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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}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 $\left( \overline{x^{\alpha}}\right) _{\alpha\in\mathbb{N}%
^{k};\ \alpha_{i}x_{2}>\cdots>x_k$.
\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}