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\title[A quotient of the ring of symmetric functions]{A quotient of the ring of symmetric functions generalizing quantum cohomology}

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\author[Darij Grinberg]{Darij Grinberg\thanks{\href{mailto:darijgrinberg@gmail.com}{darijgrinberg@gmail.com}}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, University of Minnesota, Twin Cities}

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\abstract{Consider the ring $\mathcal{S}$ of symmetric polynomials in
$k$ variables over an arbitrary base ring $\mathbf{k}$.
Fix $k$ scalars $a_1, a_2, \ldots, a_k \in \mathbf{k}$.
Let $I$ be the ideal of $\mathcal{S}$ generated by
$h_{n-k+1}-a_1, h_{n-k+2}-a_2, \ldots, h_n-a_k$,
where $h_i$ is the $i$-th complete homogeneous symmetric polynomial.

The quotient ring $\mathcal{S}/I$ generalizes both the usual and the quantum
cohomology of the Grassmannian.

We show that $\mathcal{S}/I$ has a $\mathbf{k}$-module basis consisting of
(residue classes of) Schur polynomials fitting into a
$k \times \left(  n-k\right)$-rectangle;
and that its multiplicative structure constants satisfy
the same $S_3$-symmetry as those of the Grassmannian cohomology. We
conjecture the existence of a Pieri rule (proven in two
particular cases) and a positivity property generalizing that of Gromov-Witten invariants.
}

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\keywords{symmetric functions, partitions, Schur functions, Gr\"obner bases, Grassmannian, cohomology}

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\begin{document}

\maketitle
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\section{Introduction}

Schubert calculus -- the quantitative study of the cohomology ring
$\operatorname*{H}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  $ of the Grassmannian $\operatorname*{Gr}\left(
k,n\right)  $ -- is one of the origins of much historical interest in
symmetric functions (see, e.g., \cite[Part III]{Fulton97} and \cite[Chp.
3]{Manive01}). From a modern point of view, this cohomology ring can be
regarded as a quotient of the ring $\mathcal{S}$ of symmetric polynomials in
$k$ variables $x_{1},x_{2},\ldots,x_{k}$ modulo the ideal generated by the
complete homogeneous symmetric polynomials $h_{n-k+1},h_{n-k+2},\ldots,h_{n}$.
The last three decades have seen this cohomology ring generalized and refined
in several ways, one of which is the \textit{(small) quantum cohomology ring}
$\operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  $ originating from 1993 work of Witten. Bertram,
Ciocan-Fontanine, Fulton, Postnikov and others have identified the
combinatorial structure of this ring (see \cite{Postni05} for a survey); it
turns out to be the quotient of the same ring $\mathcal{S}$ modulo the ideal
generated by $h_{n-k+1},h_{n-k+2},\ldots,h_{n-1},h_{n}+\left(  -1\right)
^{k}q$, where $q$ is a polynomial indeterminate adjoined to the base ring
(i.e., we are working over $\mathbb{Z}\left[  q\right]  $ rather than over
$\mathbb{Z}$). Various properties of $\operatorname*{H}\nolimits^{\ast}\left(
\operatorname*{Gr}\left(  k,n\right)  \right)  $ have been extended to
$\operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  $, usually with new subtleties and complexities appearing
in the process.

Combinatorialists working with $\operatorname*{H}\nolimits^{\ast}\left(
\operatorname*{Gr}\left(  k,n\right)  \right)  $ and $\operatorname*{QH}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $
commonly rely on geometry (in the case of $\operatorname*{QH}\nolimits^{\ast
}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $, fairly recent and
deep algebraic geometry) to justify the basic properties of these rings (such
as the fact that they are free as modules over the base ring, with a basis
consisting of projected Schur polynomials $\overline{s_{\lambda}}$
corresponding to partitions $\lambda$ that \textquotedblleft fit inside a
$k\times\left(  n-k\right)  $-rectangle\textquotedblright). While both of
these rings are easily defined algebraically as quotients of $\mathcal{S}$,
the literature does not seem to contain purely algebraic proofs of their basic
structural properties. One purpose of this work is to supply such proofs,
establishing two bases of these rings (defined as quotients of $\mathcal{S}$)
and the $S_{3}$-symmetry of their structure constants in the Schur basis
(which are the Littlewood-Richardson numbers in the case of $\operatorname*{H}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $, and
the Gromov-Witten invariants in the case of $\operatorname*{QH}\nolimits^{\ast
}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $). Our proofs rely on
identities for symmetric functions (see, e.g., \cite{GriRei18}).

However, we will work in far greater generality, and study the quotient of
$\mathcal{S}$ modulo the ideal $I$ generated by $h_{n-k+1}-a_{1}%
,h_{n-k+2}-a_{2},\ldots,h_{n}-a_{k}$, where $a_{1},a_{2},\ldots,a_{k}$ are
arbitrary elements of our base ring $\mathbf{k}$. (And on occasion, we will
work in even more general settings.) The resulting quotient ring
$\mathcal{S}/I$ generalizes both $\operatorname*{H}\nolimits^{\ast}\left(
\operatorname*{Gr}\left(  k,n\right)  \right)  $ and $\operatorname*{QH}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $, and
serves as the setting of an \textquotedblleft abstract Schubert
calculus\textquotedblright, which (so far) has not found any geometric
meaning. We shall exhibit two $\mathbf{k}$-module bases of this ring
$\mathcal{S}/I$ -- one consisting of projections $\overline{s_{\lambda}}$ of
Schur polynomials $s_{\lambda}$ for partitions $\lambda$ \textquotedblleft
fitting inside the $k\times\left(  n-k\right)  $-rectangle\textquotedblright%
\ (Theorem \ref{thm.S/I}), and another consisting of analogous projections
$\overline{h_{\lambda}}$ of complete homogeneous symmetric polynomials
(Theorem \ref{thm.S/J-h-basis}). We shall show that the structure constants of
$\mathcal{S}/I$ with respect to the first of these two bases exhibit the same
$S_{3}$-symmetry (Theorem \ref{cor.S3sym}) as the Littlewood-Richardson
numbers and the Gromov-Witten invariants, as well as a Pieri rule (Theorem
\ref{thm.pieri-hall-LR}) subtler than the latter. Finally, we shall conjecture
a positivity property (Conjecture \ref{conj.pos}) for these structure
constants, which hints at some hidden geometry or combinatorics behind this
\textquotedblleft abstract Schubert calculus\textquotedblright.

No proofs are given in this extended abstract; the reader should consult
\cite{Grinbe18} for them.

\section{The theorems}

\subsection{The rings $\mathcal{P}$ and $\mathcal{S}$ in general, and the
quotient $\mathcal{P}/J$}

Let $\mathbf{k}$ be a commutative ring. Let $\mathbb{N}=\left\{
0,1,2,\ldots\right\}  $. Let $k\in\mathbb{N}$. Let $\mathcal{P}$ denote the
polynomial ring $\mathbf{k}\left[  x_{1},x_{2},\ldots,x_{k}\right]  $. For
each $\alpha\in\mathbb{N}^{k}$ and each $i\in\left\{  1,2,\ldots,k\right\}  $,
we denote the $i$-th entry of $\alpha$ by $\alpha_{i}$ (so that $\alpha
=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $). For each
$\alpha\in\mathbb{N}^{k}$, we define a monomial $x^{\alpha}$ by $x^{\alpha
}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{k}^{\alpha_{k}}$.

Let $\mathcal{S}$ denote the ring of symmetric polynomials in $\mathcal{P}$;
in other words, $\mathcal{S}$ is the ring of invariants of the symmetric group
$S_{k}$ acting on $\mathcal{P}$. The following fact goes back
to Emil Artin and is proven (e.g.) in \cite[(DIFF.1.3)]{LLPT95}:

\begin{proposition}
\label{prop.artin}The $\mathcal{S}$-module $\mathcal{P}$ is free with basis
$\left(  x^{\alpha}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha_{i}<i\text{ for
each }i}$.
\end{proposition}

Now, fix an integer $n\geq k$. For each $i\in\left\{  1,2,\ldots,k\right\}  $,
let $a_{i}$ be an element of $\mathcal{P}$ with degree $<n-k+i$. (This is
clearly satisfied when $a_{1},a_{2},\ldots,a_{k}$ are constants in
$\mathbf{k}$, but also in some other cases.)

For each $\alpha\in\mathbb{N}^{k}$, we let $\left\vert \alpha\right\vert $
denote the sum of the entries of the $k$-tuple $\alpha$ (that is, $\left\vert
\alpha\right\vert =\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k}$).

For each $m\in\mathbb{Z}$, we let $h_{m}$ denote the $m$-th complete
homogeneous symmetric polynomial; this is the element of $\mathcal{S}$ defined
by%
\begin{equation}
h_{m}=\sum_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{m}\leq k}x_{i_{1}}x_{i_{2}%
}\cdots x_{i_{m}}=\sum_{\substack{\alpha\in\mathbb{N}^{k};\\\left\vert
\alpha\right\vert =m}}x^{\alpha}. \label{eq.hm}%
\end{equation}
(Thus, $h_{0}=1$, and $h_{m}=0$ when $m<0$.)

Let $J$ be the ideal of $\mathcal{P}$ generated by the $k$ differences
$h_{n-k+1}-a_{1},h_{n-k+2}-a_{2},\ldots,h_{n}-a_{k}$.

If $M$ is a $\mathbf{k}$-module and $N$ is a submodule of $M$, then the
projection of any $m\in M$ onto the quotient $M/N$ (that is, the congruence
class of $m$ modulo $N$) will be denoted by $\overline{m}$.

\begin{theorem}
\label{thm.P/J}The $\mathbf{k}$-module $\mathcal{P}/J$ is free with basis
$\left(  \overline{x^{\alpha}}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha
_{i}<n-k+i\text{ for each }i}$.
\end{theorem}

\begin{example}
Let $n=5$ and $k=2$. Then, $\mathcal{P}=\mathbf{k}\left[  x_{1},x_{2}\right]
$, and $J$ is the ideal of $\mathcal{P}$ generated by the $2$ differences
\begin{align*}
h_{4}-a_{1}  &  =\left(  x_{1}^{4}+x_{1}^{3}x_{2}+x_{1}^{2}x_{2}^{2}%
+x_{1}x_{2}^{3}+x_{2}^{4}\right)  -a_{1}\ \ \ \ \ \ \ \ \ \ \text{and}\\
h_{5}-a_{2}  &  =\left(  x_{1}^{5}+x_{1}^{4}x_{2}+x_{1}^{3}x_{2}^{2}+x_{1}%
^{2}x_{2}^{3}+x_{1}x_{2}^{4}+x_{2}^{5}\right)  -a_{2}.
\end{align*}
Theorem \ref{thm.P/J} yields that the $\mathbf{k}$-module $\mathcal{P}/J$ is
free with basis $\left(  \overline{x^{\alpha}}\right)  _{\alpha\in
\mathbb{N}^{2};\ a_{i}<3+i\text{ for each }i}$; this basis can also be
rewritten as $\left(  \overline{x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}}\right)
_{\alpha_{1}\in\left\{  0,1,2,3\right\}  ;\ \alpha_{2}\in\left\{
0,1,2,3,4\right\}  }$. As a consequence, any monomial in $\overline{x_{1}}$
and $\overline{x_{2}}$ can be written as a linear combination of elements of
this basis. For example,%
\[
\overline{x_{1}^{4}}=a_{1}-\overline{x_{1}^{3}x_{2}}-\overline{x_{1}^{2}%
x_{2}^{2}}-\overline{x_{1}x_{2}^{3}}-\overline{x_{2}^{4}}%
\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \overline{x_{2}^{5}}%
=a_{2}-a_{1}\overline{x_{1}}.
\]
These expressions will become more complicated for higher values of $n$ and
$k$.
\end{example}

Theorem \ref{thm.P/J} is related to the second part of \cite[Proposition
2.9]{CoKrWa09}.
The $\mathbf{k}$-algebra $\mathcal{P}/J$ somewhat resembles
the \textquotedblleft splitting algebra\textquotedblright%
\ $\operatorname*{Split}\nolimits_{A}^{d}\left(  p\right)  $ from
\cite[\S 1.3]{LakTho12}; the basis in Theorem \ref{thm.P/J} is similar to the
basis in \cite[(1.5)]{LakTho12}. Whether there is more to this analogy has yet
to be understood.

\subsection{The case of symmetric $a_{1},a_{2},\ldots,a_{k}$ and the quotient
$\mathcal{S}/I$}

To state our next result, we need some more notations. We define the notion of
a \textit{partition} as in \cite[Chapter 2]{GriRei18}. A
\textit{part} of a partition $\lambda$ means a nonzero entry of $\lambda$. If
$\lambda$ is a partition and $p$ is a positive integer, then $\lambda_{p}$
will denote the $p$-th entry of $\lambda$ (so that $\lambda=\left(
\lambda_{1},\lambda_{2},\lambda_{3},\ldots\right)  $). If $\lambda$ and $\mu$
are two partitions, then we say that $\mu\subseteq\lambda$ if and only if each
positive integer $p$ satisfies $\mu_{p}\leq\lambda_{p}$. A \textit{skew
partition} means a pair $\left(  \lambda,\mu\right)  $ of two partitions
satisfying $\mu\subseteq\lambda$; such a pair is denoted by $\lambda/\mu$. We
refer to \cite[\S 2.7]{GriRei18} for the definition of a \textit{vertical }%
$i$\textit{-strip} (where $i\in\mathbb{N}$). The \textit{size} of a partition
$\lambda=\left(  \lambda_{1},\lambda_{2},\lambda_{3},\ldots\right)  $ is
defined as $\lambda_{1}+\lambda_{2}+\lambda_{3}+\cdots$, and is denoted by
$\left\vert \lambda\right\vert $.

We let $\omega$ be the partition $\left(  n-k,n-k,\ldots,n-k\right)  $ with
$k$ entries equal to $n-k$. This partition $\omega$ is called the
\textit{$k\times\left(  n-k\right)  $-rectangle},
%(since this is how its Young diagram looks like)
and often denoted by $\left(  \left(n-k\right)^k \right)  $.

We let $P_{k,n}$ denote the set of all partitions that have at most $k$ parts
and have the property that each of their parts is $\leq n-k$. Equivalently,
$P_{k,n}$ is the set of all partitions $\lambda$ satisfying $\lambda
\subseteq\omega$.

For any partition $\lambda$, we let $s_{\lambda}$ denote the Schur polynomial
(in $x_{1},x_{2},\ldots,x_{k}$) corresponding to the partition $\lambda$.
(This is what is called $s_{\lambda}\left(  x_{1},x_{2},\ldots,x_{k}\right)  $
in \cite[Chapter 2]{GriRei18}.) Note that $s_{\lambda}=0$ if $\lambda$ has
more than $k$ parts.

\textbf{Assume from now on that }$a_{1},a_{2},\ldots,a_{k}$\textbf{ belong to
}$\mathcal{S}$\textbf{.}

Let $I$ be the ideal of $\mathcal{S}$ generated by the $k$ differences
$h_{n-k+1}-a_{1},h_{n-k+2}-a_{2},\ldots,h_{n}-a_{k}$.
Thus, $I$ has the same
generators as $J$, but is an ideal of $\mathcal{S}$ rather than of
$\mathcal{P}$.

\begin{theorem}
\label{thm.S/I}The $\mathbf{k}$-module $\mathcal{S}/I$ is free with basis
$\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$.
\end{theorem}

The $\mathbf{k}$-algebra $\mathcal{S}/I$ generalizes several constructions in
the literature:

\begin{itemize}
\item If $\mathbf{k}=\mathbb{Z}$ and $a_{1}=a_{2}=\cdots=a_{k}=0$, then
$\mathcal{S}/I$ becomes the cohomology ring $\operatorname*{H}\nolimits^{\ast
}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $ of the Grassmannian
$\operatorname*{Gr}\left(  k,n\right)  $ of $k$-dimensional subspaces in an
$n$-dimensional space (see, e.g., \cite[\S 9.4]{Fulton97}); the elements of
the basis $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$
correspond to the Schubert classes.

\item If $\mathbf{k}=\mathbb{Z}\left[  q\right]  $ and $a_{1}=a_{2}%
=\cdots=a_{k-1}=0$ and $a_{k}=-\left(  -1\right)  ^{k}q$, then $\mathcal{S}/I$
becomes isomorphic to the quantum cohomology ring $\operatorname*{QH}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $ of
the same Grassmannian (see \cite{Postni05}). Indeed, our ideal $I$ becomes the
$J_{kn}^{q}$ of \cite[(6)]{Postni05} in this case, and Theorem \ref{thm.S/I}
generalizes the fact that the quotient $\left(  \Lambda_{k}\otimes
\mathbb{Z}\left[  q\right]  \right)  /J_{kn}^{q}$ in \cite[(6)]{Postni05} has
basis $\left(  s_{\lambda}\right)  _{\lambda\in P_{kn}}$.
\end{itemize}

\textbf{Assume from now on that }$a_{1},a_{2},\ldots,a_{k}$\textbf{ belong to
}$\mathbf{k}$.

Let us now study the structure constants of the $\mathbf{k}$-algebra
$\mathcal{S}/I$ in the basis $\left(  \overline{s_{\lambda}}\right)
_{\lambda\in P_{k,n}}$. For each $\mu\in P_{k,n}$, let $\operatorname*{coeff}%
\nolimits_{\mu}:\mathcal{S}/I\rightarrow\mathbf{k}$ be the $\mathbf{k}$-linear
map that sends $\overline{s_{\mu}}$ to $1$ while sending all other
$\overline{s_{\lambda}}$ (with $\lambda\in P_{k,n}$) to $0$. Thus, $\left(
\operatorname*{coeff}\nolimits_{\mu}\right)  _{\mu\in P_{k,n}}$ is the dual
basis to the basis $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in
P_{k,n}}$ of $\mathcal{S}/I$.

For every partition $\nu=\left(  \nu_{1},\nu_{2},\ldots,\nu_{k}\right)  \in
P_{k,n}$, we let $\nu^{\vee}$ denote the partition \newline$\left(
n-k-\nu_{k},n-k-\nu_{k-1},\ldots,n-k-\nu_{1}\right)  \in P_{k,n}$. This
partition $\nu^{\vee}$ is called the \textit{complement} of $\nu$.

For any three partitions $\alpha,\beta,\gamma\in P_{k,n}$, let $g_{\alpha
,\beta,\gamma}=\operatorname*{coeff}\nolimits_{\gamma^{\vee}}\left(
\overline{s_{\alpha}}\overline{s_{\beta}}\right)  \in\mathbf{k}$. These
scalars $g_{\alpha,\beta,\gamma}$ are thus the structure constants of the
$\mathbf{k}$-algebra $\mathcal{S}/I$ in the basis $\left(  \overline
{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$ (although slightly reindexed).
They satisfy the following $S_{3}$-symmetry:

\begin{theorem}
\label{cor.S3sym}For any $\alpha,\beta,\gamma\in P_{k,n}$, we have%
\[
g_{\alpha,\beta,\gamma}=g_{\alpha,\gamma,\beta}=g_{\beta,\alpha,\gamma
}=g_{\beta,\gamma,\alpha}=g_{\gamma,\alpha,\beta}=g_{\gamma,\beta,\alpha
}=\operatorname*{coeff}\nolimits_{\omega}\left(  \overline{s_{\alpha}s_{\beta
}s_{\gamma}}\right)  .
\]

\end{theorem}

\begin{theorem}
\label{thm.coeffw}Each $\nu\in P_{k,n}$ and $f\in\mathcal{S}/I$ satisfy
$\operatorname*{coeff}\nolimits_{\omega}\left(  \overline{s_{\nu}}f\right)
=\operatorname*{coeff}\nolimits_{\nu^{\vee}}\left(  f\right)  $.
\end{theorem}

\subsection{Complete homogeneous symmetric polynomials}

Another basis of $\mathcal{S}/I$ can be obtained from the complete homogeneous
symmetric polynomials. For each $\ell\in\mathbb{N}$ and each $\ell$-tuple
$\nu=\left(  \nu_{1},\nu_{2},\ldots,\nu_{\ell}\right)  \in\mathbb{Z}^{\ell}$
of integers, we define the symmetric polynomial $h_{\nu}\in\mathcal{S}$ by
$h_{\nu}=h_{\nu_{1}}h_{\nu_{2}}\cdots h_{\nu_{\ell}}$. Thus, in particular,
$h_{\nu}$ is defined for any partition $\nu$.

\begin{theorem}
\label{thm.S/J-h-basis}The family $\left(  \overline{h_{\lambda}}\right)
_{\lambda\in P_{k,n}}$ is a basis of the $\mathbf{k}$-module $\mathcal{S}/I$.
\end{theorem}

The transfer matrix between the two bases $\left(  \overline{s_{\lambda}%
}\right)  _{\lambda\in P_{k,n}}$ and $\left(  \overline{h_{\lambda}}\right)
_{\lambda\in P_{k,n}}$ does not appear easy to describe. One remarkable result
that helps relating these bases is the following formula:

\begin{proposition}
\label{prop.redh.1}Let $m$ be a positive integer. Then,%
\[
h_{n+m}\equiv\sum_{j=0}^{k-1}\left(  -1\right)  ^{j}a_{k-j}s_{\left(
m,1^{j}\right)  }\operatorname{mod}I\text{,}%
\]
where $\left(  m,1^{j}\right)  $ stands for the partition $\left(
m,\underbrace{1,1,\ldots,1}_{j\text{ ones}}\right)  $.
% (whose Young diagram is a hook).
\end{proposition}

Let us return to the basis $\left(  \overline{s_{\lambda}}\right)
_{\lambda\in P_{k,n}}$ of \textquotedblleft abstract Schubert
classes\textquotedblright. A natural task to attempt when faced with such a
basis is finding a Pieri rule -- i.e., a formula for expanding a product
$\overline{s_{\lambda}h_{j}}$ in this basis, where $\lambda\in P_{k,n}$ and
$j\in\left\{  0,1,\ldots,n-k\right\}  $. Such a rule indeed exists.
To state it, we shall use the notation
$c_{\alpha,\beta}^{\gamma}$ for the
Littlewood--Richardson coefficients (see, e.g.,
\cite[Definition 2.5.8]{GriRei18} for their definition).
Our Pieri rule now says the following:

\begin{theorem}
\label{thm.pieri-hall-LR}Let $\lambda\in P_{k,n}$. Let $j\in\left\{
0,1,\ldots,n-k\right\}  $. Then,%
\[
\overline{s_{\lambda}h_{j}}=\sum_{\substack{\mu\in P_{k,n};\\\mu/\lambda\text{
is a horizontal }j\text{-strip}}}\overline{s_{\mu}}-\sum_{i=1}^{k}\left(
-1\right)  ^{i}a_{i}\sum_{\substack{\nu\in P_{k,n};\\\nu\subseteq\lambda
}}c_{\left(  n-k-j+1,1^{i-1}\right)  ,\nu}^{\lambda}\overline{s_{\nu}}.
\]
\end{theorem}

Note that the Littlewood--Richardson coefficients $c_{\left(  n-k-j+1,1^{i-1}%
\right)  ,\nu}^{\lambda}$ are nonnegative, but may be $>1$. For example, if
$n=7$ and $k=3$, then%
\begin{align*}
\overline{s_{\left(  4,3,2\right)  }h_{2}}  &  =\overline{s_{\left(
4,4,3\right)  }}+a_{1}\left(  \overline{s_{\left(  4,2\right)  }}%
+\overline{s_{\left(  3,2,1\right)  }}+\overline{s_{\left(  3,3\right)  }%
}\right)  -a_{2}\left(  \overline{s_{\left(  4,1\right)  }}+\overline
{s_{\left(  2,2,1\right)  }}+\overline{s_{\left(  3,1,1\right)  }}%
+2\overline{s_{\left(  3,2\right)  }}\right) \\
&  \ \ \ \ \ \ \ \ \ \ +a_{3}\left(  \overline{s_{\left(  2,2\right)  }%
}+\overline{s_{\left(  2,1,1\right)  }}+\overline{s_{\left(  3,1\right)  }%
}\right)  .
\end{align*}


Note that Theorem \ref{thm.pieri-hall-LR} generalizes \cite[(22)]{BeCiFu99}.
Also, in the cases $j=1$ and $j=k-1$, Theorem \ref{thm.pieri-hall-LR} takes
the following simpler forms:

\begin{proposition}
\label{prop.pieri-h1}Let $\lambda\in P_{k,n}$. Assume that $k>0$.

\textbf{(a)} If $\lambda_{1}<n-k$, then%
\[
\overline{s_{\lambda}h_{1}}=\sum_{\substack{\mu\in P_{k,n};\\\mu/\lambda\text{
is a single box}}}\overline{s_{\mu}}.
\]


\textbf{(b)} Let $\overline{\lambda}$ be the partition $\left(  \lambda
_{2},\lambda_{3},\lambda_{4},\ldots\right)  $. If $\lambda_{1}=n-k$, then%
\[
\overline{s_{\lambda}h_{1}}=\sum_{\substack{\mu\in P_{k,n};\\\mu/\lambda\text{
is a single box}}}\overline{s_{\mu}}+\sum_{i=0}^{k-1}\left(  -1\right)
^{i}a_{1+i}\sum_{\substack{\mu\in P_{k,n};\\\overline{\lambda}/\mu\text{ is a
vertical }i\text{-strip}}}\overline{s_{\mu}}.
\]

\end{proposition}

\begin{proposition}
\label{prop.pieri-hlast}Let $\lambda\in P_{k,n}$. Assume that $k>0$.

\textbf{(a)} We have%
\[
\overline{s_{\lambda}h_{n-k}}=\overline{s_{\left(  n-k,\lambda_{1},\lambda
_{2},\lambda_{3},\ldots\right)  }}-\sum_{i=1}^{k}\left(  -1\right)  ^{i}%
a_{i}\sum_{\substack{\mu\in P_{k,n};\\\lambda/\mu\text{ is a vertical
}i\text{-strip}}}\overline{s_{\mu}}.
\]


\textbf{(b)} If $\lambda_{k}>0$, then%
\[
\overline{s_{\lambda}h_{n-k}}=-\sum_{i=1}^{k}\left(  -1\right)  ^{i}a_{i}%
\sum_{\substack{\mu\in P_{k,n};\\\lambda/\mu\text{ is a vertical
}i\text{-strip}}}\overline{s_{\mu}}.
\]

\end{proposition}

Pieri rules for multiplication by $\overline{e_{j}}$ (instead of
$\overline{h_{j}}$) appear to be more complicated: For example, for $n=5$ and
$k=3$, we have%
\[
\overline{s_{\left(  2,2,1\right)  }e_{2}}=a_{1}\overline{s_{\left(
2,2\right)  }}-2a_{2}\overline{s_{\left(  2,1\right)  }}+a_{3}\left(
\overline{s_{\left(  2\right)  }}+\overline{s_{\left(  1,1\right)  }}\right)
+a_{1}^{2}\overline{s_{\left(  1\right)  }}-2a_{1}a_{2}\overline{s_{\left(
{}\right)  }}.
\]

We also have shown a ``straightening law'' that writes an
$\overline{s_\mu}$ with $\mu \notin P_{k, n}$ as a linear combination
of ``smaller'' Schur polynomials (i.e., of $\overline{s_\nu}$ with
$\left|\nu\right| < \left|\mu\right|$); this generalizes the ``rim hook algorithm''
of \cite{BeCiFu99}. We refer to \cite[\S 10]{Grinbe18} for the details.

\subsection{Positivity?}

The structure constants of the $\mathbb{Z}\left[  q\right]  $-algebra
$\operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  $ are polynomials in the indeterminate $q$, whose
coefficients are the famous Gromov-Witten invariants $C_{\lambda\mu\nu}^{d}$.
These Gromov-Witten invariants $C_{\lambda\mu\nu}^{d}$ are nonnegative
integers (as follows from their geometric interpretation, but also from the
\textquotedblleft Quantum Littlewood-Richardson Rule\textquotedblright%
\ \cite[Theorem 2]{BKPT16}). This appears to generalize to the general case of
$\mathcal{S}/I$:

\begin{conjecture}
\label{conj.pos}Let $b_{i}=\left(  -1\right)  ^{n-k-1}a_{i}$ for each
$i\in\left\{  1,2,\ldots,k\right\}  $. Let $\lambda,\mu,\nu\in P_{k,n}$. Then,
$\left(  -1\right)  ^{\left\vert \lambda\right\vert +\left\vert \mu\right\vert
-\left\vert \nu\right\vert }\operatorname*{coeff}\nolimits_{\nu}\left(
\overline{s_{\lambda}s_{\mu}}\right)  $ is a polynomial in $b_{1},b_{2}%
,\ldots,b_{k}$ with coefficients in $\mathbb{N}$.
\end{conjecture}

We have verified this conjecture for all $n\leq8$ using SageMath.

\subsection{Deforming symmetric functions}

We have so far studied a quotient $\mathcal{S}/I$ of the ring $\mathcal{S}$ of
symmetric polynomials in $k$ variables $x_{1},x_{2},\ldots,x_{k}$. But
$\mathcal{S}$ itself is a quotient of a larger ring -- the ring $\Lambda$ of
symmetric functions in infinitely many variables. Let us briefly introduce
$\Lambda$ and then state a version of Theorem \ref{thm.S/I} for this larger
ring (requiring, however, that $a_{1},a_{2},\ldots,a_{k}$ lie in $\mathbf{k}$
rather than in $\mathcal{S}$).

Let $\Lambda$ be the ring of symmetric functions in infinitely many
indeterminates $x_{1},x_{2},x_{3},\ldots$ over $\mathbf{k}$, defined as in
\cite[Chapter 2]{GriRei18}. We shall use boldfaced notations for symmetric
functions in $\Lambda$ in order to distinguish them from symmetric polynomials
in $\mathcal{S}$. In particular:

\begin{itemize}
\item For any $i\in\mathbb{Z}$, we let $\mathbf{h}_{i}$ be the $i$-th complete
homogeneous symmetric function in $\Lambda$. (This is called $h_{i}$ in
\cite[Definition 2.2.1]{GriRei18}.)

\item For any $i\in\mathbb{Z}$, we let $\mathbf{e}_{i}$ be the $i$-th
elementary symmetric function in $\Lambda$. (This is called $e_{i}$ in
\cite[Definition 2.2.1]{GriRei18}.)

\item For any partition $\lambda$, we let $\mathbf{s}_{\lambda}$ be the
corresponding Schur function in $\Lambda$. (This is called $s_{\lambda}$ in
\cite[Definition 2.2.1]{GriRei18}.)
\end{itemize}

If $\mathbf{f}\in\Lambda$ is a symmetric function, then $\mathbf{f}\left(
x_{1},x_{2},\ldots,x_{k}\right)  $ is a symmetric polynomial in $\mathcal{S}$;
the map%
\[
\Lambda\rightarrow\mathcal{S},\qquad\mathbf{f}\mapsto\mathbf{f}\left(
x_{1},x_{2},\ldots,x_{k}\right)
\]
is a surjective $\mathbf{k}$-algebra homomorphism. The kernel of this
homomorphism is the ideal $\left\langle \mathbf{e}_{k+1},\mathbf{e}%
_{k+2},\mathbf{e}_{k+3},\ldots\right\rangle $ of $\Lambda$ (where the notation
$\left\langle \cdot\right\rangle $ means the ideal generated by whatever is
inside the brackets). Hence, $\mathcal{S}\cong\Lambda/\left\langle
\mathbf{e}_{k+1},\mathbf{e}_{k+2},\mathbf{e}_{k+3},\ldots\right\rangle $, so
that%
\[
\mathcal{S}/I\cong\Lambda/\left(  \left\langle \mathbf{h}_{n-k+1}%
-a_{1},\mathbf{h}_{n-k+2}-a_{2},\ldots,\mathbf{h}_{n}-a_{k}\right\rangle
+\left\langle \mathbf{e}_{k+1},\mathbf{e}_{k+2},\mathbf{e}_{k+3}%
,\ldots\right\rangle \right)  .
\]
This suggests a further generalization: What if we replace $\mathbf{e}%
_{k+1},\mathbf{e}_{k+2},\mathbf{e}_{k+3},\ldots$ by $\mathbf{e}_{k+1}%
-b_{1},\mathbf{e}_{k+2}-b_{2},\mathbf{e}_{k+3}-b_{3},\ldots$ for some
constants $b_{1},b_{2},b_{3},\ldots$? At least in the case when $a_{1}%
,a_{2},\ldots,a_{k}\in\mathbf{k}$, this generalization still satisfies an
analogue of Theorem \ref{thm.S/I}:

\begin{theorem}
\label{thm.bigquot}Assume that $a_{1},a_{2},\ldots,a_{k}$ as well as
$b_{1},b_{2},b_{3},\ldots$ are elements of $\mathbf{k}$. Let $K$ be the ideal
$\left\langle \mathbf{h}_{n-k+1}-a_{1},\mathbf{h}_{n-k+2}-a_{2},\ldots
,\mathbf{h}_{n}-a_{k}\right\rangle +\left\langle \mathbf{e}_{k+1}%
-b_{1},\mathbf{e}_{k+2}-b_{2},\mathbf{e}_{k+3}-b_{3},\ldots\right\rangle $ of
$\Lambda$. Then, $\Lambda/K$ is a free $\mathbf{k}$-module with basis $\left(
\overline{\mathbf{s}_{\lambda}}\right)  _{\lambda\in P_{k,n}}$.
\end{theorem}

\section{Proof methods}

We shall now give a brief overview of the techniques that enter into the
proofs of the above results.
See \cite{Grinbe18} for the missing details.

\subsection{A fundamental identity}

The first step towards understanding $I$ and $J$ is an identity between
certain polynomials (Lemma \ref{lem.heh-id} below). To state it, we shall use
the notations $h_{m}$ and $e_{m}$ for complete homogeneous symmetric
polynomials and elementary symmetric polynomials in general. Thus, for any
$m\in\mathbb{Z}$ and any $p$ elements $y_{1},y_{2},\ldots,y_{p}$ of a
commutative ring, we set%
\begin{align}
h_{m}\left(  y_{1},y_{2},\ldots,y_{p}\right)   &  =\sum_{1\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{m}\leq p}y_{i_{1}}y_{i_{2}}\cdots y_{i_{m}%
}\ \ \ \ \ \ \ \ \ \ \text{and}\label{eq.def.hm}\\
e_{m}\left(  y_{1},y_{2},\ldots,y_{p}\right)   &  =\sum_{1\leq i_{1}%
<i_{2}<\cdots<i_{m}\leq p}y_{i_{1}}y_{i_{2}}\cdots y_{i_{m}}.
\label{eq.def.em}%
\end{align}
Thus, $h_{0}\left(  y_{1},y_{2},\ldots,y_{p}\right)  =1$ and $e_{0}\left(
y_{1},y_{2},\ldots,y_{p}\right)  =1$. Also, $e_{m}\left(  y_{1},y_{2}%
,\ldots,y_{p}\right)  =0$ for all $m>p$. Also, for any $m<0$, we have
$h_{m}\left(  y_{1},y_{2},\ldots,y_{p}\right)  =0$ and $e_{m}\left(
y_{1},y_{2},\ldots,y_{p}\right)  =0$. Finally, what we have previously called
$h_{m}$ without any arguments can now be rewritten as $h_{m}\left(
x_{1},x_{2},\ldots,x_{k}\right)  $.

\begin{lemma}
\label{lem.heh-id}Let $i\in\left\{  1,2,\ldots,k+1\right\}  $ and
$p\in\mathbb{N}$. Then,%
\[
h_{p}\left(  x_{i},x_{i+1},\ldots,x_{k}\right)  =\sum_{t=0}^{i-1}\left(
-1\right)  ^{t}e_{t}\left(  x_{1},x_{2},\ldots,x_{i-1}\right)  h_{p-t}\left(
x_{1},x_{2},\ldots,x_{k}\right)  .
\]
(If $i=k+1$, then the term $h_{p}\left(  x_{i},x_{i+1},\ldots,x_{k}\right)  $
on the left hand side is understood to be $h_{p}$ of an empty alphabet; this
is $1$ when $p=0$ and $0$ otherwise.)
\end{lemma}

This is easily proven (e.g., using generating functions) and likely
folklore.

\subsection{Proving Theorem \ref{thm.P/J} and Theorem \ref{thm.S/I}}

We refer to \cite[detailed version, \S 3]{Grinbe17} for the definition of a
Gr\"{o}bner basis over $\mathbf{k}$. (When $\mathbf{k}$ is a field, this
definition is classical and appears throughout the literature.) We endow the
set of monomials in $\mathcal{P}$ with a degree-lexicographic term order,
where the variables are ordered by $x_{1}>x_{2}>\cdots>x_{n}$.

\begin{proposition}
\label{prop.J.grobas}The family
\[
\left(  h_{n-k+i}\left(  x_{i},x_{i+1},\ldots,x_{k}\right)  -\sum_{t=0}%
^{i-1}\left(  -1\right)  ^{t}e_{t}\left(  x_{1},x_{2},\ldots,x_{i-1}\right)
a_{i-t}\right)  _{i\in\left\{  1,2,\ldots,k\right\}  }%
\]
is a Gr\"{o}bner basis of the ideal $J$.
\end{proposition}

This is not hard to show using Lemma \ref{lem.heh-id} and Buchberger's first
criterion (see, e.g., \cite[detailed version, Proposition 3.9]{Grinbe17}). See
\cite[Theorem 1.2.7]{Sturmf08} for a similar result. Now, Theorem
\ref{thm.P/J} easily follows from Proposition \ref{prop.J.grobas} using the
Macaulay-Buchberger basis theorem (e.g., \cite[detailed version, Proposition
3.10]{Grinbe17}).

Let us now assume that $a_{1},a_{2},\ldots,a_{k}$ belong to $\mathcal{S}$. A
simple induction argument shows the following:

\begin{lemma}
\label{lem.I.hi-red}Let $i$ be an integer such that $i>n-k$. Then,%
\[
h_{i}\equiv\left(  \text{some symmetric polynomial of degree }<i\right)
\operatorname{mod}I.
\]

\end{lemma}

Using this lemma and the Jacobi-Trudi identity, we can easily obtain:

\begin{lemma}
\label{lem.I.sl-red}Let $P_{k}$ denote the set of all partitions with at most
$k$ parts. Let $\lambda\in P_{k}$ be a partition such that $\lambda\notin
P_{k,n}$. Then,%
\[
s_{\lambda}\equiv\left(  \text{some symmetric polynomial of degree
}<\left\vert \lambda\right\vert \right)  \operatorname{mod}I.
\]

\end{lemma}

On the other hand, a well-known fact from commutative algebra states the following:

\begin{lemma}
\label{lem.freemod-span-basis}Let $M$ be a free $\mathbf{k}$-module with a
finite basis $\left(  b_{s}\right)  _{s\in S}$. Let $\left(  a_{u}\right)
_{u\in U} \in M^U$ be a family that spans $M$. Assume that $\left\vert U\right\vert
=\left\vert S\right\vert $. Then, $\left(  a_{u}\right)  _{u\in U}$ is a basis
of the $\mathbf{k}$-module $M$. (In other words: A spanning family of $M$
whose size equals the size of a basis must itself be a basis, as long as the
sizes are finite.)
\end{lemma}

Now, Theorem \ref{thm.S/I} can be proven by the following strategy: The
family
$\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$ spans the
$\mathbf{k}$-module $\mathcal{S}/I$ (by Lemma \ref{lem.I.sl-red}), while the
family $\left(  x^{\alpha}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha
_{i}<i\text{ for each }i}$ spans the $\mathcal{S}$-module $\mathcal{P}$ (by
Proposition \ref{prop.artin}). Hence, the family $\left(  \overline
{s_{\lambda}x^{\alpha}}\right)  _{\lambda\in P_{k,n};\ \alpha\in\mathbb{N}%
^{k};\ \alpha_{i}<i\text{ for each }i}$ spans the $\mathbf{k}$-module
$\mathcal{P}/J$. But its size is easily seen to be $\dbinom{n}{k}\cdot
k!=n\left(  n-1\right)  \cdots\left(  n-k+1\right)  $, which is exactly the
size of the basis
$\left(  \overline{x^{\alpha}}\right)  _{\alpha\in
\mathbb{N}^{k};\ \alpha_{i}<n-k+i\text{ for each }i}$ of the $\mathbf{k}%
$-module $\mathcal{P}/J$ (see Theorem \ref{thm.P/J}). Hence, Lemma
\ref{lem.freemod-span-basis} shows that the former family is a basis of
$\mathcal{P}/J$ as well. This readily yields Theorem \ref{thm.S/I}.

Theorem \ref{thm.bigquot} is not hard to obtain from Theorem \ref{thm.S/I}
by defining a $\mathbf{k}$-algebra endomorphism $\varphi$ of $\Lambda$ that
sends each $\mathbf{e}_{i}$ to $\mathbf{e}_{i}+b_{i-k}$
(where we set $b_{i-k} = 0$ for all $i \leq k$).
% $%
% \begin{cases}
% \mathbf{e}_{i}, & \text{if }i\leq k;\\
% \mathbf{e}_{i}+b_{i-k}, & \text{if }i>k.
% \end{cases}
% $

\subsection{Proving the $S_{3}$-symmetry}

We have two proofs of Theorem \ref{thm.coeffw}; both require significant
amounts of work. Referring again to \cite{Grinbe18} for the details, we shall
just telegraph the main ingredients of the first proof here. One of them are
the \textit{skewing operators} as defined (e.g.) in \cite[\S 2.8]{GriRei18}.
They have the property that $\left(  \mathbf{s}_{\mu}\right)  ^{\perp}\left(
\mathbf{s}_{\lambda}\right)  =\mathbf{s}_{\lambda/\mu}$ for any two partitions
$\lambda$ and $\mu$, where $\mathbf{s}_{\lambda/\mu}\in\Lambda$ is the skew
Schur function (understood to be $0$ unless $\mu\subseteq\lambda$).
We also recall the two Pieri rules for $\mathbf{e}_{i}$ in $\Lambda$
(\cite[(2.7.2)]{GriRei18} and \cite[version with solutions (ancillary file),
Lemma 12.83.3(b)]{GriRei18}):

\begin{proposition}
\label{prop.pieri.e}Let $\lambda$ be a partition, and let $i\in\mathbb{N}$.
Then, $\mathbf{s}_{\lambda}\mathbf{e}_{i}=\sum\limits_{\substack{\mu\text{ is
a partition;}\\\mu/\lambda\text{ is a vertical }i\text{-strip}}}\mathbf{s}%
_{\mu}$.
\end{proposition}

\begin{corollary}
\label{cor.pieri.eskew}Let $\lambda$ be a partition, and let $i\in\mathbb{N}$.
Then, $\mathbf{e}_{i}^{\perp}\mathbf{s}_{\lambda}=\sum\limits_{\substack{\mu
\text{ is a partition;}\\\lambda/\mu\text{ is a vertical }i\text{-strip}%
}}\mathbf{s}_{\mu}$.
\end{corollary}

An
identity of Bernstein (\cite[Exercise 2.9.1(b)]{GriRei18}) serves a crucial
role as well:

\begin{proposition}
\label{prop.bernstein.1}Let $\lambda$ be a partition. Let $m\in\mathbb{Z}$ be
such that $m\geq\lambda_{1}$. Then,%
\[
\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}\mathbf{h}_{m+i}\mathbf{e}%
_{i}^{\perp}\mathbf{s}_{\lambda}=\mathbf{s}_{\left(  m,\lambda_{1},\lambda
_{2},\lambda_{3},\ldots\right)  }.
\]

\end{proposition}

Proposition \ref{prop.bernstein.1} leads quickly to Proposition
\ref{prop.pieri-hlast}, but also (via Corollary \ref{cor.pieri.eskew})
to the following corollary:

\begin{corollary}
\label{cor.bernstein.2}Let $\lambda$ be a partition with at most $k$ parts.
Let $\overline{\lambda}$ be the partition $\left(  \lambda_{2},\lambda
_{3},\lambda_{4},\ldots\right)  $. Then,%
\[
\mathbf{s}_{\lambda}=\sum_{i=0}^{k-1}\left(  -1\right)  ^{i}\mathbf{h}%
_{\lambda_{1}+i}\sum_{\substack{\mu\text{ is a partition;}\\\overline{\lambda
}/\mu\text{ is a vertical }i\text{-strip}}}\mathbf{s}_{\mu}.
\]

\end{corollary}

Now, for each $p\in\mathbb{Z}$, we let $Q_{p}$ be the $\mathbf{k}$-submodule
of $\mathcal{S}/I$ spanned by the $\overline{s_{\lambda}}$ with $\lambda\in
P_{k,n}$ satisfying $\lambda_{k}\leq p$. Clearly, $Q_{0}\subseteq
Q_{1}\subseteq Q_{2}\subseteq\cdots$ and $\operatorname*{coeff}%
\nolimits_{\omega}\left(  Q_{n-k-1}\right)  =0$.

From Corollary \ref{cor.bernstein.2}, it is not hard to obtain Proposition
\ref{prop.pieri-h1}, but also the following lemma:

\begin{lemma}
\label{lem.coeffw.0}Let $\lambda$ be a partition with at most $k$ parts.
If $\lambda_{1}=n-k+1$, then
$\overline{s_{\lambda}}\in Q_{0}$.
\end{lemma}

Using Proposition \ref{prop.pieri.e} and Lemma \ref{lem.coeffw.0}, we can find:

\begin{lemma}
\label{lem.coeffw.eisl}Let $i\in\mathbb{N}$ and $\lambda\in P_{k,n}$. Then,
$\overline{e_{i}}\overline{s_{\lambda}}\equiv\sum\limits_{\substack{\mu\in
P_{k,n};\\\mu/\lambda\text{ is a vertical }i\text{-strip}}}\overline{s_{\mu}%
}\operatorname{mod}Q_{0}$.
\end{lemma}

Hence:

\begin{lemma}
\label{lem.coeffw.eiQ}Let $i\in\mathbb{N}$ and $p\in\mathbb{Z}$. Then,
$\overline{e_{i}}Q_{p}\subseteq Q_{p+1}$.
\end{lemma}

We furthermore recall a property of the Littlewood--Richardson
coefficients $c_{\mu,\nu}^\lambda$ (part of \cite[Remark
2.5.9]{GriRei18}):

\begin{proposition}
\label{prop.LR.props}Let $\lambda$ and $\mu$ be two partitions.

\textbf{(a)} We have $\mathbf{s}_{\lambda/\mu}=\sum\limits_{\nu\text{ is a
partition}}c_{\mu,\nu}^{\lambda}\mathbf{s}_{\nu}$.

\textbf{(b)} If $\nu$ is a partition, then $c_{\mu,\nu}^{\lambda}=0$ unless
$\left(  \nu\subseteq\lambda\text{ and }\left\vert \mu\right\vert +\left\vert
\nu\right\vert =\left\vert \lambda\right\vert \right)  $.
\end{proposition}

Next, let $\mathcal{Z}$ be the $\mathbf{k}$-submodule of $\Lambda$ with basis
$\left(  \mathbf{s}_{\lambda}\right)  _{\lambda\in P_{k,n}}$. We thus can
define a $\mathbf{k}$-linear map $\delta:\mathcal{Z}\rightarrow\mathcal{S}/I$
by setting%
\[
\delta\left(  \mathbf{s}_{\lambda}\right)  =\overline{s_{\lambda^{\vee}}%
}\ \ \ \ \ \ \ \ \ \ \text{for every }\lambda\in P_{k,n}.
\]
The following is easy to see from Proposition~\ref{prop.LR.props}:

\begin{lemma}
\label{lem.coeffw.Z-invar-skew}We have $\mathbf{f}^{\perp}\left(
\mathcal{Z}\right)  \subseteq\mathcal{Z}$ for each $\mathbf{f}\in\Lambda$.
\end{lemma}

Comparing Corollary \ref{cor.pieri.eskew} with Lemma \ref{lem.coeffw.eisl}, we
easily find:

\begin{lemma}
\label{lem.coeffw.del1}Let $i\in\mathbb{Z}$ and $\mathbf{f}\in\mathcal{Z}$.
Then, $\delta\left(  \mathbf{e}_{i}^{\perp}\mathbf{f}\right)  \equiv
\overline{e_{i}}\delta\left(  \mathbf{f}\right)  \operatorname{mod}Q_{0}$.
\end{lemma}

From this lemma, we obtain by induction (using
Lemmas~\ref{lem.coeffw.eiQ} and \ref{lem.coeffw.Z-invar-skew} as well as
the identity $\left(  \mathbf{fg}\right)
^{\perp}=\mathbf{g}^{\perp}\circ\mathbf{f}^{\perp}$ for skewing operators)
the following fact:

\begin{lemma}
\label{lem.coeffw.del2}Let $p\in\mathbb{N}$. Let $i_{1},i_{2},\ldots,i_{p}%
\in\mathbb{Z}$ and $\mathbf{f}\in\mathcal{Z}$. Then, $\delta\left(  \left(
\mathbf{e}_{i_{1}}\mathbf{e}_{i_{2}}\cdots\mathbf{e}_{i_{p}}\right)  ^{\perp
}\mathbf{f}\right)  \equiv\overline{e_{i_{1}}e_{i_{2}}\cdots e_{i_{p}}}%
\delta\left(  \mathbf{f}\right)  \operatorname{mod}Q_{p-1}$.
\end{lemma}

But if $\lambda\in P_{k,n}$, then the dual Jacobi-Trudi formula can be used to
express $\mathbf{s}_{\lambda}$ as a sum of products of the form $\mathbf{e}%
_{i_{1}}\mathbf{e}_{i_{2}}\cdots\mathbf{e}_{i_{n-k}}$ with $i_{1},i_{2}%
,\ldots,i_{n-k}\in\mathbb{Z}$. Hence, Lemma \ref{lem.coeffw.del2} yields:

\begin{lemma}
\label{lem.coeffw.del3}Let $\lambda\in P_{k,n}$ and $\mathbf{f}\in\mathcal{Z}%
$. Then, $\delta\left(  \left(  \mathbf{s}_{\lambda}\right)  ^{\perp
}\mathbf{f}\right)  \equiv\overline{s_{\lambda}}\delta\left(  \mathbf{f}%
\right)  \operatorname{mod}Q_{n-k-1}$ and thus $\operatorname*{coeff}%
\nolimits_{\omega}\left(  \delta\left(  \left(  \mathbf{s}_{\lambda}\right)
^{\perp}\mathbf{f}\right)  \right)  =\operatorname*{coeff}\nolimits_{\omega
}\left(  \overline{s_{\lambda}}\delta\left(  \mathbf{f}\right)  \right)  $
(since $\operatorname*{coeff}\nolimits_{\omega}\left(  Q_{n-k-1}\right)  =0$).
\end{lemma}

Fixing $\mu\in P_{k,n}$ and applying Lemma \ref{lem.coeffw.del3} to
$\mathbf{f}=\mathbf{s}_{\mu^{\vee}}$, we obtain (after some work):

\begin{lemma}
\label{lem.coeffw.prod2}Let $\lambda\in P_{k,n}$ and $\mu\in P_{k,n}$. Then,
$\operatorname*{coeff}\nolimits_{\omega}\left(  \overline{s_{\lambda}s_{\mu}%
}\right)  =\operatorname*{coeff}\nolimits_{\omega}\left(  \delta\left(
\left(  \mathbf{s}_{\lambda}\right)  ^{\perp}\mathbf{s}_{\mu^{\vee}}\right)
\right)  =\operatorname*{coeff}\nolimits_{\omega}\left(  \delta\left(
\mathbf{s}_{\mu^{\vee}/\lambda}\right)  \right)  =%
\begin{cases}
1, & \text{if }\lambda=\mu^{\vee};\\
0, & \text{if }\lambda\neq\mu^{\vee}%
\end{cases}
$.
\end{lemma}

This quickly yields Theorem \ref{thm.coeffw}. Theorem \ref{cor.S3sym}, in
turn, easily follows from Theorem \ref{thm.coeffw}.

\section*{Acknowledgments}

We thank Alex Postnikov, Victor Reiner, Mark Shimozono, Josh Swanson, Kaisa
Taipale, and Anders Thorup for enlightening conversations. Discovery was
greatly aided by the SageMath CAS \cite{SageMath}.

\begin{thebibliography}{99}                                                                                               %


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\end{thebibliography}


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