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\begin{document}

\title{Natural endomorphisms of connected graded bialgebras\\{\large [talk slides: CATMI 2023, Lie-St\o rmer Center, Bergen 2023-06-26]}}
\author{Darij Grinberg}
\maketitle
\tableofcontents

\begin{itemize}
\item This is an attempt at a systematic study of identities that hold for
connected graded bialgebras.

\item We start by recounting the definitions, then give some new examples of
such identities, then develop the early sprouts of a theory.

\item Lots of questions here. Some might have already been solved. The
literature is fragmented (topologists have been around for the longest but I
don't quite speak their language), so surprises are possible.

\item \textbf{Preprints:}

\url{https://www.cip.ifi.lmu.de/~grinberg/algebra/aphae-proj.pdf} (new
material, still a very rough sketch);\newline%
\url{https://www.cip.ifi.lmu.de/~grinberg/algebra/bernsteinproof.pdf} (old
treatment of the commutative case using base change; in a way obsolete, but
interesting for the methods).
\end{itemize}

\section{Bialgebras}

\subsection{General conventions}

\begin{itemize}
\item We fix a commutative ring $\mathbf{k}$. (No assumptions on characteristic!)

\item $\otimes$ always means $\otimes_{\mathbf{k}}$ by default.

\item Read \textquotedblleft$\mathbf{k}$-\textquotedblright\ in front of each
of the nouns \textquotedblleft module\textquotedblright, \textquotedblleft
algebra\textquotedblright, \textquotedblleft coalgebra\textquotedblright,
\textquotedblleft bialgebra\textquotedblright\ or \textquotedblleft
linear\textquotedblright\ by default.
\end{itemize}

\subsection{Algebras and coalgebras}

\begin{itemize}
\item \textbf{Definition.} An \textbf{algebra} means a module $A$ equipped
with a multiplication map%
\[
m:A\otimes A\rightarrow A\ \ \ \ \ \ \ \ \ \ \left(  a\otimes b\mapsto
ab\right)
\]
and a unity map%
\[
u:\mathbf{k}\rightarrow A\ \ \ \ \ \ \ \ \ \ \left(  1_{\mathbf{k}}%
\mapsto1_{A}\right)
\]
(both linear) such that the diagrams%
\[
\xymatrix{
& A  \otimes A \otimes A \ar[dl]_{m \otimes \id} \ar[dr]^{\id \otimes m} &  \\
A \otimes A \ar[dr]^{m} & &  A \otimes A \ar[dl]_{m} \\
& A &
}\qquad\qquad\xymatrix{
A \otimes \kk \ar[d]_{\id \otimes u} & A \ar[l] \ar[d]_{\id} \ar[r] & \kk \otimes A \ar[d]^{u \otimes \id} \\
A \otimes A \ar[r]^-{m} & A & A \otimes A \ar[l]_-{m}
}
\]
commute.

One usually writes $m_{A}$ and $u_{A}$ for $m$ and $u$ (and similarly elsewhere).

\item Dualizing this definition, one gets \textquotedblleft
coalgebras\textquotedblright:

\textbf{Definition.} A \textbf{coalgebra} means a module $C$ equipped with a
comultiplication map%
\[
\Delta:C\rightarrow C\otimes C\ \ \ \ \ \ \ \ \ \ \left(  c\mapsto\sum
_{i}c_{1,i}\otimes c_{2,i}\right)
\]
and a counit map%
\[
\epsilon:C\rightarrow\mathbf{k}%
\]
(both linear) such that the diagrams%
\[
\xymatrix{
& C  \otimes C \otimes C &  \\
C \otimes C \ar[ur]^{\Delta \otimes \id} & &  C \otimes C \ar[ul]_{\id \otimes \Delta} \\
& C \ar[ur]^\Delta \ar[ul]_\Delta&
} \qquad\qquad\xymatrix{
C \otimes \kk \ar[r] & C & \kk \otimes C \ar[l] \\
C \otimes C \ar[u]^{\id \otimes \epsilon} & C \ar[l]^-{\Delta} \ar[u]^\id \ar[r]_-{\Delta} & C \otimes C \ar[u]^{\epsilon \otimes \id}
}
\]
commute.

\item \textbf{Definition.} \textbf{Algebra morphisms} and \textbf{coalgebra
morphisms} are defined in the least surprising way (i.e., as linear maps that
commute with $m$ and $u$ resp. $\Delta$ and $\epsilon$ in the obvious ways).
\end{itemize}

\subsection{Bialgebras}

\begin{itemize}
\item \textbf{Definition.} A \textbf{bialgebra} is a module $H$ that is both
an algebra and a coalgebra, and that satisfies the further commutative
diagrams%
\[
\xymatrix{
& H \otimes H \ar[dl]_{\Delta \otimes \Delta} \ar[ddr]_m & \\
H \otimes H \otimes H \otimes H \ar[dd]_{\id \otimes T \otimes \id} &  & \\
& & H \ar[ddl]_\Delta\\
H \otimes H \otimes H \otimes H \ar[dr]_{m \otimes m} &  & \\
& H \otimes H &
}
\]%
\[%
\begin{array}
[c]{ccccc}%
\xymatrix{ H \otimes H \ar[r]^{\epsilon \otimes \epsilon} \ar[d]_{m} &\kk \otimes \kk \ar[d]^m \\ H \ar[r]_{\epsilon} &\kk } &
&  &  &
\xymatrix{ \kk \ar[r]^{u} \ar[d]_{\Delta} &H \ar[d]^\Delta \\ \kk \otimes \kk \ar[r]_{u \otimes u} &H \otimes H }
\end{array}
\]%
\[
\xymatrix{
\kk \ar[dr]_{u} \ar[rr]^{\id} & & \kk \\
&H \ar[ur]_{\epsilon} &
}
\]
where $T:H\otimes H\rightarrow H\otimes H$ is the \textbf{twist map }$a\otimes
b\mapsto b\otimes a$.

\item \textbf{Examples:}

\begin{itemize}
\item $\mathbf{k}$ itself is a bialgebra (with all maps being
$\operatorname*{id}:\mathbf{k}\rightarrow\mathbf{k}$).

\item If $M$ is a monoid (e.g., group), then the monoid algebra $\mathbf{k}%
\left[  M\right]  $ is a bialgebra, with%
\begin{align*}
\Delta\left(  g\right)   &  =g\otimes g\ \ \ \ \ \ \ \ \ \ \text{for all }g\in
M;\\
\epsilon\left(  g\right)   &  =1\ \ \ \ \ \ \ \ \ \ \text{for all }g\in M;\\
m\left(  g\otimes h\right)   &  =gh\ \ \ \ \ \ \ \ \ \ \text{for all }g,h\in
M;\\
u\left(  1_{\mathbf{k}}\right)   &  =e_{M}\ \ \ \ \ \ \ \ \ \ \left(
\text{that is, the unity is }e_{M}\right)  .
\end{align*}


\item If $V$ is a $\mathbf{k}$-module, then the tensor algebra $T\left(
V\right)  $ is a bialgebra, with%
\[
\Delta\left(  \underbrace{a_{1}a_{2}\cdots a_{n}}_{\text{short for }%
a_{1}\otimes a_{2}\otimes\cdots\otimes a_{n}}\right)  =\sum_{I\subseteq
\left\{  1,2,\ldots,n\right\}  }a_{I}\otimes a_{\left\{  1,2,\ldots,n\right\}
\setminus I}%
\]
for any $a_{1},a_{2},\ldots,a_{n}\in V$. Here, $a_{I}$ is the product of all
$a_{i}$ with $i\in I$ in increasing order.

\item There is also a shuffle algebra $\operatorname*{Sh}\left(  V\right)  $,
which is in some way dual to $T\left(  V\right)  $.

\item The symmetric algebra $\operatorname*{Sym}V$ (defined as a quotient of
$T\left(  V\right)  $) is also a bialgebra.

\item The ring $\Lambda$ of symmetric functions over $\mathbf{k}$ is a bialgebra.

\item The ring $\operatorname*{QSym}$ of quasisymmetric functions over
$\mathbf{k}$ is a bialgebra.

\item Various other combinatorial bialgebras such as $\operatorname*{NSym}$,
$\operatorname*{FQSym}$ (= Malvenuto--Reutenauer), posets, double posets,
graphs, hypergraphs, ....
\end{itemize}
\end{itemize}

\subsection{Graded, connected, commutative, cocommutative}

\begin{itemize}
\item \textbf{Definition.} A \textbf{graded} (co,bi)algebra is a
(co,bi)algebra $H$ that is graded (= $\mathbb{N}$-graded) as a module, and
whose operations ($m,u,\Delta,\epsilon$, whichever apply) respect the grading.
This means
\begin{align*}
H_{a}H_{b}  &  \subseteq H_{a+b}\ \ \ \ \ \ \ \ \ \ \text{for all }a,b\geq0;\\
1_{H}  &  \in H_{0};\\
\Delta\left(  H_{n}\right)   &  \subseteq\bigoplus_{k=0}^{n}H_{k}\otimes
H_{n-k}\ \ \ \ \ \ \ \ \ \ \text{for all }n\geq0;\\
\epsilon\left(  H_{n}\right)   &  =0\ \ \ \ \ \ \ \ \ \ \text{for all }n>0.
\end{align*}


\item We do \textbf{not} use topologists' sign conventions.

\item \textbf{Definition.} A graded (co,bi)algebra $H$ is \textbf{connected}
if and only if $H_{0}\cong\mathbf{k}$ as $\mathbf{k}$-modules. (For an
algebra, this automatically entails $H_{0}=\mathbf{k}\cdot1_{H}$.)

\item For example, the tensor algebra $T\left(  V\right)  $ is connected
graded, with%
\[
\left(  T\left(  V\right)  \right)  _{n}=V^{\otimes n}=\underbrace{V\otimes
V\otimes\cdots\otimes V}_{n\text{ times}}.
\]


\item \textbf{Definition.} An algebra $A$ is \textbf{commutative} if the
diagram%
\[
\xymatrix{
A \otimes A \ar[rr]^T \ar[dr]_m & & A \otimes A \ar[dl]^m \\
& A
}
\]
commutes. (Again, $T$ is the twist map.)

\item \textbf{Definition.} Dually, a coalgebra $C$ is \textbf{cocommutative}
if the diagram%
\[
\xymatrix{
C \otimes C \ar[rr]^T & & C \otimes C \\
& C \ar[ul]^{\Delta} \ar[ur]_{\Delta}
}
\]
commutes.

\item \textbf{Example:} Monoid algebras and tensor algebras are cocommutative.

Shuffle algebras and $\operatorname*{QSym}$ are commutative.

Symmetric algebras and $\Lambda$ are both.
\end{itemize}

\subsection{Convolution}

\begin{itemize}
\item \textbf{Definition.} If $C$ is a coalgebra and $A$ is an algebra, then
the module%
\[
\operatorname*{Hom}\left(  C,A\right)  :=\left\{  \text{all }\mathbf{k}%
\text{-linear maps }f:C\rightarrow A\right\}
\]
becomes an algebra itself, equipped with the \textbf{convolution product
}$\star$ defined by%
\[
f\star g=m_{A}\circ\left(  f\otimes g\right)  \circ\Delta_{C}.
\]
The unity of $\operatorname*{Hom}\left(  C,A\right)  $ is $u_{A}\circ
\epsilon_{C}$.

\item In particular, if $C$ is a coalgebra, then the dual module $C^{\ast
}=\operatorname*{Hom}\left(  C,\mathbf{k}\right)  $ is an algebra.

\item In \textbf{nice} situations, the dual statement holds: If $A$ is an
algebra that is finite free as a module, then $A^{\ast}$ is a coalgebra.

Something similar holds for graded duals in the graded-finite case (= graded,
and each degree is finite free).

\item \textbf{Duality} is a permanent theme in bialgebra theory: You can
dualize every statement, but it is not a-priori clear that the dual always
holds. Still, it is typically true and often can be derived from the primal
using some tricks.

\item \textbf{Question:} Are there general meta-theorems that guarantee this?

\item Of course, a proof that uses only element-free diagram chasing
guarantees dualizability, but not every proof is so.
\end{itemize}

\subsection{Hopf algebras}

\begin{itemize}
\item If $H$ is a bialgebra, then $\operatorname*{Hom}\left(  H,H\right)  $ is
an algebra (via convolution $\star$, as explained above).

\item The identity map $\operatorname*{id}\nolimits_{H}$ belongs to this algebra.

\item \textbf{Definition.} We call $H$ a \textbf{Hopf algebra} if
$\operatorname*{id}\nolimits_{H}$ has an inverse in this algebra.

In this case, the inverse of $\operatorname*{id}\nolimits_{H}$ is called the
\textbf{antipode} of $H$.

\item \textbf{Theorem (Takeuchi).} If $H$ is a connected graded bialgebra,
then $H$ is automatically a Hopf algebra.
\end{itemize}

\subsection{Iterated multiplications and comultiplications}

\begin{itemize}
\item Actually, the antipode can be computed explicitly.

\item \textbf{Definition.} Let $A$ be an algebra. For any integer $k\geq0$,
define the linear map%
\[
m^{\left[  k\right]  }:A^{\otimes k}\rightarrow A
\]
recursively by%
\[
m^{\left[  0\right]  }=u\ \ \ \ \ \ \ \ \ \ \text{and}%
\ \ \ \ \ \ \ \ \ \ m^{\left[  k\right]  }=m\circ\left(  m^{\left[
k-1\right]  }\otimes\operatorname*{id}\right)  .
\]


In the language of elements:%
\[
m^{\left[  k\right]  }\left(  a_{1}\otimes a_{2}\otimes\cdots\otimes
a_{k}\right)  =a_{1}a_{2}\cdots a_{k}.
\]
This map $m^{\left[  k\right]  }$ is called an \textbf{iterated
multiplication} map. (It is commonly called $m^{\left(  k-1\right)  }$, but my
indexing is better :)

\item Dually:

\textbf{Definition.} Let $C$ be a coalgebra. For any integer $k\geq0$, define
the linear map%
\[
\Delta^{\left[  k\right]  }:C\rightarrow C^{\otimes k}%
\]
recursively by%
\[
\Delta^{\left[  0\right]  }=\epsilon\ \ \ \ \ \ \ \ \ \ \text{and}%
\ \ \ \ \ \ \ \ \ \ \Delta^{\left[  k\right]  }=\left(  \Delta^{\left[
k-1\right]  }\otimes\operatorname*{id}\right)  \circ\Delta.
\]
This map $\Delta^{\left[  k\right]  }$ is called an \textbf{iterated
comultiplication} map.

\item \textbf{Proposition.} Let $A$ be an algebra, and $C$ a coalgebra. Then,
any $k$ elements $f_{1},f_{2},\ldots,f_{k}$ of the convolution algebra
$\operatorname*{Hom}\left(  C,A\right)  $ satisfy%
\[
f_{1}\star f_{2}\star\cdots\star f_{k}=m^{\left[  k\right]  }\circ\left(
f_{1}\otimes f_{2}\otimes\cdots\otimes f_{k}\right)  \circ\Delta^{\left[
k\right]  }.
\]


\item \textbf{Theorem (Takeuchi's formula for the antipode).} Let $H$ be a
connected graded bialgebra. Let
\begin{align*}
\overline{\operatorname*{id}}  &  =\operatorname*{id}\nolimits_{H}%
-u\circ\epsilon=\operatorname*{id}-\underbrace{p_{0}}%
_{\substack{\text{projection}\\H\rightarrow H_{0}}}\\
&  =\left(  \text{projection from }H=\bigoplus_{i=0}^{\infty}H_{i}\text{ onto
}\bigoplus_{i=1}^{\infty}H_{i}\right)  .
\end{align*}


Then, the antipode $S$ of $H$ is given by%
\[
S=\sum_{k=0}^{\infty}\left(  -1\right)  ^{k}\underbrace{\overline
{\operatorname*{id}}^{\star k}}_{\substack{=\overline{\operatorname*{id}}%
\star\overline{\operatorname*{id}}\star\cdots\star\overline{\operatorname*{id}%
}\\=m^{\left[  k\right]  }\circ\overline{\operatorname*{id}}^{\otimes k}%
\circ\Delta^{\left[  k\right]  }}}.
\]
The sum here converges pointwise: In fact, if $x\in H_{n}$, then
$\overline{\operatorname*{id}}^{\star k}\left(  x\right)  =0$ for all $k>n$.

\item \textbf{Proof.} Actually quite easy!
\[
\operatorname*{id}\nolimits_{H}=\underbrace{u\circ\epsilon}%
_{\substack{\text{unity of the}\\\text{convolution algebra}}}+\,\overline
{\operatorname*{id}},
\]
and $\overline{\operatorname*{id}}$ is locally nilpotent; thus, the inverse of
$\operatorname*{id}\nolimits_{H}$ can be found using $\left(  1+q\right)
^{-1}=1-q+q^{2}-q^{3}\pm\cdots$.
\end{itemize}

\section{Some identities}

\subsection{On the order of $S^{2}$}

\begin{itemize}
\item The antipode of a Hopf algebra is always called $S$.

\item \textbf{Theorem (Sweedler?).} If a Hopf algebra $H$ is commutative or
cocommutative, then its antipode is an involution: that is, $S^{2}%
=\operatorname*{id}$. (Here and in the following, $S^{2}=S\circ S$, not
$S\star S$.)

\item Not true for general $H$. (In general, $S$ may even be non-invertible.)

\item However:

\item \textbf{Theorem (Aguiar and Lauve 2014).} If $H$ is a connected graded
bialgebra, then%
\[
\left(  \operatorname*{id}-S^{2}\right)  ^{n}\left(  H_{n}\right)
=0\ \ \ \ \ \ \ \ \ \ \text{for each }n\geq0.
\]
(Thus, $S^{2}$ is $\operatorname*{id}$ \textquotedblleft up
to\textquotedblright\ a locally nilpotent \textquotedblleft error
term\textquotedblright. In other words, $S^{2}$ is locally unipotent.)

\item \textbf{Theorem (Aguiar 2017).} Even better: In the same setup,%
\[
\left(  \left(  \operatorname*{id}+S\right)  \circ\left(  \operatorname*{id}%
-S^{2}\right)  ^{n-1}\right)  \left(  H_{n}\right)
=0\ \ \ \ \ \ \ \ \ \ \text{for each }n>0.
\]


\item For some $H$ (for example, Malvenuto--Reutenauer), we even have%
\[
\left(  \operatorname*{id}-S^{2}\right)  ^{n-1}\left(  H_{n}\right)
=0\ \ \ \ \ \ \ \ \ \text{ for each }n>1.
\]


\item I generalize these in
\href{https://arxiv.org/abs/2109.02101}{arXiv:2109.02101}.
\end{itemize}

\subsection{The random-to-top operator}

\begin{itemize}
\item Here is another series of recent results (mostly unpublished -- see
\url{https://www.cip.ifi.lmu.de/~grinberg/algebra/aphae-proj.pdf} for outlined
proofs --, but related work was done by Amy Pang in arXiv:1609.04312, arXiv:2108.09097).

\item \textbf{Definition.} If $H$ is any graded module (e.g., bialgebra), and
if $n\geq0$, then $p_{n}$ shall denote the canonical projection $H\rightarrow
H_{n}$ (regarded as a map $H\rightarrow H$).

Note that $p_{0}=u\circ\epsilon$ when $H$ is connected.

\item \textbf{Definition.} If $H$ is a graded bialgebra, and if $n\geq0$, then
we set%
\[
\rho_{n}:=p_{n}\star\operatorname*{id}\in\operatorname*{Hom}\left(
H,H\right)  .
\]
In particular, $\rho_{1}=p_{1}\star\operatorname*{id}$ is called
\textbf{random-to-top operator}, since it acts on a tensor algebra $H=T\left(
V\right)  $ as follows:%
\[
\rho_{1}\left(  \underbrace{a_{1}a_{2}\cdots a_{n}}_{\text{short for }%
a_{1}\otimes a_{2}\otimes\cdots\otimes a_{n}}\right)  =\sum_{k=1}%
^{n}\underbrace{a_{k}\cdot a_{1}a_{2}\cdots\widehat{a_{k}}\cdots a_{n}%
}_{\substack{\text{this is our input tensor,}\\\text{with the }k\text{-th
factor moved to front}}}.
\]


\item \textbf{Theorem.} Let $H$ be a connected graded bialgebra.

\begin{enumerate}
\item[\textbf{(a)}] We have $\rho_{1}=0$ on $H_{0}$, and $\rho_{1}%
=\operatorname*{id}$ on $H_{1}$.

\item[\textbf{(b)}] For each $n\geq2$, we have%
\[
\left(  \rho_{1}-n\right)  \circ\left(  \rho_{1}-\left(  n-2\right)  \right)
^{2}\circ\prod_{i=0}^{n-3}\left(  \rho_{1}-i\right)  ^{n-1-i}%
=0\ \ \ \ \ \ \ \ \ \ \text{on }H_{n}.
\]
(Note: Here and below, $\prod$ is product with respect to $\circ$, not to
$\star$. Same applies to powers.)

For example,%
\begin{align*}
\left(  \rho_{1}-2\right)  \circ\rho_{1}^{2}  &
=0\ \ \ \ \ \ \ \ \ \ \text{on }H_{2},\ \ \ \ \ \ \ \ \ \ \text{and}\\
\left(  \rho_{1}-3\right)  \circ\left(  \rho_{1}-1\right)  ^{2}\circ\rho
_{1}^{2}  &  =0\ \ \ \ \ \ \ \ \ \ \text{on }H_{3}%
,\ \ \ \ \ \ \ \ \ \ \text{and}\\
\left(  \rho_{1}-4\right)  \circ\left(  \rho_{1}-2\right)  ^{2}\circ\left(
\rho_{1}-1\right)  ^{2}\circ\rho_{1}^{3}  &  =0\ \ \ \ \ \ \ \ \ \ \text{on
}H_{4}.
\end{align*}

\end{enumerate}

\item It seems that this polynomial is minimal (in general). However:

\item \textbf{Theorem.} If we assume further that $H$ is commutative, or (even
weaker) that $ab=ba$ for all $a,b\in H_{1}$, then%
\[
\left(  \rho_{1}-n\right)  \circ\prod_{i=0}^{n-2}\left(  \rho_{1}-i\right)
=0\ \ \ \ \ \ \ \ \ \ \text{on }H_{n}%
\]
for any $n\geq0$.

\item More generally:

\textbf{Theorem.} Let $k$ be a positive integer. Assume that every two
elements of $H_{1}+H_{2}+\cdots+H_{k}$ commute. Let $n$ be a positive integer.
Then,%
\[
\prod_{i\in F\left(  n,k\right)  }\left(  \rho_{k}-i\right)
=0\ \ \ \ \ \ \ \ \ \ \text{on }H_{n},
\]
where $F\left(  n,k\right)  $ is a somewhat intricate finite set of integers.

\item \textbf{Question:} Does an unconditional result hold for $\rho_{k}$,
similar to our first theorem for $\rho_{1}$?
\end{itemize}

\subsection{But what else can we say?}

\begin{itemize}
\item These are instances of identities that hold in every connected graded
bialgebra and involve only $m,u,\Delta,\epsilon$ and projections on
homogeneous components. (Recall: $\overline{\operatorname*{id}}%
=\operatorname*{id}-p_{0}$, so that Takeuchi's formula writes $S$ in these terms.)

\item \textbf{Question:} Is there a mechanical way to prove such identities?
(For a fixed $n$, say.)

(We will partly answer this below.)
\end{itemize}

\section{Natural transformations on a graded Hopf algebra}

\subsection{What is our calculus?}

\begin{itemize}
\item The operations we are working with are defined for any graded bialgebra.
They are thus \textbf{natural operations} on a graded bialgebra, i.e., natural
transformations from one of the four forgetful functors%
\begin{align*}
\left\{  \text{graded bialgebras}\right\}   &  \rightarrow\left\{
\text{modules}\right\}  ,\\
\left\{  \text{graded bialgebras}\right\}   &  \rightarrow\left\{
\text{graded modules}\right\}  ,\\
\left\{  \text{connected graded bialgebras}\right\}   &  \rightarrow\left\{
\text{modules}\right\}  ,\\
\left\{  \text{connected graded bialgebras}\right\}   &  \rightarrow\left\{
\text{graded modules}\right\}
\end{align*}
to itself. (These are four different but related settings.)
\end{itemize}

\subsection{Descent operators}

\begin{itemize}
\item How does a typical such operation look like?

\item \textbf{Definition.} A \textbf{weak composition} means a tuple
$\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ of
nonnegative integers.

\textbf{Example:} $\left(  3,0,0,5,1,0\right)  $.

\item \textbf{Definition.} A\textbf{ composition} means a tuple $\alpha
=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ of positive integers.

\textbf{Example:} $\left(  3,5,1\right)  $.

\item \textbf{Definition.} Let $\alpha=\left(  \alpha_{1},\alpha_{2}%
,\ldots,\alpha_{k}\right)  $ be a weak composition, and let $\sigma
\in\mathfrak{S}_{k}$ be a permutation of $\left\{  1,2,\ldots,k\right\}  $
(for the same $k$). Then, for any graded bialgebra $H$, we define a linear map%
\begin{align*}
p_{\alpha,\sigma}  &  =m^{\left[  k\right]  }\circ\left(  p_{\alpha_{1}%
}\otimes p_{\alpha_{2}}\otimes\cdots\otimes p_{\alpha_{k}}\right)  \circ
T_{\sigma}\circ\Delta^{\left[  k\right]  }\\
&  \in\operatorname*{Hom}\left(  H,H\right)  .
\end{align*}
Here, $T_{\sigma}$ is the $\sigma$-twist map%
\begin{align*}
H^{\otimes k}  &  \rightarrow H^{\otimes k},\\
h_{1}\otimes h_{2}\otimes\cdots\otimes h_{k}  &  \mapsto h_{\sigma\left(
1\right)  }\otimes h_{\sigma\left(  2\right)  }\otimes\cdots\otimes
h_{\sigma\left(  k\right)  }.
\end{align*}


\item \textbf{Example.} For $\alpha=\left(  3,5\right)  $ and $\sigma=t_{1,2}$
(the transposition swapping $1$ with $2$), we have%
\[
p_{\alpha,\sigma}=m\circ\left(  p_{3}\otimes p_{5}\right)  \circ
\underbrace{T}_{\text{twist}}\circ\Delta.
\]
Thus, in Sweedler notation,%
\[
p_{\alpha,\sigma}\left(  x\right)  =\sum_{\left(  x\right)  }p_{3}\left(
x_{\left(  2\right)  }\right)  p_{5}\left(  x_{\left(  1\right)  }\right)  .
\]


\item We call $p_{\alpha,\sigma}$ a \textbf{descent operator} or a
\textbf{BPPC operator} (short for \textquotedblleft break, permute, project
and combine\textquotedblright). Note that the $p_{\alpha_{1}}\otimes
p_{\alpha_{2}}\otimes\cdots\otimes p_{\alpha_{k}}$ and $T_{\sigma}$ parts can
be (quasi)commuted:%
\[
\left(  p_{\alpha_{1}}\otimes p_{\alpha_{2}}\otimes\cdots\otimes p_{\alpha
_{k}}\right)  \circ T_{\sigma}=T_{\sigma}\circ\left(  p_{\beta_{1}}\otimes
p_{\beta_{2}}\otimes\cdots\otimes p_{\beta_{k}}\right)
\]
for $\left(  \beta_{1},\beta_{2},\ldots,\beta_{k}\right)  =\left(
\alpha_{\sigma^{-1}\left(  1\right)  },\alpha_{\sigma^{-1}\left(  2\right)
},\ldots,\alpha_{\sigma^{-1}\left(  k\right)  }\right)  $.

\item \textbf{Simple observation.} Any such map $p_{\alpha,\sigma}$ vanishes
on $H_{n}$ unless $n$ is the sum of the entries of $\alpha$.

\item \textbf{Definition.} Given any weak composition $\alpha$, we set%
\[
p_{\alpha}:=p_{\alpha,\operatorname*{id}}%
\]
(where $\operatorname*{id}$ is the identity permutation).

\item \textbf{Simple observation.}

\begin{enumerate}
\item[\textbf{(a)}] We have%
\[
p_{\alpha,\sigma}=p_{\sigma\cdot\alpha}\ \ \ \ \ \ \ \ \ \ \text{if }H\text{
is commutative.}%
\]


\item[\textbf{(b)}] We have
\[
p_{\alpha,\sigma}=p_{\alpha}\ \ \ \ \ \ \ \ \ \ \text{if }H\text{ is
cocommutative.}%
\]

\end{enumerate}

\item Operators of the form $p_{\alpha}$ were studied by Patras and Reutenauer
for commutative or cocommutative $H$. They showed that the span of such
$p_{\alpha}$ operators is closed under both $\circ$ and $\star$. But this is
not true for general $H$. Instead, we need all $p_{\alpha,\sigma}$.

\item \textbf{Simple observation.} Let $\alpha$ be a weak composition of
length $k$, and let $\alpha^{\operatorname*{red}}$ be the result of removing
all zero entries from $\alpha$. Let $\sigma\in\mathfrak{S}_{k}$ be any
permutation. If $H$ is connected, then
\[
p_{\alpha,\sigma}=p_{\alpha^{\operatorname*{red}},\tau}%
\]
for an appropriate permutation $\tau$. (To get $\tau$, find all $i$ such that
$\alpha_{i}=0$, and remove the respective $\sigma\left(  i\right)  $ from
$\sigma$; then standardize.)

\item Thus, if $H$ is connected, all descent operators can be written as
$p_{\alpha,\sigma}$ for (non-weak) compositions $\alpha$.

\item \textbf{Question:} Is it true that any reasonable natural transformation
from the forgetful functor%
\[
\left\{  \text{connected graded bialgebras}\right\}  \rightarrow\left\{
\text{graded modules}\right\}
\]
to itself is an infinite linear combination of $p_{\alpha,\sigma}$'s?

\item \textbf{Remark.} The \textquotedblleft infinite\textquotedblright\ part
is technical; we can always restrict to a given $H_{n}$, and then the
combination will be finite.

\item \textbf{Remark.} \textquotedblleft Graded\textquotedblright\ is
important: Otherwise, for $\mathbf{k}=\mathbb{F}_{p}$, the Frobenius $x\mapsto
x^{p}$ would enter the stage.

\item \textbf{Question:} But perhaps we can still characterize these natural
transformations without gradedness if $\mathbf{k}$ is a field?

\item In practice, all the identities we know can be stated in terms of
$p_{\alpha,\sigma}$ and $\circ$ and $\star$, unless they are conditional.
\end{itemize}

\subsection{Formulas for general descent operators}

\begin{itemize}
\item \textbf{Theorem (product formulas).} Let $\alpha=\left(  \alpha
_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ be a weak composition, and let
$\sigma\in\mathfrak{S}_{k}$ be a permutation.

Let $\beta=\left(  \beta_{1},\beta_{2},\ldots,\beta_{\ell}\right)  $ be a weak
composition, and let $\tau\in\mathfrak{S}_{\ell}$ be a permutation. Then:

\begin{enumerate}
\item[\textbf{(a)}] We have
\[
p_{\alpha,\sigma}\star p_{\beta,\tau}=p_{\alpha\beta,\sigma\oplus\tau},
\]
where $\alpha\beta$ is the concatenation of $\alpha$ and $\beta$ (that is, the
weak composition $\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k},\beta
_{1},\beta_{2},\ldots,\beta_{\ell}\right)  $), whereas $\sigma\oplus\tau$ is
the image of $\left(  \sigma,\tau\right)  $ under the obvious map
$\mathfrak{S}_{k}\times\mathfrak{S}_{\ell}\rightarrow\mathfrak{S}_{k+\ell}$.

\item[\textbf{(b)}] We have%
\[
p_{\alpha,\sigma}\circ p_{\beta,\tau}=\sum_{\substack{\gamma_{i,j}%
\in\mathbb{N}\text{ for all }i\in\left[  k\right]  \text{ and }j\in\left[
\ell\right]  ;\\\gamma_{i,1}+\gamma_{i,2}+\cdots+\gamma_{i,\ell}=\alpha
_{i}\text{ for all }i\in\left[  k\right]  ;\\\gamma_{1,j}+\gamma_{2,j}%
+\cdots+\gamma_{k,j}=\beta_{j}\text{ for all }j\in\left[  \ell\right]
}}p_{\left(  \gamma_{1,1},\gamma_{1,2},\ldots,\gamma_{k,\ell}\right)
,\tau\left[  \sigma\right]  },
\]
where $\tau\left[  \sigma\right]  \in\mathfrak{S}_{k\ell}$ is the permutation
that sends each $\ell\left(  i-1\right)  +j$ (with $i\in\left[  k\right]  $
and $j\in\left[  \ell\right]  $) to $k\left(  \tau\left(  j\right)  -1\right)
+\sigma\left(  i\right)  $.

\textbf{Example:} Using one-line notation for permutations,
\[
p_{\left(  a,b\right)  ,\left[  2,1\right]  }\circ p_{\left(  c,d\right)
,\left[  2,1\right]  }=\sum_{\substack{c_{1}+d_{1}=a;\\c_{2}+d_{2}%
=b;\\c_{1}+c_{2}=c;\\d_{1}+d_{2}=d}}p_{\left(  c_{1},d_{1},c_{2},d_{2}\right)
,\left[  4,2,3,1\right]  }.
\]

\end{enumerate}

\item \textbf{Proof:} easy computation for \textbf{(a)}; multi-page
computation using several lemmas for \textbf{(b)}.

(Featuring the Zolotarev shuffle, known from quadratic reciprocity.)

\item Particular cases of these formulas were found by Patras in 1993 for
commutative $H$ and for cocommutative $H$. A Hopf monoid variant was found by
Aguiar and Mahajan (Chapters 10--11 in \textit{Bimonoids for Hyperplane
Arrangements}, 2020).
\end{itemize}

\subsection{The $p_{\alpha,\sigma}$ are linearly independent}

\begin{itemize}
\item \textbf{Theorem. }Generically, the $p_{\alpha,\sigma}$ are linearly
independent. That is: There is a connected graded Hopf algebra $H$ such that
the family%
\[
\left(  p_{\alpha,\sigma}\right)  _{\substack{k\in\mathbb{N}\text{;}%
\\\alpha\text{ is a composition of length }k\text{;}\\\sigma\in\mathfrak{S}%
_{k}}}
\]
(of endomorphisms of $H$) is $\mathbf{k}$-linearly independent.

\item This $H$ is the free $\mathbf{k}$-algebra with generators
\[
x_{i,j}\qquad\text{for all }i,j\in\mathbb{Z}\text{ satisfying }1\leq i<j,
\]
which are understood to be homogeneous of degree $j-i$. The comultiplication
$\Delta:H\rightarrow H\otimes H$ is given by%
\[
\Delta\left(  x_{i,j}\right)  =\sum_{k=i}^{j}x_{i,k}\otimes x_{k,j},
\]
where $x_{k,k}:=1$.

(\textbf{Remark:} This is a noncommutative version of a unipotent Schur algebra.)
\end{itemize}

\subsection{Universal calculus of $p_{\alpha,\sigma}$ maps}

\begin{itemize}
\item The above theorems allow for mechanical verification of identities for
connected graded bialgebras: Expand in terms of $p_{\alpha,\sigma}$'s (using
the product formulas), and compare coefficients. Of course, this gets more
laborious the higher $n$ is.

\item \textbf{Question.} What about non-connected graded bialgebras?

(This includes bialgebras in general, as those are trivially graded with
$H_{0}=H$.)

\item \textbf{Note.} Such questions would be easy if the respective categories
had free objects. Do they? I don't think so (but the real question is
\textquotedblleft how close can we get\textquotedblright)...
\end{itemize}

\section{The combinatorial Hopf algebra behind this}

\subsection{NSym}

\begin{itemize}
\item Patras's formulas for $p_{\alpha}\circ p_{\beta}$ when $H$ is
commutative or cocommutative can be restated in terms of a combinatorial Hopf
algebra called $\operatorname*{NSym}$.

\item \textbf{Definition.} Let $\operatorname*{NSym}$ be the free algebra with
generators $H_{1},H_{2},H_{3},\ldots$ (that is, the tensor algebra of the free
$\mathbf{k}$-module with basis $\left(  H_{1},H_{2},H_{3},\ldots\right)  $).

(Sorry -- this is standard notation, unrelated to our old $H_{i}$ for the
$i$-th degree component of $H$.)

We make $\operatorname*{NSym}$ into a graded algebra by setting each $H_{i}$
homogeneous of degree $i$.

We make $\operatorname*{NSym}$ into a connected graded bialgebra by setting%
\begin{align*}
\Delta\left(  H_{n}\right)   &  =\sum_{i=0}^{n}H_{i}\otimes H_{n-i}%
\ \ \ \ \ \ \ \ \ \ \text{and}\\
\epsilon\left(  H_{n}\right)   &  =0\ \ \ \ \ \ \ \ \ \ \text{for each }%
n\geq1.
\end{align*}
Here, $H_{0}:=1$.

\item This connected graded bialgebra (thus Hopf algebra)
$\operatorname*{NSym}$ is called the \textbf{Hopf algebra of noncommutative
symmetric functions}, since its abelianization $\operatorname*{NSym}%
\nolimits^{\operatorname*{ab}}$ is the Hopf algebra $\Lambda
=\operatorname*{Sym}$ of symmetric functions.

($\operatorname*{NSym}$ is also called the \textbf{Leibniz--Hopf algebra} by
Hazewinkel, and is denoted $NCSF$ by the French school.)

\item There is a second multiplication defined on $\operatorname*{NSym}$,
called the \textbf{internal product} or \textbf{Kronecker product}. Its
definition needs a notation:

\item \textbf{Definition.} We set%
\[
H_{\alpha}:=H_{\alpha_{1}}H_{\alpha_{2}}\cdots H_{\alpha_{k}}%
\ \ \ \ \ \ \ \ \ \ \text{for any composition }\alpha=\left(  \alpha
_{1},\alpha_{2},\ldots,\alpha_{k}\right)  ,
\]
so that $\left(  H_{\alpha}\right)  _{\alpha\text{ is a composition}}$ is a
basis of the module $\operatorname*{NSym}$.

\item \textbf{Definition. }We define a bilinear operation $\ast$ on
$\operatorname*{NSym}$, called \textbf{internal product}, by setting%
\begin{align*}
&  H_{\beta}\ast H_{\gamma}=\sum_{\substack{A\in\mathbb{N}^{k\times\ell
};\\\operatorname*{row}A=\beta;\\\operatorname*{column}A=\gamma}}H_{\left(
\operatorname*{read}A\right)  ^{\operatorname*{red}}}\\
&  \ \ \ \ \ \ \ \ \ \ \text{for all compositions }\beta=\left(  \beta
_{1},\beta_{2},\ldots,\beta_{k}\right) \\
&  \ \ \ \ \ \ \ \ \ \ \text{and }\gamma=\left(  \gamma_{1},\gamma_{2}%
,\ldots,\gamma_{\ell}\right)  .
\end{align*}
Here, the sum ranges over all $k\times\ell$-matrices $A$ with nonnegative
integer entries such that the row sums of $A$ are $\beta_{1},\beta_{2}%
,\ldots,\beta_{k}$ and the column sums of $A$ are $\gamma_{1},\gamma
_{2},\ldots,\gamma_{\ell}$. The notation $\operatorname*{read}A$ denotes the
weak composition obtained by concatenating the rows of $A$ from top to bottom.

\item This internal product $\ast$ is called \textquotedblleft
internal\textquotedblright\ since it is not graded but rather stays inside a
given degree: i.e.,%
\begin{align*}
\operatorname*{NSym}\nolimits_{n}\ast\operatorname*{NSym}\nolimits_{m}  &
=0\ \ \ \ \ \ \ \ \ \ \text{for }n\neq m,\ \ \ \ \ \ \ \ \ \ \text{and}\\
\operatorname*{NSym}\nolimits_{n}\ast\operatorname*{NSym}\nolimits_{n}  &
\subseteq\operatorname*{NSym}\nolimits_{n}.
\end{align*}


\item As a consequence, $\ast$ has no unity in $\operatorname*{NSym}$, but one
in each component $\operatorname*{NSym}\nolimits_{n}$ and one in the
completion $\widehat{\operatorname*{NSym}}$ (namely, $H_{0}+H_{1}+H_{2}%
+\cdots$).

\item We let $\operatorname*{NSym}\nolimits^{\left(  2\right)  }$ denote the
non-unital algebra $\operatorname*{NSym}$ with product $\ast$.

\item \textbf{Theorem (Patras 1993).} Let $H$ be a cocommutative graded
bialgebra (sorry -- no relation to the $H_{i}\in\operatorname*{NSym}$). Then,
$H$ becomes a left $\operatorname*{NSym}\nolimits^{\left(  2\right)  }%
$-module, by having $H_{\alpha}\in\operatorname*{NSym}\nolimits^{\left(
2\right)  }$ act as $p_{\alpha}$ for every composition $\alpha$.

\item The same applies to commutative $H$ instead of cocommutative $H$; just
replace \textquotedblleft left\textquotedblright\ by \textquotedblleft
right\textquotedblright.

\item This can also be reinterpreted in terms of $\operatorname*{QSym}%
\nolimits^{\left(  2\right)  }$-comodules (this is called the
\textquotedblleft Bernstein homomorphism\textquotedblright\ in Hazewinkel's
terms; see
\url{https://www.cip.ifi.lmu.de/~grinberg/algebra/bernsteinproof.pdf} ).
\end{itemize}

\subsection{The general case}

\begin{itemize}
\item What if $H$ is neither commutative nor cocommutative?

\item \textbf{Definition.} A \textbf{mopiscotion} (please find a better name
for this!) is a pair $\left(  \alpha,\sigma\right)  $, where $\alpha$ is a
composition of length $k$ (for some $k\in\mathbb{N}$) and $\sigma$ is a
permutation in $\mathfrak{S}_{k}$.

Let $\operatorname*{PNSym}$ be the free $\mathbf{k}$-module with basis
$\left(  F_{\alpha,\sigma}\right)  _{\left(  \alpha,\sigma\right)  \text{ is a
mopiscotion}}$.

If $\alpha=\left(  \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right)  $ is a weak
composition and $\sigma\in\mathfrak{S}_{k}$, then we set%
\[
F_{\alpha,\sigma}:=F_{\alpha^{\operatorname*{red}},\tau},
\]
where $\tau$ is obtained from $\sigma$ by removing all $\sigma\left(
i\right)  $ for which $\alpha_{i}=0$ (and standardizing).

Define two multiplications on $\operatorname*{PNSym}$: one \textquotedblleft
external multiplication\textquotedblright\ (which mirrors convolution of
$p_{\alpha,\sigma}$'s) given by%
\[
F_{\alpha,\sigma}\cdot F_{\beta,\tau}=F_{\alpha\beta,\sigma\oplus\tau};
\]
and another \textquotedblleft internal multiplication\textquotedblright%
\ (which mirrors composition of $p_{\alpha,\sigma}$'s) given by%
\[
F_{\alpha,\sigma}\ast F_{\beta,\tau}=\sum_{\substack{\gamma_{i,j}\in
\mathbb{N}\text{ for all }i\in\left[  k\right]  \text{ and }j\in\left[
\ell\right]  ;\\\gamma_{i,1}+\gamma_{i,2}+\cdots+\gamma_{i,\ell}=\alpha
_{i}\text{ for all }i\in\left[  k\right]  ;\\\gamma_{1,j}+\gamma_{2,j}%
+\cdots+\gamma_{k,j}=\beta_{j}\text{ for all }j\in\left[  \ell\right]
}}F_{\left(  \gamma_{1,1},\gamma_{1,2},\ldots,\gamma_{k,\ell}\right)
,\tau\left[  \sigma\right]  }.
\]
Also, define a comultiplication $\Delta$ on $\operatorname*{PNSym}$ by%
\[
\Delta\left(  F_{\alpha,\sigma}\right)  =\sum_{\substack{\beta,\gamma\text{
weak compositions;}\\\text{entrywise sum }\beta+\gamma=\alpha}}F_{\beta
,\sigma}\otimes F_{\gamma,\sigma},
\]
mirroring the formula%
\[
\left(  p_{\alpha,\sigma}\text{ for }H\otimes G\right)  =\sum_{\substack{\beta
,\gamma\text{ weak compositions;}\\\text{entrywise sum }\beta+\gamma=\alpha
}}\left(  p_{\beta,\sigma}\text{ for }H\right)  \otimes\left(  p_{\gamma
,\sigma}\text{ for }G\right)
\]
that holds for any two graded bialgebras $H$ and $G$.

\item If I have not made any mistakes, then:

\textbf{Theorem.} $\operatorname*{PNSym}$ becomes a connected graded Hopf
algebra when equipped with the external multiplication $\cdot$, and a
(non-graded) bialgebra when equipped with the internal multiplication $\ast$.

\item \textbf{Theorem.} Let $\operatorname*{PNSym}\nolimits^{\left(  2\right)
}$ be the nonunital algebra $\operatorname*{PNSym}$ with multiplication $\ast
$. Then, every connected graded bialgebra $H$ becomes a $\operatorname*{PNSym}%
\nolimits^{\left(  2\right)  }$-module, with $F_{\alpha,\sigma}$ acting as
$p_{\alpha,\sigma}$.

\item \textbf{Question:} Check this all.

\item \textbf{Question:} What is the combinatorial meaning of
$\operatorname*{PNSym}$ ?

\item \textbf{Question:} Is there a cancellation-free formula for the antipode
of $\operatorname*{PNSym}$ ?

\item \textbf{Question:} Should we expect any identities that connect the
internal multiplication\ with the external\ multiplication and the coproduct?
Some kind of \textquotedblleft splitting formula\textquotedblright?

\item \textbf{Question:} Does $\operatorname*{PNSym}$ embed into
noncommutative formal power series?

\item \textbf{Remark:} An analogue of $\operatorname*{PNSym}\nolimits^{\left(
2\right)  }$ for Hopf monoids is the \textbf{Janus algebra} of Marcelo Aguiar.
Is there a way to translate results between Hopf monoids and Hopf algebras?
\end{itemize}

\section{Thanks}

\begin{itemize}
\item ... to \textbf{Marcelo Aguiar}, \textbf{Amy Pang}, \textbf{Victor
Reiner} and \textbf{Christophe Reutenauer} for conversations;

\item ... to \textbf{Kurusch Ebrahimi-Fard} and \textbf{Gunnar Fl\o ystad} for
the invitation;

\item ... to \textbf{you} for your patience.

\item Extra kudos if you can make progress on some of the questions!
\end{itemize}


\end{document}