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\begin{document}

\author{Darij Grinberg (Drexel University)}
\title{Three questions on symmetric group algebras}
\date{2024-06-04, Harvard University, Cambridge, MA}

\frame{\titlepage\textbf{slides:} {\color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/harvard2024.pdf}}}

\begin{frame}
\frametitle{\ \ \ \ Are Specht modules pure?}

\begin{itemize}
\item Let $\mathcal{A}=\mathbf{k}\left[  S_{n}\right]  $ be the group algebra
of the symmetric group $S_{n}$ (aka $\mathfrak{S}_{n}$) over a commutative
ring $\mathbf{k}$.

\item Let $D$ be a diagram with $n$ cells. For instance, for $n=9$, we can
have%
\[
D_{9}=
\begin{tikzpicture}[scale=0.5]
\draw[fill=red!50] (1, 0) rectangle (2, 1);
\draw[fill=red!50] (0, 1) rectangle (1, 2);
\draw[fill=red!50] (1, 1) rectangle (2, 2);
\draw[fill=red!50] (0, 1) rectangle (-1, 0);
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\draw[fill=red!50] (4, 0) rectangle (5, 1);
\draw[fill=red!50] (5, 1) rectangle (6, 2);
\end{tikzpicture}
\ \ .
\]


\item Let $\mathcal{S}^{D}$ be its Specht module.
\only<1-3>{This is a left $\mathcal{A}$-module
defined in any of the following equivalent ways:

\begin{itemize}
\only<1>{
\item as the span of polytabloids
\[
\mathbf{e}_{T}=\sum_{\substack{w \in S_n\text{ preserves}\\\text{the columns of }%
T}}\left(  -1\right)  ^{w}
\underbrace{\overline{wT}}_{\substack{\text{This means the} \\ \text{tabloid of $wT$.}}}
\]
within the Young module $\mathcal{M}^{D}$ (free $\mathbf{k}$-module on tabloids);}

\only<2>{
\item as the left ideal of $\mathcal{A}$ generated by $\mathbf{N}%
_{T}\mathbf{P}_{T}$, where $T$ is any filling of $D$ with the numbers
$1,2,\ldots,n$, and where%
\[
\hspace{-2pc}
\mathbf{P}_{T}=\sum_{\substack{w \in S_n \text{ preserves}\\\text{the rows of }%
T}}w\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ \mathbf{N}_{T}%
=\sum_{\substack{w \in S_n \text{ preserves}\\\text{the columns of }T}}\left(
-1\right)  ^{w}w;
\]
}

\only<3>{
\item as the span of certain determinants in a polynomial ring (many options here).}
\end{itemize}
}

\pause \pause \pause
\item \textbf{Question.} Is $\mathcal{S}^{D}$ a direct addend of the Young
module $\mathcal{M}^{D}$ as a $\mathbf{k}$-module?

\only<4>{
\item Note that proving this for $\mathbf{k}=\mathbb{Z}$ would suffice.

\item \textbf{Equivalent question.} If $\mathbf{k}$ is a finite field, is
$\dim_{\mathbf{k}}\mathcal{S}^{D}$ independent on $\mathbf{k}$ ?

\item \textbf{Better hope:} Does $\mathcal{S}^{D}$ have a combinatorially
meaningful basis?
}

\only<5>{
\item Well-known positive answer when $D$ is a skew Young diagram (Garnir's
standard basis theorem).

\item I think the answer is still positive when $D$ is row-convex
(Reiner/Shimozono 1993).

\item Same questions exist for Schur and Weyl modules (over
$\operatorname*{GL}\nolimits_{n}$), but not sure if still equivalent.}
\end{itemize}

\vspace{9pc}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ The Gelfand--Tsetlin subalgebra}

\begin{itemize}
\item Let $\mathbf{m}_{k}:=t_{1,k}+t_{2,k}+\cdots+t_{k-1,k}$ be the $k$-th
\emph{Young--Jucys--Murphy element} for each $k\in\left[  n\right]  $ (where
$t_{i,j}$ means the transposition $i\leftrightarrow j$).

\item The $\mathbf{k}$-subalgebra of $\mathcal{A}=\mathbf{k}\left[
S_{n}\right]  $ generated by $\mathbf{m}_1, \mathbf{m}_2, \ldots,
\mathbf{m}_n$ is commutative, and known as the \emph{Gelfand--Tsetlin
algebra}.

\item \textbf{Question.} Is it free as a $\mathbf{k}$-module? (dimension = \#
of involutions = \# of straight-shaped standard tableaux.)

\item Again, proving it for $\mathbf{k}=\mathbb{Z}$ is enough.

\item True for $n\leq6$.

\item Well-known positive answer for $\mathbf{k}=\mathbb{Q}$ (explicit basis:
the diagonal vectors $e_{T,T}$ of the seminormal basis of $\mathbf{k}\left[
S_{n}\right]  $).

\item Partial result: For all $i_{1}<i_{2}<\cdots<i_{k}$, we have%
\[
\mathbf{m}_{i_{1}}\mathbf{m}_{i_{2}}\cdots\mathbf{m}_{i_{k}}=\sum
_{\substack{w\in S_{n};\\\operatorname*{NoSt}w=\left\{  i_{1},i_{2}%
,\ldots,i_{k}\right\}  }}w.
\]

\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ The simplest-looking open question you'll see today}

\begin{itemize}
\item For any permutation $w\in S_{n}$, define%
\begin{align*}
\operatorname*{exc}w &  :=\left(  \text{\# of }i\in\left[  n\right]  \text{
such that }w\left(  i\right)  >i\right)  \ \ \ \ \ \ \ \ \ \ \text{and}\\
\operatorname*{anxc}w &  :=\left(  \text{\# of }i\in\left[  n\right]  \text{
such that }w\left(  i\right)  <i\right)  .
\end{align*}


\item For any $a,b\in\mathbb{N}$, define%
\[
\mathbf{X}_{a,b}:=\sum_{\substack{w\in S_{n};\\\operatorname*{exc}%
w=a;\\\operatorname*{anxc}w=b}}w\in\mathbf{k}\left[  S_{n}\right]  .
\]


\item \textbf{Conjecture.} These elements $\mathbf{X}_{a,b}$ for all
$a,b\in\mathbb{N}$ commute (for fixed $n$). In other words, $\mathbf{X}%
_{a,b}\mathbf{X}_{c,d}=\mathbf{X}_{c,d}\mathbf{X}_{a,b}$ for all
$a,b,c,d\in\mathbb{N}$.

\item Checked for all $n\leq7$.

\item This generalizes a limiting case of the Bethe subalgebra (Mukhin/Tarasov/Varchenko).
\end{itemize}
\end{frame}


\end{document}