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\begin{document}
\title{The Petrie symmetrie functions and Murnaghan--Nakayama rules}
\author{\href{http://www.cip.ifi.lmu.de/~grinberg/}{Darij Grinberg}}
\date{4 February 2020\\
Institut Mittag--Leffler, Djursholm, Sweden}
\frame{\titlepage\textbf{slides: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/djursholm2020.pdf}}
\newline\textbf{paper: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/petriesym.pdf}}
\newline\textbf{overview: \color{red}
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/fps20pet.pdf}} \newline}
\begin{frame}
\frametitle{\ \ \ \ \ Manifest}
\begin{itemize}
\item What you are going to see:
\begin{itemize}
\item A new family $\left( G\left( k,m\right) \right) _{m\geq0}$ of
symmetric functions for each $k>0$. (So, a family of families.)
\item It \textquotedblleft interpolates\textquotedblright\ between the $e$'s
and the $h$'s in a sense.
\item Various nice properties if I do say so myself.
\item A proof (sketch) of a conjecture coming from algebraic groups.
\item A source of homework exercises for your symmetric functions
class.\pause
\end{itemize}
\item What you are \textbf{not} going to see:
\begin{itemize}
\item Meaning.
\item Theories.
\item (mostly) actual combinatorics (algorithms, bijections, etc.).
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: notation, 1}
\begin{itemize}
\item We will use standard notations for symmetric functions, such as used in:
\begin{itemize}
\item {\color{red}
\href{https://www.cambridge.org/core/books/enumerative-combinatorics/D8DDDFF7E8EBF0BCFE99F5E6918CE2A8}{Richard
Stanley, \textit{Enumerative Combinatorics, volume 2}, CUP 2001}}.
\item {\color{red}
\href{http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf}{D.G. and
Victor Reiner, \textit{Hopf algebras in Combinatorics}, 2012-2020+}}. \pause
\end{itemize}
\item Let $\mathbf{k}$ be a commutative ring ($\mathbb{Z}$ and $\mathbb{Q}$
will suffice).
\item Let $\mathbb{N}:=\left\{ 0,1,2,\ldots\right\} $.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: notation, 2}
\begin{itemize}
\item A \defn{weak composition} means a sequence $\left( \alpha_{1}%
,\alpha_{2},\alpha_{3},\ldots\right) \in\mathbb{N}^{\infty}$ such that all
$i\gg0$ satisfy $\alpha_{i}=0$.
\item We let \defn{$\operatorname*{WC}$} be the set of all weak compositions.
\item We write \defn{$\alpha_{i}$} for the $i$-th entry of a weak composition
$\alpha$.
\item The \defn{size} of a weak composition $\alpha$ is defined to be
$\left\vert \alpha\right\vert :=\alpha_{1}+\alpha_{2}+\alpha_{3}+\cdots$.
\pause
\item A \defn{partition} means a weak composition $\alpha$ satisfying
$\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}\geq\cdots$.
\item A \defn{partition of $n$} means a partition $\alpha$ with $\left\vert
\alpha\right\vert =n$.
\item We let \defn{$\operatorname*{Par}$} denote the set of all partitions.
For each $n\in\mathbb{Z}$, we let \defn{$\operatorname*{Par}\nolimits_{n}$}
denote the set of all partitions of $n$. \pause
\item We often omit trailing zeroes from partitions: e.g., $\left(
3,2,1,0,0,0,\ldots\right) =\left( 3,2,1\right) =\left( 3,2,1,0\right) $.
\item The partition $\left( 0,0,0,\ldots\right) =\left( {}\right) $ is
called the \defn{empty partition} and denoted by \defn{$\varnothing$}.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: notation, 3}
\begin{itemize}
\item We will use the notation \defn{$m^{k}$} for \textquotedblleft%
$\underbrace{m,m,\ldots,m}_{k\text{ times}}$\textquotedblright\ in partitions.
(For example, $\left( 2,1^{4}\right) =\left( 2,1,1,1,1\right) $.) \pause
\item For any weak composition $\alpha$, we let \defn{$\mathbf{x}^{\alpha}$}
denote the monomial $x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}x_{3}^{\alpha_{3}%
}\cdots$. It has degree $\left\vert \alpha\right\vert $. \pause
\item The ring
\defn{$\mathbf{k}\left[ \left[ x_{1},x_{2},x_{3},\ldots\right]
\right] $} consists of formal infinite $\mathbf{k}$-linear combinations of monomials
$\mathbf{x}^{\alpha}$.
These combinations are called \defn{formal power series}.
\item The \defn{symmetric functions} are the formal power series
$f\in\mathbf{k}\left[ \left[ x_{1},x_{2},x_{3},\ldots\right] \right] $
that are
\begin{itemize}
\item \defn{of bounded degree} (i.e., all monomials in $f$ have degrees
$0$ and $m\in\mathbb{N}%
$)\textbf{:}
\begin{itemize}
\item
\[
G\left( k\right) =\sum_{\substack{\lambda\in\operatorname*{Par}%
;\\\lambda_{i}m$. \pause
\item $G\left( m,m\right) =h_{m}-p_{m}$.
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ Petrie functions and the coproduct of $\Lambda$}
\begin{itemize}
\item This is for the friends of Hopf algebras:%
\[
\Delta\left( G\left( k,m\right) \right) =\sum_{i=0}^{m}G\left(
k,i\right) \otimes G\left( k,m-i\right)
\]
for each $k>0$ and $m\in\mathbb{N}$.
Here, \defn{$\Delta$} is the \defn{comultiplication} of $\Lambda$, defined to
be the $\mathbf{k}$-algebra homomorphism%
\begin{align*}
\Delta:\Lambda & \rightarrow\Lambda\otimes\Lambda,\\
e_{n} & \mapsto\sum_{i=0}^{n}e_{i}\otimes e_{n-i}.
\end{align*}
\pause
\item In terms of alphabets, this says%
\begin{align*}
& \left( G\left( k,m\right) \right) \left( x_{1},x_{2},x_{3}%
,\ldots,y_{1},y_{2},y_{3},\ldots\right) \\
& =\sum_{i=0}^{m}\left( G\left( k,i\right) \right) \left( x_{1}%
,x_{2},x_{3},\ldots\right) \cdot\left( G\left( k,m-i\right) \right)
\left( y_{1},y_{2},y_{3},\ldots\right) .
\end{align*}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ Expanding Petries in the Schur basis}
\begin{itemize}
\item We can expand the $G\left( k,m\right) $ in the Schur basis $\left(
s_{\lambda}\right) _{\lambda\in\operatorname*{Par}}$: e.g.,%
\[
G\left( 4,6\right) =s_{\left( 2,1,1,1,1\right) }-s_{\left( 2,2,1,1\right)
}+s_{\left( 3,3\right) }.
\]
\pause
\item Surprisingly, it turns out that all coefficients are in $\left\{
0,1,-1\right\} $. \pause
\item Better yet: Any product $G\left( k,m\right) \cdot s_{\mu}$ expands in
the Schur basis with coefficients in $\left\{ 0,1,-1\right\} $. \pause
\item Let us see what the coefficients are.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ Petrie numbers}
\begin{itemize}
\item We let $\left[ \mathcal{A}\right] $ denote the \defn{truth value} of a
statement $\mathcal{A}$ (that is, $1$ if $\mathcal{A}$ is true, and $0$ if
$\mathcal{A}$ is false). \pause
\item Let $\lambda=\left( \lambda_{1},\lambda_{2},\ldots,\lambda_{\ell
}\right) \in\operatorname*{Par}$ and $\mu=\left( \mu_{1},\mu_{2},\ldots
,\mu_{\ell}\right) \in\operatorname*{Par}$, and let $k$ be a positive
integer. Then, the
\defn{$k$-Petrie number $\operatorname*{pet}\nolimits_{k}\left( \lambda,\mu\right) $}
of $\lambda$ and $\mu$ is the integer defined by%
\[
\operatorname*{pet}\nolimits_{k}\left( \lambda,\mu\right) =\det\left(
\left( \left[ 0\leq\lambda_{i}-\mu_{j}-i+j{ For example, for $\ell=3$, we have%
\begin{align*}
& \operatorname*{pet}\nolimits_{k}\left( \lambda,\mu\right) \\
& =\det
\resizebox{3in}{!}{$\left( \begin{array} [c]{ccc}\left[ 0\leq\lambda_{1}-\mu_{1}{\textbf{Corollary:} Let $k$ be a positive integer. Then,%
\[
G\left( k\right) =\sum_{\lambda\in\operatorname*{Par}}\operatorname*{pet}%
\nolimits_{k}\left( \lambda,\varnothing\right) s_{\lambda}.
\]
Thus, for each $m\in\mathbb{N}$, we have%
\[
G\left( k,m\right) =\sum_{\lambda\in\operatorname*{Par}\nolimits_{m}%
}\operatorname*{pet}\nolimits_{k}\left( \lambda,\varnothing\right)
s_{\lambda}.
\]
}
\pause One proof of the Theorem uses alternants; the other uses the
\textquotedblleft semi-skew Cauchy identity\textquotedblright%
\begin{align*}
\sum_{\lambda\in\operatorname*{Par}}s_{\lambda}\left( \mathbf{x}\right)
s_{\lambda/\mu}\left( \mathbf{y}\right) & =s_{\mu}\left( \mathbf{x}%
\right) \cdot\prod_{i,j=1}^{\infty}\left( 1-x_{i}y_{j}\right) ^{-1}\\
& =s_{\mu}\left( \mathbf{x}\right) \cdot\sum_{\lambda\in\operatorname*{Par}%
}h_{\lambda}\left( \mathbf{x}\right) m_{\lambda}\left( \mathbf{y}\right)
\end{align*}
(for any $\mu\in\operatorname*{Par}$ and for two sets of indeterminates
$\mathbf{x}=\left( x_{1},x_{2},x_{3},\ldots\right) $ and $\mathbf{y}=\left(
y_{1},y_{2},y_{3},\ldots\right) $).
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ What are the Petrie numbers?}
\begin{itemize}
\item We have shown that $\operatorname*{pet}\nolimits_{k}\left( \lambda
,\mu\right) \in\left\{ 0,1,-1\right\} $, but what exactly is it? \pause
\item
{\color{red}{\href{https://projecteuclid.org/euclid.pjm/1102912464}{Gordon and Wilkinson
1974}}} prove that Petrie matrices have determinants $\in\left\{
0,1,-1\right\} $ by induction. This is little help to us.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ What are the Petrie numbers? The easy case}
\begin{itemize}
\item \textbf{Proposition:} Let $\lambda\in\operatorname*{Par}$ and $k>0$ be
such that $\lambda_{1}\geq k$. Then, $\operatorname*{pet}\nolimits_{k}\left(
\lambda,\varnothing\right) =0$. \pause
\item To get a description in all other cases, recall the definition of \defn{transpose (aka conjugate) partitions}:
Given a partition $\lambda\in\operatorname*{Par}$, we define the
\defn{transpose partition $\lambda^t$} of $\lambda$ to be the partition $\mu$
given by%
\[
\mu_{i}=\left\vert \left\{ j\in\left\{ 1,2,3,\ldots\right\} \ \mid
\ \lambda_{j}\geq i\right\} \right\vert \ \ \ \ \ \ \ \ \ \ \text{for all
}i\geq1.
\]
In terms of Young diagrams, this is just flipping the diagram of $\lambda$
across the diagonal.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ What are the Petrie numbers? Formula for $\operatorname{pet}_k\left(\lambda,\varnothing\right)$}
\begin{itemize}
\item \textbf{Theorem:} Let $\lambda\in\operatorname*{Par}$ and $k>0$ be such
that $\lambda_{1}0$, we define a map $\defnm{\mathbf{f}_{k}}:\Lambda
\rightarrow\Lambda$ by setting%
\[
\mathbf{f}_{k}\left( a\right) =a\left( x_{1}^{k},x_{2}^{k},x_{3}^{k}%
,\ldots\right) \ \ \ \ \ \ \ \ \ \ \text{for each }a\in\Lambda.
\]
This map $\mathbf{f}_{k}$ is called the \defn{$k$-th Frobenius endomorphism}
of $\Lambda$. (Also known as plethysm by $p_{k}$. Perhaps the nicest
plethysm!) \pause
\item \textbf{Theorem:} Let $k$ be a positive integer. Let $m\in\mathbb{N}$.
Then,%
\[
G\left( k,m\right) =\sum_{i\in\mathbb{N}}\left( -1\right) ^{i}%
h_{m-ki}\cdot\mathbf{f}_{k}\left( e_{i}\right) .
\]
\pause
\item \textbf{Theorem:} Fix a positive integer $k$. Assume that $1-k$ is
invertible in $\mathbf{k}$. Then, the family $\left( G\left( k,m\right)
\right) _{m\geq1}=\left( G\left( k,1\right) ,G\left( k,2\right)
,G\left( k,3\right) ,\ldots\right) $ is an algebraically independent
generating set of the commutative $\mathbf{k}$-algebra $\Lambda$.
\item Thus, products of several elements of this family form a basis of
$\Lambda$ (if $1-k$ is invertible in $\mathbf{k}$). These bases remain to be studied.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ The Liu--Polo conjecture}
\begin{itemize}
\item This all begin with the following conjecture
(\href{https://arxiv.org/abs/1908.08432}{Liu and Polo, arXiv:1908.08432}):%
\[
\sum_{\substack{\lambda\in\operatorname*{Par}\nolimits_{2n-1};\\\left(
n-1,n-1,1\right) \triangleright\lambda}}m_{\lambda}=\sum_{i=0}^{n-2}\left(
-1\right) ^{i}s_{\left( n-1,n-1-i,1^{i+1}\right) }\ \ \ \ \ \ \ \text{for
any }n>1.
\]
Here, the symbol $\triangleright$ stands for \defn{dominance} of partitions
(also known as majorization); i.e., for two partitions $\lambda$ and $\mu$, we
have%
\begin{align*}
& \lambda\triangleright\mu\ \ \ \ \text{if and only if}\\
& \left( \lambda_{1}+\lambda_{2}+\cdots+\lambda_{i}\geq\mu_{1}+\mu
_{2}+\cdots+\mu_{i}\text{ for all }i\right) .
\end{align*}
\item Let me briefly outline how this conjecture can be proved.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ The Liu--Polo conjecture, proof: 1}
\begin{itemize}
\item The partitions $\lambda\in\operatorname*{Par}\nolimits_{2n-1}$
satisfying $\left( n-1,n-1,1\right) \triangleright\lambda$ are precisely the
partitions $\lambda\in\operatorname*{Par}\nolimits_{2n-1}$ satisfying
$\lambda_{i}0$ and each $m\in\left\{ 0,1,\ldots,n\right\} $. \pause
Hence,%
\[
\mathbf{B}_{n-1}\left( h_{n}-p_{n}\right) =h_{2n-1}-h_{n-1}p_{n}=G\left(
n,2n-1\right) .
\]
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ The Liu--Polo conjecture, proof: Applying Murnaghan--Nakayama}
\begin{itemize}
\item The Murnaghan--Nakayama rule yields%
\[
p_{n}=\sum_{i=0}^{n-1}\left( -1\right) ^{i}s_{\left( n-i,1^{i}\right) }.
\]
Subtracting this from $h_{n}=s_{\left( n\right) }=s_{\left( n-0,1^{0}%
\right) }$, we find
\[
h_{n}-p_{n}=\sum_{i=0}^{n-2}\left( -1\right) ^{i}s_{\left( n-1-i,1^{i+1}%
\right) }.
\]
Hence,%
\begin{align*}
\mathbf{B}_{n-1}\left( h_{n}-p_{n}\right) & =\sum_{i=0}^{n-2}\left(
-1\right) ^{i}\mathbf{B}_{n-1}\left( s_{\left( n-1-i,1^{i+1}\right)
}\right) \\
& =\sum_{i=0}^{n-2}\left( -1\right) ^{i}s_{\left( n-1,n-1-i,1^{i+1}%
\right) }%
\end{align*}
(by $\mathbf{B}_{m}\left( s_{\lambda}\right) =s_{\left( m,\lambda
_{1},\lambda_{2},\lambda_{3},\ldots\right) }$).
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ The Liu--Polo conjecture, proof: Applying Murnaghan--Nakayama}
\begin{itemize}
\item Since $\mathbf{B}_{n-1}\left( h_{n}-p_{n}\right) =G\left(
n,2n-1\right) $, we now get%
\[
G\left( n,2n-1\right) =\mathbf{B}_{n-1}\left( h_{n}-p_{n}\right)
=\sum_{i=0}^{n-2}\left( -1\right) ^{i}s_{\left( n-1,n-1-i,1^{i+1}\right)
}.
\]
This proves the conjecture from Liu/Polo.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ MNable symmetric functions}
\begin{itemize}
\item Now to something different.
Recall our formula%
\[
G\left( k,m\right) \cdot s_{\mu}=\sum_{\lambda\in\operatorname*{Par}%
\nolimits_{m+\left\vert \mu\right\vert }}\underbrace{\operatorname*{pet}%
\nolimits_{k}\left( \lambda,\mu\right) }_{\in\left\{ 0,1,-1\right\}
}s_{\lambda}.
\]
\pause
\item \textbf{Problem:} What other functions can we replace $G\left(
k,m\right) $ by and still get such a formula?
In other words, what other $f\in\Lambda$ satisfy%
\[
f\cdot s_{\mu}=\sum_{\lambda\in\operatorname*{Par}}\left( \text{something in
}\left\{ 0,1,-1\right\} \right) s_{\lambda}\ \ \ ?
\]
\pause
\item Let us restate this more formally.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ The Hall inner product}
\begin{itemize}
\item We recall the \defn{Hall inner product $\left( \cdot,\cdot\right)
:\Lambda\times\Lambda\rightarrow\mathbf{k}$}; it is the unique $\mathbf{k}%
$-bilinear form on $\Lambda$ that satisfies
\[
\left( s_{\lambda},s_{\mu}\right) =\delta_{\lambda,\mu}%
\ \ \ \ \ \ \ \ \ \ \text{for all }\lambda,\mu\in\operatorname*{Par}.
\]
It also is symmetric and nondegenerate and satisfies%
\[
\left( h_{\lambda},m_{\mu}\right) =\delta_{\lambda,\mu}%
\ \ \ \ \ \ \ \ \ \ \text{for all }\lambda,\mu\in\operatorname*{Par}.
\]
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ MNable symmetric functions: definition}
\begin{itemize}
\item \textbf{Definition:} Let $\mathbf{k}=\mathbb{Z}$ from now on.
\begin{itemize}
\item A symmetric function $f\in\Lambda$ will be called \defn{signed
multiplicity-free} if $f$ can be expanded as a linear combination of distinct
Schur functions with all coefficients in $\left\{ -1,0,1\right\} $. (That
is, if the Hall inner product $\left( f,s_{\mu}\right) $ is $-1$ or $0$ or
$1$ for each partition $\mu$.) \pause
\item A symmetric function $f\in\Lambda$ will be called \defn{MNable} if for
each partition $\mu$, the product $fs_{\mu}$ is signed multiplicity-free.
\pause
\end{itemize}
\item For example, $h_{3}p_{2}$ is signed multiplicity-free, since
\[
h_{3}p_{2}=s_{\left( 5\right) }+s_{\left( 3,2\right) }-s_{\left(
3,1,1\right) };
\]
\pause but it is not MNable, since the product
\begin{align*}
h_{3}p_{2}s_{\left( 2\right) } & =-s_{\left( 3,2,1,1\right) }+s_{\left(
3,2,2\right) }-s_{\left( 4,1,1,1\right) }+s_{\left( 4,3\right) }\\
& \ \ \ \ \ \ \ \ \ \ -s_{\left( 5,1,1\right) }+2s_{\left( 5,2\right)
}+s_{\left( 6,1\right) }+s_{\left( 7\right) }%
\end{align*}
is not signed multiplicity-free (due to the coefficient of $s_{\left(
5,2\right) }$ being $2$).
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ MNable symmetric functions: examples}
\begin{itemize}
\item \textbf{Definition:} Let $\mathbf{k}=\mathbb{Z}$ from now on.
\begin{itemize}
\item A symmetric function $f\in\Lambda$ will be called \defn{signed
multiplicity-free} if $f$ can be expanded as a linear combination of distinct
Schur functions with all coefficients in $\left\{ -1,0,1\right\} $. (That
is, if the Hall inner product $\left( f,s_{\mu}\right) $ is $-1$ or $0$ or
$1$ for each partition $\mu$.)
\item A symmetric function $f\in\Lambda$ will be called \defn{MNable} if for
each partition $\mu$, the product $fs_{\mu}$ is signed multiplicity-free.
\end{itemize}
\item \only<1>{\textbf{First Pieri rule:} Each $\mu\in\operatorname*{Par}$ and
$i\in\mathbb{N}$ satisfy%
\[
h_{i}s_{\mu}=\sum_{\substack{\lambda\in\operatorname*{Par};\\\lambda/\mu\text{
is a horizontal }i\text{-strip}}}s_{\lambda}.
\]
The right hand side is signed multiplicity-free (without any $-1$'s). Thus,
$h_{i}$ is MNable.}
\only<2>{\textbf{Second Pieri rule:} Each $\mu\in\operatorname*{Par}$ and
$i\in\mathbb{N}$ satisfy%
\[
e_{i}s_{\mu}=\sum_{\substack{\lambda\in\operatorname*{Par};\\\lambda/\mu\text{
is a vertical }i\text{-strip}}}s_{\lambda}.
\]
The right hand side is signed multiplicity-free (without any $-1$'s). Thus,
$e_{i}$ is MNable.}
\only<3-4>{\textbf{Murnaghan--Nakayama rule:} Each $\mu\in\operatorname*{Par}$
and $i>0$ satisfy
\[
p_{i}s_{\mu}=\sum_{\substack{\lambda\in\operatorname*{Par};\\\lambda/\mu\text{
is a rim hook of size }i}} \pm s_{\lambda}.
\]
The right hand side is signed multiplicity-free. Thus, $p_{i}$ is MNable.}
\pause \pause \pause
\item Roughly speaking, an $f\in\Lambda$ is MNable if and only if there is a
\textbf{M}urnaghan-\textbf{N}akayama-like rule for $fs_{\mu}$. Thus, the name
\textquotedblleft\textbf{MN}able\textquotedblright.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ MNable symmetric functions: results, 1}
\begin{itemize}
\item \textbf{Question:} Which symmetric functions are MNable? \pause
\item \textbf{Theorem:}
\begin{itemize}
\item The functions $h_{i}$ and $e_{i}$ are MNable for each $i\in\mathbb{N}$.
\item The function $p_{i}$ is MNable for each positive integer $i$. \pause
\item The Petrie function $G\left( k,m\right) $ and the difference $G\left(
k,m\right) -h_{m}$ are MNable for any integers $k\geq1$ and $m\geq0$.
\pause
\item The differences $h_{i}-p_{i}$ and $h_{i}-e_{i}$ are MNable for each
positive integer $i$. (This includes $h_{1}-e_{1}=0$.) \pause
\item The difference $h_{i}-p_{i}-e_{i}$ is MNable for each \textbf{even}
positive integer $i$. \pause
\item The product $p_{i}p_{j}$ is MNable whenever $i>j>0$. \pause
\item The function $m_{\left( k^{n}\right) }$ as well as the differences
$h_{nk}-m_{(k^{n})}$ and $e_{nk}-\left( -1\right) ^{\left( k-1\right)
n}m_{\left( k^{n}\right) }$ are MNable for any positive integers $n$ and $k$
(where $\left( k^{n}\right) $ denotes the $n$-tuple $\left( k,k,\ldots
,k\right) $).
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ MNable symmetric functions: results, 2}
\begin{itemize}
\item \textbf{Theorem (continued):}
\begin{itemize}
\item If some $f\in\Lambda$ is MNable, then so are $-f$ and $\omega\left(
f\right) $, where $\omega:\Lambda\rightarrow\Lambda$ is the
\defn{fundamental involution} of $\Lambda$ (that is, the $\mathbf{k}$-algebra
automorphism sending $e_{n}\mapsto h_{n}$). \pause
\item A symmetric function $f\in\Lambda$ is MNable if and only if all its
homogeneous components are MNable. \pause
\item If $f\in\Lambda$ is MNable and $k$ is a positive integer, then
$\mathbf{f}_{k}\left( f\right) $ is MNable. \pause
\item A symmetric function $f\in\Lambda$ is MNable if and only if $\left(
f,s_{\lambda/\mu}\right) \in\left\{ -1,0,1\right\} $ for each skew
partition $\lambda/\mu$.
\end{itemize}
\pause
\item \only<5>{The proofs use various techniques; the coefficients are not always easy
to describe.}
\only<6>{The MNability of a symmetric function can be tested in finite time using
the last bullet point.}
\only<7>{The families listed above cover all MNable homogeneous symmetric
functions of degree $<4$. In degree $4$, we also have%
\[
s_{\left( 1,1,1,1\right) }-s_{\left( 3,1\right) }+s_{\left( 4\right)
}\ \ \ \ \ \ \ \ \ \ \text{and}\ \ \ \ \ \ \ \ \ \ s_{\left( 4\right)
}-s_{\left( 2,2\right) }.
\]
}
\only<8>{All MNable $s_{\lambda}$, $m_{\lambda}$, $h_{\lambda}$ and $e_{\lambda}$
appear in the list above. Not sure if all MNable $p_{\lambda}$.}
\end{itemize}
\vspace{10pc}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ \ MNable symmetric functions: question}
\begin{itemize}
\item \textbf{Question:} What symmetric functions are MNable?
\begin{itemize}
\item Any hope of a full classification?
\item Any more infinite families?
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\fti{Bonus problem}
\begin{center}
{\LARGE \bf Bonus problem} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf Dual stable Grothendieck polynomials}
\end{center}
\vspace{1cm}
\end{frame}
\begin{frame}
\fti{Reminder on Schur functions}
\begin{itemize}
\item Here is a conjecture I'm curious to hear ideas about.
\pause \item Fix a commutative ring $\kk$. \\
Recall that for any skew partition $\lm$, the
\defn{(skew) Schur function $s_{\lm}$} is defined as the power series
\[
\sum_{T \text{ is an SST of shape } \lambda / \mu} \xx^{\cont T} \in \kk\left[\left[x_1, x_2, x_3, \ldots\right]\right] ,
\]
where ``SST'' is short for ``semistandard Young tableau'', and where
\[
\xx^{\cont T}
= \prod_{k \geq 1} x_k^{\text{number of times } T \text{ contains entry } k} .
\]
\pause \item Let us generalize this by extending the sum and introducing extra parameters.
\end{itemize}
\end{frame}
\begin{frame}
\fti{Dual stable Grothendieck polynomials, 1: RPPs}
\begin{itemize}
\item A \defn{reverse plane partition (RPP)} is defined like an SST
(semistandard Young tableau),
but entries increase \textbf{weakly} both along rows and down columns. For example,
\begin{ytableau}
\none & 1 & 2 & 2 \\
\none & 2 & 2 \\
2 & 4
\end{ytableau}
is an RPP.
\pause \\
(In detail: An RPP is a map $T$ from a skew Young diagram to
$\set{\text{positive integers}}$ such that
$T\tup{i, j} \leq T\tup{i, j+1}$ and
$T\tup{i, j} \leq T\tup{i+1, j}$ whenever
these are defined.)
\pause
\item Let $\kk$ be a commutative ring, and fix any elements
$t_1, t_2, t_3, \ldots \in \kk$.
\end{itemize}
\vspace{10cm}
\end{frame}
\begin{frame}
\fti{Dual stable Grothendieck polynomials, 2: definition}
\begin{itemize}
\item Given a skew partition $\lambda / \mu$, we define the \defn{refined dual stable Grothendieck polynomial $\wtg_{\lambda / \mu}$} to be the formal power series
\[
\sum_{T \text{ is an RPP of shape } \lambda / \mu} \xx^{\ircont T} \ttt^{\ceq T} \in \kk\left[\left[x_1, x_2, x_3, \ldots\right]\right] ,
\]
where
\[
\xx^{\ircont T}
= \prod_{k \geq 1} x_k^{\text{number of columns of } T \text{ containing entry } k}
\]
and
\[
\ttt^{\ceq T}
= \prod_{i \geq 1} t_i^{\text{number of } j \text{ such that } T\left(i, j\right) = T\left(i+1, j\right)}
\]
(where $T\left(i, j\right) = T\left(i+1, j\right)$ implies, in particular, that both $\left(i, j\right)$ and $\left(i+1, j\right)$ are cells of $T$).
\newline
This is a formal power series in $x_1, x_2, x_3, \ldots$ (despite the name ``polynomial'').
\end{itemize}
\vspace{10cm}
\end{frame}
\begin{frame}
\fti{Dual stable Grothendieck polynomials, 3: examples on $\xx^{\ircont T}$}
%\textbf{Examples on $\xx^{\ircont T}$:}
\begin{itemize}
\item Recall:
\[
\xx^{\ircont T}
= \prod_{k \geq 1} x_k^{\text{number of columns of } T \text{ containing entry } k} .
\]
\item If $T = \begin{ytableau}
\none & 1 & 2 & 2 \\
\none & 2 & 2 \\
2 & 3
\end{ytableau}$, then $\xx^{\ircont T} = x_1 x_2^4 x_3$.
The $x_2$ has exponent $4$, not $5$, because the two $2$'s in column $3$ count only once.
\pause
\item If $T$ is an SST, then $\xx^{\ircont T} = \xx^{\cont T}$.
\end{itemize}
\end{frame}
\begin{frame}
\fti{Dual stable Grothendieck polynomials, 3: examples on $\ttt^{\ceq T}$}
%\textbf{Examples on $\ttt^{\ceq T}$:}
\begin{itemize}
\item Recall that
\[
\ttt^{\ceq T}
= \prod_{i \geq 1} t_i^{\text{number of } j \text{ such that } T\left(i, j\right) = T\left(i+1, j\right)}
\]
\item If $T = \begin{ytableau}
\none & 1 & 2 & 2 \\
\none & 2 & 2 \\
2 & 3
\end{ytableau}$, then $\ttt^{\ceq T} = t_1$, due to $T\left(1, 3\right) = T\left(2, 3\right)$.
\pause
\item If $T$ is an SST, then $\ttt^{\ceq T} = 1$.
\item In general, $\ttt^{\ceq T}$ measures ``how often'' $T$ breaks the SST condition.
\end{itemize}
\end{frame}
\begin{frame}
\fti{Dual stable Grothendieck polynomials, 5}
\begin{itemize}
\item If we set $t_1 = t_2 = t_3 = \cdots = 0$, then $\wtg_{\lm} = s_{\lm}$.
\pause
\item If we set $t_1 = t_2 = t_3 = \cdots = 1$, then $\wtg_{\lm} = g_{\lm}$, the \defn{dual stable Grothendieck polynomial} of Lam and Pylyavskyy ({\red \arxiv{0705.2189}}).
\item The general case, to our knowledge, is new.
\pause
\item \textbf{Theorem (Galashin, G., Liu, {\red \arxiv{1509.03803}}):} The power series $\wtg_{\lm}$ is symmetric in the $x_i$ (not in the $t_i$).
\pause
\item \textbf{Example 1:} If $\lambda = \left(n\right)$ and $\mu = \left(\right)$, then $\wtg_{\lm} = h_n$, the $n$-th complete homogeneous symmetric function.
\pause
\item \textbf{Example 2:} If $\lambda = \left(\underbrace{1, 1, \ldots, 1}_{n \text{ ones}}\right)$ and $\mu = \left(\right)$, then $\wtg_{\lm} = e_n\left(t_1, t_2, \ldots, t_{n-1}, x_1, x_2, x_3, \ldots\right)$, where $e_n$ is the $n$-th elementary symmetric function.
\pause
\item \textbf{Example 3:} If $\lambda = \left(2,1\right)$ and $\mu = \left(\right)$, then $\wtg_{\lm} = \sum\limits_{a\leq b; \ a < c} x_a x_b x_c + t_1 \sum\limits_{a \leq b} x_a x_b = s_{(2,1)} + t_1 s_{(2)}$.
\end{itemize}
\end{frame}
\begin{frame}
\fti{Jacobi-Trudi identity?}
\begin{itemize}
\item \textbf{Conjecture:}
Let the conjugate partitions of $\lambda$ and
$\mu$ be $\lambda^{t}=\left( \left( \lambda^{t}\right) _{1},\left(
\lambda^{t}\right) _{2},\ldots,\left( \lambda^{t}\right) _{N}\right) $ and
$\mu^{t}=\left( \left( \mu^{t}\right) _{1},\left( \mu^{t}\right)
_{2},\ldots,\left( \mu^{t}\right) _{N}\right) $. Then,%
\begin{align*}
& \widetilde{g}_{\lambda/\mu}\\
& =\det\left( \left( e_{\left( \lambda^{t}\right) _{i}-i-\left( \mu
^{t}\right) _{j}+j}\left( \mathbf{x},\mathbf{t}\left[ \left( \mu
^{t}\right) _{j}+1:\left( \lambda^{t}\right) _{i}\right] \right) \right)
_{1\leq i\leq N,\ 1\leq j\leq N}\right) .
\end{align*}
Here, $\left( \mathbf{x},\mathbf{t}\left[ k:\ell\right] \right) $ denotes
the alphabet $\left( x_{1},x_{2},x_{3},\ldots,t_{k},t_{k+1},\ldots,t_{\ell
-1}\right) $.
\textbf{Warning:} If $\ell\leq k$, then $t_{k},t_{k+1},\ldots,t_{\ell-1}$
means nothing. No \textquotedblleft antimatter\textquotedblright\ variables!
\pause
\item This would generalize the Jacobi-Trudi identity for Schur
functions in terms of $e_i$'s.
\pause
\item I have some even stronger conjectures, with less evidence...
\pause
\item The case $\mu = \varnothing$ has been proven by Damir Yeliussizov
in {\red \arxiv{1601.01581}}.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{\ \ \ \ Thank you}
\begin{itemize}
\item \textbf{Linyuan Liu, Patrick Polo} for the original motivation.
\item \textbf{Ira Gessel, Jim Haglund, Christopher Ryba, Richard Stanley and
Mark Wildon} for interesting discussions.
\item \textbf{the Mathematisches Forschungsinstitut Oberwolfach and the
Institut Mittag--Leffler} for hosting me.
\item \textbf{you} for your patience and corrections.
\end{itemize}
\end{frame}
\end{document}