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\begin{document}

\title{A quotient of the ring of symmetric functions generalizing quantum cohomology}
\author{\href{http://www.cip.ifi.lmu.de/~grinberg/}{Darij Grinberg}}
\date{1 March 2019 \\
University of Minnesota, Minneapolis, MN}

\frame{\titlepage\textbf{slides: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/umn2019.pdf}}
\newline\textbf{paper: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/basisquot.pdf}}
\newline\textbf{overview: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/fpsac19.pdf}} \newline}

\begin{frame}
\frametitle{\ \ \ \ \ What is this about?}

\begin{itemize}
\item From a modern point of view, \textbf{Schubert calculus} (a.k.a.
classical enumerative geometry, or Hilbert's 15th problem) is about two
cohomology rings:
\[
\operatorname*{H}\nolimits^{\ast}\left(  \underbrace{\operatorname*{Gr}\left(
k,n\right)  }_{\text{Grassmannian}}\right)  \text{ and }\operatorname*{H}%
\nolimits^{\ast}\left(  \underbrace{\operatorname*{Fl}\left(  n\right)
}_{\text{flag variety}}\right)
\]
(both varieties over $\mathbb{C}$). \pause


\item In this talk, we are concerned with the first. \pause


\item Classical result: as rings,%
\begin{align*}
&  \operatorname*{H}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  \\
&  \cong\left(  \text{symmetric polynomials in }x_{1},x_{2},\ldots,x_{k}\text{
over }\mathbb{Z}\right)  \\
&  \qquad\diagup\left(  h_{n-k+1},h_{n-k+2},\ldots,h_{n}\right)
_{\operatorname*{ideal}},
\end{align*}
where the $h_{i}$ are complete homogeneous symmetric polynomials (to be
defined soon).
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Quantum cohomology of $\Gr(k, n)$}

\begin{itemize}
% \only<2>{\item For
% comparison, the \textbf{classical cohomology} of the Grassmannian is
% \begin{align*}
% &  \operatorname*{H}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
% k,n\right)  \right)  \\
% &  \cong\left(  \text{symmetric polynomials in }x_{1},x_{2},\ldots,x_{k}\text{
% over }\mathbb{Z}\right)
% \phantom{bla}
% \\
% &  \qquad\diagup\left(  h_{n-k+1},h_{n-k+2},\ldots,h_{n}\right)
% _{\operatorname*{ideal}}.
% \end{align*}
% }
% \only<1->{
\item (Small) \textbf{Quantum cohomology}
is a deformation of cohomology from the
1980--90s. For the Grassmannian, it is
\begin{align*}
&  \operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  \\
&  \cong\left(  \text{symmetric polynomials in }x_{1},x_{2},\ldots,x_{k}\text{
over }\mathbb{Z}\left[  q\right]  \right)  \\
&  \qquad\diagup\left(  h_{n-k+1},h_{n-k+2},\ldots,h_{n-1},h_{n}+\left(
-1\right)  ^{k}q\right)  _{\operatorname*{ideal}}.
\end{align*}
% }
% \only<2>{\item So $\QHo^*\tup{\Gr\tup{k,n}}$ differs from
% $\Ho^*\tup{\Gr\tup{k,n}}$ only in the last generator
% of the ideal (which is now $h_n + \tup{-1}^k q$, not
% $h_n$).}

\pause
\item Many properties of classical cohomology still hold here.
\\
In particular:
$\operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  $ has a $\mathbb{Z}\left[  q\right]  $-module basis
$\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$ of (projected)
Schur polynomials (to be defined soon), with $\lambda$ ranging over all
partitions with $\leq k$ parts and each part $\leq n-k$. The structure
constants are the \textbf{Gromov--Witten invariants}.
References:

\begin{itemize}
\item {\color{red} \href{https://doi.org/10.1006/jabr.1999.7960}{Aaron
Bertram, Ionut Ciocan-Fontanine, William Fulton, \textit{Quantum
multiplication of Schur polynomials}, 1999}}.

\item {\color{red}
\href{https://math.mit.edu/~apost/papers/affine_approach.pdf}{Alexander
Postnikov, \textit{Affine approach to quantum Schubert calculus}, 2005}}.
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Where are we going?}

\begin{itemize}
\item \textbf{Goal:} Deform $\operatorname*{H}\nolimits^{\ast}\left(
\operatorname*{Gr}\left(  k,n\right)  \right)  $ using $k$ parameters instead
of one, generalizing $\operatorname*{QH}\nolimits^{\ast}\left(
\operatorname*{Gr}\left(  k,n\right)  \right)  $. \pause

\item The new ring has no geometric interpretation known so far,
but various properties suggesting such an interpretation likely exists.
\pause

\item I will now start from scratch and define standard notations around
symmetric polynomials, then introduce the deformed cohomology ring
algebraically.
\pause

\item {\red There is a number of open questions and things to explore.}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ A more general setting: $\mathcal{P}$ and $\mathcal{S}$}

\begin{itemize}
\item Let \defn{$\mathbf{k}$} be a commutative ring. \\
Let $\defnm{\mathbb{N}} = \left\{ 0,1,2,\ldots\right\}  $. % \\
Let $\defnm{k} \in \NN$. %Let $n\geq k\geq0$ be integers.
\pause

\item Let $\defnm{\mathcal{P}} = \mathbf{k}\left[  x_{1},x_{2},\ldots,x_{k}\right]  $
be the polynomial ring in $k$ indeterminates over $\kk$. \pause

\item For each $k$-tuple $\alpha\in\mathbb{N}^{k}$ and each $i\in\left\{  1,2,\ldots
,k\right\}  $, let \defn{$\alpha_i$} be the $i$-th entry of $\alpha$.
Same for infinite sequences. \pause

\item For each $\alpha\in\mathbb{N}^{k}$, let \defn{$x^{\alpha}$} be the monomial
$x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{k}^{\alpha_{k}}$,
and let \defn{$\abs{\alpha}$} be the degree $\alpha_1 + \alpha_2 + \cdots + \alpha_k$
of this monomial. \pause

\item Let \defn{$\mathcal{S}$} denote the ring of \defn{symmetric} polynomials in
$\mathcal{P}$. \\
These are the polynomials $f \in \mathcal{P}$ satisfying
\[
 f \tup{x_1, x_2, \ldots, x_k}
 = f \tup{x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(k)}}
\]
for all permutations $\sigma$ of $\set{1, 2, \ldots, k}$.
\pause

\item \textbf{Theorem (Artin $\leq$1944):} The $\mathcal{S}$-module
$\mathcal{P}$ is free with basis
\[
\left(  x^{\alpha}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha_{i}<i\text{ for
each }i}
\qquad \text{(or, informally: ``$\tup{x_1^{<1} x_2^{<2} \cdots x_k^{<k}}$'')}.
\]
\pause
\textbf{Example:} For $k = 3$, this basis is
$\tup{ 1, x_3, x_3^2, x_2, x_2 x_3, x_2 x_3^2 }$.

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric polynomials}

\begin{itemize}
\item The ring $\mathcal{S}$ of symmetric polynomials in $\mathcal{P}%
=\mathbf{k}\left[  x_{1},x_{2},\ldots,x_{k}\right]  $ has several bases,
usually indexed by certain sets of (integer) partitions.

First, let us recall what partitions are:
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ $k$-partitions: definition}

\begin{itemize}
\only<1-2>{%
\item A \defn{partition} means a \textbf{weakly decreasing} sequence of
nonnegative integers that has only finitely many nonzero entries. \\}
\only<2>{
\textbf{Examples:} $\tup{4, 2, 2, 0, 0, 0, \ldots}$ and $\tup{3, 2, 0, 0, 0, 0, \ldots}$ and $\tup{5, 0, 0, 0, 0, 0, \ldots}$
are three partitions. \\
$\tup{2, 3, 2, 0, 0, 0, \ldots}$ and $\tup{2, 1, 1, 1, \ldots}$ are not.
}%
\only<3->{%
\item A \defn{$k$-partition} means a \textbf{weakly decreasing} $k$-tuple
$\tup{\lambda_1, \lambda_2, \ldots, \lambda_k} \in \NN^k$. \\}
\only<4->{
\textbf{Examples:} $\tup{4, 2, 2}$ and $\tup{3, 2, 0}$ and $\tup{5, 0, 0}$
are three $3$-partitions. \\
$\tup{2, 3, 2}$ is not.
}

\pause \pause \pause \pause
\only<5>{
\item Thus there is a bijection
\begin{align*}
 \set{\text{$k$-partitions}} &\to \set{\text{partitions with at most $k$ nonzero entries}} , \\
 \lambda &\mapsto \tup{\lambda_1, \lambda_2, \ldots, \lambda_k, 0, 0, 0, \ldots} .
\end{align*}
}

\pause

\item If $\lambda \in \NN^k$ is a $k$-partition, then its
\defn{Young diagram $Y\tup{\lambda}$} is defined as a table made out
of $k$ left-aligned rows,
where the $i$-th row has $\lambda_i$ boxes.
\\
\textbf{Example:} If $k = 6$ and $\lambda = \tup{5, 5, 3, 2, 0, 0}$, then
\[
 Y \tup{\lambda} = \ydiagram{5, 5, 3, 2} \quad .
\]
(Empty rows are invisible.)

\pause

\item The same convention applies to partitions.
\end{itemize}

\vspace{9cm}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric polynomials: the $e$-basis}

\begin{itemize}
\item For each $m\in\mathbb{Z}$, we let $e_{m}$ denote the $m$-th
\defn{elementary symmetric polynomial}:%
\[
\defnm{e_m}
=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{m}\leq k}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}=\sum_{\substack{\alpha\in\left\{  0,1\right\}  ^{k};\\\left\vert
\alpha\right\vert =m}}x^{\alpha}\in\mathcal{S}.
\]
(Thus, $e_{0}=1$, and $e_{m}=0$ when $m<0$.) \pause

\item For each $\nu=\left(  \nu_{1},\nu_{2},\ldots,\nu_{\ell}\right)
\in\mathbb{Z}^{\ell}$ (e.g., a $k$-partition when $\ell = k$), set%
\[
\defnm{e_{\nu}}
=e_{\nu_{1}}e_{\nu_{2}}\cdots e_{\nu_{\ell}}\in\mathcal{S}.
\]


\pause

\item \textbf{Theorem (Gauss):} The commutative $\kk$-algebra $\calS$ is freely
generated by the elementary symmetric polynomials $e_1, e_2, \ldots, e_k$.
(That is, it is generated by them, and they are algebraically independent.)

\pause

\item \textbf{Equivalent restatement:}
$\left(  e_{\lambda}\right)  _{\lambda \text{ is a partition whose entries
are $\leq k$}}$
is a basis of the $\mathbf{k}$-module $\mathcal{S}$.

\pause

\item Note that $e_{m}=0$ when $m>k$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric polynomials: the $h$-bases}

\begin{itemize}
\item For each $m\in\mathbb{Z}$, we let $h_{m}$ denote the $m$-th
\defn{complete homogeneous symmetric polynomial}:
\[
\defnm{h_{m}}
=\sum_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{m}\leq k}x_{i_{1}}x_{i_{2}%
}\cdots x_{i_{m}}=\sum_{\substack{\alpha\in\mathbb{N}^{k};\\\left\vert
\alpha\right\vert =m}}x^{\alpha}\in\mathcal{S}.
\]
(Thus, $h_{0}=1$, and $h_{m}=0$ when $m<0$.) \pause


\item For each $\nu=\left(  \nu_{1},\nu_{2},\ldots,\nu_{\ell}\right)
\in\mathbb{Z}^{\ell}$ (e.g., a $k$-partition when $\ell = k$), set%
\[
\defnm{h_{\nu}}=h_{\nu_{1}}h_{\nu_{2}}\cdots h_{\nu_{\ell}}\in\mathcal{S}.
\]


\pause


\item \textbf{Theorem:} The commutative $\kk$-algebra $\calS$ is freely
generated by the complete homogeneous
symmetric polynomials $h_1, h_2, \ldots, h_k$.
% (That is, it is generated by them, and they are algebraically independent.)

\pause

\item \textbf{Equivalent restatement:}
$\left( h_{\lambda}\right)  _{\lambda \text{ is a partition whose entries
are $\leq k$}}$
is a basis of the $\mathbf{k}$-module $\mathcal{S}$.

\pause
\item \textbf{Theorem:} $\left(  h_{\lambda}\right)  _{\lambda\text{ is a
$k$-partition}}$ is a basis of the $\mathbf{k}$-module
$\mathcal{S}$.
(Another basis!)
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric polynomials: the $s$-basis (Schur polynomials)}

\begin{itemize}
\item For each $k$-partition $\lambda$, we let
$s_{\lambda}$ be the \defn{$\lambda$-th Schur polynomial}:
\begin{align*}
\defnm{s_{\lambda}}
&  =
\dfrac{\det\left(  \left(  x_{i}^{\lambda_{j}+k-j}\right)  _{1\leq
i\leq k,\ 1\leq j\leq k}\right)  }{\det\left(  \left(  x_{i}^{k-j}\right)
_{1\leq i\leq k,\ 1\leq j\leq k}\right)  }\qquad\left(  \text{alternant
formula}\right)  \\
&  =\det\left(  \left(  h_{\lambda_{i}-i+j}\right)  _{1\leq i\leq k,
\ 1\leq j\leq k  }\right)
\qquad\left(  \text{Jacobi-Trudi}\right)  .
\end{align*}

\pause

\item \textbf{Theorem:} The equality above holds, and
$s_\lambda$ is a symmetric polynomial with
nonnegative coefficients.
\only<2>{
Explicitly, \vspace{-0.5pc}
\[
s_\lambda =
\sum_{\substack{T\text{ is a semistandard $\lambda$-tableau}%
\\\text{with entries }1,2,\ldots,k}}\ \ \prod_{i=1}^{k}x_{i}^{\left(
\text{number of }i\text{'s in }T\right)  } ,
\]
where a \defn{semistandard $\lambda$-tableau with entries $1,2,\ldots,k$}
is a way of putting an integer $i \in \set{1, 2, \ldots, k}$ into each box of
$Y\tup{\lambda}$ such that the entries \textbf{weakly} increase along
rows and \textbf{strictly} increase along columns. 
}

\pause

\item \textbf{Theorem:} $\left(  s_{\lambda}\right)  _{\lambda\text{ is a
$k$-partition}}$ is a basis of the $\mathbf{k}$-module
$\mathcal{S}$.
\end{itemize}
\vspace{9cm}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric polynomials: Littlewood-Richardson coefficients}

\begin{itemize}
\item If $\lambda$ and $\mu$ are two $k$-partitions, then the product
$s_\lambda s_\mu$ can be again written as a $\kk$-linear combination of
Schur polynomials (since these form a basis):
\[
s_\lambda s_\mu
= \sum_{\nu \text{ is a $k$-partition}} \defnm{c^\nu_{\lambda, \mu}} s_\nu ,
\]
where the $c^\nu_{\lambda, \mu}$ lie in $\kk$.
These $c^\nu_{\lambda, \mu}$ are called the \defn{Littlewood-Richardson
coefficients}.

\pause

\item \textbf{Theorem:} These Littlewood-Richardson coefficients
$c^\nu_{\lambda, \mu}$ are nonnegative integers (and count something).
\end{itemize}

\vspace{9cm}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric polynomials: Schur polynomials for non-partitions}

\begin{itemize}
\item We have defined
\begin{align*}
\defnm{s_{\lambda}}
&  =\det\left(  \left(  h_{\lambda_{i}-i+j}\right)  _{1\leq i\leq k,
\ 1\leq j\leq k  }\right)
\end{align*}
for $k$-partitions $\lambda$. \\
Apply the same definition to arbitrary $\lambda \in \ZZ^k$.
\pause

\item \textbf{Proposition:}
If $\alpha \in \ZZ^k$, then $s_\alpha$ is either $0$ or equals
$\pm s_\lambda$ for some $k$-partition $\lambda$.
\only<2>{ \\ (So we get nothing really new.)}
\pause \\
More precisely:
Let $\beta = \tup{\alpha_1 + \tup{k-1},
                  \alpha_2 + \tup{k-2},
                  \ldots,
                  \alpha_k + \tup{k-k}}$.
\pause
\begin{itemize}
\item If $\beta$ has a negative entry, then $s_\alpha = 0$.
\item If $\beta$ has two equal entries, then $s_\alpha = 0$.
\item Otherwise, let $\gamma$ be the $k$-tuple obtained by
	   sorting $\beta$ in decreasing order, and let $\sigma$
	   be the permutation of the indices that causes this
	   sorting.
	   Let $\lambda$ be the $k$-partition
	   $\tup{\gamma_1 - \tup{k-1},
             \gamma_2 - \tup{k-2},
             \ldots,
             \gamma_k - \tup{k-k}}$.
      Then, $s_\alpha = \tup{-1}^\sigma s_\lambda$. 
\end{itemize}

\pause
\item Also, the alternant formula still holds if
all $\lambda_i + \tup{k-i}$ are $\geq 0$.
\end{itemize}

\vspace{9cm}

\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ A more general setting: $a_1, a_2, \ldots, a_k$ and $J$}

\begin{itemize}
\item Pick any integer $\defnm{n} \geq k$.

\pause

\item Let $\defnm{a_{1},a_{2},\ldots,a_{k}} \in \mathcal{P}$ such that $\deg
a_{i}<n-k+i$ for all $i$. (For example, this holds if $a_{i}\in\mathbf{k}$.)
\pause

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


\pause


\item \textbf{Theorem (G.):} The $\mathbf{k}$-module $\mathcal{P}\diagup J$ is
free with basis
\begin{align*}
& \left(  \overline{x^{\alpha}}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha
_{i}<n-k+i\text{ for each }i} \\
& \qquad \tup{\text{informally: ``$\tup{\overline{x_1^{<n-k+1} x_2^{<n-k+2} \cdots x_n^{<n}}}$''}} 
\end{align*}
where the overline $\overline{\ \vphantom{g}\ \ }$ means \textquotedblleft
projection\textquotedblright\ onto whatever quotient we need (here: from
$\mathcal{P}$ onto $\mathcal{P}\diagup J$).

(This basis has $n\left(  n-1\right)  \cdots\left(  n-k+1\right)  $ elements.)
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ A slightly less general setting: symmetric $a_1, a_2, \ldots, a_k$ and $J$}

\begin{itemize}
\item \textbf{FROM NOW ON, assume that} $a_{1},a_{2},\ldots,a_{k}%
\in\mathcal{S}$. \pause


\item Let \defn{$I$} be the ideal of $\mathcal{S}$ generated by the $k$ differences%
\[
h_{n-k+1}-a_{1},\ \ h_{n-k+2}-a_{2},\ \ \ldots,\ \ h_{n}-a_{k}.
\]
(Same differences as for $J$, but we are generating an ideal of $\mathcal{S}$
now.) \pause


\item Let
$\defnm{\omega} = \underbrace{\left(  n-k,n-k,\ldots,n-k\right)  }_{k\text{
entries}}$
and
\begin{align*}
\defnm{P_{k,n}}  &  =\left\{  \lambda\text{ is a $k$-partition }\mid\ \lambda_{1}\leq
n-k \right\} \\
&  =\left\{  \text{$k$-partitions }\lambda\subseteq\omega\right\}  .
\end{align*}

\item Here, for two $k$-partitions $\alpha$ and $\beta$, we say that
\defn{$\alpha \subseteq \beta$} if and only if $\alpha_i \leq \beta_i$
for all $i$.


\item \textbf{Theorem (G.):} The $\mathbf{k}$-module $\mathcal{S}\diagup I$ is
free with basis
\[
\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}.
\]

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ An even less general setting: constant $a_1, a_2, \ldots, a_k$}

\begin{itemize}
\item \textbf{FROM NOW ON, assume that} $a_{1},a_{2},\ldots,a_{k}\in
\mathbf{k}$. \pause


\item This setting still is general enough to encompass ...

\begin{itemize}
\item \textbf{classical cohomology:}
If $\mathbf{k}=\mathbb{Z}$ and $a_{1}=a_{2}=\cdots=a_{k}=0$, then
$\mathcal{S}\diagup I$ becomes the cohomology ring $\operatorname*{H}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $; the
basis $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$
corresponds to the Schubert classes.

\item \textbf{quantum cohomology:}
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 $\mathcal{S}%
\diagup I$ becomes the quantum cohomology ring $\operatorname*{QH}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $.
\pause

\end{itemize}

\item The above theorem lets us work in these rings (and more generally)
without relying on geometry.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ $S_3$-symmetry of the Gromov--Witten invariants}

\begin{itemize}
\item Recall that $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in
P_{k,n}}$ is a basis of the $\mathbf{k}$-module $\mathcal{S}\diagup I$.
\pause


For each $\mu\in P_{k,n}$, let $\defnm{\operatorname*{coeff}\nolimits_{\mu
}}:\mathcal{S}\diagup I\rightarrow\mathbf{k}$ send each element to its
$\overline{s_{\mu}}$-coordinate wrt this basis. \pause


\item For every $k$-partition $\nu=\left(  \nu_{1},\nu_{2},\ldots,\nu_{k}\right)
\in P_{k,n}$, we define%
\[
\defnm{\nu^{\vee}}
:=\left(  n-k-\nu_{k},n-k-\nu_{k-1},\ldots,n-k-\nu_{1}\right)  \in
P_{k,n}.
\]
This $k$-partition $\nu^{\vee}$ is called the \defn{complement} of $\nu$.
\pause


\item For any three $k$-partitions $\alpha,\beta,\gamma\in P_{k,n}$, let
\[
\defnm{g_{\alpha,\beta,\gamma}}
:=\operatorname*{coeff}\nolimits_{\gamma^{\vee}}\left(
\overline{s_{\alpha}}\overline{s_{\beta}}\right)  \in\mathbf{k}.
\]
These generalize the Littlewood--Richardson coefficients and (3-point)
Gromov--Witten invariants. \pause


\item \textbf{Theorem (G.):} For any $\alpha,\beta,\gamma\in P_{k,n}$, we have%
\begin{align*}
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{align*}


\pause


\item \textbf{Equivalent restatement:} Each $\nu\in P_{k,n}$ and
$f\in\mathcal{S}\diagup I$ satisfy $\operatorname*{coeff}\nolimits_{\omega
}\left(  \overline{s_{\nu}}f\right)  =\operatorname*{coeff}\nolimits_{\nu
^{\vee}}\left(  f\right)  $.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ The $h$-basis}

\begin{itemize}
\item \textbf{Theorem (G.):} The $\mathbf{k}$-module $\mathcal{S}\diagup I$ is
free with basis
\[
\left(  \overline{h_{\lambda}}\right)  _{\lambda\in P_{k,n}}.
\]


\pause


\item The transfer matrix between the two bases $\left(  \overline{s_{\lambda
}}\right)  _{\lambda\in P_{k,n}}$ and $\left(  \overline{h_{\lambda}}\right)
_{\lambda\in P_{k,n}}$ is unitriangular wrt the \textquotedblleft
size-then-anti-dominance\textquotedblright\ order, but seems hard to describe.
\pause


\item \textbf{Proposition (G.):} Let $m$ be a positive integer. Then,%
\[
\overline{h_{n+m}}=\sum_{j=0}^{k-1}\left(  -1\right)  ^{j}a_{k-j}%
\overline{s_{\left(  m,1^{j}\right)  }},
\]
where $\left(  m,1^{j}\right)  :=(m,\underbrace{1,1,\ldots,1}_{j\text{ ones}%
},0,0,0,\ldots)$ (a hook-shaped $k$-partition).
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ The Pieri rule for symmetric polynomials}

\begin{itemize}
\item If $\alpha$ and $\beta$ are two $k$-partitions, then we say that
\defn{$\alpha \diagup \beta$ is a horizontal strip} if and only if
the Young diagram $Y\tup{\alpha}$ is obtained from $Y\tup{\beta}$
by adding some (possibly none) extra boxes with no two of these new
boxes lying in the same column.
\only<1>{
\\ \textbf{Example:}
If $k = 4$ and $\alpha = \tup{5,3,2,1}$ and $\beta = \tup{3,2,2,0}$,
then $\alpha \diagup \beta$ is a horizontal strip, since
\[
Y\tup{\beta}
= \ydiagram{3,2,2,0} \subseteq
\begin{ytableau}
\ & \ & \ & X & X \\
\ & \  & X \\
\ & \\
X
\end{ytableau}
= Y\tup{\alpha}
\]
with no two $X$'s in the same column.
}
\pause

\item Equivalently, $\alpha \diagup \beta$ is a horizontal strip
if and only if
\[
\alpha_1 \geq \beta_1 \geq \alpha_2 \geq \beta_2 \geq \alpha_3
\geq \cdots \geq \alpha_k \geq \beta_k .
\]

\pause

\item Furthermore, given $j \in \NN$, we say that
\defn{$\alpha \diagup \beta$ is a horizontal $j$-strip}
if $\alpha \diagup \beta$ is a horizontal strip and
$\abs{\alpha} - \abs{\beta} = j$.

\pause

\item \textbf{Theorem (Pieri).}
Let $\lambda$ be a $k$-partition.
Let $j \in \NN$.
Then,
\[
s_{\lambda}h_{j}
= \sum_{\substack{\mu\text{ is a $k$-partition};\\\mu\diagup
\lambda\text{ is a}\\\text{horizontal }j\text{-strip}}}
s_\mu .
\]

\end{itemize}

\vspace{9cm}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ A Pieri rule for $\calS \diagup I$}

\begin{itemize}
\item \textbf{Theorem (G.):} Let $\lambda\in P_{k,n}$. Let $j\in\left\{
0,1,\ldots,n-k\right\}  $. Then,%
\[
\overline{s_{\lambda}h_{j}}=\sum_{\substack{\mu\in P_{k,n};\\\mu\diagup
\lambda\text{ is a}\\\text{horizontal }j\text{-strip}}}\overline{s_{\mu}}%
-\sum_{i=1}^{k}\left(  -1\right)  ^{i}a_{i}\sum_{\nu\subseteq\lambda
}c_{\left(  n-k-j+1,1^{i-1}\right)  ,\nu}^{\lambda} \overline{s_{\nu}}.
\]
% where $c_{\alpha,\beta}^{\gamma}$ are the usual Littlewood--Richardson coefficients.

\pause


\item This generalizes the h-Pieri rule from
{\href{https://doi.org/10.1006/jabr.1999.7960}{Bertram, Ciocan-Fontanine and
Fulton}}, but note that $c_{\left(  n-k-j+1,1^{i-1}\right)  ,\nu}^{\lambda}$
may be $>1$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ A Pieri rule for $\calS \diagup I$: example}

\begin{itemize}

\item \textbf{Example:}
For $n = 7$ and $k = 3$, we have
\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*}


\pause


\item Multiplying by $e_{j}$ appears harder:
For $n = 5$ and $k = 3$, we have
\[
\overline{s_{\left(  2,2,1\right)  }e_{2}}
= a_{1}\overline{s_{\left(
2,2\right)  }}-2a_{2}\overline{s_{\left(  2,1\right)  }}+a_{3}\left(
\overline{s_{\left(  2\right)  }}+\overline{s_{\left(  1,1\right)  }}\right)
+a_{1}^{2}\overline{s_{\left(  1\right)  }}-2a_{1}a_{2}\overline{s_{\left(
{}\right)  }}.
\]

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ A ``rim hook algorithm''}

\begin{itemize}
\item For $\operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Gr}\left(
k,n\right)  \right)  $,
{\href{https://doi.org/10.1006/jabr.1999.7960}{Bertram, Ciocan-Fontanine and
Fulton} give a \textquotedblleft rim hook algorithm\textquotedblright\ that
rewrites an arbitrary }$\overline{s_{\mu}}$ as $\left(  -1\right)
^{\text{something}}q^{\text{something}}\overline{s_{\lambda}}$ with
$\lambda\in P_{k,n}$.

Is there such a thing for $\mathcal{S}\diagup I$? \pause


If $n=6$ and $k=3$, then%
\[
\overline{s_{\left(  4,4,3\right)  }}=a_{2}^{2}\overline{s_{\left(  1\right)
}}-2a_{1}a_{2}\overline{s_{\left(  2\right)  }}+a_{1}^{2}\overline{s_{\left(
3\right)  }}+a_{3}\overline{s_{\left(  3,2\right)  }}-a_{2}\overline
{s_{\left(  3,3\right)  }}.
\]
\only<2>{Looks hopeless...} %
\pause

\item \textbf{Theorem (G.):} Let $\mu$ be a $k$-partition with $\mu_{1}>n-k$.
Let
\begin{align*}
W  & =\left\{ \lambda = \left(  \lambda_{1},\lambda_{2},\ldots,\lambda_{k}\right) \in \ZZ^k
\ \mid\ \lambda_{1}=\mu_{1}-n\right.  \\
& \qquad\left.  \text{and }\lambda_{i}-\mu_{i}\in\left\{  0,1\right\}  \text{
for all }i\in\left\{  2,3,\ldots,k\right\}  \right\} .
% \subseteq \NN^k .
\end{align*}
\only<3>{(Not all elements of $W$ are $k$-partitions, but all belong to $\ZZ^{k}$,
so we know how to define $s_\lambda$ for them.)}%
\pause
Then,
\[
\overline{s_\mu}
= \sum_{j=1}^k \tup{-1}^{k-j} a_j
\sum_{\substack{\lambda \in W; \\ \abs{\lambda} = \abs{\mu} - \tup{n-k+j}}} \overline{s_\lambda} .
\]

% \pause

% Note that this is only one step of the reduction;
% the $\lambda \in W$ may not be in $P_{k,n}$ even after
% ``normalization''.

% Note: There are no rim hooks visible in this rule!
% But the definition of $W$ might remind you of rim hook
% deletion (specifically, deleting the rim hook starting in
% row $1$ if it exists) and vertical strip addition
% (specifically, inserting all possible vertical strips).
% Nevertheless, the $k$-partitions obtained from the set $W$
% (even after we remove the ones that give $0$ and reorder
% the rest into actual $k$-partitions) are NOT the ones
% obtained from $\mu$ by removing a rim hook and then adding
% a vertical strip, and are also NOT the ones obtained from
% $\mu$ by adding a vertical strip and then removing a rim
% hook. You have to somehow add the strip and remove the
% rim hook "simultaneously", whatever this means (I don't
% have a formal way of defining this, except by defining
% the set $W$).

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Positivity?}

\begin{itemize}
\item \textbf{Conjecture:} 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 $\mathbb{N}$.

\item Verified for all $n\leq 8$ using SageMath.

\item This would generalize positivity of Gromov--Witten invariants.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Other bases?}

\begin{itemize}
\item \textbf{Theorem (G.):} The $\mathbf{k}$-module $\mathcal{S}\diagup I$ is
free with basis
\[
\left(  \overline{m_{\lambda}}\right)  _{\lambda\in P_{k,n}},
\]
where
\begin{align*}
\defnm{m_\lambda}
&= \left(\text{the sum of all distinct permutations of }\right. \\
& \qquad \left. \text{ the monomial }
x_1^{\lambda_1} x_2^{\lambda_2} \cdots x_k^{\lambda_k} \right)
\end{align*}
is a \defn{monomial symmetric polynomial}.
%\newline
%Here, \defn{$\ell\tup{\lambda}$} means the number of nonempty
%entries of $\lambda$.

\pause

\item What are the structure constants?

\pause

\item The family
$\left(  \overline{p_{\lambda}}\right)  _{\lambda\in P_{k,n}}$
built of the power-sum symmetric functions $p_\lambda$ is not
generally a basis (not even if $\kk = \QQ$ and $a_i = 0$).

\pause

\item What about other bases? Forgotten symmetric functions?

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ More questions}

\begin{itemize}
\item \textbf{Question:} Does $\mathcal{S}\diagup I$ have a geometric meaning?
If not, why does it behave so nicely?

\pause


\item \textbf{Question:} What if we replace the generators
$h_{n-k+i} - a_i$ of our ideals by $p_{n-k+i} - a_i$ ?

\pause


\item \textbf{Question:} Do other properties of $\operatorname*{QH}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $
generalize to $\mathcal{S}\diagup I$?

\pause Computations show that Postnikov's ``curious duality'' and
``cyclic hidden symmetry'' and
\href{https://doi.org/10.1006/jabr.1999.7960}{Bertram et al's}
$\Gr\tup{k,n} \leftrightarrow \Gr\tup{n-k,n}$ duality do not generalize
(at least not in any straightforward way).

\pause


\item \textbf{Question:} Is there an analogous generalization of
$\operatorname*{QH}\nolimits^{\ast}\left(  \operatorname*{Fl}\left(  n\right)
\right)  $ ? Is it connected to
\href{https://arxiv.org/abs/alg-geom/9702012}{Fulton's
``universal Schubert polynomials''?}

\pause


\item \textbf{Question:} Is there an equivariant analogue?

\pause

\item \textbf{Question:} What about quotients of the quasisymmetric
polynomials?

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ $S_k$-module structure}

\begin{itemize}
\item The symmetric group $S_{k}$ acts on $\mathcal{P}$, with invariant ring
$\mathcal{S}$.

\item What is the $S_{k}$-module structure on $\mathcal{P}\diagup J$ ?

\pause


\item \textbf{Almost-theorem (G., needs to be checked):} Assume that
$\mathbf{k}$ is a $\mathbb{Q}$-algebra. Then, as $S_{k}$-modules,
\[
\mathcal{P}\diagup J\cong\left(  \mathcal{P}\diagup\mathcal{PS}^{+}\right)
^{\times\dbinom{n}{k}}\cong\left(  \underbrace{\mathbf{k}S_{k}}_{\text{regular
rep}}\right)  ^{\times\dbinom{n}{k}},
\]
where $\mathcal{PS}^{+}$ is the ideal of $\mathcal{P}$ generated by symmetric
polynomials with constant term $0$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Deforming symmetric functions, 1}

\begin{itemize}
\item Let us recall symmetric \textbf{functions} (not polynomials) now; we'll
need them soon anyway.
\begin{align*}
\mathcal{S}  &  :=\left\{  \text{symmetric polynomials in }x_{1},x_{2}%
,\ldots,x_{k}\right\}  ;\\
\defnm{\Lambda} &  :=\left\{  \text{symmetric functions in }x_{1},x_{2},x_{3}%
,\ldots\right\}  .
\end{align*}


\pause


\item We use standard notations for symmetric functions, but in boldface:%
\begin{align*}
\mathbf{e}  &  =\text{elementary symmetric,}\\
\mathbf{h}  &  =\text{complete homogeneous,}\\
\mathbf{s}  &  =\text{Schur (or skew Schur).}%
\end{align*}


\pause


\item We have%
\begin{align*}
\mathcal{S}  &  \cong\Lambda\diagup\left(  \mathbf{e}_{k+1},\ \ \mathbf{e}%
_{k+2},\ \ \mathbf{e}_{k+3},\ \ \ldots\right)  _{\operatorname*{ideal}}%
,\qquad\text{thus}\\
\mathcal{S}\diagup I  &  \cong\Lambda\diagup\left(  \mathbf{h}_{n-k+1}%
-a_{1},\ \ \mathbf{h}_{n-k+2}-a_{2},\ \ \ldots,\ \ \mathbf{h}_{n}%
-a_{k},\right. \\
&  \qquad\qquad\left.  \mathbf{e}_{k+1},\ \ \mathbf{e}_{k+2},\ \ \mathbf{e}%
_{k+3},\ \ \ldots\right)  _{\operatorname*{ideal}}.
\end{align*}


\pause


\item So why not replace the $\mathbf{e}_{j}$ by $\mathbf{e}_{j}-b_{j}$ too?
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Deforming symmetric functions, 2}

\begin{itemize}
\item \textbf{Theorem (G.):} Assume that $a_{1},a_{2},\ldots,a_{k}$ as well as
$b_{1},b_{2},b_{3},\ldots$ are elements of $\mathbf{k}$. Then,%
\begin{align*}
&  \Lambda\diagup\left(  \mathbf{h}_{n-k+1}-a_{1},\ \ \mathbf{h}_{n-k+2}%
-a_{2},\ \ \ldots,\ \ \mathbf{h}_{n}-a_{k},\right. \\
&  \qquad\qquad\left.  \mathbf{e}_{k+1}-b_{1},\ \ \mathbf{e}_{k+2}%
-b_{2},\ \ \mathbf{e}_{k+3}-b_{3},\ \ \ldots\right)  _{\operatorname*{ideal}}%
\end{align*}
is a free $\mathbf{k}$-module with basis $\left(  \overline{\mathbf{s}%
_{\lambda}}\right)  _{\lambda\in P_{k,n}}$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ On the proofs, 1}

\begin{itemize}
\item Proofs of all the above (except for the $S_{k}$-action and the
$\overline{m_{\lambda}}$-basis) can be found in

\begin{itemize}
\item {\color{red} Darij Grinberg, \textit{A basis for a quotient of symmetric
polynomials (draft)},
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/basisquot.pdf}} .
\end{itemize}

\pause


\item \textbf{Main ideas:}

\begin{itemize}
\item Use Gr\"{o}bner bases to show that $\mathcal{P}\diagup J$ is free with
basis $\left(  \overline{x^{\alpha}}\right)  _{\alpha\in\mathbb{N}%
^{k};\ \alpha_{i}<n-k+i\text{ for each }i}$.

(This was already outlined in {\color{red}
\href{http://www.numdam.org/article/RSMUP_2009__121__179_0.pdf}{Aldo Conca,
Christian Krattenthaler, Junzo Watanabe, \textit{Regular Sequences of
Symmetric Polynomials}, 2009}}.)

\pause


\item Using that + Jacobi--Trudi, show that $\mathcal{S}\diagup I$ is free
with basis $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$.

\pause


\item As for the rest, compute in $\Lambda$... a lot.
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Gr\"obner bases, 1: the degree-lexicographic order}

\begin{itemize}
\only<1-2>{
\item A brief introduction to Gr\"obner bases is appropriate here.
\pause
\item Gr\"obner bases are ``particularly uncomplicated''
      generating sets for ideals in polynomial rings. \\
      (But take the word ``basis'' with a grain of salt
      -- they can have redundant elements, for example.)
}
\pause

\item A \defn{monomial order} is a total order on the
monomials in $\calP$ with the properties that
\begin{itemize}
\item $1 \leq \mathfrak{m}$ for each monomial $\mathfrak{m}$;
\item $\mathfrak{a} \leq \mathfrak{b}$ implies
      $\mathfrak{a} \mathfrak{m} \leq \mathfrak{b} \mathfrak{m}$
      for any monomials $\mathfrak{a}, \mathfrak{b}, \mathfrak{m}$;
\item the order is well-founded (i.e., we can do induction
      over it).
\end{itemize}
\pause\pause

\item The \defn{degree-lexicographic order} is the monomial order
      defined as follows:
      Two monomials
      $\mathfrak{a} = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_k^{\alpha_k}$
      and
      $\mathfrak{b} = x_1^{\beta_1} x_2^{\beta_2} \cdots x_k^{\beta_k}$
      satisfy $\mathfrak{a} > \mathfrak{b}$
      if and only if
      \begin{itemize}
      \item either $\deg \mathfrak{a} > \deg \mathfrak{b}$
      \item or $\deg \mathfrak{a} = \deg \mathfrak{b}$ and
            the smallest $i$ satisfying $\alpha_i \neq \beta_i$
            satisfies $\alpha_i > \beta_i$.
      \end{itemize}
\pause

\item Given a monomial order,
      \begin{itemize}
      \item each nonzero polynomial $f \in \calP$
      has a well-defined \defn{leading monomial}
      (= the highest monomial appearing in $f$).
      \pause
      \item a polynomial $f$ is called \defn{quasi-monic} if the coefficient
      of its leading term in $f$ is invertible.
      \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Gr\"obner bases, 2: What is a Gr\"obner basis?}

\begin{itemize}
\item If $\calI$ is an ideal of $\calP$, then a
      \defn{Gr\"obner basis} of $\calI$ (for a fixed
      monomial order) means a family
      $\left(f_i\right)_{i \in G}$ of quasi-monic polynomials
      that
      \begin{itemize}
      \item generates $\calI$, and
      \item has the property that the leading monomial of
            any nonzero $f \in \calI$ is divisible by the
            leading monomial of some $f_i$.
      \end{itemize}
      \pause

\item \textbf{Example:} Let $k = 3$, and rename $x_1, x_2, x_3$ as $x, y, z$.
      Use the degree-lexicographic order.
      Let $\calI$ be the ideal generated by $x^2 - yz, y^2 - zx, z^2 - xy$.
      Then: \pause
      \begin{itemize}
      \item The triple $\tup{x^2 - yz, y^2 - zx, z^2 - xy}$ is not
            a Gr\"obner basis of $\calI$, since its leading monomials are
            $x^2, xz, xy$, but the leading term $y^3$ of the
            polynomial $y^3 - z^3 \in \calI$ is not divisible by
            any of them. \pause
      \item The quadruple
            $\left(y^{3} -  z^{3}, x^{2} -  y z, x y -  z^{2}, x z -  y^{2}\right)$
            is a Gr\"obner basis of $\calI$.
            (Thanks SageMath, and whatever packages it uses for this.)
            % P.<x,y,z> = PolynomialRing(QQ, order="deglex")
            % I = P.ideal(x**2 - y*z, y**2 - z*x, z**2 - x*y)
            % latex(I.groebner_basis())
      \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Gr\"obner bases, 3: Buchberger's first criterion}

\begin{itemize}
\item Note: Our definition of Gr\"obner basis is a straightforward
      generalization of the usual one, since $\kk$ may not be a
      field. \\
      Note that some texts use different generalizations!
      \pause

\item \textbf{Theorem (Buchberger's first criterion).}
      Let $\calI$ be an ideal of $\calP$. \\ Let
      $\left(f_i\right)_{i \in G}$ be a family of quasi-monic polynomials
      that generates $\calI$. \\
      Assume that the leading monomials of all the $f_i$
      are mutually coprime (i.e., each indeterminate appears in
      the leading monomial of $f_i$ for at most one $i \in G$). \\
      Then, $\left(f_i\right)_{i \in G}$ is a Gr\"obner basis
      of $\calI$.
      \pause

\item \textbf{Example:} Let $k = 3$, and rename $x_1, x_2, x_3$ as $x, y, z$.
      Use the degree-lexicographic order.
      Let $\calI$ be the ideal generated by $x^3 - yz, y^3 - zx, z^3 - xy$.
      Then, $\tup{x^3 - yz, y^3 - zx, z^3 - xy}$ is a Gr\"obner basis
      of $\calI$, since its leading monomials $x^3, y^3, z^3$
      are mutually coprime.

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Gr\"obner bases, 4: Macaulay's basis theorem}

\begin{itemize}
\item \textbf{Theorem (Macaulay's basis theorem).}
      Let $\calI$ be an ideal of $\calP$ that has a Gr\"obner
      basis $\left(f_i\right)_{i \in G}$.
      A monomial $\mathfrak{m}$ will be called \defn{reduced} if
      it is not divisible by the leading term of any $f_i$.
      Then, the projections of the reduced monomials form
      a basis of the $\kk$-module $\calP \diagup \calI$.
      \pause

\item \textbf{Example:} Let $k = 3$, and rename $x_1, x_2, x_3$ as $x, y, z$.
      Use the degree-lexicographic order.
      Let $\calI$ be the ideal generated by $x^3 - yz, y^3 - zx, z^3 - xy$.
      Then, $\tup{x^3 - yz, y^3 - zx, z^3 - xy}$ is a Gr\"obner basis
      of $\calI$.
      \\
      Thus, $\tup{\overline{x^i y^j z^\ell}}_{i, j, \ell < 3}$ is a basis
      of $\calP \diagup \calI$.
      \pause

\item \textbf{Example:} Let $k = 3$, and rename $x_1, x_2, x_3$ as $x, y, z$.
      Use the degree-lexicographic order.
      Let $\calI$ be the ideal generated by $x^2 - yz, y^2 - zx, z^2 - xy$.
      Then, $\left(y^{3} -  z^{3}, x^{2} -  y z, x y -  z^{2}, x z -  y^{2}\right)$
      is a Gr\"obner basis of $\calI$. \\
      Thus, $\tup{\overline{x}} \cup \tup{\overline{y^j z^\ell}}_{j < 3}$
      is a basis of $\calP \diagup \calI$.

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ On the proofs, 2: the Gr\"obner basis argument}

\begin{itemize}
\item It is easy to prove the identity
\[
h_{p}\left(  x_{i .. k}\right)  =\sum_{t=0}^{i-1}\left(  -1\right)  ^{t}%
e_{t}\left(  x_{1 .. i-1}\right)  h_{p-t}\left(  x_{1 .. k}\right)
\]
for all $i\in\left\{  1,2,\ldots,k+1\right\}  $ and $p\in\mathbb{N}$.

Here, $\defnm{x_{a .. b}}$ means $x_{a},x_{a+1},\ldots,x_{b}$.

\item Use this to show that
\[
\left(  h_{n-k+i}\left(  x_{i .. k}\right)  -\sum_{t=0}^{i-1}\left(
-1\right)  ^{t}e_{t}\left(  x_{1 .. i-1}\right)  a_{i-t}\right)
_{i\in\left\{  1,2,\ldots,k\right\}  }%
\]
is a Gr\"{o}bner basis of the ideal $J$ wrt the degree-lexicographic
order. %, where the variables are ordered by $x_{1}>x_{2}>\cdots>x_k$.

\item Thus, Macaulay's basis theorem shows that
$\left(  \overline{x^{\alpha}}\right)  _{\alpha\in
\mathbb{N}^{k};\ \alpha_{i}<n-k+i\text{ for each }i}$
is a basis of the $\kk$-module $\calP \diagup J$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ On the proofs, 3: the first basis of $\mathcal{S} \diagup I$}

\begin{itemize}
\item How to prove that $\mathcal{S}\diagup I$ is free with basis $\left(
\overline{s_{\lambda}}\right)  _{\lambda\in P_{k,n}}$ ? \pause


\item Jacobi--Trudi lets you recursively reduce each $\overline{s_{\lambda}}$
with $\lambda\notin P_{k,n}$ into smaller $\overline{s_{\mu}}$'s. \pause


$\Longrightarrow$ $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in
P_{k,n}}$ spans $\mathcal{S}\diagup I$. \pause


\item On the other hand, $\left(  x^{\alpha}\right)  _{\alpha\in\mathbb{N}%
^{k};\ \alpha_{i}<i\text{ for each }i}$ spans $\mathcal{P}$ as an
$\mathcal{S}$-module (Artin). \pause


\item Combining these yields that $\left(  \overline{s_{\lambda}x^{\alpha}%
}\right)  _{\lambda\in P_{k,n};\ \alpha\in\mathbb{N}^{k};\ \alpha_{i}<i\text{
for each }i}$ spans $\mathcal{P}\diagup I\mathcal{P}=\mathcal{P}\diagup J$.
\pause


\item But we also know that the family $\left(  \overline{x^{\alpha}}\right)
_{\alpha\in\mathbb{N}^{k};\ \alpha_{i}<n-k+i\text{ for each }i}$ is a basis of
$\mathcal{P}\diagup J$. \pause


\item What can you say if a $\mathbf{k}$-module has a basis $\left(
a_{v}\right)  _{v\in V}$ and a spanning family $\left(  b_{u}\right)  _{u\in
U}$ of the same finite size ($\left\vert U\right\vert =\left\vert V\right\vert
<\infty$)? \pause


Easy exercise: You can say that $\left(  b_{u}\right)  _{u\in U}$ is also a
basis. \pause


\item Thus, $\left(  \overline{s_{\lambda}x^{\alpha}}\right)  _{\lambda\in
P_{k,n};\ \alpha\in\mathbb{N}^{k};\ \alpha_{i}<i\text{ for each }i}$ is a
basis of $\mathcal{P}\diagup J$. \pause


\item $\Longrightarrow$ $\left(  \overline{s_{\lambda}}\right)  _{\lambda\in
P_{k,n}}$ is a basis of $\mathcal{S}\diagup I$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ On the proofs, 4: Bernstein's identity}

\begin{itemize}
\item The rest of the proofs are long computations inside $\Lambda$, using
various identities for symmetric functions. \pause


\item Maybe the most important one:

\textbf{Bernstein's identity:} Let $\lambda$ be a partition. Let
$m\in\mathbb{Z}$ be such that $m\geq\lambda_{1}$. Then,%
\[
\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}\mathbf{h}_{m+i}\left(
\mathbf{e}_{i}\right)  ^{\perp}\mathbf{s}_{\lambda}=\mathbf{s}_{\left(
m,\lambda_{1},\lambda_{2},\lambda_{3},\ldots\right)  }.
\]
Here, $\defnm{\mathbf{f}^{\perp}\mathbf{g}}$
means \textquotedblleft$\mathbf{g}$
skewed by $\mathbf{f}$\textquotedblright\ (so that $\left(  \mathbf{s}_{\mu
}\right)  ^{\perp}\mathbf{s}_{\lambda}=\mathbf{s}_{\lambda\diagup\mu}$).
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Thank you}

\begin{itemize}
\item \textbf{Sasha Postnikov} for the paper that gave rise
to this project 5 years ago.

\item \textbf{Victor Reiner, Tom Roby, Travis Scrimshaw,
Mark Shimozono, Josh Swanson, Kaisa Taipale, and
Anders Thorup} for enlightening discussions.

\item \textbf{you} for your patience.
\end{itemize}
\end{frame}


\end{document}