that form an important basis for quasisymmetric functions, given by
\begin{equation}
  F_{\alpha}(X) = \sum_{\mathrm{flat}(b) \ \mathrm{refines} \ \alpha} x_1^{b_1} x_2^{b_2} \cdots,
  \label{e:gessel}
\end{equation}
where the sum is over weak compositions $b$ whose nonzero parts, $\mathrm{flat}(b)$, refine $\alpha$.
  
For a standard Young tableau $T$, say that $i$ is a \newword{descent} of $T$ if $i+1$ lies weakly left of $i$. The \newword{descent composition of $T$}, denoted by $\Des(T)$, is the strong composition given by maximal length runs between descents.

For example, the left tableau $T$ in \cref{fig:SYT} has a descent only at $2$, and so $\Des(T)=(2,1)$, whereas the right tableau $U$ has a descent only at $1$, and so $\Des(U) = (1,2)$.

By \cite{Ges84}, we may define a Schur function as follows using the fundamental basis,
\begin{equation}
  s_{\lambda}(X) = \sum_{T \in \SYT(\lambda)} F_{\Des(T)}(X).
  \label{e:schur}
\end{equation}
For example, from \cref{fig:SYT}, we have $s_{(2,1)}(X) = F_{(2,1)}(X) + F_{(1,2)}(X)$.

Using this paradigm, Young's lattice becomes a powerful tool in studying Schur functions as well as the many contexts in which they arise.

Based on the \newword{quasi-Yamanouchi tableaux} of Assaf and Searles \cite{AS18}, Assaf defined \newword{standard key tableaux} as follows \cite[Definition~3.10]{Ass-W}.

\begin{definition}[\cite{Ass-W}]
  A \newword{standard key tableau} is a bijective filling of a key diagram with $1,2,\ldots,n$ such that rows weakly decrease from left to right, and if some entry $i$ is above and in the same column as an entry $k$ with $i<k$, then there is an entry $j$ immediately right of $k$ and $i<j$. Denote the set of standard key tableaux of shape $a$ by $\SKT(a)$.
  \label{def:SKT}
\end{definition}

Parallel to the case for Young's lattice, saturated chains from $\varnothing$ to $a$ in the key poset precisely correspond to standard key tableaux of shape $a$.

\begin{theorem}
  Saturated chains from $\varnothing$ to $a$ in the key poset are in bijection with standard key tableaux of shape $a$ by the correspondence placing $n-i+1$ into the unique cell of $a^{(i)}\setminus a^{(i-1)}$. 
  \label{thm:chains}
\end{theorem}

\begin{figure}[ht]
  \begin{displaymath}
    \arraycolsep=2pt
    \begin{array}{ccccccc}
      \varnothing & \cover & \vline\smtab{ 3 \\ & \\ \\\hline } & \cover & \vline\smtab{ 3 \\ 2 & \\ \\\hline } & \cover & \vline\smtab{ 3 \\ 2 & 1 \\ \\\hline }
    \end{array} \hspace{4\cellsize}
    \arraycolsep=2pt
    \begin{array}{ccccccc}
    \varnothing & \cover & \vline\smtab{ \\ 3 & \\ \\\hline } & \cover & \vline\smtab{ \\ 3 & 2 \\ \\\hline } & \cover & \vline\smtab{ 1 \\ 3 & 2\\ \\\hline } 
    \end{array}
  \end{displaymath}
  \caption{\label{fig:SKT}Two saturated chains in the key poset from $\varnothing$ to $(0,2,1)$.}
\end{figure}

For example, \cref{fig:SKT} shows the two saturated chains in the key poset from $\varnothing$ to $(0,2,1)$. Notice, as well, the two saturated chains in \cref{fig:SYT} are also saturated chains in the key poset from $\varnothing$ to $(0,1,2)$ under the label reversing map $i \mapsto n-i+1$.

The \newword{key polynomials}, indexed by weak compositions, form an important basis for the full polynomial ring. Key polynomials arise as characters of Demazure modules \cite{Dem74a} for the general linear group and coincide with Schubert polynomials \cite{LS82} in the vexillary case \cite{LS90}. Key polynomials are nonsymmetric generalizations of Schur functions, studied combinatorially by Reiner and Shimozono \cite{RS95} and later by Mason \cite{Mas09}, though our perspective follows that of Assaf and Searles \cite{AS18} and Assaf \cite{Ass-W} who define them as the \newword{fundamental slide generating polynomial} for standard key tableaux.

Assaf and Searles introduced the \newword{fundamental slide polynomials} \cite{AS17}, indexed by weak compositions, that form a basis for the full polynomial ring.

\begin{definition}[\cite{AS17}]
  The \newword{fundamental slide polynomial} $\fund_{a}$ is given by
  \begin{equation}
    \fund_{a} = \sum_{\substack{\mathrm{flat}(b) \ \mathrm{refines} \ \mathrm{flat}(a) \\ b_1+\cdots+b_k \geq a_1+\cdots+a_k \ \forall k}} x_1^{b_1} x_2^{b_2} \cdots x_n^{b_n},
    \label{e:slide}
  \end{equation}
  where the sum is over weak compositions $b$ that dominate $a$ in lexicographic order.
  \label{def:slide}
\end{definition}

For a standard key tableau $T$, say that $i$ is a \newword{descent} of $T$ if $i+1$ lies weakly \emph{right} of $i$. We assign a \newword{weak descent composition} for $T$, defined in \cite[Definition~3.12]{Ass-W}, that will index the corresponding fundamental slide polynomial. 

\begin{definition}[\cite{Ass-W}]
  For a standard key tableau $T$, let $(\tau^{(k)}|\ldots|\tau^{(1)})$ be the partitioning of the decreasing word $n \cdots 2 1$ broken between $i+1$ and $i$ precisely whenever $i$ is a descent of $T$. Set $t^{\prime}_i$ to be the lowest row index in $T$ of a letter in $\tau^{(i)}$. Set $t_k = t^{\prime}_k$ and, for $i<k$, set $t_i = \min(t^{\prime}_i , t_{i+1}-1)$. Define the \newword{weak descent composition of $T$}, denoted by $\des(T)$, by $\des(T)_{t_i} = |\tau^{(i)}|$ and all other parts are zero if all $t_i>0$; and $\des(T) = \varnothing$ otherwise.
  \label{def:des}
\end{definition}

For example, the left tableau $T$ in \cref{fig:SKT} has a descent only at $2$, so $(\tau^{(2)} | \tau^{(1)}) = (3 | 21)$. Thus $t_2 = t^{\prime}_2 = 3$, $t^{\prime}_1 = 2$, and $t_1 = \min(t^{\prime}_1,t_2-1) = \min(2,2) = 2$, giving $\des(T) = (0,|\tau^{(1)}|,|\tau^{(2)}|) = (0,2,1)$. The right tableau $U$ in \cref{fig:SKT} has a descent only at $1$, so $(\tau^{(2)} | \tau^{(1)}) = (32 | 1)$. Thus $t_2 = t^{\prime}_2 = 2$, $t^{\prime}_1 = 3$, and $t_1 = \min(t^{\prime}_1,t_2-1) = \min(3,1) = 1$, giving $\des(U) = (|\tau^{(1)}|,|\tau^{(2)}|,0) = (1,2,0)$.

We take \cite[Corollary~3.16]{Ass-W} as our definition for key polynomials.

\begin{definition}[\cite{Ass-W}]
  For a weak composition $a$, the \newword{key polynomial} $\key_{a}$ is given by
  \begin{equation}
    \key_{a} = \sum_{T \in \SKT(a)} \fund_{\des(T)},
    \label{e:key}
  \end{equation}
  where the sum is over $T\in\SKT(a)$ for which $\des(T)\neq\varnothing$.
  \label{def:key}  
\end{definition}

For example, from \cref{fig:SKT}, we have $\kappa_{(0,2,1)} = \fund_{(0,2,1)} + \fund_{(1,2,0)}$.

The key polynomials generalize Schur polynomials in that Schur polynomials are key polynomials and Schur functions are the \newword{stable limits} of key polynomials, 
\begin{eqnarray}
  \key_{(0,\ldots,0,\lambda_{\ell},\ldots,\lambda_1)}(x_1,x_2,\ldots,x_n) & = & s_{\lambda}(x_1,x_2,\ldots,x_n) \\
  \lim_{m \rightarrow\infty} \key_{0^m\times a}(x_1,\ldots,x_m,0,\ldots,0) & = & s_{\mathrm{sort}(a)}(X) .
\end{eqnarray}

Thus we hope to use the key poset similar to Young's lattice to study important properties of key polynomials that relate them to representation theory and geometry.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Positivity}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:skew}

Standard Young tableaux are saturated chains in Young's lattice beginning at $\varnothing$. We may define a more general class of objects, called standard skew Young tableaux, as saturated chains between any two partition $\lambda \subset \nu$. The direct combinatorial characterization is completely analogous: the \newword{standard skew Young tableaux} of shape $\nu\setminus\lambda$ are bijective fillings of the set-theoretic difference $\nu\setminus\lambda$, called the \newword{skew diagram}, with $1,2,\ldots,n$ such that rows increase left to right and columns increase bottom to top.

Directly extending the notion of descents and descent compositions allows us to define \newword{skew Schur functions} for partitions $\lambda\subset \nu$ by
\begin{equation}
  s_{\nu\setminus\lambda}(X) = \sum_{T \in \SYT(\nu\setminus\lambda)} F_{\Des(T)}(X).
  \label{e:skew_schur}
\end{equation}

Skew Schur functions are symmetric functions, and so we may consider the \newword{Littlewood--Richardson coefficients} $c_{\lambda,\mu}^{\nu}$ that give their Schur expansion,
\begin{equation}
  s_{\nu\setminus\lambda}(X) = \sum_{\mu} c_{\lambda,\mu}^{\nu} s_{\mu}(X).
\end{equation}
A priori, these coefficients are integers. It is a deep result in algebraic combinatorics, with myriad beautiful proofs, that these coefficients are \emph{nonnegative}. This nonnegativity manifests in representation theory multiplicities of irreducible representations in tensor products of polynomial representations for the general linear group and in geometry as intersection numbers for Grassmannian Schubert varieties.

One (of many) proofs of the nonnegativity of $c_{\lambda,\mu}^{\nu}$ from the skew Schur function perspective utilizes \newword{dual equivalence} \cite[Definition~4.1]{Ass15} to consolidate skew standard Young tableaux into equivalence classes, each corresponding to a single Schur function. %Loosely, a dual equivalence for a set of objects is a collection of involutions indexed by $2 \leq i \leq n-1$ that commute when indices are far apart and whose local equivalence classes are Schur positive. Formally, we have \cite[Definition~4.1]{Ass15}.

Given a strong composition $\alpha$ of $n$ and integers $1 \leq h \leq i\leq n$, let $\alpha_{(h,i)}$ be the composition obtained by deleting the first $h-1$ and last $n-i$ pieces from $\alpha$.

For example, for $\alpha = (3,2,3,1)$, a strong composition of $9$, we have $\alpha_{(3,7)} = (1,2,2)$ by deleting the first $2$ pieces from $\alpha_1$ and the last $2$ pieces from $\alpha_4$ and $\alpha_3$.

\begin{definition}[\cite{Ass15}]
  Let $\mathcal{A}$ be a finite set, and $\Des$ a map from $\mathcal{A}$ to strong compositions of $n$. A \newword{dual equivalence for $(\mathcal{A},\Des)$} is a family of involutions $\{\varphi_i\}_{1<i<n}$ on $\mathcal{A}$ such that
  \renewcommand{\theenumi}{\roman{enumi}}
  \begin{enumerate}
  \item For all $0 \leq i-h \leq 3$ and all $T \in \mathcal{A}$, there exists a partition $\lambda$ such that
    \[ \sum_{U \in [T]_{(h,i)}} F_{\Des_{(h-1,i+1)}(U)}(X) = s_{\lambda}(X), \]
    where $[T]_{(h,i)}$ is the equivalence class generated by $\varphi_h,\ldots,\varphi_i$.
    
  \item For all $|i-j| \geq 3$ and all $T \in\mathcal{A}$, we have $\varphi_{j} \varphi_{i}(T) = \varphi_{i} \varphi_{j}(T)$.

  \end{enumerate}
  \label{def:deg}
\end{definition}

Define \newword{simple involutions} $s_i$ for $i=1,2,\ldots,n-1$ on standard fillings of rank $n$ that interchange $i$ and $i+1$. On standard Young tableaux, we combine these into \newword{elementary dual equivalence involutions}, denoted by $d_i$, that act by
\begin{equation}
  d_i(T) = \left\{ \begin{array}{rl}
    T & \text{if $i$ lies between $i-1$ and $i+1$ in reading order, } \\
    s_{i-1} \cdot T & \text{if $i+1$ lies between $i$ and $i-1$ in reading order, } \\
    s_{i} \cdot T & \text{if $i-1$ lies between $i$ and $i+1$ in reading order, }
  \end{array} \right.
  \label{e:ede}
\end{equation}
where we may take either row or column reading order in any direction; see \cref{fig:deg}.

\begin{figure}[ht]  
  \begin{center}
    \begin{tikzpicture}[xscale=2,yscale=1]
      \node at (0,0)   (a) {$\tableau{5 \\ 2 & 4 \\ 1 & 3 }$}; 
      \node at (1,0)   (b) {$\tableau{5 \\ 3 & 4 \\ 1 & 2}$}; 
      \node at (2,0)   (c) {$\tableau{4 \\ 3 & 5 \\ 1 & 2}$}; 
      \node at (3,0)   (d) {$\tableau{4 \\ 2 & 5 \\ 1 & 3}$}; 
      \node at (4,0)   (e) {$\tableau{3 \\ 2 & 5 \\ 1 & 4}$}; 
      \draw[thick,color=blue  ,<->] (a.008) -- (b.172) node[midway,above] {$d_2$} ;
      \draw[thick,color=violet,<->] (a.352) -- (b.188) node[midway,below] {$d_3$} ;
      \draw[thick,color=purple,<->] (b) -- (c) node[midway,above] {$d_4$} ;
      \draw[thick,color=blue  ,<->] (c) -- (d) node[midway,above] {$d_2$} ;
      \draw[thick,color=violet,<->] (d.008) -- (e.172) node[midway,above] {$d_3$} ;
      \draw[thick,color=purple,<->] (d.352) -- (e.188) node[midway,below] {$d_4$} ;
    \end{tikzpicture}
  \end{center}
  \caption{\label{fig:deg}The elementary dual equivalence involutions on $\SYT(2,2,1)$.}
\end{figure}

Haiman \cite{Hai92} showed that these are well-defined on standard Young tableaux and that all standard Young tableaux of fixed shape fall into a single equivalence class. Assaf \cite[Proposition~3.3]{Ass15} showed that they give an example of a dual equivalence, and, more importantly, the converse holds. That is, by \cite[Theorem~3.7]{Ass15}, any dual equivalence is essentially this. At the level of generating functions, we have \cite[Corollary~4.4]{Ass15}.

\begin{theorem}[\cite{Ass15}]
  If there exists a dual equivalence for $(\mathcal{A},\mathrm{Des})$, then $\sum_{T \in \mathcal{A}} F_{\mathrm{Des}(T)}(X)$ is symmetric and Schur positive.
  \label{thm:deg}
\end{theorem}

It follows directly from the local nature of \cref{def:deg} that extending the elementary dual equivalences to standard skew Young tableaux using Eq.~\eqref{e:ede} results in a dual equivalence for $\SYT(\nu\setminus\lambda)$, thus giving a combinatorial proof of the following.

\begin{corollary}[\cite{Ass15}]
  For $\lambda\subset\nu$, $c_{\lambda,\mu}^{\nu}$ is the number of dual equivalence classes of $\SYT(\nu\setminus\lambda)$ isomorphic to $\SYT(\mu)$. In particular, skew Schur functions are Schur positive.
  \label{cor:skewSYT}
\end{corollary}

Assaf considered \newword{standard skew key tableaux} in \cite[Definition~4.7]{Ass-W}, defined for any pair of weak compositions $a \subset d$. However, the positivity result for the corresponding \newword{skew key polynomials} \cite[Theorem~4.10]{Ass-W}, generalizing \cref{cor:skewSYT}, is proved only for the limited case when the smaller weak composition is a \emph{partition}. Examples show that these are not the only cases where nonnegativity holds, with many overlooked examples arising naturally from geometric contexts. As we shall see, the key to positivity lies in the key poset. We begin by generalizing \cref{thm:chains} to skew diagrams.

\begin{theorem}
  Saturated chains from $a$ to $d$ in the key poset are in bijection with $\SKT(d\setminus a)$ by the correspondence placing $n-i+1$ into the unique cell of $a^{(i)}\setminus a^{(i-1)}$. 
  \label{thm:skew-chains}
\end{theorem}

Following \cite{Ass-W}, extend \cref{def:des} directly to skew standard key tableaux.

\begin{definition}
  For weak compositions $a \prec d$, the \newword{skew key polynomial} $\key_{d\setminus a}$ is given by
  \begin{equation}
    \key_{d \setminus a} = \sum_{T \in \SKT(d\setminus a)} \fund_{\des(T)} .
    \label{e:skewkey}
  \end{equation}
  \label{def:skewkey}
\end{definition}

Skew key polynomials generalize skew Schur polynomials in that every skew Schur polynomial is a skew key polynomial and every skew Schur function is the stable limit of a skew key polynomial. However, skew key polynomials are more general, with examples of stable limits of skew key polynomials that are not skew Schur functions.

Note that unlike \cite[Definition~4.8]{Ass-W}, we define skew key polynomials only for comparable elements of the key poset. However, the special case of skewing by an increasing composition, for which the positivity in \cite[Theorem~4.10]{Ass-W} holds, conforms with this more restrictive definition by \cref{prop:keypart}. 

Since key polynomials are a basis for all polynomials, we may define \newword{weak composition Littlewood--Richardson coefficients} $c_{a,b}^{d}$ as the key expansion of skew key polynomials,
\begin{equation}
  \key_{d \setminus a} = \sum_{b} c_{a,b}^{d} \key_{b} .
\end{equation}
A priori, these coefficients are \emph{integers}. To prove nonnegativity of the weak composition Littlewood--Richardson coefficients, we utilize \newword{weak dual equivalence} \cite{Ass-W}, a polynomial generalization of dual equivalence, to consolidate skew standard key tableaux into equivalence classes, each of which corresponds to a single key polynomial. 

For a weak composition $a$ of rank $n$ and integers $1\leq h \leq i\leq n$, let $a_{(h,i)}$ be the weak composition obtained by deleting the first $h-1$ and last $n-i$ pieces from $a$.

For example, let $a = (0,3,2,0,3,1)$, a weak composition of rank $9$. Then $a_{(3,7)} = (0,1,2,0,2,0)$ by deleting the first $2$ pieces from $a_2$ and the last $2$ pieces from $a_6$ and $a_5$.

\begin{definition}
  Let $\mathcal{A}$ be a finite set and $\des$ a map from $\mathcal{A}$ to weak compositions of $n$. A \newword{weak dual equivalence for $(\mathcal{A},\des)$} is a family of involutions $\{\psi_i\}_{1<i<n}$ on $\mathcal{A}$ such that
  \renewcommand{\theenumi}{\roman{enumi}}
  \begin{enumerate}
  \item For all $i-h \leq 3$ and all $T \in \mathcal{A}$, there exists a weak composition $a$ such that
    \[ \sum_{U \in [T]_{(h,i)}} \fund_{\des_{(h-1,i+1)}(U)} = \key_{a}, \]
    where $[T]_{(h,i)}$ is the equivalence class generated by $\psi_h,\ldots,\psi_i$.
    
  \item For all $|i-j| \geq 3$ and all $T \in\mathcal{A}$, we have $\psi_{j} \psi_{i}(T) = \psi_{i} \psi_{j}(T)$.
  \end{enumerate}
  \label{def:wdeg}
\end{definition}

Define the \newword{braid involutions} $b_i$ for $i=2,\ldots,n-1$ on standard fillings of rank $n$ for which exactly one of $i-1$ or $i+1$ lies in the same row as $i$ by cycling entries $i-1,i,i+1$ in the unique way that maintains this condition. For example, $b_2$ will exchange the two standard key tableaux of shape $(0,2,1)$ shown in \cref{fig:SKT}. Using these, we define the elementary weak dual equivalence involutions from \cite[Definition~3.21]{Ass-W}.

\begin{definition}[\cite{Ass-W}]
  Define \newword{elementary weak dual equivalence involutions}, denoted by $d_i$, on skew standard key tableaux that act by
  \begin{equation}
    d_i(T) = \left\{ \begin{array}{rl}
      b_{i} \cdot T & \text{if exactly one of $i-1,i+1$ lies in the row of $i$, } \\
      s_{i-1} \cdot T & \text{else if $i+1$ lies between $i$ and $i-1$ in reading order, } \\
      s_{i} \cdot T & \text{else if $i-1$ lies between $i$ and $i+1$ in reading order, } \\
      T & \text{otherwise} 
    \end{array} \right.
    \label{e:wede}
  \end{equation}
  where we take \emph{column} reading order, bottom to top and left to right.
  \label{def:wede}
\end{definition}

For examples of the elementary weak involutions, see \cref{fig:wdeg}.

\begin{figure}[ht]  
  \begin{center}
    \begin{tikzpicture}[xscale=2,yscale=1]
      \node at (0,0)   (a) {$\tableau{5 & 2 \\ 4 \\ 3 & 1 }$}; 
      \node at (1,0)   (b) {$\tableau{5 & 3 \\ 4 \\ 2 & 1 }$}; 
      \node at (2,0)   (c) {$\tableau{5 & 4 \\ 3 \\ 2 & 1 }$}; 
      \node at (3,0)   (d) {$\tableau{5 & 4 \\ 1 \\ 3 & 2 }$}; 
      \node at (4,0)   (e) {$\tableau{5 & 3 \\ 1 \\ 4 & 2 }$}; 
      \draw[thick,color=blue  ,<->] (a.008) -- (b.172) node[midway,above] {$d_2$} ;
      \draw[thick,color=violet,<->] (a.352) -- (b.188) node[midway,below] {$d_3$} ;
      \draw[thick,color=purple,<->] (b) -- (c) node[midway,above] {$d_4$} ;
      \draw[thick,color=blue  ,<->] (c) -- (d) node[midway,above] {$d_2$} ;
      \draw[thick,color=violet,<->] (d.008) -- (e.172) node[midway,above] {$d_3$} ;
      \draw[thick,color=purple,<->] (d.352) -- (e.188) node[midway,below] {$d_4$} ;
    \end{tikzpicture}
  \end{center}
  \caption{\label{fig:wdeg}The elementary dual equivalence involutions on $\SKT(2,1,2)$.}
\end{figure}

Assaf \cite[Theorem~3.25]{Ass-W} showed  these are well-defined involutions on standard key tableaux, all standard key tableaux of fixed shape fall into a single equivalence class, and this is an example of a weak dual equivalence. Moreover, under certain stability conditions, the converse holds. That is, by \cite[Theorem~3.29]{Ass-W}, any weak dual equivalence is essentially this and, on the level of generating polynomials, we have the following.

\begin{theorem}[\cite{Ass-W}]
  If there exists a weak dual equivalence for $(\mathcal{A},\mathrm{des})$ for which $\des(T)\neq\varnothing$ for every $T\in\mathcal{A}$, then $\sum_{T \in \mathcal{A}} \fund_{\mathrm{des}(T)}$ is a nonnegative sum of key polynomials.
  \label{thm:wdeg}
\end{theorem}

The condition that the weak descent composition is nonempty for every element can often be circumvented if the polynomials under consideration stabilize. Using this along with the same elementary weak dual equivalence involutions, the local nature of \cref{def:wdeg} allows us to give a combinatorial proof of the following.

\begin{theorem}
  For $a \prec d$ in the key poset, $c_{a,b}^{d}$ is the number of weak dual equivalence classes of $\SKT(d\setminus a)$ isomorphic to $\SKT(b)$. In particular, skew key polynomials are key positive.
  \label{thm:skewpos}
\end{theorem}

The special case when $a$ is weakly increasing is proved in \cite[Theorem~4.10]{Ass-W} and follows from \cref{thm:skewpos} by \cref{prop:keypart}. In fact, we use the key poset to prove that \cref{thm:skewpos} is tight, outside of a few accidental cases.

\begin{theorem}
  For $a \subset d$ for which $a \not\prec d$ in the key poset, if $\des(T)\neq\varnothing$ for every $T \in \SKT(d\setminus a)$, then there exists a weak composition $b$ for which $c_{a,b}^{d}$ is negative. 
  \label{thm:skewneg}
\end{theorem}

Thus the key poset precisely characterizes nonnegativity of skew key polynomials.

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