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\author{Darij Grinberg (UMN)
\\ \textit{joint work with Victor Reiner (UMN)
            and Jia Huang (UNK)}
}
\title[Critical groups for Hopf algebra modules]{Critical groups for Hopf algebra modules}

\date{17 April 2017 \\
University of Wisconsin, Madison}

\begin{document}

\frame{\titlepage
\textbf{slides:} \\ {\red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/madison17.pdf}} \\
\textbf{paper:} \\ {\red \url{http://www.cip.ifi.lmu.de/~grinberg/algebra/McKayTensor.pdf}}
            or {\red \arxiv{1704.03778v1}}
}

\begin{frame}
\fti{1. Chip-firing on digraphs}
\begin{center}
{\Huge \bf \sc 1} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf Chip-firing on digraphs}
\end{center}
\vspace{1cm}
References:
\begin{itemize}
\item \href{http://arxiv.org/abs/0801.3306}{\red Alexander E. Holroyd, Lionel Levine, Karola M\'esz\'aros, Yuval Peres, James Propp, David B. Wilson, \textit{Chip-Firing and Rotor-Routing on Directed Graphs}, arXiv:0801.3306}.
\item \href{https://arxiv.org/abs/1601.06849}{\red Georgia Benkart, Caroline Klivans, Victor Reiner, \textit{Chip firing on Dynkin diagrams and McKay quivers}, arXiv:1601.06849}.
%\item \href{http://www.mit.edu/~gaetz/docs/final_thesis.pdf}{\red Christian Gaetz, \textit{Critical groups of McKay-Cartan matrices}, honors thesis 2016}.
\end{itemize}
\end{frame}

\begin{frame}
\fti{Chip-firing on digraphs and the critical group}

\begin{itemize}

\item \textit{Chip-firing} on a loopless
      digraph $D$ is a ``solitaire
      game'' (rigorously: rewriting system, or
      finite state machine).
      A brief definition:
      \begin{itemize}
      \item Start with a finite (nonnegative, integer)
            number of (undistinguishable)
            game chips on each vertex on $D$.
      \item Each move (i.e., step) consists of picking a
            vertex $v$ that has at least as many chips as it
            has outgoing arcs,
            and ``distributing'' chips to its
            out-neighbors
            (i.e., for each arc $a$ having source $v$,
            we move a chip from $v$ to the target of $a$).
            This is called ``firing $v$''.
      \end{itemize}

\item \only<1-5>{\textbf{Example}: \\}
      \only<1>{
      Start with
      \[
      \xymatrix{
       & \red{3} \ar@/^/[rd] \ar@/_/[rd] \\
       \red{1} \ar[ru] & & \red{1} \ar[ll]
      } .
      \]
      (The vertices drawn in red are the ones that
      can be fired.)
      \par Let us fire the top vertex.
      } \only<2>{
      After firing the top vertex, obtain
      \[
      \xymatrix{
       & 1 \ar@/^/[rd] \ar@/_/[rd] \\
       \red{1} \ar[ru] & & \red{3} \ar[ll]
      } .
      \]
      \par Let us fire the bottom left vertex.
      } \only<3>{
      After then firing the bottom left vertex, get
      \[
      \xymatrix{
       & \red{2} \ar@/^/[rd] \ar@/_/[rd] \\
       0 \ar[ru] & & \red{3} \ar[ll]
      } .
      \]
      \par Let us fire the bottom right vertex thrice.
      } \only<4-5>{
      After then firing the bottom right vertex
      thrice, get
      \[
      \xymatrix{
       & \red{2} \ar@/^/[rd] \ar@/_/[rd] \\
       \red{3} \ar[ru] & & 0 \ar[ll]
      } .
      \]
      } \only<5>{
      And so on... this game can (and will) go on
      forever.
      }
      \only<6-8>{\textbf{Another example}: \\}
      \only<6>{
      Start with
      \[
      \xymatrix{
       1 \arebi[r] \arebi[d] & 0 \arebi[d] \\
       \red{2} \arebi[r] & 0
      } .
      \]
      (The vertices drawn in red are the ones that
      can be fired.)
      } \only<7>{
      After firing the bottom left vertex, obtain
      \[
      \xymatrix{
       \red{2} \arebi[r] \arebi[d] & 0 \arebi[d] \\
       0 \arebi[r] & 1
      } .
      \]
      } \only<8>{
      After then firing the top left vertex, get
      \[
      \xymatrix{
       0 \arebi[r] \arebi[d] & 1 \arebi[d] \\
       1 \arebi[r] & 1
      } .
      \]
      No more firing is possible here; the game has
      \textit{terminated}.
      }
      
      \only<9>{
      We see that the chip-firing game will sometimes
      terminate after finitely many steps, but
      sometimes never will.
      There are some nontrivial results
      (Bj\"orner, Lovasz, Shor and others):
      \begin{itemize}
      \item Whether it terminates depends only on the
            starting configuration (not on the choices
            of vertices to fire).
      \item If it terminates, the configuration
            obtained in the end depends only on the
            starting configuration.
      \end{itemize}
      }
      
      \only<10-11>{
      A neater situation is obtained if we fix a
      ``global sink'' $q$ (a vertex reachable from
      every vertex), and disallow firing $q$.
      Then, the game \textbf{always} terminates.
      }
      \only<11>{
      Again, there are remarkable properties
      (see {\red \href{http://arxiv.org/abs/0801.3306}{Holroyd et al.,
      arXiv:0801.3306}}):
      \begin{itemize}
      \item The configuration
            obtained in the end depends only on the
            starting configuration.
      \item ``Sandpile monoid'' and ``sandpile group''.
      \item Relations to Eulerian walks and to
            spanning trees.
      \end{itemize}
      }
\vspace{10cm}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Chip-firing on digraphs: the matrix point of view, 1}

\begin{itemize}

\item We can describe chip-firing on a loopless
      digraph $D$ via the Laplacian of $D$.

\item Label the vertices of $D$ by $1, 2, \ldots, n$.
      %in such a way that the global sink $q$ is $n$.

\item The \textit{Laplacian} of $D$ is the
      $n \times n$-matrix $L$ whose $\tup{i, j}$-th entry is
      \[
      L_{i, j} =
      \begin{cases}
        \deg^+ i , & \text{if } j = i ; \\
         - a_{i, j}, & \text{if } j \neq i
        \end{cases} ,
      \]
      where $\deg^+ i$ is the outdegree of the vertex $i$,
      and $a_{i, j}$ is the number of arcs from $i$ to $j$.

\pause
\item A configuration (i.e., placement of chips on the
      vertices of $D$) is modelled by a row vector
      with $n$ entries (the $i$-th entry being the number
      of chips at vertex $i$).

\pause
\item Firing the vertex $i$ modifies such a vector by
      subtracting the $i$-th row of $L$.

\pause
\item The same holds for the variant where we fix a global
      sink $q$ and never fire it...
\end{itemize}

\end{frame}

\begin{frame}
\fti{Chip-firing on digraphs: the matrix point of view, 2}

\begin{itemize}

\item We can describe chip-firing on a loopless
      digraph $D$ with a global sink $q$
      via the \textbf{reduced} Laplacian of $D$.

\item Label the vertices of $D$ by $1, 2, \ldots, n$
      in such a way that the global sink $q$ is $n$.

\item The \textit{reduced Laplacian} of $D$ is the
      $\tup{n-1} \times \tup{n-1}$-matrix $\overline{L}$
      obtained from $L$ by removing the last row and
      the last column.

\pause
\item A configuration (i.e., placement of chips on the
      vertices of $D$) is modelled by a row vector
      with $n-1$ entries (the $i$-th entry being the number
      of chips at vertex $i$).
      \par We forget the number of chips on the sink here.

\pause
\item Firing the vertex $i$ modifies such a vector by
      subtracting the $i$-th row of $\overline{L}$.
\end{itemize}

\end{frame}

\begin{frame}
\fti{Nonsingular M-matrices, 1}

\begin{itemize}

\item Restating everything in terms of the Laplacian $L$
      and forgetting about the digraph allows us to
      crystallize the important parts of the argument
      and gain further generality.
\end{itemize}

\end{frame}

\begin{frame}
\fti{Nonsingular M-matrices, 2}

\begin{itemize}

\item A \textit{Z-matrix} is an $\ell\times\ell$-matrix
      $C \in \ZZ^{\ell\times\ell}$
      whose off-diagonal entries $C_{i,j}$ (with $i \neq j$)
      are all $\leq 0$.

\pause
\item A \textit{nonsingular M-matrix} is a Z-matrix $C$ whose inverse
      $C^{-1}$ exists and satisfies $C^{-1} \geq 0$. \par
      Here, inequalities between matrices are entrywise.

\pause
\item \textbf{Theorem (Gabrielov, Benkart, Klivans, Reiner, ...?):}
      For a Z-matrix $C$, the following are equivalent:
      \begin{itemize}
      \item $C$ is a nonsingular M-matrix.
      \item $C^T$ is a nonsingular M-matrix.
      \item There exists a column vector $x \in \QQ^\ell$ with
            $x > 0$ and $C x > 0$. (Again, entrywise.)
      \item The ``generalized chip-firing game'' in which we start
            with a row vector $r \geq 0$ and keep subtracting
            rows of $C$ while keeping the vector $\geq 0$
            is confluent (i.e., terminates, and the final state
            depends only on the starting state).
      \end{itemize}

\pause
\item Actually, ``depends only on the starting state'' follows from
      ``Z-matrix'', but termination requires ``nonsingular M-matrix''.
\end{itemize}

\end{frame}

\begin{frame}
\fti{The sandpile monoid}

\begin{itemize}

\item Given a digraph $D$ with a chosen global sink $q$,
      we can define a finite abelian monoid as follows:
      \begin{itemize}
      \item A \textit{chip configuration} is a placement
            of finitely many chips on the vertices of $D$.
            \par
            (Rigorously: a nonnegative integer vector.)
      \item Chips placed on $q$ are ignored.
      \pause
      \item Configurations are added entrywise.
      \item The \textit{stabilization} of a configuration
            $x$ is the configuration obtained from $x$ by
            repeatedly firing vertices ($\neq q$) until
            this no longer becomes possible.
            We call this stabilization $x^\circ$.
      \item A configuration is \textit{stable} if no
            vertex can be fired in it.
      \pause
      \item The \textit{sandpile monoid} of
            $\tup{D, q}$ is the monoid of all stable
            configurations, with monoid operation
            given by $\tup{f, g} \mapsto \tup{f+g}^\circ$.
      \end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\fti{The critical group}

\begin{itemize}

\item Given a digraph $D$ with a chosen global sink $q$,
      we can define a finite abelian group as follows:
      \begin{itemize}
      \item If $M$ is a finite abelian monoid, then the
            intersection of all (nonempty) ideals of $M$
            is a group. (Neat exercise.)
      \item Applied to $M$ being the sandpile monoid of
            $\tup{D, q}$, this yields the
            \textit{critical group} of $\tup{D, q}$.
            (Also known as the \textit{sandpile group}.)
      \end{itemize}

\pause
\item But again, we can also define this in terms of
      the Laplacian:
      Namely, the critical group of $\tup{D, q}$ is
      $K \tup{D, q}
      = \operatorname{coker} \tup{\overline{L}^T}
      = \ZZ^{n-1} / \tup{\overline{L}^T \ZZ^{n-1}}$.

\pause
\item When $D$ is Eulerian, this group does not depend on
      $q$ (up to iso).
      Thus, we call it just $K \tup{D}$.

\item When $D$ is Eulerian, we have
      $\operatorname{coker} \tup{L^T}
      = \underbrace{\ZZ}_{\text{free part}} \oplus
        \underbrace{K \tup{D}}_{\text{torsion part}}$.

\pause
\item Much of chip-firing theory doesn't need a digraph.
      A square matrix over $\ZZ$ is enough...
      and a nonsingular M-matrix is particularly helpful.

\end{itemize}

\end{frame}

\begin{frame}
\fti{2. The critical group of a group character}
\begin{center}
{\Huge \bf \sc 2} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf The critical group of a group character}
\end{center}
\vspace{1cm}
References:
\begin{itemize}
\item \href{https://arxiv.org/abs/1601.06849}{\red Georgia Benkart, Caroline Klivans, Victor Reiner, \textit{Chip firing on Dynkin diagrams and McKay quivers}, arXiv:1601.06849}.
\item \href{http://www.mit.edu/~gaetz/docs/final_thesis.pdf}{\red Christian Gaetz, \textit{Critical groups of McKay-Cartan matrices}, honors thesis 2016}.
\item \href{http://www-users.math.umn.edu/~reiner/Talks/SandpilesAndHopfAlgebras.pdf}{\red Victor Reiner's talk slides}.
\end{itemize}
\end{frame}

\begin{frame}
\fti{The McKay matrix of a representation, 1}

\begin{itemize}

\only<1>{\item Where else can we get nonsingular M-matrices from?}

\pause
\item Let $G$ be a finite group. \par
      Let $S_1, S_2, \ldots, S_{\ell+1}$ be the irreps
      (= irreducible representations) of $G$ over $\CC$.
      Let $\chi_1, \chi_2, \ldots, \chi_{\ell+1}$ be
      their characters.

\pause
\item Fix any representation $V$ of $G$ over $\CC$
      (not necessarily irreducible), and let $\chi_V$ be its character.
      Set $n = \dim V  = \chi_V (e)$.

\pause
\item The \textit{McKay matrix} of $V$ is the
      $\tup{\ell+1}\times\tup{\ell+1}$-matrix $M_V$
      whose $\tup{i, j}$-th entry is
      the coefficient $m_{i, j}$ in the expansion
      \[
      \chi_{S_i \otimes V}
      = \chi_i \chi_V
      = \sum_{j=1}^{\ell+1} m_{i, j} \chi_j .
      \]

\pause
\item We define a further $\tup{\ell+1}\times\tup{\ell+1}$-matrix
      $L_V$ (our ``Laplacian'') by
      $L_V = n I - M_V$. \par
      {\blue \textbf{Warning:} Unlike the digraph case, the matrix
      $L_V$ neither has row sums $0$ nor has column sums $0$!}
\end{itemize}

\end{frame}

\begin{frame}
\fti{The McKay matrix of a representation, 2a: example}

\begin{itemize}

\item \textbf{Example:}
      The symmetric group $\mathfrak{S}_4$ has $5$ irreps
      $S_1, S_2, S_3, S_4, S_5$, corresponding to the partitions
      $\tup{4}, \tup{3, 1}, \tup{2, 2}, \tup{2, 1, 1},
      \tup{1, 1, 1, 1}$, respectively.
      We shall just call them
      $D^4, D^{31}, D^{22}, D^{211}, D^{1111}$ for clarity. \par
      Their characters
      $\chi_0 = \chi_{D^4},
      \ \chi_1 = \chi_{D^{31}},
      \ \chi_2 = \chi_{D^{22}},
      \ \chi_3 = \chi_{D^{211}},
      \ \chi_4 = \chi_{D^{1111}}$ are the rows of the following
      character table:
        $$
        \bordermatrix{
        ~ &e&(ij)&(ij)(kl)&(ijk)&(ijkl) \cr
        \chi_{D^{4}}&1&1&1&1&1 \cr
        \chi_{D^{31}}&3&1&0&-1&-1\cr 
        \chi_{D^{22}}&2&0&-1&2&0 \cr
        \chi_{D^{211}}&3&-1&0&-1&1\cr
        \chi_{D^{1111}}&1&-1&1&1&-1}
        $$
      (these are given by weighted counting of rim hook tableaux,
      according to the Murnaghan-Nakayama rule).

\end{itemize}
\end{frame}

\begin{frame}
\fti{The McKay matrix of a representation, 2b: example}

\begin{itemize}

\item \textbf{Example (cont'd):}
      Let $V = D^{31}$.
      Then, the McKay matrix $M_V$ is
      $$M_V = \left(\begin{matrix}
        0 & 1 & 0 & 0 & 0\\
        \only<1,3,4>{1 & 1 & 1 & 1 & 0}
        \only<2>{\red{1} & \red{1} & \red{1} & \red{1} & \red{0}}
        \\
        \only<1,2,4>{0 & 1 & 0 & 1 & 0}
        \only<3>{\red{0} & \red{1} & \red{0} & \red{1} & \red{0}}
        \\
        0 & 1 & 1 & 1 & 1\\
        0 & 0 & 0 & 1 & 0\\
        \end{matrix}\right) \only<2,3,4>{.}$$
      \only<1>{(these are Kronecker coefficients, since $D^{31}$
      too is irreducible).}
      \only<2>{For example, the {\red second row} is because
      \[
      \chi_{{\red D^{31}} \otimes D^{31}}
      = {\red 1} \chi_{D^4} + {\red 1} \chi_{D^{31}}
        + {\red 1} \chi_{D^{22}} + {\red 1} \chi_{D^{211}}
        + {\red 0} \chi_{D^{1111}} .
      \]
      }
      \only<3>{For example, the {\red third row} is because
      \[
      \chi_{{\red D^{22}} \otimes D^{31}}
      = {\red 0} \chi_{D^4} + {\red 1} \chi_{D^{31}}
        + {\red 0} \chi_{D^{22}} + {\red 1} \chi_{D^{211}}
        + {\red 0} \chi_{D^{1111}} .
      \]
      }
      \only<4>{Hence,
      \[
      L_V = \underbrace{n}_{= \dim V = 3} I - M_V
      = \tup{ \begin{matrix}
      3 & -1 & 0 & 0 & 0\\
        -1 & 2 & -1 & -1 & 0\\
        0 & -1 & 3 & -1 & 0\\
        0 & -1 & -1 & 2 & -1\\
        0 & 0 & 0 & -1 & 3\\
        \end{matrix} }.
      \]
      }

\vspace{10cm}

\end{itemize}

\end{frame}

\begin{frame}
\fti{The critical group of a representation}

\begin{itemize}

\item Let $\overline{L_V}$ be the matrix $L_V$ with
      its row and column corresponding to the trivial irrep removed.
      This is an $\ell \times \ell$-matrix.

\item Define the \textit{critical group} $K \tup{V}$ of $V$ by
      $K \tup{V} = \coker \tup{\overline{L_V}}$.

\item Also,
      $\operatorname{coker} \tup{L_V}
      \cong \ZZ \oplus K \tup{V}$.
      \par That said, $K \tup{V}$ is not always torsion.

\pause
\item In our above \textbf{example},
      \[
      \hspace{-1cm}
      L_V = \tup{ \begin{matrix}
      3 & -1 & 0 & 0 & 0\\
        -1 & 2 & -1 & -1 & 0\\
        0 & -1 & 3 & -1 & 0\\
        0 & -1 & -1 & 2 & -1\\
        0 & 0 & 0 & -1 & 3\\
        \end{matrix} }
      \Longrightarrow
      \overline{L_V}
      = \tup{ \begin{matrix}
         2 & -1 & -1 & 0\\
         -1 & 3 & -1 & 0\\
         -1 & -1 & 2 & -1\\
         0 & 0 & -1 & 3\\
        \end{matrix} } .
      \]
      \only<2>{(Here, we removed the $1$-st row and $1$-st column,
      since they index the trivial irrep.)}
      \only<3>{Hence,
      $K \tup{V} = \coker \tup{\overline{L_V}}
      \cong \ZZ / 4 \ZZ$.}

\pause
\item (Recall that the cokernel of a square matrix $M \in \ZZ^{N \times N}$ is
      $\cong \bigoplus_i \tup{\ZZ / m_i \ZZ}$, where the $m_i$ are the
      diagonal entries in the Smith normal form of $M$.
      This is how the above was computed.)

\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{Facts on the critical group}

\begin{itemize}

\item \textbf{Theorems (Benkart, Klivans, Reiner, Gaetz):}
      \begin{itemize}
      \only<1,2>{\item The column vector
            $\sss = \tup{\dim S_1, \dim S_2, \ldots, \dim S_{\ell+1}}^T$
            belongs to $\ker \tup{L_V}$.
            \par It spans the $\ZZ$-module $\ker \tup{L_V}$ if and
            only if the $G$-representation $V$ is faithful.}
      \pause
      \item \only<2>{Actually, $M_V$ and $L_V$ can be diagonalized: \par
            For each $g \in G$, the vector $\sss\tup{g}
            = \tup{ \chi_{S_1}\tup{g}, \chi_{S_2}\tup{g}, \ldots,
                    \chi_{S_{\ell+1}}\tup{g} }^T$
            (a column of the character table of $G$)
            is an eigenvector of $M_V$ (with eigenvalue
            $\chi_V \tup{g}$)
            and of $L_V$ (with eigenvalue $n - \chi_V \tup{g}$).}
            \pause
            If the $G$-representation $V$ is faithful, then
            $\overline{L_V}$ is a nonsingular M-matrix. \par
            \only<3>{(Hence, a theory of ``chip-firing'' exists.
            Benkart, Klivans and Reiner have further results on this,
            but much is still unexplored. \par
            For \textbf{some} groups $G$ and representations $V$,
            this ``chip-firing'' is equivalent to actual chip-firing
            on certain specific digraphs.
            See Benkart-Klivans-Reiner paper.)}
      \pause
      \item If the $G$-representation $V$ is faithful, then
            \[
            \card{ K\tup{V} }
            = \dfrac{1}{\card{G}}
              \prod\limits_{G\text{-conjugacy class } \ive{g} \neq \ive{e}}
              \tup{n - \chi_V \tup{g}}.
            \]
      \pause
      \item For the regular $G$-representation $\CC G$, we have
            \[
            K \tup{\CC G}
            \cong \tup{\ZZ / n \ZZ}^{\ell-1} .
            \]
            Here, $n = \dim \tup{\CC G} = \card{G}$ and
            $\ell = \tup{\text{number of }G\text{-conjugacy classes}}-1$.
      \end{itemize}

\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{Generalizing?}

\begin{itemize}

\item How to generalize this picture?

\pause
\begin{itemize}
\item \only<1-5>{characteristic-$0$ representations
      $\to$ modular representations.}
      \only<6>{{\blue characteristic-$0$ representations
      $\to$ modular representations.}}

\pause
\item algebraically closed field $\CC$ $\to$ any field.

\pause
\item finite-dimensional $\to$ arbitrary dimension.

\pause
\item \only<1-5>{finite group $\to$ finite-dimensional Hopf algebra.}
      \only<6>{{\blue finite group $\to$ finite-dimensional Hopf algebra.}}
\end{itemize}
      
\pause
\item We shall only study the two {\blue blue} directions.
      (The others are interesting, too!)

\end{itemize}

\end{frame}

\begin{frame}
\fti{3. The critical group of a Hopf algebra module}
\begin{center}
{\Huge \bf \sc 3} \\
\noindent\rule[0.5ex]{\linewidth}{1pt}
{\Large \bf The critical group of a Hopf algebra module}
\end{center}
\vspace{1cm}
References:
\begin{itemize}
\item \href{{http://www.cip.ifi.lmu.de/~grinberg/algebra/McKayTensor.pdf}}{\red Darij Grinberg, Jia Huang, Victor Reiner, \textit{Critical groups for Hopf algebra modules}, arXiv:1704.03778}.
\end{itemize}
\end{frame}


\begin{frame}
\fti{Introducing the Hopf algebra $A$}

\begin{itemize}
\item Let $\FF$ be any algebraically closed field of any characteristic.

\item All $\FF$-vector spaces in the following are
      finite-dimensional. \par
      $\dim$ always means $\FF$-vector space dimension. \par
      $\otimes$ always means $\otimes_\FF$.

\item Let $A$ be a finite-dimensional Hopf algebra over $\FF$.
      This means:
      \begin{itemize}
      \item First of all, $A$ is an $\FF$-algebra.
      \item Also, $A$ is finite-dimensional as an $\FF$-vector space.
      \item Also, $A$ is equipped with
      \begin{itemize}
        \item a comultiplication $\Delta : A \to A \otimes A$,
        \item a counit $\epsilon : A \to \FF$,
        \item an antipode $\alpha : A \to A$
      \end{itemize}
      satisfying certain axioms.
      \end{itemize}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Representations of $A$: generalities}

\begin{itemize}
\item In the following, ``$A$-module'' means ``left $A$-module''.

\item Classical results on representations of $A$:
      \begin{itemize}
      \item There are finitely many simple $A$-modules
            $S_1, S_2, \ldots, S_{\ell+1}$, \par
            and finitely many indecomposable projective $A$-modules
            $P_1, P_2, \ldots, P_{\ell+1}$, \par
            and they can (and will) be labelled in such a way that
            $P_i$ is the projective cover of $S_i$.
      \pause
      \item The left-regular $A$-module $A$ decomposes as a direct sum
            \begin{equation*}
            %\label{left-regular-decomposition}
            A
            \cong \bigoplus_{i=1}^{\ell+1} P_i^{\dim{S_i}}.
            \end{equation*}
      \end{itemize}
      \pause
      \item For an $A$-module $V$, if $[V: S_i]$ denotes the
            multiplicity of $S_i$ as a 
            composition factor of $V$, then  
            \begin{equation*}
            %\label{eq.alg-reps.V:Si}
            [V:S_i] = \dim \Hom_A ( P_i, V ).
            \end{equation*}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Representations of $A$: tensor category structure}

\begin{itemize}
\item So far we have just used the $\FF$-algebra structure on $A$
      (and the algebraic closedness of $\FF$). \par
      What if we take into account the Hopf algebra structure too?

\pause
\item The Hopf algebra structure on $A$ allows us to
      \begin{itemize}
      \item make the tensor product $V \otimes W$ of two $A$-modules $V$
            and $W$ into an $A$-module as well
            (using $\Delta$);
      \pause
      \item define a ``trivial $A$-module'' called $\epsilon$
            (using $\epsilon$); \par
            this trivial $A$-module is $\FF$ as a vector space,
            and thus is simple.
      \pause
      \item make the homspace $\Hom(V, W) = \Hom_\FF(V, W)$
            (any unadorned $\Hom$ sign means $\Hom_\FF$ here and
            henceforth) of two $A$-modules $V$ and $W$ into an
            $A$-module as well (using $\Delta$ and $\alpha$),
            \par
            \pause
            and thus in particular define a ``dual $A$-module''
            $V^\ast$ of an $A$-module $V$ (without switching
            sides).
      \end{itemize}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Hopf algebra examples, 1: the group algebra}

\begin{itemize}
\item \textbf{Example 1:}
      Let $A$ be the group algebra $\FF G$ of a finite group $G$.
      \par This becomes a Hopf algebra by setting
      \[
        \begin{aligned}
        \epsilon(g)&=1,\\
        \Delta(g)&=g \otimes g,\\
        \alpha(g)&=g^{-1}
        \end{aligned}
      \]
      for all $g \in G$.
      \pause
      \par
      The $A$-modules are precisely the representations of $G$;
      the notions of tensor product, trivial module, etc. are the ones
      we know from group representation theory.
      \par
      \pause
      Note that if $\operatorname{char} \FF = 0$, then $A$ is semisimple, so that
      the theory dramatically simplifies (e.g., we have
      $P_i = S_i$ for all $i$).
\end{itemize}

\end{frame}

\begin{frame}
\fti{Hopf algebra examples, 0: the universal enveloping algebra}

\begin{itemize}
\item \textbf{``Example 0'':}
      This example does not really fit into our framework (yet?), but
      is too good to omit: \par
      Let $A$ be the universal enveloping algebra $U \tup{\mathfrak{g}}$ of a Lie algebra $\mathfrak{g}$.
      \par This becomes a Hopf algebra by setting
      \[
        \begin{aligned}
        \epsilon(x)&=0,\\
        \Delta(x)&=x \otimes 1 + 1 \otimes x,\\
        \alpha(x)&=-x
        \end{aligned}
      \]
      for all $x \in \mathfrak{g}$.
      \pause
      \par
      The $A$-modules are precisely the representations of $\mathfrak{g}$;
      the notions of tensor product, trivial module, etc. are the ones
      we know from Lie algebra representation theory.
      \par
      \pause
      Sadly, $A$ is usually infinite-dimensional, and our theory is not
      ready for this.
      (Restricted universal enveloping algebras in characteristic $p$
      do work, though.)
\end{itemize}

\end{frame}

\begin{frame}
\fti{Hopf algebra examples, 2: the generalized Taft Hopf algebra}

\begin{itemize}
\item \textbf{Example 2:}
      Fix integers $m \geq 0$ and $n > 0$ with $m \mid n$.
      Fix a primitive $n$-th root of unity $\omega \in \FF$.
      (Recall we assumed $\FF$ algebraically closed!)
      \pause
      \par
      As an $\FF$-algebra,
      the {\it generalized Taft Hopf algebra} $A=H_{n,m}$
      is given by
      \begin{align*}
      \text{generators} \qquad & g, x; \\
      \text{relations} \qquad & g^n = 1, \qquad x^m = 0, \qquad
                         xg = \omega g x .
      \end{align*}
      \pause
      \only<3>{More conceptual definition: $A$
      is a skew group ring
        $$
        H_{n,m} =\FF[\ZZ/n\ZZ] \ltimes \FF[x]/(x^m)
        $$
        for the cyclic group $\ZZ/n\ZZ=\{e,g,g^2,\ldots,g^{n-1}\}$
        acting on coefficients in a truncated polynomial algebra $\FF[x]/(x^m)$,
        via $gx g^{-1}=\omega^{-1} x$.}
      \pause
      \par
      \only<4>{This $A$ has $\FF$-basis 
        $\{g^i x^j : 0 \leq i < n \text{ and }0 \leq j <m\}$,
        whence $\dim A = mn$.

        It becomes a Hopf algebra by setting
        \[
        \begin{array}{rclcrcl}
        \epsilon(g)&=&1,&            &\epsilon(x)&=&0,\\
        \Delta(g)&=&g \otimes g,&    &\Delta(x)&=&1\otimes x + x \otimes g,\\
        \alpha(g)&=&g^{-1},& &\alpha(x)&=&-\omega^{-1}g^{-1}x.
        \end{array}
        \]}
      \only<5>{This $A$ has $n$ projective indecomposable modules,
        each of dimension $m$, whereas its $n$ simple modules are all
        $1$-dimensional.}

\vspace{10cm}
\end{itemize}

\end{frame}

\begin{frame}
\fti{Hopf algebra examples, 3: a weird one from Radford}

\begin{itemize}
\item \textbf{Example 3:}
      Fix integers $m \geq 0$ and $n > 0$ such that $n$ is
      even and $n$ lies in $\FF^\times$.
      Fix a primitive $n$-th root of unity $\omega \in \FF$.
      (Recall we assumed $\FF$ algebraically closed!)
      \pause
      \par
      As an $\FF$-algebra, $A$
      is given by
      \begin{align*}
      \text{generators} \qquad & g, x_1, x_2, \ldots, x_m; \\
      \text{relations} \qquad & g^n=1, \qquad x_i^2=0, \\
                             & x_i x_j=-x_j x_i, \qquad
                             gx_i g^{-1}= \omega x_i.
      \end{align*}
      \pause
      \only<3>{More conceptual definition: $A$
      is a skew group ring
        $$
        A(n,m) =\FF[\ZZ/n\ZZ] \ltimes \bigwedge_\FF[x_1,\ldots,x_m],
        $$
        for the cyclic group $\ZZ/n\ZZ=\{e,g,g^2,\ldots,g^{n-1}\}$
        acting this time on coefficients in an exterior algebra 
        $\bigwedge_\FF[x_1,\ldots,x_m]$,
        via $gx_i g^{-1}=\omega x_i$.
        }
      \pause
      \par
      \only<4>{This $A$ has $\FF$-basis 
        $\left\{
        g^i x_J:  0 \leq i < n, \  J \subseteq \{1,2,\ldots,m\}
        \right\}
        $
        (where $x_J:=x_{j_1} x_{j_2} \cdots x_{j_k}$ if $J=\{j_1 < j_2<\cdots< j_k\}$),
        whence $\dim A = n 2^m$.

        It becomes a Hopf algebra by setting
        \begin{align*}
        \epsilon\left(g\right) &= 1,
        \qquad
        & \epsilon\left(x_i\right) &= 0, \\
        \Delta\left(g\right) &= g \otimes g,
        \qquad
        & \Delta\left(x_i\right) &= 1 \otimes x_i + x_i \otimes g^{n/2}, \\
        \alpha\left(g\right) &= g^{-1},
        \qquad
        & \alpha\left(x_i\right) &= - x_i g^{n/2}.
        \end{align*}
        }

\vspace{10cm}
\end{itemize}

\end{frame}

\begin{frame}
\fti{The Grothendieck ring $G_0 \tup{A}$}

\begin{itemize}
\item We want to define a ``critical group'' $K \tup{V}$
      of an $A$-module $V$.

\pause
\item Unlike in the case of group representations,
      characters are not very useful;
      in particular, they don't ``characterize'' modules
      much any more.

\pause
\item Instead, there is the \textit{Grothendieck group}
      $G_0(A)$. \par
      It is defined as the $\ZZ$-module with
      \begin{itemize}
      \item generators
      $\ive{V}$ corresponding to all $A$-modules $V$
      (keep in mind: finite-dimensional!),
      \item relations $[U]-[V]+[W] = 0$ for each
      short exact sequence $0 \to U \to V \to W \to 0$
      of $A$-modules.
      \end{itemize}
\pause
\item This $\ZZ$-module $G_0(A)$ has a basis
      $\tup{[S_1],\ldots,[S_{\ell+1}]}$, due to the Jordan-H\"older
      theorem.
\pause
\item The $\ZZ$-module $G_0(A)$ becomes a ring (not always commutative)
      by setting $[V] \cdot [W] = [V \otimes W]$
      and $1 = [\epsilon]$.
\end{itemize}
\end{frame}

\begin{frame}
\fti{The McKay matrix of an $A$-module, 1}

\begin{itemize}

\only<1>{\item Now, let us generalize the McKay matrix of a group
         representation to the case of an $A$-module.}

\pause
\item Fix any $A$-module $V$, and set $n = \dim V$.

\pause
\item The \textit{McKay matrix} of $V$ is the
      $\tup{\ell+1}\times\tup{\ell+1}$-matrix $M_V$
      whose $\tup{i, j}$-th entry is
      the multiplicity $\ive{S_j \otimes V : S_i}$ of the simple
      $A$-module $S_i$ in (a composition series of) $S_j \otimes V$.
      \par In other words, its entries are chosen to satisfy
      \[
      \ive{S_i \otimes V}
      = \ive{S_i} \ive{V}
      = \sum_{j=1}^{\ell+1} \tup{M_V}_{i, j} \ive{S_j}.
      \]
      \pause
      More conceptually:
      The matrix $M_V$ represents the right multiplication by
      $\ive{V}$ as an endomorphism of $G_0\tup{A}$ with respect to
      the basis $\tup{[S_1],\ldots,[S_{\ell+1}]}$.

\pause
\item We define a further $\tup{\ell+1}\times\tup{\ell+1}$-matrix
      $L_V$ (our ``Laplacian'') by
      $L_V = n I - M_V$.
\end{itemize}

\end{frame}

\begin{frame}
\fti{The critical group of an $A$-module}

\begin{itemize}

\item We want to define our critical group $K \tup{V}$ in such a way
      that
      $\operatorname{coker} \tup{L_V}
      \cong \ZZ \oplus K \tup{V}$.
      \par
      How do we pick up a canonical complement to $\ZZ$ ?

\pause
\only<2>{\item It is tempting to again define $\overline{L_V}$ by
      removing the trivial row and trivial column from $L_V$, and set
      $K \tup{V} = \coker \tup{\overline{L_V}}$.
      But that does not work.}

\pause
\item In matrix terms: \par
      Consider the column vector
      $\sss = \tup{\dim S_1, \dim S_2, \ldots, \dim S_{\ell+1}}^T$.
      Then, set $K \tup{V} = \sss^\perp / \im \tup{L_V}$, where
      $\sss^\perp = \left\{\xx \in \ZZ^{\ell+1} \mid \sss^T \xx = 0 \right\}$.
      
\pause
\item More conceptually: \par
      There is an \textit{augmentation map} on $G_0(A) \to \ZZ$.
      This is the ring homomorphism sending each $\ive{V}$ to $\dim V$.
      \par
      Let $I$ be its kernel.

\item Set $K\tup{V} = I / G_0\tup{A} \tup{n - \ive{V}}$. \par
      Note that multiplication by $n - \ive{V}$ corresponds to the
      action of $M_V$, so this makes sense.
\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{Facts on the critical group}

\begin{itemize}

\item \textbf{Theorems (G., Huang, Reiner):}
      \begin{itemize}
      \only<1>{\item
            $\ppp = \tup{\dim P_1, \dim P_2, \ldots, \dim P_{\ell+1}}^T
            \in \ker \tup{L_V}$, \\
            whereas
            $\sss = \tup{\dim S_1, \dim S_2, \ldots, \dim S_{\ell+1}}^T
            \in \ker \tup{\tup{L_V}^T}$.
            \par These vectors span the respective kernels over $\QQ$ if and
            only if the $A$-module $V$ is \textit{tensor-rich}
            (which is our way to say that each simple $A$-module
            appears in a composition series of $V^{\otimes k}$ for
            at least one $k$).}
      \pause
      \item \only<2>{If $A = \FF G$ is a group algebra,
            then $M_V$ and $L_V$ can be diagonalized: \par
            \begin{itemize}
            \item Fix a \textit{Brauer character} $\chi_W$ for each
                  $A$-module $W$.
            \item Given a $p$-regular element $g \in G$, let
                  $\sss(g)= \tup{ \chi_{S_1}(g),\ldots,\chi_{S_{\ell+1}}(g)}^T$
                  be the Brauer character values of the simple
                  $\FF G$-modules at $g$.
                  Then, $\sss(g)$ is an eigenvector of $\tup{M_V}^T$ (with eigenvalue
                  $\chi_V \tup{g}$)
                  and of $\tup{L_V}^T$ (with eigenvalue $n - \chi_V \tup{g}$).
            \item Given a $p$-regular element $g \in G$, let
                  $\ppp(g)= \tup{ \chi_{P_1}(g),\ldots,\chi_{P_{\ell+1}}(g)}^T$
                  be the Brauer character values of the indecomposable
                  projective $\FF G$-modules at $g$.
                  Then, $\ppp(g)$ is an eigenvector of $M_V$ (with eigenvalue
                  $\chi_V \tup{g}$)
                  and of $L_V$ (with eigenvalue $n - \chi_V \tup{g}$).
            \end{itemize}}
            \pause
            If the $A$-module $V$ is tensor-rich, then
            $\overline{L_V}$ is a nonsingular M-matrix. \par
            \only<3>{(Hence, a theory of ``chip-firing'' exists.
            We have not studied it.
            It is complicated by the fact that
            $K\tup{V}$ is not generally isomorphic to
            $\coker \tup{\overline{\tup{L_V}^T}}$ any more.)}
      \pause
      \only<4>{\item Actually, the following are equivalent:
            \begin{itemize}
            \item[(i)] The matrix $\overline{L_V}$ (obtained from
                        $L_V$ by removing the row and the column
                        corresponding to the trivial $A$-module
                        $\varepsilon$) is a nonsingular M-matrix.
            \item[(ii)] The matrix $\overline{L_V}$ is nonsingular.
            \item[(iii)] $L_V$ has rank $\ell$, so nullity $1$.
            \item[(iv)] The critical group $K(V)$ is finite.
            \item[(v)] The $A$-module $V$ is tensor-rich.
            \end{itemize}}
      \pause
      \only<5,6>{\item If the $A$-module $V$ is tensor-rich, then
            \begin{align*}
            & \card{ K\tup{V} }\\
            &= \left| \dfrac{\gamma}{\dim A}
              \tup{\text{product of the nonzero eigenvalues of } L_V}
              \right| ,
            \end{align*}
            where $\gamma = \gcd \tup{ \dim P_1, \dim P_2, \ldots, \dim P_{\ell+1} }$.
            \pause
            \par If $A = \FF G$ is a group algebra, then this can be
            rewritten in a more explicit way using Brauer characters
            as well:
            $$
            \card{K(V)} = \frac{\gamma}{\card{G}} 
            \prod\limits_{\substack{[g] \neq [e] \text{ is a }p\text{-regular} \\ \text{conjugacy class in } G}} \left(n-\chi_V(g)\right).
            $$
            \par Also, $\gamma$ is the size of the
            $p$-Sylow subgroups of $G$ if $\FF$ has characteristic $p > 0$.}
      \pause
      \item For the regular $A$-module $A$, we have
            \begin{align*}
            K \tup{A}
            \cong \tup{\ZZ / \gamma \ZZ} \oplus \tup{\ZZ / n \ZZ}^{\ell-1} .
            \end{align*}
            Here, $n = \dim A$,
            $\gamma = \gcd \tup{ \dim P_1, \dim P_2, \ldots, \dim P_{\ell+1} }$
            and
            $\ell = \tup{\text{number of simple }A\text{-modules}} - 1$.
      \end{itemize}

\end{itemize}

\vspace{10cm}

\end{frame}

\begin{frame}
\fti{Questions}

\begin{itemize}

\item We diagonalized $L_V$ and $M_V$ when $A$ is a group algebra.
      Can we do it in general?

\pause
\item What is the meaning of
      $\gamma = \gcd \tup{ \dim P_1, \dim P_2, \ldots, \dim P_{\ell+1} }$
      for an arbitrary Hopf algebra $A$ ? \par
      Is there a notion of ``Sylow subalgebras'' (presumably
      subalgebras of dimension $\gamma$, which have the form
      $\FF \oplus \tup{\text{nilpotents}}$ and over which $A$ is free
      as a module)?

\pause
\item What can we say about $L_V$ when $V$ is not tensor-rich?

\pause
\item What can we say about $A$ and $V$ if $V$ is not tensor-rich?
      Does $A \to \End V$ factor through a quotient Hopf algebra then?

\pause
\item Does the theory extend to infinite-dimensional $A$ ?
      (Think of universal enveloping algebras -- the finite-dimensional
      $A$-modules can be fairly tame.)

\end{itemize}

\end{frame}

\begin{frame}
\fti{Thank you!}

\textbf{Thanks} to

\begin{itemize}
\item you all for being here,
\item Paul Terwilliger for the invitation,
\item Georgia Benkart, Pavel Etingof, Jim Humphreys, Radha Kessar, Peter J. McNamara, Hans-J\"urgen
Schneider, Peter Webb, and Sarah Witherspoon.
\end{itemize}

\end{frame}

\end{document}
