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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{5 December 2018 \\
University of Connecticut, Storrs}

\frame{\titlepage
\textbf{slides: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/uconn2018.pdf}}
\newline\textbf{paper: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/basisquot.pdf}} \newline}

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

\begin{itemize}
\item From a modern point of view, \textbf{Schubert calculus} 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).
\end{itemize}
\end{frame}

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

\begin{itemize}
\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*}
\pause


\item Many properties from classical cohomology still hold. 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, 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}.

\item 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{\ \ \ \ \ A more general setting: $\mathcal{P}$ and $\mathcal{S}$}

\begin{itemize}
\item We will now 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 Let $\mathbf{k}$ be a commutative ring. Let $\mathbb{N}=\left\{
0,1,2,\ldots\right\}  $. Let $n\geq k\geq0$ be integers. \pause


\item Let $\mathcal{P}=\mathbf{k}\left[  x_{1},x_{2},\ldots,x_{k}\right]  $.
\pause


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


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

\item Let $\mathcal{S}$ denote the ring of \textbf{symmetric} polynomials in
$\mathcal{P}$. \pause


\item \textbf{Theorem (Artin }$\leq$\textbf{1944):} The $\mathcal{S}$-module
$\mathcal{P}$ is free with basis
\[
\left(  x^{\alpha}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha_{i}<i\text{ for
each }i}.
\]

\end{itemize}
\end{frame}

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

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


% \item For each $\alpha\in\mathbb{N}^{k}$, we let $\left\vert \alpha\right\vert
% =\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k}$. \pause


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


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


\pause


\item \textbf{Theorem (G.):} The $\mathbf{k}$-module $\mathcal{P}\diagup J$ is
free with basis
\[
\left(  \overline{x^{\alpha}}\right)  _{\alpha\in\mathbb{N}^{k};\ \alpha
_{i}<n-k+i\text{ for each }i},
\]
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 $I$ 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 For each partition $\lambda$, let $s_{\lambda}\in\mathcal{S}$ be the
corresponding Schur polynomial. \pause


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


\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 several that we know:

\begin{itemize}
\item 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 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 $\operatorname*{coeff}\nolimits_{\mu
}:\mathcal{S}\diagup I\rightarrow\mathbf{k}$ send each element to its
$\overline{s_{\mu}}$-coordinate wrt this basis. \pause


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


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


\pause


\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 $\left(  m,1^{j}\right)  :=(m,\underbrace{1,1,\ldots,1}_{j\text{ ones}%
})$ (a hook-shaped partition).
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ A Pieri rule}

\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 Bertram/Ciocan-Fontanine/Fulton Pieri rule, but
note that $c_{\left(  n-k-j+1,1^{i-1}\right)  ,\nu}^{\lambda}$ may be $>1$.

\pause


\item Example:%
\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:%
\[
\overline{s_{\left(  2,2,1\right)  }e_{2}}=a_{1}\overline{s_{\left(
2,2\right)  }}-2a_{2}\overline{s_{\left(  2,1\right)  }}+a_{3}\left(
\overline{s_{\left(  2\right)  }}+\overline{s_{\left(  1,1\right)  }}\right)
+a_{1}^{2}\overline{s_{\left(  1\right)  }}-2a_{1}a_{2}\overline{s_{\left(
{}\right)  }}.
\]

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ 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 $n\leq7$ using SageMath.

\item This would generalize positivity of Gromov--Witten invariants.
\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 other bases does $\mathcal{S}\diagup I$ have?
Monomial symmetric? Power-sum?

\pause


\item \textbf{Question:} Do other properties of $\operatorname*{QH}%
\nolimits^{\ast}\left(  \operatorname*{Gr}\left(  k,n\right)  \right)  $ (such
as \textquotedblleft curious duality\textquotedblright\ and \textquotedblleft
cyclic hidden symmetry\textquotedblright) generalize to $\mathcal{S}\diagup I$?

{\footnotesize \grey (The $\Gr(k, n) \to \Gr(n-k, n)$
duality isomorphism does not
exist in general: If $\mathbf{k}=\mathbb{C}$ and $a_{1}=6$ and $a_{2}=16$,
then
$\left(\mathcal{S}\diagup I\right)_{k = 2, \  n = 3,\ a_1 = 6,\ a_2 = 16}
\cong\mathbb{C}\left[  x\right]  /\left(  \left(
x-10\right)  \left(  x+2\right)  ^{2}\right)  $, which can never be a
$\left(\mathcal{S}\diagup I\right)_{k = 1, \  n = 3}$,
since
$\left(\mathcal{S}\diagup I\right)_{k = 1, \  n = 3} \cong
\mathbb{C}\left[x\right] / \left(x^3 - a_1\right)$.) }

\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 Fulton's \textquotedblleft universal Schubert
polynomials\textquotedblright?

\pause


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

\pause

\item \textbf{Question:} \textquotedblleft Straightening
rules\textquotedblright\ for $\overline{s_{\lambda}}$ when $\lambda\notin
P_{k,n}$, similar to the Bertram/{Ciocan-Fontanine/Fulton \textquotedblleft
rim hook algorithm\textquotedblright?}
\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\}  ;\\
\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) 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{\ \ \ \ \ On the proofs, 2: the Gr\"obner basis argument}

\begin{itemize}
\item The Gr\"{o}bner basis argument relies on the easy 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, $x_{a .. b}$ means $x_{a},x_{a+1},\ldots,x_{b}$.

\item Use it 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 term
order, where the variables are ordered by $x_{1}>x_{2}>\cdots>x_k$.

\item This Gr\"{o}bner basis leads to a basis of $\mathcal{P}\diagup J$, which
is precisely our $\left(  \overline{x^{\alpha}}\right)  _{\alpha\in
\mathbb{N}^{k};\ \alpha_{i}<n-k+i\text{ for each }i}$.
\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, $\mathbf{f}^{\perp}\mathbf{g}$ 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{Tom Roby} for the invitation.

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

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

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


\end{document}