grad home
projects home
up
| Introductory notes on partition categories
| WORK IN PROGRESS!!!
%\setcounter{section}{1}
%\setcounter{subsection}{4}
\subsection{Partitions}
%\newcommand{\EE}{{\mathbf E}}
Here $S$ is a set and $2^S$ the power set of $S$
(so $2^{S \times S}$ is the set of relations on $S$ to $S$).
Let
\[
TC: 2^{ S\times S } \rightarrow 2^{ S\times S }
\]
be reflexive, symmetric transitive (RST) closure.
That is, $TC(\rho )$ is the smallest equivalence relation containing
relation $\rho$ on $S$ to $S$.
$\EE(S)$ denotes the set of partitions on set $S$.
Note that $\EE(S)$ is in natural bijection
with the set of equivalence relations on $S$. It will be convenient to
treat this bijection as an identification.
Example: The partition $\{\{ 1,2 \} , \{ 3 \}\} $
is written $ \{ (1,2) , (2,1), (1,1) ,(2,2) , (3,3) \}$ as a relation.
$\EE(S,T) := \EE(S^1 \cup T^0)$
Recall that $\underline{n} := \{ 1,2,..., n \}$ and define
$\EE(n,m) = \EE(\underline{n} , \underline{m})$.
Example: $\EE(2,0) = \{ \{ \{ 1^1, 2^1 \} \} , \{ \{ 1^1 \} , \{ 2^1 \}\}\}$.
If $T \subset S$ and $p \in \EE(S)$ then $p|_T$
(or $p$ regarded as a partition of $T$) denotes the restriction of $p$ to $T$.
%$\nle$
\subsubsection{On graphs and partitions}
Given a set $S$, let
\[
\pp: \Gamma(S) \rightarrow \EE(S)
\]
denote the map which takes graph $g$ to the partition into connected
components.
For $T \subset S$ let
\[
\pp: \Gamma(S) \rightarrow \EE(T)
\]
be $\pp(g) = \pp(g)|_T$.
\subsubsection{On pictorial realisations of partitions}
A diagram for $q \in \EE(S)$ is any
relative graph $g$ such that $\pp(g) =q$,
or $\pp(g) |_S = q$.
(Usually we have in mind a plane drawn picture of $g$ as
above.)
Among the graphs $g$ in $\pp^{-1} (q)$ are ones for which the structure map from $S$ is injective.
In `drawing' partitions
we will usually use pictures of graphs of this type,
so that each $s \in S$ labels a different vertex.
(Although of course if $s,t \in S$ did `label' the same vertex this would simply put them in the same component and hence in the same part.)
Indeed, by a picture of a partition $q$ we will generally mean
such a picture.
\begin{figure}
\caption{Two pictures encoding the same example partition from $\EE(7,3) = \EE(\{ 1,2,..., 7 \} , \{ 1,2,3 \} )$.}
\includegraphics{xfig/partit-low2ab.gif}
\end{figure}
%\includegraphics{xfig/partit-low2a.gif}
Let us unpack the drawing conventions
a little further in case of a partition from some $\EE(n,m)$.
In this case each of the subsets of vertices has a natural order on it.
If we are indeed drawing pictures of (a graph representing) the
partition in the plane
then we will generally draw the labelled vertices in the natural order
(left to right) on the frame - as in the Figure.
We will call this the frame drawing convention for naturally
ordered vertex sets.
(It is worth emphasising that this _is_ nothing more than a convention.)
\subsection{Composition of partitions}
Define
\[
* : \EE(S) \times \EE(T) \rightarrow \EE(S \cup T)
\]
by $\rho * \nu = TC( \rho \cup \nu )$
(on the right we regard $\rho,\nu$ as equivalence relations,
by the identification convention, so that
$\rho \cup \nu$ is a relation).
Note that $\rho \cup \nu$ is a reflexive relation, but not transitive
in general.
Note the repeated use of the identification convention here.
$TC(\rho \cup \nu)$ is nominally a relation, but $\rho * \nu$ is a partition,
so it is the corresponding partition.
Exercise: What does $*$ look like at the level of pictures?
\begin{figure}
\caption{Two pictures encoding example partitions from
$\EE(7,3) $
and $\EE(3,4)$. \label{figpic2}}
\includegraphics{xfig/partit-low2aa.gif}
\end{figure}
\subsection{The partition categories}
We start with the simplest partition category: $\caP^1$.
Here the object set is $\N_0$ and the hom-sets take the form $\EE(n,m)$.
The binary operation
$.: \EE(n,m) \times \EE(m,l) \rightarrow \EE(n,l)$
is given by
$$
p.q =
\iota^{\sigma_-} (( \iota^{\sigma} (p) * q )|_{{\underline{n}}^2 \cup {\underline{l}}^0} )
$$
where $\iota^{\sigma_-} , \iota^\sigma$ act on label indices as before.
Note that the underlying set of $\iota^\sigma (p)$ is
$\underline{n}^2 \cup \underline{m}^1$;
while the underlying set of $q$ is
$\underline{m}^1 \cup \underline{l}^0$,
so they meet in $\underline{m}^1$.
If we apply $\iota^\sigma$ to the upper picture in Fig.\ref{figpic2}
then this picture meets the lower picture in $\{ 1^1 , 2^1 , 3^1 \}$.
\begin{figure}
\caption{Composing example partitions from
$\EE(7,3) $
and $\EE(3,4)$ as in the composition $p.q$. \label{figpic2anim}}
\includegraphics{xfig/partit-anim2.gif}
\end{figure}
Consider the triple
%$( \N_0 , \caP^1(-,-) , .)$,
$\caP^1 = ( \N_0 , \EE(-,-) , .)$.
%where ...
%$\caP^{1}(n,m) = \EE(n,m)$ and
%$.: \caP^1(n,m) \times \caP^1(m,l) \rightarrow \caP^1(n,l)$
Caveat: WEBPAGE CONSTRUCTION IN PROGRESS - CHECK details!
\begin{proposition} $\caP^1$ is a category. \end{proposition}
Exercise: Check that $\caP^1$ is a category.
Hints: We need to check that the composition is associative,
and that there are appropriate identity elements.
There are various ways
to show associativity. For example, note that the concatenation of two pictures
(so that they meet at the matching labels) makes a usable picture
for the composite. It is a realisation containing more than the
minimum number of vertices needed, but this is not an issue.
The other caveat is that the nominal labels on the top row are wrong,
but again this is clearly not an issue.
With this in mind one can then make a picture of a double composite,
$p*(q*r)$ say, in the same way; and also a picture for $(p*q)*r$.
However these are the same picture, and we are done.
The identity elements will be clear
...
%Next we construct
\subsubsection{The partition base category $\caP^\circ$}
Here the hom-sets take the form
$\EE(n,m) \times \N_0$
(or we may take $\EE(S,T) \times \N_0$, but it is sufficient
representation theoretically to work with a skeleton).
...
\subsubsection{The partition category $\caP$}
Next we construct the `ordinary' partition category $\caP$.
This is a linear category over a commutative ring $K$ containing an element
$\delta$.
(Typically $K = \C$.)
The hom-sets take the form $K \EE(n,m)$. The object set is $\N_0$.
The composition is as for $\caP^1$, extended linearly,
except that if the concatenation of diagrams includes
...
Next:
\section{Representation theory}