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% Dimension formulae for the Lie superalgebra $sl(m/n)$
% J. Van der Jeugt
% J. Math. Phys. 36 (1995), 605-611.
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\def\et{\eta}
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\begin{document}
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\begin{center}
{\Large \bf Dimension formulae for the Lie superalgebra $sl(m/n)$}\\[2cm]
J.~Van der Jeugt\footnote{Senior Research Associate N.F.W.O. (National Fund
for Scientific Research of Belgium)} \\[8mm]
Department of Applied Mathematics and Computer Science,
University of Ghent,\\
Krijgslaan 281--S9, B9000 Gent, Belgium\\
E-mail~: Joris.VanderJeugt@rug.ac.be
\end{center}

\vskip 3cm
\begin{abstract}
Although character formulae for simple finite-dimensional modules
of the Lie superalgebra $sl(m/n)$ are not known in general, they are
known in the cases of so-called typical and of singly atypical modules.
For singly atypical modules, however, the character formula is such
that the classical limiting procedure yielding a dimension formula
from a character formula does, in general, not work. In this paper,
another approach is presented yielding an explicit dimension formula
for all singly atypical modules of $sl(m/n)$ in terms of the highest
weight labels.
\end{abstract}
\vskip 5mm
\noindent PACS~: 02.20b
\vskip 5mm
\noindent Short title~: Dimension formulae for $sl(m/n)$.
\newpage
%
\section{Introduction}  \label{sec-intro}

Despite the fact that Lie superalgebras and their representations (modules)
have been the subject of much attention in both the
mathematical~\cite{k75,k77a,k77b,sch} and the
physics~\cite{bal,cor,don,hur,snr} literature
there remains no complete account of the
finite-dimensional simple modules of even the simplest family of basic
classical Lie superalgebras, namely $sl(m/n)$.
Kac~\cite{k77a} provided a labelling scheme for simple modules, and went
on to distinguish between two types of simple modules~: typical and
atypical~\cite{k77b}. The former are well understood~: their characters
and dimensions are easily obtained. The latter are more difficult.
In general, their characters are not known, but recently there have
been a number of breakthroughs in this area~: for singly atypical modules
(for which the highest weight $\La$ is atypical with respect to only
one single odd root) a formula has been constructed~\cite{van2};
for all atypical modules a formula has been
conjectured~\cite{van1,pen,ser};
for atypical Kac modules (see Section~II), the composition
series has been conjectured~\cite{hug} and partially shown to be
correct~\cite{su}.

It would be natural to expect that when a character
formula has been proved, such as in the singly atypical case which is
the subject of this paper, it would require little effort to
obtain a dimension formula from it. Indeed, putting $\lim e^\mu =1$
in the formal character $\cha V(\La)=\sum_\mu \hbox{mult}(\mu) e^\mu$
of a module $V(\La)$ with highest weight $\La$ gives immediately
the dimension $\dim V(\La)$. However, in the case under consideration
here, this limit cannot be taken explicitly (although it can be
performed in specific examples, after lengthy calculations) due
to the form of the character formula itself.

In this paper, we use another approach to find explicit dimension formulae
for singly atypical modules as a polynomial in the highest weight labels.
We still use the character formula,
but perform no limiting procedure, and we also rely heavily
on the structure of the Kac module itself described in~\cite{van2}.
This reduces the problem to solving a polynomial equation,
where we can make use of a result due to Schlosser~\cite{slo}.

The structure is as follows~: in Section~II some definitions for
$sl(m/n)$ and its modules are recalled, including the character
formulae that will be needed. In Section~III the dimension formula
is deduced, using Schlosser's result. Finally, in Section~IV the
formula is illustrated for the case $sl(2/3)$. The results presented
here are basically mathematical; on the other hand when groups or
Lie (super)algebras are used in physics one of the most important
features of the irreducible representations (simple modules) is
the dimension. With this in mind, the explicit formulae
derived in this paper should prove to be useful.

\section{The Lie superalgebra $sl(m/n)$ and its modules} \label{sec-slmn}
\setcounter{equation}{0}

The general linear Lie superalgebra
$gl(m/n)$ with $m,n\in\Nat$ is defined by~:
\beq
\begin{array}{rl}
gl(m/n)=\Bigl\lbrace x=\left(\matrix{A&B\cr C&D\cr}\right) \mid &
 A\in M_{m\times m}, B\in M_{m\times n}, \\
 & C\in M_{n\times m}, D\in M_{n\times n} \Bigr\rbrace ,
\end{array}
\label{defgl}
\eeq
where $M_{p\times q}$ is the space of all $p\times q$ complex
matrices. The even subspace $gl(m/n)_{\bar 0}$ has $B=0$ and
$C=0$; the odd subspace $gl(m/n)_{\bar 1}$ has $A=0$ and
$D=0$. The Lie superalgebra bracket is determined by
\beq
[a,b]=ab-(-1)^{\al\be}ba,\qquad\forall a\in G_\al\hbox{ and }
\forall b\in G_\be.
\label{comm-rel}
\eeq
One defines the {\it supertrace} $\hbox{str}(x)$
of $x=\bigl({A\atop C}{B\atop D}\bigr)$ as~\cite{k77a,sch}
$\hbox{str}(x)=\hbox{tr}(A)-\hbox{tr}(D)$,
where {\sl tr} is the ordinary trace.
The {\it special linear Lie superalgebra} $G=sl(m/n)$ is:
\beq
sl(m/n)=\lbrace x\in gl(m/n)\mid \hbox{str}(x)=0\rbrace.
\eeq
A Cartan subalgebra $H$ of $G$ has dimension $m+n-1$ and is the space of
diagonal matrices in~(\ref{defgl}). The dual space $H^*$ is described in
the basis of forms $\ep_i$ and $\de_j$ ($i=1,\ldots,m$, $j=1,\ldots,n$), where
$\ep_i:x\rightarrow A_{ii}$ and $\de_j:x\rightarrow D_{jj}$.
In terms of these, the roots of $G$ are given by the elements $\ep_i-\ep_j$,
$\de_i-\de_j$ (even roots) and  $\pm\be_{ij}=\pm(\ep_i-\de_j)$ (odd roots).
The non-degenerate bilinear form $\langle\;|\;\rangle$ on $H^*$ is determined
by \cite{van2} $\langle\ep_i | \ep_j\rangle=-\de_{ij}$,
$\langle\ep_i | \de_j\rangle=0$, and
$\langle\de_i | \de_j\rangle=\de_{ij}$.
Denote by $\De$ the set of all roots, by $\De_0$ the set of even
roots, and by $\De_1$ the set of odd roots.
A set of simple roots $\{\al_1,\al_2,\ldots,\al_{m+n-1}\}$
of $\De$ may be chosen as follows~\cite{k77a}: $\al_i=\ep_i-\ep_{i+1}$
($1\leq i\leq m-1$), $\al_m=\ep_m-\de_1$, $\al_{m+j}=\de_j-\de_{j+1}$
($1\leq j\leq n-1$).
The even and odd {\it positive} roots of $G$ are then given by
\begin{eqnarray}
\De_0^+ & = & \{\ep_i-\ep_j\;(i<j),\;\de_i-\de_j\;(i<j)\},\nonumber\\
\De_1^+ & = & \{\be_{ij}=\ep_i-\de_j \}.
\label{deltas}
\end{eqnarray}

An element $\La\in H^*$
with $\La=\sum_i\la_i\ep_i+\sum_j\mu_j\de_j$ can be written
as $\La=(\la_1\ldots\la_m$ $|\mu_1$ $\ldots$ $\mu_n)$
in terms of its {\it components} in the $\ep\de$-basis,
or in terms of its {\it Dynkin
labels} $\La=[a_1,\ldots,a_{m-1}$; $a_m$; $a_{m+1},\ldots,a_{m+n-1}]$, where
$a_i=\la_i-\la_{i+1}$
($1\leq i\leq m-1$), $a_m=\la_m+\mu_1$, $a_{m+j}=\mu_j-\mu_{j+1}$
($1\leq j\leq n-1$). The $m+n-1$ Dynkin labels uniquely characterize a weight
from $sl(m/n)$; it will often be useful to work with the $m+n$ $(\la|\mu)$
components which uniquely label a $gl(m/n)$ weight (just as a partition $\la$,
used for $gl(n)$, is commonly used to label the corresponding weight in
$sl(n)$ too).

Let $G=G_{-1}\oplus G_0\oplus G_{+1}$ be the $\Zah$ grading of $G$ in which
$G_0=G_{\bar 0}$ and $G_{+1}$, $G_{-1}$ are spanned by, respectively,
positive and negative odd root vectors. For every dominant weight
$\La\in H^*$, $V^0(\La)$ denotes the simple $G_0$ module with highest
weight $\La$. This module can be extended to a $G_0\oplus G_{+1}$
module by putting $G_{+1}V^0(\La)=0$. The induced $G$ module
$\VK$, first introduced by Kac~\cite{k77b} and referred to as the
Kac module, is defined by
\beq
\VK={\rm Ind}_{G_0\oplus G_{+1}}^G\;V^0(\La)=
U(G)\otimes_{G_0\oplus G_{+1}}V^0(\La) \cong
U(G_{-1})\otimes V^0(\La),
\label{induced}
\eeq
where $U(G_{-1})$ is the universal enveloping algebra of $G_{-1}$.
Since the character of $V^0(\La)$ is well known, the character of
$\VK$ can be written in the form
 \beq
 \cha \VK={L_1\over L_0} \sum_{w\in W}
             \vep(w)e^{w(\La+\rh)},
 \label{cha-Kac}
 \eeq
or, equivalently, by
 \beq
 \cha \VK = {L_0}^{-1} \sum_{w\in W}\vep(w)
   w\Bigl\lbrace e^{\La+\rh_0}
   \prod_{\be\in\De_1^+}\left(1+e^{-\be}\right)\Bigr\rbrace,
 \label{cha-Kac-alt}
 \eeq
where $\rh=\rh_0-\rh_1$ (with $2\rh_0$ and $2\rh_1$ the sums of
the elements of $\De_0^+$ and $\De_1^+$ respectively),
$ L_0=\prod_{\al\in\De_0^+}(e^{\al/2}-e^{-\al/2})$,
$ L_1=\prod_{\be\in\De_1^+}(e^{\be/2}+e^{-\be/2})$,
$W$ the Weyl group of $G_0$ and therefore also of $G$, and $\vep(w)$
the signature of $w\in W$.
The notation $e^\la$ ($\la\in H^*$) is the {\em formal exponential},
i.e.\ the exponential function defined on $H^*$ by $e^\la (\mu)=
e^{\langle\la |\mu\rangle}$. A {\em regular exponential function}
\cite{kacw} is a finite linear combination of exponentials $e^\la$;
a {\em rational exponential function} \cite{kacw} is a ratio $P/Q$ where
$P$ and $Q$ are regular exponential functions and $Q\ne 0$. If $V$ is
a $G$ module its character is a regular exponential function and
$\dim (V)=\cha (V) (0)$, i.e.\ the evaluation of the regular exponential
function $\cha (V)$ at 0. It is useful to introduce the following
functions~:
\bea
 \ch_K(\La)&=&{L_1\over L_0} \sum_{w\in W}
             \vep(w)e^{w(\La+\rh)}, \\
 \ch^0(\La)&=&{1\over L_0} \sum_{w\in W}
             \vep(w)e^{w(\La+\rh_0)};
\eea
when $\La$ is integral dominant, $\cha \VK = \ch_K(\La)$ and
$\cha V^0(\La) = \ch^0(\La)$.

The Kac module $\VK$ is an indecomposable $G$ module, however, it is
not always simple. Kac showed that $\VK$ is simple if and only if
 \beq
 \langle\La+\rh\mid\be\rangle\not=0,\quad\forall\be\in\De_1^+,
 \label{cond-typ}
 \eeq
and in this case $\VK$ is called a typical module. In general,
$\VK$ has a unique maximal submodule $M$, and the quotient module
$V(\La)=\VK/M$ is the simple module with highest weight $\La$. If
$ \langle\La+\rh\mid\be\rangle =0$ for some $\be$ in $\De_1^+$,
then $\La$, $V(\La)$ and $\VK$ are called atypical.
Recently, much attention has gone to the study of atypical modules
and to the composition factors of atypical Kac modules~\cite{hug}.
As far as characters is concerned, a character formula for all
simple modules of $G$ has been conjectured in~\cite{van1}. On the other hand,
a character formula has been proved for singly atypical modules,
that is those for which the highest weight $\La$ satisfies
$\#\{\be\in\De_1^+ |  \langle\La+\rh\mid\be\rangle=0\}=1$.
In this case, the maximal submodule $M$ of $\VK$ is itself a simple
module $V(\Si)$, the highest weight of which can be determined
with the techniques described in~\cite{van2}. Let $\ga=\be_{kl}$
($1\leq k\leq m$, $1\leq l\leq n$) be the unique odd root for which
$ \langle\La+\rh\mid\ga\rangle = 0$, then it was shown in~\cite{van2}
(see also \cite{kacw}) that $\cha V(\La)=\ch_\ga(\La)$, where
 \beq
 \ch_\ga (\La)={L_1\over L_0} \sum_{w\in W}
             \vep(w)w\left({e^{\La+\rh}\over 1+e^{-\ga}}\right) .
 \label{cha-sing}
 \eeq
The module $V(\La)$ will be referred to as a simple module
singly atypical with respect to $\ga$.

Usually, dimension formulae can be derived from character formulae~\cite{hum}
simply by taking a limit (here, it would require taking the
limit for $e^{\ep_i}$ and $e^{\de_j} \rightarrow 1$), corresponding to
the evaluation of the character function at 0. However,
it turns out that taking such a direct limit in (\ref{cha-sing})
is not conceivable, and one is forced to consider an indirect technique,
as will be described in the following section.

We conclude this section by a Lemma.
\begin{lemm}
Let $\al\in\De_0^+$, $\ga\in\De_1^+$ and $\La\in H^*$.
 \begin{itemize}
 \item[(a)] If $\langle\La+\rh_0|\al\rangle=0$ then $\ch_K(\La)=0$.
 \item[(b)] If furthermore $\langle\ga|\al\rangle=0$ then $\ch_\ga(\La)=0$.
 \end{itemize}
\label{lem1}
\end{lemm}
\noindent {\bf Proof.} \\
(a) If $\langle\La+\rh_0|\al\rangle=0$ then $r_\al (\La+\rh_0)=\La+\rh_0$
with $r_\al\in W$ the reflection with respect to the root $\al$ and
$\vep(r_\al)=-1$. The statement then follows from the properties of the
Weyl group and the fact (obtained by using the Weyl invariance of $\rh_1$)
that $\ch_K(\La)=e^{-\rh_1}L_1 \ch^0(\La)$
(see also Reference~\cite{van2}, Lemma~6.2).\\
(b) In this case $r_\al$ leaves $e^{\La+\rh_0} \over 1+e^{-\ga}$ invariant.
Thus it follows again from the Weyl group structure that the
expression $\sum_{w} \vep(w)w\left({e^{\La+\rh_0}\over 1+e^{-\ga}}\right)$
is equal to zero. \mybox

\section{Dimension formulae}
\setcounter{equation}{0}

Since the dimension of a simple $G_0$ module is well known, i.e.
\beq
\dim V^0(\La) = \ch^0(\La)(0) =
 \prod_{\al\in\De_0^+} { \langle\La+\rh_0|\al\rangle \over
\langle\rh_0|\al\rangle},
\eeq
it follows immediately from~(\ref{cha-Kac}) that
\beq
\dim \VK = 2^{mn} \dim V^0(\La)
= 2^{mn} \prod_{\al\in\De_0^+} { \langle\La+\rh_0|\al\rangle \over
\langle\rh_0|\al\rangle},
\eeq
or, explicitly,
\beq
\dim \VK = 2^{mn}\prod_{i<j}{(\la_i-\la_j+j-i)\over(j-i)}
 \prod_{i<j}{(\mu_i-\mu_j+j-i)\over(j-i)}.
\label{dimtyp}
\eeq

Next, we focuss our attention to singly atypical modules, and
make the following two observations.
From the character formula
(\ref{cha-sing}), it follows that $\ch_\ga(\La)+\ch_\ga(\La-\ga)=\cha\VK$. Since
$\ch_\ga=\cha V(\La)$ and because of the structure of a singly atypical
Kac module $\VK$ described in the previous section, we have that
$\ch_\ga(\La-\ga)=\cha M =\cha V(\Si)$. Note that $\La-\ga$ need
not be dominant; if it is not dominant, then $\Si$ is different from
$\La-\ga$ and it is singly atypical with respect to a root different
from $\ga$~\cite{van2}.

\begin{prop}
Let $V(\La)$ be singly atypical with respect to $\ga$. Then
$\dim V(\La)$ is a {\em polynomial expression} in the variables
$\la_i-\la_j$ ($1\leq i<j\leq m$) and $\mu_i-\mu_j$ ($1\leq i<j\leq n$).
\label{propo}
\end{prop}
\noindent {\bf Proof.} The character of $V(\La)$ can be rewritten as
follows~:
\bea
\cha V(\La)&=& \ch_\ga(\La) =
{L_0}^{-1} \sum_{w\in W}\vep(w)
   w\Bigl\lbrace e^{\La+\rh_0}
   {\prod_{\be\in\De_1^+}\left(1+e^{-\be}\right) \over (1+e^{-\ga})}
   \Bigr\rbrace \nn\\
&=&{L_0}^{-1} \sum_{w\in W}\vep(w)
   w\Bigl\lbrace e^{\La+\rh_0}
   \prod_{\be\in\De_1^+,\; \be\ne\ga}\left(1+e^{-\be}\right)
   \Bigr\rbrace
\eea
Expanding the product in the last expression yields $2^{mn-1}$ terms,
implying that $\ch_\ga(\La)$ is equal to $2^{mn-1}$ terms of the form
$\ch^0(\et)$, where $\et=\La - (\hbox{sum }\hbox{over }\hbox{some }
\be{\rm 's})$,
with $\be\in\De_1^+/\{\ga\}$. Thus every $\et$ is integral and hence
$\ch^0(\et)(0)=\prod_{\al\in\De_0^+} { \langle\et+\rh_0|\al\rangle \over
\langle\rh_0|\al\rangle}$, which is a polynomial expression in the
$\la_i-\la_j$ and the $\mu_i-\mu_j$. \mybox

\begin{prop}
Let $V(\La)$ be singly atypical with respect to $\ga$. Then $\dim V(\La)$
is of the following form~:
\bea
\dim V(\La)&=&d_\ga(\La)\equiv 2^{mn}\left(\prod_{i<j;\, i\& j\ne k}{(\la_i-\la_j+j-i)\over(j-i)}
 \prod_{i<j;\, i\& j\ne l}{(\mu_i-\mu_j+j-i)\over(j-i)}\right)\nn\\
&\times& f(\la_1,\ldots,\la_m,\mu_1,\ldots,\mu_n),
\label{33}
\eea
where $f$ is again a polynomial in the $\la_i-\la_j$ and $\mu_i-\mu_j$.
\end{prop}
\noindent {\bf Proof.} We have $\dim V(\La)= \ch_\ga (\La) (0)$ with
$\ch_\ga(\La)(0)$ a polynomial in the variables $\la_i-\la_j$ and
$\mu_i-\mu_j$ (according to Proposition~\ref{propo}). But when
$\langle\La+\rh_0|\al\rangle=0$ with $\langle\ga|\al\rangle=0$,
then $\ch_\ga(\La)=0$ (see Lemma~\ref{lem1}) and then also
$\ch_\ga(\La)(0)=0$. Thus this polynomial must have $\langle\La+\rh_0|
\al\rangle$ as a factor when $\langle\ga|\al\rangle=0$. The elements
$\al\in\De_0^+$ orthogonal to $\ga=\be_{kl}$ are all $\ep_i-\ep_j$ with
$i$ and $j$ different from $k$, and all $\de_i-\de_j$ with $i$ and $j$
different from $l$. The remaining numerical factors in (\ref{33}) are
used for later convenience. \mybox

It is useful to introduce the following variables~:
\beq
(x_1,\ldots,x_{m-1})=(\la_k-\la_1+1-k,\la_k-\la_2+2-k,\ldots,\la_k-\la_m+m-k),
\label{x}
\eeq
where, of course, the component $\la_k-\la_k+k-k$ is missing in the rhs, and
\beq
(y_1,\ldots,y_{n-1})=(\mu_1-\mu_l+l-1,\mu_2-\mu_l+l-2,\ldots,\mu_n-\mu_l+l-n).
\label{y}
\eeq
Then
$f(\la_1,\ldots,\mu_n)= p(x_1,\ldots,x_{m-1},y_1,\ldots,y_{n-1},z)$,
where $z$ stands for the remaining $\la_i-\la_j$ and $\mu_i-\mu_j$.

Since $\cha M=\ch_\ga(\La-\ga)$, and the final dimension formula can
be obtained by taking a limit, we must have that $\dim M=d_\ga(\La-\ga)$.
Then, it follows from $d_\ga(\La)+d_\ga(\La-\ga)=\dim\VK$ that
the unknown polynomial $p$ should satisfy
\bea
p(x_1,\ldots,x_{m-1},y_1,\ldots,y_{n-1},z)&+&
p(x_1-1,\ldots,x_{m-1}-1,y_1-1,\ldots,y_{n-1}-1,z)\nn\\
&& =\qquad{(-1)^{n-k-l-1}\prod_{i=1}^{m-1} x_i\prod_{j=1}^{n-1} y_j \over
 (m-k)!(k-1)!(n-l)!(l-1)! }.
\label{eqp}
\eea
Herein, the $z$-variables are the same for $\La$ and for $\La-\ga$.
One concludes that $p$ depends only upon the variables $x_1,\ldots,y_{n-1}$.
Finally, to solve (\ref{eqp}), we can make use of a proposition due
to Schlosser~\cite{slo}~:
\begin{prop}
The unique polynomial solution of the equation
\beq
 g(u_1,\ldots,u_s)+g(u_1-1,\ldots,u_s-1)= u_1\cdots u_s
\eeq
is given by
\beq
 g(u_1,\ldots,u_s)={1\over 2} \sum_{r=0}^s H_{s-r} e_r(u_1,\ldots,u_s),
\eeq
where $e_r$ are the elementary symmetric functions~\cite{mac} and the
coefficients $H_i$ are given in terms of the Bernoulli numbers~:
\beq
H_i={2(2^{i+1}-1)\over i+1}B_{i+1}.
\eeq
\end{prop}

The coefficients $H_i$ can be easily calculated using their generating
function
\beq
\sum_{i=0}^\infty H_i {t^i\over i!}= 2{ e^t\over 1+e^t}.
\eeq
The first few values are given in the following table~:
$$
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$i$&0&1&2&3&4&5&6&7&8&9&10&11 \\ \hline
$H_i$&1&$1/2$&0&$-1/4$&0&$1/2$&$0$&$-17/8$&0&$31/2$&0&$-691/4$ \\ \hline
\end{tabular}
$$
\vskip 5mm
Putting all information together, we have the following result~:
\begin{theo}
Let $\La=(\la_1,\ldots,\la_m|\mu_1,\ldots,\mu_n)$ be a dominant
highest weight singly atypical of type $\ga=\be_{kl}$. Then
\bea
\dim V(\La)&=&
2^{mn-1}\left(\prod_{i<j;\, i\& j\ne k}{(\la_i-\la_j+j-i)\over(j-i)}
 \prod_{i<j;\, i\& j\ne l}{(\mu_i-\mu_j+j-i)\over(j-i)}\right) \nn\\
 &\times&{(-1)^{n-k-l-1}\over(m-k)!(k-1)!(n-l)!(l-1)!}\nn\\
 &\times&\sum_{r=0}^{m+n-2} H_{m+n-2-r} e_r(x_1,\ldots,x_{m-1},y_1,\ldots,y_{n-1}),
\label{dim}
\eea
with $x_i$ and $y_j$ given in (\ref{x}) and (\ref{y}).
\end{theo}

\section{Discussion and example}
\setcounter{equation}{0}

The formula given in (\ref{dim}) is valid for simple modules $V(\La)$ which are
singly atypical with respect to the root $\ga=\be_{kl}$. In terms of
the $(\la|\mu)$ components or the Dynkin labels $a_i$, this means
that~:
\bea
0&=&\langle \La+\rh|\be_{kl}\rangle = \la_k+\mu_l+m-k-l+1\nn\\
 &=&\sum_{p=k}^{m-1}a_p+a_m-\sum_{q=1}^{l-1}a_{m+q}+m-k-l+1,
\eea
and $\langle\La+\rh|\be\rangle\ne 0$ for all $\be\in\De_1^+/\{\be_{kl}\}$.

Consider as an example $G=sl(2/3)$, and take first $k=l=1$.
The first product in the rhs of (\ref{dim}) is simply $(\mu_2-\mu_3+1)$;
the summation part in (\ref{dim}) becomes
\beq
x_1y_1y_2+{1\over 2}(x_1y_1+x_1y_2+y_1y_2)-{1\over 4},
\eeq
where $x_1=\la_1-\la_2+1$, $y_1=\mu_2-\mu_1-1$ and $y_2=\mu_3-\mu_1-2$.
The dimension formula is most appropriately expressed in terms of
the Dynkin labels $[a_1;a_2;a_3,a_4]$, which, in this case, satisfy the
condition $a_1+a_2+1=0$, following from the atypicality condition~:
\bea
\dim V(\La)&=& 4(a_4+1)(4a_1a_3^2+4a_1a_3a_4+8a_1a_3+2a_1a_4+6a_3^2\nn\\
& &+6a_3a_4+2a_1+14a_3+4a_4+5).
\eea
For completeness, we list here the dimensions of all other singly
atypical $sl(2/3)$ modules as well. When $\La$ is atypical with
respect to $\be_{12}$, then
\beq
\dim V(\La)= 4(a_3+a_4+2)(4a_1a_3a_4+2a_1a_3+6a_1a_4+6a_3a_4+4a_1+4a_3+
8a_4+7).
\eeq
If $\La$ is singly atypical with respect to $\be_{13}$, then
\bea
\dim V(\La)&=& 4(a_3+1)(4a_1a_3a_4+4a_1a_4^2+6a_1a_3+16a_1a_4+6a_3a_4\nn\\
& &+6a_4^2+14a_1+8a_3+22a_4+17).
\eea
For $\La$ singly atypical with respect to $\be_{21}$, we have
\bea
\dim V(\La)&=& 4(a_4+1)(2a_1a_3^2+4a_1a_3a_4+8a_1a_3+2a_1a_4+2a_3^2\nn\\
& &+2a_3a_4+2a_1+2a_3-1).
\eea
For modules singly atypical with respect to $\be_{22}$, the dimension is
\beq
\dim V(\La)= 4(a_3+a_4+2)(4a_1a_3a_4+2a_1a_3+6a_1a_4+2a_3a_4+4a_1+
4a_4+1).
\eeq
And finally, for modules atypical with respect to $\be_{23}$, we have
\bea
\dim V(\La)&=& 4(a_3+1)(4a_1a_3a_4+4a_1a_4^2+6a_1a_3+16a_1a_4+2a_3a_4\nn\\
& &+2a_4^2+14a_1+4a_3+10a_4+11).
\eea

The dimensions of Thierry-Mieg's tables~\cite{t-m} can be verified against the
formulae given here, and there is complete agreement.

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%
\end{document}

