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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
$<N$ for some $N=N_{f}$);

\item \defn{symmetric} (i.e., permuting the $x_{i}$ does not change
$f$). \pause

\end{itemize}

\item We let
\[
\defnm{\Lambda} =\left\{  \text{symmetric functions }f\in\mathbf{k}\left[
\left[  x_{1},x_{2},x_{3},\ldots\right]  \right]  \right\}  .
\]
This is a $\mathbf{k}$-subalgebra of $\mathbf{k}\left[  \left[  x_{1}%
,x_{2},x_{3},\ldots\right]  \right]  $, graded by the degree.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: $m$}

\begin{itemize}
\item The $\mathbf{k}$-module $\Lambda$ has several bases indexed by the set
$\operatorname*{Par}$. \pause


\item The \defn{monomial basis} $\left(  m_{\lambda}\right)  _{\lambda
\in\operatorname*{Par}}$:

For each partition $\lambda$, we define the
\defn{monomial symmetric function
}$m_{\lambda}\in\Lambda$ by
\[
m_{\lambda}=\sum_{\substack{\alpha\text{ is a weak composition;}\\\alpha\text{
can be obtained from }\lambda\\\text{by permuting entries}}}\mathbf{x}%
^{\alpha}.
\]
\pause For example:%
\[
m_{\left(  2,2,1\right)  }=\sum_{i<j<k}x_{i}^{2}x_{j}^{2}x_{k}+\sum
_{i<j<k}x_{i}^{2}x_{j}x_{k}^{2}+\sum_{i<j<k}x_{i}x_{j}^{2}x_{k}^{2}.
\]
\pause


The family $\left(  m_{\lambda}\right)  _{\lambda\in\operatorname*{Par}}$ is a
basis of the $\mathbf{k}$-module $\Lambda$, called the \defn{monomial basis}.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: $h$}

\begin{itemize}
\item The \defn{complete basis} $\left(  h_{\lambda}\right)  _{\lambda
\in\operatorname*{Par}}$:

For each $n\in\mathbb{Z}$, define the \defn{complete homogeneous symmetric
function $h_{n}$} by%
\[
h_{n}=\sum_{i_{1}\leq i_{2}\leq\cdots\leq i_{n}}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n}}=\sum_{\substack{\alpha\in\operatorname*{WC};\\\left\vert
\alpha\right\vert =n}}\mathbf{x}^{\alpha}=\sum_{\lambda\in\operatorname*{Par}%
\nolimits_{n}}m_{\lambda}.
\]
\pause For example,%
\begin{align*}
h_{1}  &  =x_{1}+x_{2}+x_{3}+\cdots;\\
h_{2}  &  =\sum_{i\leq j}x_{i}x_{j}=\sum_{i}x_{i}^{2}+\sum_{i<j}x_{i}x_{j};\\
h_{0}  &  =1;\\
h_{n}  &  =0\ \ \ \ \ \ \ \ \ \ \text{for all }n<0.
\end{align*}
\pause For each partition $\lambda$, we define
\[
\defnm{h_{\lambda}}=h_{\lambda_{1}}h_{\lambda_{2}}h_{\lambda_{3}}\cdots\in\Lambda.
\]


The family $\left(  h_{\lambda}\right)  _{\lambda\in\operatorname*{Par}}$ is a
basis of the $\mathbf{k}$-module $\Lambda$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: $e$}

\begin{itemize}
\item The \defn{elementary basis} $\left(  e_{\lambda}\right)  _{\lambda
\in\operatorname*{Par}}$:

For each $n\in\mathbb{Z}$, define the \defn{elementary symmetric function
$e_{n}$} by%
\[
e_{n}=\sum_{i_{1}<i_{2}<\cdots<i_{n}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}%
=\sum_{\substack{\alpha\in\operatorname*{WC}\cap\left\{  0,1\right\}
^{\infty};\\\left\vert \alpha\right\vert =n}}\mathbf{x}^{\alpha}=m_{\left(
1^{n}\right)  }.
\]
\pause For example,%
\begin{align*}
e_{1}  &  =x_{1}+x_{2}+x_{3}+\cdots;\\
e_{2}  &  =\sum_{i<j}x_{i}x_{j};\\
e_{0}  &  =1;\\
e_{n}  &  =0\ \ \ \ \ \ \ \ \ \ \text{for all }n<0.
\end{align*}
\pause For each partition $\lambda$, we define
\[
\defnm{e_{\lambda}}=e_{\lambda_{1}}e_{\lambda_{2}}e_{\lambda_{3}}\cdots\in\Lambda.
\]


The family $\left(  e_{\lambda}\right)  _{\lambda\in\operatorname*{Par}}$ is a
basis of the $\mathbf{k}$-module $\Lambda$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: $p$}

\begin{itemize}
\item The \defn{power-sum symmetric functions} $p_{n}$:

For each positive integer $n$, define the \defn{power-sum symmetric function
$p_{n}$} by%
\[
p_{n}=x_{1}^{n}+x_{2}^{n}+x_{3}^{n}+\cdots=m_{\left(  n\right)  }.
\]


\pause We can make a basis out of (products of) $p_{n}$'s when $\mathbf{k}$ is
a $\mathbb{Q}$-algebra.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Symmetric functions: $s$}

\begin{itemize}
\item The \defn{Schur basis }$\left(  s_{\lambda}\right)  _{\lambda
\in\operatorname*{Par}}$:

For each partition $\lambda$, we can define the \defn{Schur function}
$s_{\lambda}$ in many equivalent ways, e.g.:

\begin{itemize}
\item We have%
\[
s_{\lambda}=\sum_{\substack{T\text{ is a semistandard}\\\text{Young tableau of
shape }\lambda}}\mathbf{x}_{T},
\]
where $\mathbf{x}_{T}$ denotes the monomial obtained by multiplying the
$x_{i}$ for all entries $i$ of $T$. \pause


\item If $\lambda=\left(  \lambda_{1},\lambda_{2},\ldots,\lambda_{\ell
}\right)  $, then%
\[
s_{\lambda}=\det\left(  \left(  h_{\lambda_{i}-i+j}\right)  _{1\leq i\leq
\ell,\ 1\leq j\leq\ell}\right)
\]
(the \defn{first Jacobi--Trudi formula}).
\end{itemize}

The family $\left(  s_{\lambda}\right)  _{\lambda\in\operatorname*{Par}}$ is a
basis of the $\mathbf{k}$-module $\Lambda$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Petrie functions: definition of $G(k)$}

\begin{itemize}
\item For any positive integer $k$, set%
\begin{align*}
&  \defnm{G\left( k\right)}\\
&  =\sum_{\substack{\alpha\in\operatorname*{WC};\\\alpha_{i}<k\text{ for all
}i}}\mathbf{x}^{\alpha}\\
&  =\sum\left(  \text{all monomials whose exponents are all }<k\right) \\
&  \in\mathbf{k}\left[  \left[  x_{1},x_{2},x_{3},\ldots\right]  \right]
\ \ \ \ \ \ \ \ \ \ \left(  \text{not }\in\Lambda\text{ in general}\right)  .
\end{align*}
\vspace{10pc}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Petrie functions: definition of $G(k,m)$}

\begin{itemize}
\item For any positive integer $k$ and any $m\in\mathbb{N}$, we let%
\begin{align*}
&  \defnm{G\left( k,m\right)}\\
&  =\sum_{\substack{\alpha\in\operatorname*{WC};\\\left\vert \alpha\right\vert
=m;\\\alpha_{i}<k\text{ for all }i}}\mathbf{x}^{\alpha}\\
&  =\sum\left(  \text{all degree-}m\text{ monomials whose exponents are all
}<k\right) \\
&  \in\Lambda.
\end{align*}


\pause For example,%
\begin{align*}
G\left(  3,4\right)   &  =\sum_{i<j<k<\ell}x_{i}x_{j}x_{k}x_{\ell}%
+\sum_{i<j<k}x_{i}^{2}x_{j}x_{k}+\sum_{i<j<k}x_{i}x_{j}^{2}x_{k}\\
&  \ \ \ \ \ \ \ \ \ \ +\sum_{i<j<k}x_{i}x_{j}x_{k}^{2}+\sum_{i<j}x_{i}%
^{2}x_{j}^{2}\\
&  =m_{\left(  1,1,1,1\right)  }+m_{\left(  2,1,1\right)  }+m_{\left(
2,2\right)  }.
\end{align*}

\end{itemize}

\vspace{10pc}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Petrie functions: basic properties}

\begin{itemize}
\item I named $G\left(  k\right)  $ and $G\left(  k,m\right)  $ the
\defn{Petrie functions}, for reasons that will become clear eventually.
\pause


\item \textbf{Basic properties} (for arbitrary $k>0$ and $m\in\mathbb{N}%
$)\textbf{:}

\begin{itemize}
\item
\[
G\left(  k\right)  =\sum_{\substack{\lambda\in\operatorname*{Par}%
;\\\lambda_{i}<k\text{ for all }i}}m_{\lambda}=\prod_{i=1}^{\infty}\left(
x_{i}^{0}+x_{i}^{1}+\cdots+x_{i}^{k-1}\right)  .
\]
\pause


\item $G\left(  k,m\right)  $ is the $m$-th degree component of $G\left(
k\right)  $. \pause


\item
\[
G\left(  k,m\right)  =\sum_{\substack{\lambda\in\operatorname*{Par}%
;\\\left\vert \lambda\right\vert =m;\\\lambda_{i}<k\text{ for all }%
i}}m_{\lambda}.
\]
\pause


\item $G\left(  2,m\right)  =e_{m}$. \pause


\item $G\left(  k,m\right)  =h_{m}$ whenever $k>m$. \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%
\[
\operatorname*{pet}\nolimits_{k}\left(  \lambda,\mu\right)  =\det\left(
\left(  \left[  0\leq\lambda_{i}-\mu_{j}-i+j<k\right]  \right)  _{1\leq
i\leq\ell,\ 1\leq j\leq\ell}\right)  .
\]


\pause
\only<3>{ 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}<k\right] & \left[ 0\leq\lambda_{1}-\mu _{2}+1<k\right] & \left[ 0\leq\lambda_{1}-\mu_{3}+2<k\right] \\ \left[ 0\leq\lambda_{2}-\mu_{1}-1<k\right] & \left[ 0\leq\lambda_{2}-\mu_{2}<k\right] & \left[ 0\leq\lambda_{2}-\mu_{3}+1<k\right] \\ \left[ 0\leq\lambda_{3}-\mu_{1}-2<k\right] & \left[ 0\leq\lambda_{3}-\mu_{2}-1<k\right] & \left[ 0\leq\lambda_{3}-\mu_{3}<k\right] \end{array} \right) $}
.
\end{align*}
For example,%
\[
\operatorname*{pet}\nolimits_{4}\left(  \left(  3,1,1\right)  ,\left(
2,1\right)  \right)  =\det\left(
\begin{array}
[c]{ccc}%
1 & 1 & 0\\
0 & 1 & 1\\
0 & 0 & 1
\end{array}
\right)  =1.
\]
} \pause


\item \textbf{Proposition:} We have $\operatorname*{pet}\nolimits_{k}\left(
\lambda,\mu\right)  \in\left\{  0,1,-1\right\}  $ for all $\lambda$ and $\mu$.
\pause


\item \textit{Proof idea.} Each row of the matrix $\left(  \left[
0\leq\lambda_{i}-\mu_j-i+j<k\right]  \right)  _{1\leq i\leq\ell,\ 1\leq j\leq\ell}$
has the form
\[
(\underbrace{0,0,\ldots,0}_{a\text{ zeroes}},\underbrace{1,1,\ldots
,1}_{b\text{ ones}},\underbrace{0,0,\ldots,0}_{c\text{ zeroes}}%
)\ \ \ \ \ \ \text{for some }a,b,c\in\mathbb{N}.
\]
Thus, this matrix is the transpose of a \defn{Petrie
matrix}. Hence, its determinant is $\in\left\{  -1,0,1\right\}  $ (by
{\color{red}{\href{https://projecteuclid.org/euclid.pjm/1102912464}{Gordon and Wilkinson
1974}}}).
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Expanding Petries in the Schur basis: the formula}

\begin{itemize}
\item \textbf{Theorem:} Let $k$ be a positive integer. Let $\mu\in
\operatorname*{Par}$. Then,%
\[
G\left(  k\right)  \cdot s_{\mu}=\sum_{\lambda\in\operatorname*{Par}%
}\operatorname*{pet}\nolimits_{k}\left(  \lambda,\mu\right)  s_{\lambda}.
\]
\pause Thus, for each $m\in\mathbb{N}$, we have%
\[
G\left(  k,m\right)  \cdot s_{\mu}=\sum_{\lambda\in\operatorname*{Par}%
\nolimits_{m+\left\vert \mu\right\vert }}\operatorname*{pet}\nolimits_{k}%
\left(  \lambda,\mu\right)  s_{\lambda}.
\]
\pause


\item \only<3>{\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 $\lambda_{1}<k$. Let $\mu=\lambda^{t}$ (the transpose partition of
$\lambda$). Thus, $\mu_{k}=0$.

For each $i\in\left\{  1,2,\ldots,k-1\right\}  $, set
\[
\beta_{i}=\mu_{i}-i\qquad\text{and}\qquad\gamma_{i}=1+\underbrace{\left(
\beta_{i}-1\right)  \%k}_{\substack{\text{remainder of }\beta_{i}%
-1\\\text{modulo }k}}.
\]


\textbf{(a)} If the $k-1$ numbers $\gamma_{1},\gamma_{2},\ldots,\gamma_{k-1}$
are not distinct, then $\operatorname*{pet}\nolimits_{k}\left(  \lambda
,\varnothing\right)  =0$.

\textbf{(b)} If the $k-1$ numbers $\gamma_{1},\gamma_{2},\ldots,\gamma_{k-1}$
are distinct, then%
\[
\operatorname*{pet}\nolimits_{k}\left(  \lambda,\varnothing\right)  =\left(
-1\right)  ^{\left(  \beta_{1}+\beta_{2}+\cdots+\beta_{k-1}\right)  +g+\left(
\gamma_{1}+\gamma_{2}+\cdots+\gamma_{k-1}\right)  },
\]
where%
\[
g=\left\vert \left\{  \left(  i,j\right)  \in\left\{  1,2,\ldots,k-1\right\}
^{2}\ \mid\ i<j\text{ and }\gamma_{i}<\gamma_{j}\right\}  \right\vert .
\]


\pause


\item \textbf{Question:} Is there such a description for $\operatorname*{pet}%
\nolimits_{k}\left(  \lambda,\mu\right)  $ ?
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ Other properties}

\begin{itemize}
\item For any $k>0$, 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
$\lambda_{i}<n$ for all $i$. \pause


\item Thus,
\[
\sum_{\substack{\lambda\in\operatorname*{Par}\nolimits_{2n-1};\\\left(
n-1,n-1,1\right)  \triangleright\lambda}}m_{\lambda}=G\left(  n,2n-1\right)
.
\]
\pause


\item So it remains to show that%
\[
G\left(  n,2n-1\right)  =\sum_{i=0}^{n-2}\left(  -1\right)  ^{i}s_{\left(
n-1,n-1-i,1^{i+1}\right)  }.
\]
\pause


\item The formula for $\operatorname*{pet}\nolimits_{k}\left(  \lambda
,\varnothing\right)  $ should be useful here, but the combinatorics is tortuous.

Instead, we can work algebraically:
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ The Liu--Polo conjecture, proof: $G(n, 2n-1)$ explicitly}

\begin{itemize}
\item We can easily see that
\[
G\left(  n,n+k\right)  =h_{n+k}-h_{k}p_{n}\ \ \ \ \ \ \ \text{for each }%
k\in\left\{  0,1,\ldots,n-1\right\}  .
\]
\pause Thus, in particular, $G\left(  n,2n-1\right)  =h_{2n-1}-h_{n-1}p_{n}$.
\pause


\item By the way, this is also a particular case of the
\[
G\left(  k,m\right)  =\sum_{i\in\mathbb{N}}\left(  -1\right)  ^{i}%
h_{m-ki}\cdot\mathbf{f}_{k}\left(  e_{i}\right)
\]
formula.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ \ The Liu--Polo conjecture, proof: Bernstein operators}

\begin{itemize}
\item Recall the \defn{skewing operations} $f^{\perp}:\Lambda\rightarrow
\Lambda$ for all $f\in\Lambda$. \pause


\item For any $m\in\mathbb{N}$, we define a map $\defnm{\mathbf{B}_{m}}:\Lambda
\rightarrow\Lambda$ (known as a \defn{$m$-th Bernstein operator} in
Zelevinsky's language, or as a \defn{Schur row-adder} in Garsia's) by setting%
\[
\mathbf{B}_{m}\left(  f\right)  =\sum_{i\in\mathbb{N}}\left(  -1\right)
^{i}h_{m+i}e_{i}^{\perp}f\ \ \ \ \ \ \ \ \ \ \text{for all }f\in
\Lambda\text{.}%
\]
\pause \vspace{-0.5pc}

\item \textbf{Theorem} (implicit in Zelevinsky's book; solved exercise in
G./Reiner)\textbf{:} If $\lambda\in\operatorname{Par}$ and $m\in\mathbb{Z}$
satisfy $m\geq\lambda_{1}$, then%
\[
\mathbf{B}_{m}\left(  s_{\lambda}\right)  =s_{\left(  m,\lambda_{1}%
,\lambda_{2},\lambda_{3},\ldots\right)  }.
\]
\pause \vspace{-1pc}

\item On the other hand, it is not hard to see that%
\begin{align*}
\mathbf{B}_{m}\left(  h_{n}\right)   &  =h_{m}h_{n}-h_{m+1}h_{n-1}%
\ \ \ \ \ \ \ \ \ \ \text{and}\\
\mathbf{B}_{m}\left(  p_{n}\right)   &  =h_{m}p_{n}-h_{m+n}%
\end{align*}
for each $n>0$ 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}