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\author{Darij Grinberg (UMN)}
\title[$\FqT$-symmetric functions]{Function-field symmetric functions: \\ In search of an $\FqT$-combinatorics}

\date{%27 January 2017, CAAC 2017, Montr\'eal; \\
27 February 2017, Cornell}

\begin{document}

\frame{\titlepage
\textbf{slides:} \\ {\red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/cornell-feb17.pdf}} \\
\textbf{preprint (WIP, and currently a mess):} \\ {\red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/schur-ore.pdf}}
}

\begin{frame}
\fti{Symmetric functions and Witt vectors}

\begin{itemize}
\item The connection between symmetric functions and (big) Witt
vectors is due to Cartier around 1970 (vaguely; made explicit by
Reutenauer in 1995), and can be used to
the benefit of either.

\item Modern references: e.g., Hazewinkel's \textit{Witt vectors,
part 1} ({\red \arxiv{0804.3888v1}}, see also
{\red \href{http://www.cip.ifi.lmu.de/~grinberg/algebra/typos1short.pdf}{errata}}),
and works of James Borger (mainly {\red \arxiv{0801.1691v6}}, as well as
{\red \arxiv{math/0407227v1}} joint with Wieland).

\pause

\item Let $\Nplus = \set{1, 2, 3, \ldots}$. The \textit{(big) Witt
vector functor} is a functor $W : \CRing \to \CRing$, sending any
commutative ring $A$ to a new commutative ring $W(A)$ with some extra
structure.

\item Note that $W(A)$ is a ring, not an $A$-algebra.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of Witt vectors, 1: ghost maps}

\begin{itemize}
\item Let $A$ be a commutative ring.

We abbreviate a family $\tup{a_k}_{k \in \Nplus} \in A^\Nplus$ as
$\mathbf{a}$.
Similarly for other letters.

\item For each $n \in \Nplus$, define a map $w_n : A^\Nplus \to A$ by
\[
w_n \tup{ \mathbf{a} } = \sum_{d \mid n} d a_d^{n/d} .
\]
The map $w_n$ is called the \textit{$n$-th ghost projection}.

\only<1>{
\item \textbf{Examples:}
\begin{itemize}
\item $w_1 = a_1$.

\item If $p$ is a prime, then $w_p = a_1^p + p a_p$.

\item $w_6 = a_1^6 + 2 a_2^3 + 3 a_3^2 + 6 a_6$.
\end{itemize}
}
\only<2-3>{
\item Let $w : A^\Nplus \to A^\Nplus$ be the map given by
\[
w \tup{\mathbf{a}} = \tup{ w_n\tup{\mathbf{a}} }_{n \in \Nplus} .
\]
We call $w$ the \textit{ghost map}.
\pause\pause
\item This ghost map $w$ is not linear and in general not injective or
surjective. \textbf{However}, its image turns out to be a subring of
$A^\Nplus$. It is called the \textit{ring of ghost-Witt vectors}.
}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of Witt vectors, 2: addition}

\begin{itemize}
\item For example, for any $\mathbf{a}, \mathbf{b} \in A^\Nplus$,
we have $w \tup{\mathbf{a}} + w \tup{\mathbf{b}} = w \tup{\mathbf{c}}$
for some $\mathbf{c} \in A^\Nplus$. How to compute this
$\mathbf{c}$ ?

\pause
\item Good news:
\begin{itemize}
\item $w$ is injective if $A$ is torsionfree (as $\ZZ$-module).
\item $w$ is bijective if $A$ is a $\QQ$-vector space.
\end{itemize}

\pause
\item Hence, we can compute $\mathbf{c}$ back from $w \tup{\mathbf{c}}$
by recursion (coordinate by coordinate). Miraculously, the denominators
vanish.

\textbf{Examples:}
\begin{itemize}
\item $w_1(\mathbf{c}) = w_1(\mathbf{a}) + w_1(\mathbf{b})$
$\Longleftrightarrow$
$c_1 = a_1 + b_1$.

\item $w_2(\mathbf{c}) = w_2(\mathbf{a}) + w_2(\mathbf{b})$
$\Longleftrightarrow$
$c_1^2 + 2 c_2 = \tup{a_1^2 + 2 a_2} + \tup{b_1^2 + 2 b_2}$
$\overset{\text{naturality}}{\Longleftrightarrow}$
$c_2 = a_2 + b_2 + \dfrac12 \tup{a_1^2 + b_1^2 - \tup{a_1+b_1}^2}$,
and the RHS is indeed a $\ZZ$-polynomial.
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of Witt vectors, 3: $W(A)$}

\begin{itemize}
\item Let's make a new ring out of this: We define $W(A)$ to be the
ring that equals $A^\Nplus$ as a set, but whose ring structure is such
that $W : \CRing \to \CRing$ is a functor, and
$w$ is a natural (in $A$) ring homomorphism from $W(A)$ to
$A^\Nplus$.

\pause

\item This looks abstract and confusing, but the underlying idea is
simple: Define addition on $W(A)$ so that
$w_n \tup{\mathbf{a} + \mathbf{b}}
= w_n (\mathbf{a}) + w_n (\mathbf{b})$ for all $n$.

Thus, $\mathbf{a} + \mathbf{b}$ is the $\mathbf{c}$ from last page.

\item Functoriality is needed, because there might be several choices for
a given $A$ (if $A$ is not torsionfree), but only one consistent
choice for all rings $A$. Functoriality forces us to pick the consistent
choice.

\pause

\item If $\mathbf{a} \in W(A)$, then the $a_n$ are called the
\textit{Witt coordinates} of $\mathbf{a}$, while the $w_n(\mathbf{a})$
are called the \textit{ghost coordinates} of $\mathbf{a}$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of Witt vectors, 4: coda}

\begin{itemize}

\item The ring $W(A)$ is called the \textit{ring of (big) Witt vectors
over $A$}.

\item The functor $\CRing \to \CRing, \  A \mapsto W(A)$ is called
the \textit{(big) Witt vector functor}.

\pause

\item For any given prime $p$, there is a canonical quotient 
$W_p(A)$ of
$W(A)$ called the \textit{ring of $p$-typical Witt vectors of $A$}.
Number theorists usually care about the latter ring. For example,
$W_p\tup{\Fp} = \ZZ_p$ (the $p$-adics). We have nothing to say about it
here.

\item $W(A)$ comes with more structure: Frobenius and Verschiebung
endomorphisms, a comonad comultiplication map $W(A) \to W(W(A))$, etc.
\end{itemize}

\end{frame}

\begin{frame}
\fti{Avatars of Witt vectors, 1: Power series}

\begin{itemize}

\item There are some equivalent ways to define $W(A)$. Let me show two.

\item One is the Grothendieck construction using power series (see, again,
Hazewinkel, or Rabinoff's {\red \arxiv{1409.7445}}):

\item Let $\Lambda(A)$ be the topological ring defined as
follows:

\begin{itemize}
\item As topological spaces,
$\Lambda(A) = 1 + t A\ive{\ive{t}} = \set{\text{power
series with constant term } 1}$.
\item Addition $\widehat{+}$ in $\Lambda(A)$ is multiplication of
power series.
\item Multiplication $\widehat{\cdot}$ in $\Lambda(A)$ is given by
\[
\tup{1 - at} \widehat{\cdot} \tup{1 - bt} = 1 - abt
\]
(and distributivity and continuity, and naturality in $A$).

\end{itemize}

\pause

\item Canonical ring isomorphism
\[
W(A) \to \Lambda(A),
\qquad
\mathbf{a} \mapsto \prod_{n=1}^\infty \tup{1 - a_n t^n} .
\]

\end{itemize}

\end{frame}

\begin{frame}
\fti{Avatars of Witt vectors, 2: Characters of $\Lambda$ (virtual
alphabets)}

\begin{itemize}

\item Here is another: Let $\Lambda$ be the Hopf algebra of symmetric
functions over $\ZZ$. (No direct relation to $\Lambda(A)$; just traditional
notations clashing.)

\item Define ring $\Alg(\Lambda, A)$ as follows:
\begin{itemize}
\item As set, $\Alg(\Lambda, A) = \set{\text{algebra homomorphisms }
\Lambda \to A}$.
\item Addition = convolution.
\item Multiplication = convolution using the
\textit{second comultiplication} on $\Lambda$ (= Kronecker comultiplication
= Hall dual of Kronecker multiplication).
\end{itemize}

\pause

\only<1-2>{
\item The elements of $\Alg(\Lambda, A)$ are known as
\textit{characters} of $\Lambda$ (as in
{\red \href{http://www.math.cornell.edu/~maguiar/CHalgebra.pdf}{Aguiar-Bergeron-Sottile}})
or \textit{virtual alphabets} (to the Lascoux school) or as
\textit{specializations of symmetric functions} (as in Stanley's EC2).
}
\only<3>{
\item There is a unique family $\tup{w_n}_{n \in \Nplus}$ of symmetric
functions satisfying $p_n = \sum_{d \mid n} d w_d^{n/d}$ for all
$n \in \Nplus$. (Equivalently, it is determined by
$h_n = \sum_{\lambda \vdash n} w_\lambda$, where
$w_\lambda = w_{\lambda_1} w_{\lambda_2} \cdots$.) These are called the
\textit{Witt coordinates}.

\pause

\item We have a ring isomorphism
\[
\Alg(\Lambda, A) \to W(A),
\qquad
f \mapsto \tup{f\tup{w_n}}_{n \in \Nplus} .
\]
}

\end{itemize}

\end{frame}

\begin{frame}
\fti{Avatars of Witt vectors, 2: Characters of $\Lambda$, cont'd}

\begin{itemize}

\item There is a unique family $\tup{w_n}_{n \in \Nplus}$ of symmetric
functions satisfying $p_n = \sum_{d \mid n} d w_d^{n/d}$ for all
$n \in \Nplus$. (Equivalently, it is determined by
$h_n = \sum_{\lambda \vdash n} w_\lambda$, where
$w_\lambda = w_{\lambda_1} w_{\lambda_2} \cdots$.) These generate
$\Lambda$ as a ring, are called the
\textit{Witt coordinates}, and were
first \href{http://dx.doi.org/10.1006/aima.1995.1009}{introduced in 1995 by Reutenauer}.

\item We have a ring isomorphism
\[
\Alg(\Lambda, A) \to W(A),
\qquad
f \mapsto \tup{f\tup{w_n}}_{n \in \Nplus} .
\]

\pause

\item We also have a ring homomorphism (isomorphism when $A$ is a
$\QQ$-algebra)
\[
\Alg(\Lambda, A) \to A^\Nplus,
\qquad
f \mapsto \tup{f\tup{p_n}}_{n \in \Nplus} .
\]

These form a commutative diagram
\[
\xymatrix{
\Alg(\Lambda, A) \ar[r]^\cong \ar[rd] & W(A) \ar[d]^w \\
& A^\Nplus
}
\]

\end{itemize}

\end{frame}

\begin{frame}
\fti{Reconstructing $\Lambda$ from $W = \Alg(\Lambda, -)$}

\begin{itemize}

\item This also works in reverse: We can reconstruct $\Lambda$ from
the functor $W$, as its representing object. Namely:

\begin{itemize}
\item The functor $\operatorname{Forget} \circ W : \CRing \to
\mathbf{Set}$ determines $\Lambda$ as a ring (by Yoneda).

\item The functor $\operatorname{Forget} \circ W : \CRing \to
\mathbf{Ab}$ (additive group of $W(A)$) determines $\Lambda$ as
a Hopf algebra.

\item The functor $W : \CRing \to \CRing$ determines $\Lambda$ as a
Hopf algebra equipped with a second comultiplication.

\item The comonad structure on $W$ additionally determines plethysm
on $\Lambda$.
\end{itemize}

\pause

\item Thus, if symmetric functions hadn't been around, Witt vectors
would have let us rediscover them.
\end{itemize}

\end{frame}

\begin{frame}
\fti{The ghost-Witt integrality theorem (aka Dwork lemma), 1}

\begin{itemize}

\item Assume you don't know about $\Lambda(A)$ or $\Lambda$. How would
you go about proving that the Witt vector functor $W$ exists?
\pause \newline
In other words, why do the denominators (e.g., in the computation of
$\mathbf{c}$ satisfying $w(\mathbf{a}) + w(\mathbf{b})
= w(\mathbf{c})$) ``miraculously'' vanish?

\pause
\item This is a consequence of the \textit{ghost-Witt integrality
theorem}, also known (in parts) as \textit{Dwork's lemma}. I shall
state a (more or less) maximalist version of it; only
the $\mathcal{C} \Longleftrightarrow \mathcal{E}$ part is actually
needed.

\end{itemize}

\end{frame}

\begin{frame}
\fti{The ghost-Witt integrality theorem (aka Dwork lemma), 2}

\begin{itemize}

\item \textbf{Ghost-Witt integrality theorem.}

Let $A$ be a commutative ring. For every $n \in \Nplus$, let
$\varphi_{n}:A\rightarrow A$ be an endomorphism of the ring
$A$. Assume that:

\begin{itemize}
\item We have $\varphi_{p}\left(  a\right)  \equiv
a^{p}\operatorname{mod}pA$ for every $a\in A$ and
every prime $p$.

\item We have $\varphi_{1}=\operatorname*{id}$, and we have
$\varphi_{n}\circ\varphi_{m}=\varphi_{nm}$ for every $n,m\in\Nplus$.
(Thus, $n \mapsto \varphi_n$ is an action of
the multiplicative monoid $\Nplus$ on $A$ by ring endomorphisms.)
\end{itemize}

\pause

[For a stupid example, let $A = \ZZ$ and $\varphi_n = \id$.
\\
For an example that is actually useful to Witt vectors,
let $A$ be a polynomial ring over $\ZZ$, and let $\varphi_n$ send each
indeterminate to its $n$-th power.]

\pause

\item Let $\mathbf{b} = \tup{ b_n }_{n \in \Nplus} \in A^{\Nplus}$ be
a sequence of elements of $A$. Then, the following assertions are
equivalent: [continued on next page]

\end{itemize}
\end{frame}

\begin{frame}
\fti{The ghost-Witt integrality theorem (aka Dwork lemma), 3}

\begin{itemize}

\item \textbf{Ghost-Witt integrality theorem, continued.}

The following are equivalent:
\begin{enumerate}
\item[$\mathcal{C}$:] Every $n \in \Nplus$ and every prime divisor $p$
of $n$ satisfy
\[
\varphi_{p}\left(  b_{n / p}\right)  \equiv b_{n}\operatorname{mod}%
p^{v_{p}\left(  n\right)  }A
\]
(where $v_p(n)$ is the multiplicity of $p$ in the factorization of $n$).

\pause
\item[$\mathcal{D}$:] There exists a sequence $\mathbf{x}
= \tup{ x_n }_{n \in \Nplus} \in A^{\Nplus}$ of elements of $A$ such
that
\[
b_{n} = \sum_{d\mid n}dx_{d}^{n / d}
= w_n \tup{\mathbf{x}}
\text{ for every }n\in \Nplus .
\]
In other words, $\mathbf{x}$ belongs to the image of the ghost map
$w$.

\pause
\item[$\mathcal{E}$:] There exists a sequence $\mathbf{y}
= \tup{ y_n }_{n \in \Nplus} \in A^{\Nplus}$ of elements of $A$ such
that 
\[
b_{n} = \sum_{d\mid n}d\varphi_{n / d}\left(  y_{d}\right)  \text{ for
every }n\in \Nplus .
\]
\end{enumerate}
\end{itemize}
\end{frame}

\begin{frame}
\fti{The ghost-Witt integrality theorem (aka Dwork lemma), 4}

\begin{itemize}

\item \textbf{Ghost-Witt integrality theorem, continued.}

\begin{enumerate}
\item[$\mathcal{F}$:] Every $n \in \Nplus$ satisfies
\[
\sum_{d\mid n}\mu\left(  d\right)  \varphi_{d}\left(  b_{n / d}\right)  \in
nA.
\]

\pause
\item[$\mathcal{G}$:] Every $n \in \Nplus$ satisfies
\[
\sum_{d\mid n}\phi\left(  d\right)  \varphi_{d}\left(  b_{n / d}\right)  \in
nA.
\]

\pause
\item[$\mathcal{J}$:] There exists a ring homomorphism from the
ring $\Lambda$ to $A$ which sends $p_{n}$ (the $n$-th power sum symmetric
function) to $b_{n}$ for every $n\in \Nplus$.
\end{enumerate}

\pause

\item Note that this theorem has various neat consequences, like the
famous necklace divisibility $n \mid \sum\limits_{d \mid n} \mu(d) q^{n/d}$
for $n \in \Nplus$ and $q \in \ZZ$. (And various generalizations.)
\end{itemize}
\end{frame}

\begin{frame}
\fti{$\ZZ$ and $\FqT$: a tale of two rings}

Now to something completely different...

\begin{itemize}

\item Fix a prime power $q$.

\item There is a famous analogy between the elements of $\ZZ$ and the
elements of $\FqT$. (This is related to $q$-enumeration, the lore of
the field with $1$ element, etc.)

All that matters to us is that
\begin{itemize}
\item \textbf{positive} integers in $\ZZ$ correspond to
\textbf{monic} polynomials in $\FqT$;
\item \textbf{primes} in $\ZZ$ correspond to \textbf{irreducible}
monic polynomials in $\FqT$.
\end{itemize}

\pause

\item Let $\FqT_+$ be the set of all \textbf{monic} polynomials in
$\FqT$.

\pause

\item Let's define an analogue of (big) Witt vectors for $\FqT$
instead of $\ZZ$.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of $\FqT$-Witt vectors, 1: ghost maps}

\begin{itemize}

\item Let $A$ be a commutative $\FqT$-algebra.

We abbreviate a family $\tup{a_N}_{N \in \FqT_+} \in A^{\FqT_+}$ as
$\mathbf{a}$.

\item For each $N \in \FqT_+$, define a map $w_N : A^{\FqT_+} \to A$ by
\[
w_N \tup{ \mathbf{a} } = \sum_{D \mid N} D a_D^{q^{\deg(N/D)}} ,
\]
where the sum is over all \textbf{monic} divisors $D$ of $N$.

\item Let $w : A^{\FqT_+} \to A^{\FqT_+}$ be the map given by
\[
w \tup{\mathbf{a}} = \tup{ w_N\tup{\mathbf{a}} }_{N \in {\FqT_+}} .
\]

\pause
\item This ``ghost map'' $w$ is $\Fq$-linear, but not $\FqT$-linear.
\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of $\FqT$-Witt vectors, 2: $W_q(A)$}

\begin{itemize}
\item Let's make a new $\FqT$-algebra out of this: We define $W_q(A)$ to be the
$\FqT$-algebra
\begin{itemize}
\item that equals $A^{\FqT_+}$ as a set, but
\item which is functorial in $A$ (that is, we are really
defining a functor
$W_q : \CRing_{\FqT} \to \CRing_{\FqT}$, where $\CRing_R$ is the
category of commutative $R$-algebras), and
\item whose $\FqT$-algebra structure is such
that $w$ is a natural (in $A$) homomorphism of $\FqT$-algebras from $W_q(A)$ to
$A^{\FqT_+}$.
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of $\FqT$-Witt vectors, 2: $W_q(A)$, cont'd}

\begin{itemize}

\item \textbf{Example:} The addition in $W_q(A)$ is the same as in
$A^{\FqT_+}$ (since $w$ is $\Fq$-linear, and so
$W_q(A) = A^{\FqT_+}$ as $\Fq$-modules), so this would be boring.
\pause
Instead, let's set $\mathbf{c} = T\mathbf{a}$ in $W_q(A)$, and compute
$w_\pi(\mathbf{c})$ for an \textbf{irreducible} $\pi$. \\
\pause
Start with $c_1 = T a_1$, which is easy to check. \\
\pause
$w_\pi(\mathbf{c}) = T w_\pi(\mathbf{a})$ \\
$\Longleftrightarrow$
$c_1^{q^{\deg \pi}} + \pi c_\pi
= T a_1^{q^{\deg \pi}} + T \pi a_\pi$ \\
$\overset{c_1 = Ta_1}{\Longleftrightarrow}$
$\tup{T a_1}^{q^{\deg \pi}} + \pi c_\pi
= T a_1^{q^{\deg \pi}} + T \pi a_\pi$ \\
$\Longleftrightarrow$
$\pi c_\pi = T \pi a_\pi - \tup{T^{q^{\deg \pi}} - T} a_1^{q^{\deg \pi}}$
\\
$\overset{\text{naturality}}{\Longleftrightarrow}$
$c_\pi = T a_\pi - \dfrac{T^{q^{\deg \pi}} - T}{\pi} a_1^{q^{\deg \pi}} $.
\\
The fraction on the RHS is a polynomial due to a known fact from
Galois theory (namely: $T^{q^k} - T = \prod\limits_{\gamma \in \FqT_+ \text{ irreducible};\  \deg \gamma \mid k} \gamma$).

\end{itemize}

\end{frame}

\begin{frame}
\fti{Definition of $\FqT$-Witt vectors, 3: coda}

\begin{itemize}

\item There is also a second construction of $W_q(A)$, using Carlitz
polynomials, yielding an isomorphic $\FqT$-algebra. (See the preprint.)

\end{itemize}

\end{frame}

\begin{frame}
\fti{Avatars of $\FqT$-Witt vectors?}

\begin{itemize}

\item Can we find anything similar to the two avatars of $W(A)$ ?

\pause

\item Power series? This appears to require a notion of power series
where the exponents are polynomials in $\FqT$. Product ill-defined
due to lack of actual ``positivity''. Seems too much to wish...

\pause
\item $\Alg(\Lambda, A)$? Well, we can try brute force: Remember how
$\Lambda$ was reconstructed from $W$, and do something similar to
``reconstruct'' a representing object from $W_q$. We'll come back to
this shortly.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Surprise: $\mathcal{F}$-modules, 1}

\begin{itemize}

\item First, a surprise...
\pause

\item We aren't using the whole $\FqT$-algebra structure on $A$ !

(This is unlike the $\ZZ$-case, where it seems that we use the
commutative ring $A$ in full.)

\end{itemize}

\end{frame}

\begin{frame}
\fti{Surprise: $\mathcal{F}$-modules, 2}

\begin{itemize}

\item Let $\mathcal{F}$ be the noncommutative ring
\[
\Fq\left< F, T \mid FT = T^q F \right> .
\]
This is an $\Fq$-vector space with basis $\tup{T^i F^j}_{\tup{i, j}
\in \NN^2}$, and is an Ore polynomial ring. It shares many properties
of usual univariate polynomials (see papers of Ore).

\pause

\only<1-2>{\item Actually,
\[
\mathcal{F} \cong \tup{\FqT\ive{X}_{q-\operatorname{lin}}, +, \circ} ,
\]
where $\FqT\ive{X}_{q-\operatorname{lin}}$ are the polynomials in $X$
over $\FqT$ where $X$ occurs only with exponents $q^k$, and where
$\circ$ is composition of polynomials.}
\only<3-4>{\item What matters to us:

Each commutative $\FqT$-algebra canonically becomes a (left)
$\mathcal{F}$-module by having
\begin{itemize}
\item $T$ act as multiplication by $T$, and
\item $F$ act as the Frobenius (i.e., taking $q$-th powers).
\end{itemize}

Thus, we have a functor $\CRing_{\FqT} \to \mathbf{Mod}_{\mathcal{F}}$.
}

\only<4>{\item There are other sources of $\mathcal{F}$-modules too
(cf. Jacobson on ``commutative restricted Lie algebras'').}

\end{itemize}

\end{frame}

\begin{frame}
\fti{$\FqT$-Witt vectors of an $\mathcal{F}$-module}

\begin{itemize}

\item Let $A$ be a (left) $\mathcal{F}$-module.

We abbreviate a family $\tup{a_N}_{N \in \FqT_+} \in A^{\FqT_+}$ as
$\mathbf{a}$.

\item For each $N \in \FqT_+$, define a map $w_N : A^{\FqT_+} \to A$ by
\[
w_N \tup{ \mathbf{a} } = \sum_{D \mid N} D F^{\deg(N/D)} a_D ,
\]
where the sum is over all \textbf{monic} divisors $D$ of $N$.

\item Let $w : A^{\FqT_+} \to A^{\FqT_+}$ be the map given by
\[
w \tup{\mathbf{a}} = \tup{ w_N\tup{\mathbf{a}} }_{N \in {\FqT_+}} .
\]

\pause

\item We define $W_q(A)$ to be the
$\mathcal{F}$-module
\begin{itemize}
\item that equals $A^{\FqT_+}$ as a set, but
\item which is functorial in $A$ (that is, we are really
defining a functor
$W_q : \mathbf{Mod}_{\mathcal{F}} \to \mathbf{Mod}_{\mathcal{F}}$), and
\item whose $\mathcal{F}$-module structure is such
that $w$ is a natural (in $A$) homomorphism of $\mathcal{F}$-modules from $W_q(A)$ to
$A^{\FqT_+}$.
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{An $\mathcal{F}$-ghost-Witt integrality theorem, 1}

\begin{itemize}

\item Again, there is a ``ghost-Witt integrality theorem'' that helps
prove the existence of the $W_q$ functors.

\end{itemize}

\end{frame}

\begin{frame}
\fti{An $\mathcal{F}$-ghost-Witt integrality theorem, 2}

\begin{itemize}

\item \textbf{$\mathcal{F}$-ghost-Witt integrality theorem.}

Let $A$ be a (left) $\mathcal{F}$-module. For every $P \in \FqT_+$, let
$\varphi_{P}:A\rightarrow A$ be an endomorphism of the $\mathcal{F}$-module
$A$. Assume that:

\begin{itemize}
\item We have $\varphi_{\pi}\left(  a\right)  \equiv
F^{\deg\pi} a \operatorname{mod} \pi A$ for every $a\in A$ and
every monic irreducible $\pi \in \FqT_+$.

\item We have $\varphi_{1}=\operatorname*{id}$, and we have
$\varphi_{N}\circ\varphi_{M}=\varphi_{NM}$ for every $N,M\in\FqT_+$.
(Thus, $N \mapsto \varphi_N$ is an action of
the multiplicative monoid $\FqT_+$ on $A$ by $\mathcal{F}$-module endomorphisms.)
\end{itemize}

\pause

\item Let $\mathbf{b} = \tup{ b_n }_{n \in \Nplus} \in A^{\FqT_+}$ be
a family of elements of $A$. Then, the following assertions are
equivalent: [continued on next page]

\end{itemize}
\end{frame}

\begin{frame}
\fti{An $\mathcal{F}$-ghost-Witt integrality theorem, 3}

\begin{itemize}

\item \textbf{Ghost-Witt integrality theorem, continued.}

The following are equivalent:
\begin{enumerate}
\item[$\mathcal{C}$:] Every $N \in \FqT_+$ and every monic irreducible
divisor $\pi$ of $N$ satisfy
\[
\varphi_{\pi}\left(  b_{N / \pi}\right)  \equiv b_N
\operatorname{mod} \pi^{v_{\pi}\left( N \right)  }A.
\]

\pause

\only<1,2,4>{
\item[$\mathcal{D}_2$:] There exists a family $\mathbf{x}
= \tup{ x_N }_{N \in \FqT_+} \in A^{\FqT_+}$ of elements of $A$ such
that
\[
b_N = \sum_{D\mid N} D F^{\deg(N/D)} x_D
= w_N \tup{\mathbf{x}}
\text{ for every } N\in \FqT_+ .
\]
In other words, $\mathbf{x}$ belongs to the image of the ghost map
$w$.}

\pause
\only<3>{
\item[$\mathcal{D}_1$:] There exists a family $\mathbf{x}
= \tup{ x_N }_{N \in \FqT_+} \in A^{\FqT_+}$ of elements of $A$ such
that
\begin{align*}
& b_N = \sum_{D\mid N} D \dfrac{N}{D}\ive{T+F} x_D
\qquad\text{ for every } N\in \FqT_+ .
\end{align*}
[This is mainly interesting due to the connection to Carlitz
polynomials.]}

\pause
\only<1,2,3,4>{
\item[$\mathcal{E}$:] There exists a family $\mathbf{y}
= \tup{ y_N }_{N \in \FqT_+} \in A^{\FqT_+}$ of elements of $A$ such
that 
\[
b_N = \sum_{D\mid N}D\varphi_{N / D}\left(  y_D\right)  \text{ for
every }N\in \FqT_+ .
\]
}
\end{enumerate}
\end{itemize}
\end{frame}

\begin{frame}
\fti{An $\mathcal{F}$-ghost-Witt integrality theorem, 4}

\begin{itemize}

\item \textbf{Ghost-Witt integrality theorem, continued.}

\begin{enumerate}
\item[$\mathcal{F}$:] Every $N \in \FqT_+$ satisfies%
\[
\sum_{D\mid N}\mu\left(  D\right)  \varphi_{D}\left(  b_{N / D}\right)  \in
NA.
\]
Here, $\mu$ is an $\FqT$-version of the M\"obius function, defined
as the usual one (i.e., squarefree $\mapsto$ number of distinct
irreducible factors; non-squarefree $\mapsto$ 0).

\pause
\item[$\mathcal{G}$:] Every $N \in \FqT_+$ satisfies%
\[
\sum_{D\mid N}\phi\left(  D\right)  \varphi_{D}\left(  b_{N / D}\right)  \in
NA ,
\]
where $\phi$ is one of two reasonable $\FqT$-versions of the Euler
totient function.

\pause
\item[$\mathcal{J}$:] ???
\end{enumerate}

\pause

\item To state $\mathcal{J}$, we need an $\FqT$-analogue of the
symmetric functions.
\end{itemize}
\end{frame}

\begin{frame}
\fti{Tinfoil, 1: What is $\Lambda_{\mathcal{F}}$ ?}

\begin{itemize}
\item Now, back to the question: We have found two functors
\begin{align*}
W_q &: \CRing_{\FqT} \to \CRing_{\FqT}
\qquad \text{and} \\
W_q &: \mathbf{Mod}_{\mathcal{F}} \to \mathbf{Mod}_{\mathcal{F}} .
\end{align*}
What are their representing objects? Call them $\Lambda'_{\mathcal{F}}$
and $\Lambda_{\mathcal{F}}$.

\pause
\item Both objects (they are distinct) have good claims on the name
``$\FqT$-symmetric functions''.

\pause
\item I shall focus on $\Lambda_{\mathcal{F}}$, since it is smaller.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Tinfoil, 2: ``Re''constructing $\Lambda_{\mathcal{F}}$}

\begin{itemize}

\item Proceed in the same way as when we reconstructed $\Lambda$ from
the functor $W$, but now reconstruct the representing object
$\Lambda_{\mathcal{F}}$ of the functor
$W_q : \mathbf{Mod}_{\mathcal{F}} \to \mathbf{Mod}_{\mathcal{F}}$:

\begin{itemize}
\item The functor $\operatorname{Forget} \circ W_q : \mathbf{Mod}_{\mathcal{F}} \to
\mathbf{Set}$ determines $\Lambda_{\mathcal{F}}$ as an $\mathcal{F}$-module
(by Yoneda).

\item The functor $W_q : \mathbf{Mod}_{\mathcal{F}} \to \mathbf{Mod}_{\mathcal{F}}$
determines $\Lambda_{\mathcal{F}}$ as an
$\mathcal{F}$-$\mathcal{F}$-bimodule.

\item There is an additional comonad structure on $W_q$, which determines
a ``plethysm'' on $\Lambda_{\mathcal{F}}$, but I know nothing
about it.
\end{itemize}

\pause

\item So what is this $\Lambda_{\mathcal{F}}$ ?

\pause

\item I don't really know.
\end{itemize}

\end{frame}

\begin{frame}
\fti{Tinfoil, 3: Some computations inside $\Lambda_{\mathcal{F}}$}

\begin{itemize}

\item At least, we can compute in $\Lambda_{\mathcal{F}}$ (in theory):

\item The left $\mathcal{F}$-module $\Lambda_{\mathcal{F}}$ has a basis
$\tup{w_N}_{N \in \FqT_+}$, similarly to the generating set
$\tup{w_n}_{n \in \Nplus}$ of the commutative ring $\Lambda$.

\item The left $\mathcal{F}$-module $\Lambda_{\mathcal{F}}$ has an
``almost-basis''
$\tup{p_N}_{N \in \FqT_+}$, similarly to the ``almost-generating set''
$\tup{p_n}_{n \in \Nplus}$ of the commutative ring $\Lambda$.

Here, ``almost-basis'' means ``basis after localizing so that
elements of $\FqT_+$ become invertible''. (Noncommutative localization,
but a harmless case thereof.)

\pause

\item The right $\mathcal{F}$-module structure is easily expressed on
the $p_N$'s (just as the second comultiplication of $\Lambda$ is easily
expressed on the $p_n$'s):
\[
p_N f = f p_N \qquad \text{for all } f \in \mathcal{F} \text{ and }
N \in \FqT_+ .
\]

\item You can thus express $q f$ for each $f \in \mathcal{F}$ and
$q \in \Lambda_{\mathcal{F}}$ by recursion (all fractions will turn out
polynomial at the end), but nothing really explicit.

\end{itemize}

\end{frame}

\begin{frame}
\fti{Tinfoil, 4: Examples}

\begin{itemize}
\item Here are some of these expressions:
\begin{align*}
w_{\pi}T &= Tw_{\pi} - \dfrac{T^{q^{\deg \pi}} - T}{\pi} F^{\deg \pi} w_1 ; \\
w_{\pi}F &= \pi^{q-1}Fw_{\pi}
\end{align*}
for any irreducible $\pi \in \FqT_+$.

\end{itemize}
\end{frame}

\begin{frame}
\fti{Bonus oddity (back in $\ZZ$): a ``ghost-Burnside theorem''}

\begin{itemize}
\item This is not $\FqT$-related, but I find it curious.

\item
\only<1,2>{Remember how the ghost-Witt equivalence theorem generalizes
the divisibility $n \mid \sum\limits_{d \mid n} \mu(d) q^{n/d}$
for $n \in \Nplus$ and $q \in \ZZ$:}
\only<3,4,5>{The following strange equivalence also generalizes the
divisibility $n \mid \sum\limits_{d \mid n} \mu(d) q^{n/d}$
for $n \in \Nplus$ and $q \in \ZZ$:}

\pause

\only<1,2>{
\item \textbf{Ghost-Witt:} The following (among others) are equivalent:
\begin{enumerate}
\item[$\mathcal{C}$:] Every $n \in \Nplus$ and every prime divisor $p$
of $n$ satisfy
\[
\varphi_{p}\left(  b_{n / p}\right)  \equiv b_{n}\operatorname{mod}%
p^{v_{p}\left(  n\right)  }A
\]
(where $v_p(n)$ is the multiplicity of $p$ in the factorization of $n$).

\item[$\mathcal{G}$:] Every $n \in \Nplus$ satisfies
\[
\sum_{d\mid n}\phi\left(  d\right)  \varphi_{d}\left(  b_{n / d}\right)  \in
nA.
\]
\end{enumerate}
[Remember that you can pick $\varphi_n = \id$ when $A = \ZZ$.]
}
\pause
\only<3,4,5>{
\item \textbf{Ghost-Burnside:} The following are equivalent:
\begin{enumerate}
\item[$\mathcal{R}$:]
\only<3,5>{Every $n \in \Nplus$, every $d \mid n$ and
every prime divisor $p$ of $d$ satisfy
\[
\phi\left(d\right) b_d^{n/d} \equiv
\phi\left(d\right) b_{d/p}^{n/(d/p)}
\mod p^{v_{p}\left(  n\right)  }A
\]
(where $v_p(n)$ is the multiplicity of $p$ in the factorization of $n$).}
\only<4>{When $A$ is ``nice'' (viz., $px \in p^k A \Longrightarrow
x \in p^{k-1} A$, and the quotient ring $A/pA$ is reduced), this simplifies
to: Every $d \in \Nplus$ and every prime divisor $p$
of $d$ satisfy
\[
b_{d/p}^p \equiv b_d \mod p A .
\]
}

\item[$\mathcal{S}$:] Every $n \in \Nplus$ satisfies
\[
\sum_{d\mid n}\phi\left(  d\right)  b_d^{n/d}
\in nA.
\]
\end{enumerate}
\pause \pause
\item This leads to a notion of ``ghost-Burnside vectors'', which also
form a subring of $A^\Nplus$. Not sure yet what they are good for...
}

\end{itemize}

\end{frame}

\begin{frame}

\textbf{Thanks} to James Borger for some inspiring discussions.
Thanks to Christophe Reutenauer for historiographical comments.

And thank you!

\end{frame}

\end{document}
