% Plain TEX file
% THE COMPOSITION FACTORS OF KAC MODULES OF $sl(M/N)$
% R.C. King, J.W.B. Hughes, J.\ Van der Jeugt
% in Proceedings of the XVIIIth International Colloquium on Group
% Theoretical Methods in Physics, Lecture Notes in Physics 382,
% eds. V.V. Dodonov and V.I. Man'ko,
% Springer, Berlin, 1991; pp. 522-526.
%
%
% MACROS
%
% The Greek symbols defined by the first two letters of their name
%
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\de{\delta}
\def\ep{\epsilon}  \def\vep{\varepsilon}
\def\ze{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\rh{\rho}
\def\si{\sigma}
\def\ta{\tau}
\def\ph{\phi}
\def\ch{\raise 2pt\hbox{$\chi$}}  % raise this a bit
\def\ps{\psi}
\def\om{\omega}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\Th{\Theta}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Ph{\Phi}
\def\Ps{\Psi}
\def\Om{\Omega}
%
% Definitions for the Young tableaux
%
\def\mystrut{\hbox{\vrule height9.6pt depth2pt width0pt}}
\def\smallstrut{\hbox{\vrule height8.6pt depth 1pt width 0pt}}
\def\norulefill{\leaders\hrule height0pt\hfill}
%
% Definitions for the Kac-module and the G_0 module:
%
\def\VK{\overline V(\La)}
\def\V0{V_0(\La)}
%
% Definition of additional maths operators
\def\tr{\mathop{\rm tr}}
\def\str{\mathop{\rm str}}
\def\ch{\mathop{\rm ch}}
%
\def\Real{{\tt R}}
\def\Nat{{\tt N}}
\def\Ha{{\tt H}}
\def\Zah{{\tt Z}}
\def\Expn{{\tt E}}
\def\C{{\tt C}}
\def\Q{{\tt Q}}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\
\vskip 12mm
\centerline{THE COMPOSITION FACTORS OF KAC MODULES OF $sl(M/N)$}
\vskip 3mm
\centerline{R.C.\ King (University of Southampton, U.K.), J.W.B.\ Hughes
(Queen Mary and}
\centerline{Westfield College, U.K.), J.\ Van der Jeugt \footnote*
{Research Associate of the NFWO Belgium}
(University of Ghent, Belgium) \footnote\dag{Talk presented by
R.C.\ King}}
\vskip 6mm

Throughout this article we adopt the notation and conventions of the
article [1] entitled {\sl Atypical modules of the Lie superalgebra $gl(m/n)$}
based on the Colloquium talk by Dr J.\ Van der Jeugt and published
elsewhere in these Proceedings. The complex Lie superalgebra
$sl(m/n)$ is the subalgebra of $gl(m/n)$ consisting of matrices
$x =\left( {A\atop C}{B\atop D} \right)$, with $A, B, C$ and $D$ matrices
of size $m\times m, m\times n, n\times m$ and $n\times n$, respectively,
with $\str x =\tr A - \tr D = 0$. $G=sl(m/n)$ admits a grading
$G=G_{-1} \oplus G_0 \oplus G_1$ with $G_0 = sl(m)\oplus \C \oplus sl(n)$, the
even subalgebra of $sl(m/n)$. The universal enveloping algebras of
$G$, $G_0$ and $G_{-1}$ are denoted by $U(G), U(G_0)$ and $U(G_{-1})$,
respectively.

The weight space $H�*$ is the dual of the Cartan subalgebra $H$ of $sl(m/n)$.
It is spanned by the forms $\ep_i$ ($i=1,\ldots,m$)
and $\de_j$ ($j=1,\ldots,n$), with $\sum_{i=1}�m \ep_i - \sum_{j=1}�n \de_j =0$,
and is equipped with an inner product such that
$\langle\ep_i|\ep_j\rangle = \de_{ij}$,
$\langle\ep_i|\de_j\rangle = 0$,
$\langle\de_i|\de_j\rangle = -\de_{ij}$,
where $\de_{ij}$ is the usual Kronecker symbol.
The set of simple roots is taken to be the distinguished set
$\{\ep_i-\ep_{i+1}, i=1,2,\ldots,m-1;\ep_m-\de_1;\de_j-\de_{j+1},
j=1,2,\ldots,n-1\}$, so that the sets of positive even and odd roots are
given by
$\De_0�+ = \{ \ep_i - \ep_j, 1 \le i<j \le m;
           \de_i - \de_j, 1 \le i<j \le n \}$ and
$\De_1�+ = \{ \be_{ij}=\ep_i-\de_j, 1 \le i \le m, 1 \le j \le n \}$,
respectively.  It is convenient to define $\rh =\rh_0 -\rh_1$ with
$\rh_0 = {1\over2}\sum_{\al \in \De_0�+} \al$, and
$\rh_1 = {1\over2}\sum_{\be \in \De_1�+} \be$.

In the $\ep\de$-basis a weight $\La\in H�*$ takes the form
$\La = \sum_{i=1}�m \mu_i\ep_i + \sum_{j=1}�n \nu_j\de_j$. The corresponding
Kac-Dynkin labels are defined by $a_i=\mu_i-\mu_{i+1}$ for $i=1,2,\ldots,m-1$;
$a_m=\mu_m+\nu_1$ and $a_{m+j} = \nu_j-\nu_{j+1}$ for $j=1,2,\ldots,n-1$.
With these two conventions we write
$\La = (\mu_1\mu_2\ldots\mu_m|\nu_1\nu_2\ldots\nu_n) =
[a_1a_2\ldots a_{m-1};a_m;a_{m+1}\ldots a_{m+n-1}]$.
The $\ep\de$-notation has a built-in redundancy thanks to the identity
$\sum_{i=1}�m \ep_i - \sum_{j=1}�n \de_j =0$. This may be exploited to
ensure that $\mu_m \ge 0$ and $\nu_1 \le 0$. In what follows it is convenient
to denote all negative integers $-k$ by $\bar k$.

A weight $\La\in H�*$ is said to be integral dominant if and only if
$a_i\in \Nat$ for $i\not= m$ and $a_m\in \C$. Corresponding to each
integral dominant weight $\La$ there exists an irreducible finite-dimensional
highest weight module $\V0 = U(G_0)v_\La$ of $sl(m)\oplus \C\oplus sl(n)$.
Extending this to a $G_0\oplus G_{+1}$ module by setting $G_{+1}V_0(\La)=0$
and inducing to $G$ then gives the Kac-module [2] $\VK$ of $sl(m/n)$. This is
isomorphic to $U(G_{-1})\otimes V_0(\La)$ and is generated through the
action on $V_0(\La)$ of the exterior algebra over $e(-\be)$ with
 $\be\in\De_1�+$.
Thus $\VK$ and $\V0$ share the same highest weight vector $v_\La$, and
$\dim \VK = 2�{mn}dim \V0$. In general the Kac-module $\VK$ of $sl(m/n)$ is
indecomposable but reducible, with composition factors isomorphic to various
irreducible modules of $sl(m/n)$. Our aim is to
present a prescription for determining all these composition factors.
First we have two theorems:

{\bf Theorem 1.} (Kac [2]) Let $M(\La)$ be the unique
maximal submodule of $\VK$, then $V(\La) = \VK /M(\La)$ is irreducible.

{\bf Theorem 2.} (Gould [3]) Let $v_{\Om} = \Pi_{\be \in \De_1�+} e(-\be)
 v_\La$,
then $X(\La)=U(G)v_\Om$ is irreducible and $X(\La)=V(\Ga)$ for some $\Ga$.

As pointed out elsewhere [4] a key construct in discussing the structure
of $\VK$ is the atypicality matrix $A(\La)$. Its matrix elements are
given by $A(\La)_{ij}=\langle \La+\rh|\be_{ij}\rangle$ with $\be_{ij}=
\ep_i - \de_j \in \De_1�+$ for $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$. The
integral dominant weight $\La$ is said to be typical if $A(\La)$ contains no
zeros, and atypical of degree $r$ if $A(\La)$ contains $r$ zeros.
With this definition we have two more theorems:

{\bf Theorem 3.} (Kac [2]) If
$\La$ is typical then $\VK = V(\La)$ is irreducible, and thus consists
of a single composition factor.

{\bf Theorem 4.} (Van der Jeugt et al. [5]) If $\La$ is singly atypical of type
$\be$ then $\VK$ is the semi-direct sum of
$V(\La)=\VK / M(\La)$ and $V(\Ph)=X(\La)=M(\La)$. The algorithm for
determining $\Ph$ is as follows: Construct the sequence $S_\be =(\be_1
\be_2\ldots\be_k)$ of positive odd roots with $\be_1=\be$, such that
$\langle \La+\rh|\be_1 \rangle=0$ with $\La-\be_1$ non-dominant;
$\langle \La+\rh-\be_1|\be_2 \rangle=0$ with $\La-\be_1-\be_2$
non-dominant; and so on, until the sequence terminates with
$\langle \La+\rh-\be_1-\be_2-\cdots-\be_{k-1}|\be_k \rangle=0$ with
$\La-\be_1-\be_2-\cdots-\be_k$ dominant. Then $\Ph = \La-S(\be)$,
where $S(\be)=\be_1+\be_2+\cdots+\be_k$.

This procedure can be implemented diagramatically [1,4,5], see for example
the singly atypical case $\La=(7663|\bar1\bar1\bar3\bar3\bar5\bar5)
=[103;2;02020]$ of $sl(4/6)$ illustrated in [1] for which $\Ph =
(5533|\bar1\bar1\bar2\bar2\bar2\bar4)=[020;2;01002]$. $S(\be)$
defines, and is defined by, the removal of a continuous boundary
strip of boxes from each portion of the composite Young diagram
$F�{\bar{\ka}�\prime;\mu}$ with the row lengths of $F�\mu$ and
the column lengths of $F�{\bar{\ka}�\prime}$ determined by the
parts of the partitions $\mu=(\mu_1\mu_2\ldots\mu_m)$ and $\ka=
(-\nu_n\ldots-\nu_2-\nu_1)$.

The problem is to find the generalisation of these results appropriate
to the multiply atypical case. Consider the case for which $\La$ is
doubly atypical of type $\be_1= \ep_i-\de_j$ and $\be_2=\ep_k-\de_l$,
with $k<i$ and $j<l$, so that $\langle\La+\rh|\be\rangle =0$ for
$\be\in \De_1�+$ if and only if $\be = \be_1$ or $\be_2$.
Let $A(\La)_{kj}=x=-A(\La)_{il}$ and $h=i-k+l-j-1$, the hook length between
the two zeros of the atypicality matrix, then $\La$
is said to be normal if $x\ge h+2$, quasi-critical if $x=h+1$ and
critical if $x=h$. On the basis of extensive investigations we conjecture
the following:

{\bf Conjecture 5.} (i) If $\La$ is doubly atypical and normal then $\VK$
contains four composition
factors isomorphic to irreducible modules with highest weights $\La$,
$\La-S(\be_1)$, $\La-S(\be_2)$ and $\La-S(\be_1)-S(\be_2)$.

(ii) If $\La$ is doubly atypical and quasi-critical then $\VK$ contains five
composition factors isomorphic to irreducible modules with highest weights
$\La$, $\La-S(\be_1)$, $\La-S(\be_2)$, $\La-S(\be_1)-S(\be_2)$ and
$\La-S(\be_1 L \be_2)$, where $S(\be_1 L \be_2)$ is defined by
the removal of
continuous boundary strips starting from the position specified by $\be_2$
and continuing until they link with and include the strips
associated with $S(\be_1)$.

(iii) If $\La$ is doubly atypical and critical then $\VK$ contains three
composition factors isomorphic to irreducible modules with highest weights
$\La$, $\La-S(\be_1)$ and $\La-S(\be_1 W \be_2)$, where $S(\be_1 W \be_2)$ is
obtained by first removing the strips defined by $S(\be1)$ and then
removing further strips starting this time from the positions specified
by $\be_2$ which wrap around the first strips and continue until the resulting
diagram is once more regular.

These three possibilities are illustrated in the following examples:

(i) The $sl(2/3)$ case $\La=(52|\bar2\bar3\bar4)=[3;0;11]$ is doubly atypical
of type $\be_1=\be_{21}$ and $\be_2=\be_{14}$. The atypicality
matrix and the starting points of the strips to be removed are
indicated in the following diagram:
$$
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan3\hrulefill\cr
   \omit& &\omit& &&2&\cr
   \multispan2\norulefill&\multispan5\hrulefill\cr
   \omit& && && &\cr
   \multispan7\hrulefill\cr
   &1&& && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&4&\omit&2&\omit&0
     && && && && &&2&\cr
   \multispan6\norulefill&\multispan{11}\hrulefill\cr
   \omit&0&\omit&\bar 2&\omit&\bar 4
     && &&1&\cr
   \multispan6\norulefill&\multispan5\hrulefill\cr
   }}
$$
$\La$ is normal since $x=4$ and $h=2$. In this case $S(\be_1)=\be_{21}$ and
$S(\be_2)=\be_{14}$. The four composition factors have highest weights:
$(52|\bar 2\bar 3\bar 4)=[3;0;11]$, $(51|\bar 1\bar 3\bar 4)=[4;0;21]$,
$(42|\bar 2\bar 3\bar 3)=[2;0;10]$ and $(41|\bar1\bar3\bar3)=[3;0;20]$
corresponding to the strip removals:
$$
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan3\hrulefill\cr
   \omit& &\omit& && &\cr
   \multispan2\norulefill&\multispan5\hrulefill\cr
   \omit& && && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&0
     && && && && && &\cr
   \multispan6\norulefill&\multispan{11}\hrulefill\cr
   \omit&0&\omit&0&\omit&0
     && && &\cr
   \multispan6\norulefill&\multispan5\hrulefill\cr
   }}
\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan3\hrulefill\cr
   \omit& &\omit& && &\cr
   \multispan2\norulefill&\multispan5\hrulefill\cr
   \omit& && && &\cr
   \multispan7\hrulefill\cr
   &1&& && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&0
     && && && && && &\cr
   \multispan6\norulefill&\multispan{11}\hrulefill\cr
   \omit&1&\omit&0&\omit&0
     && &&1&\cr
   \multispan6\norulefill&\multispan5\hrulefill\cr
   }}
\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan3\hrulefill\cr
   \omit& &\omit& &&2&\cr
   \multispan2\norulefill&\multispan5\hrulefill\cr
   \omit& && && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&2
     && && && && &&2&\cr
   \multispan6\norulefill&\multispan{11}\hrulefill\cr
   \omit&0&\omit&0&\omit&0
     && && &\cr
   \multispan6\norulefill&\multispan5\hrulefill\cr
   }}
\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan3\hrulefill\cr
   \omit& &\omit& &&2&\cr
   \multispan2\norulefill&\multispan5\hrulefill\cr
   \omit& && && &\cr
   \multispan7\hrulefill\cr
   &1&& && &\cr
   \multispan7\hrulefill\cr
   & && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&2
     && && && && &&2&\cr
   \multispan6\norulefill&\multispan{11}\hrulefill\cr
   \omit&1&\omit&0&\omit&0
     && &&1&\cr
   \multispan6\norulefill&\multispan5\hrulefill\cr
   }}
$$

(ii) The $sl(2/2)$ case $\La=(32|\bar 2\bar 3)=[1;0;1]$ is doubly atypical of
type $\be=\be_{21}$ and $\be=\be_{12}$:
$$
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan2\norulefill&\multispan3\hrulefill\cr
   \omit& &&2&\cr
   \multispan5\hrulefill\cr
   &1&& &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill&\multispan6\hrulefill\cr
   \omit&2&\omit&0
     && && &&2&\cr
   \multispan4\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&\bar 2
     && &&1&\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   }}
$$

In this case $\La$ is quasi-critical since
$x=2$ and $h=1$. Now $S(\be_1)=\be_{21}$, $S(\be_2)=\be_{12}$ and
$S(\be_1L\be_2)=\be_{11}+\be_{12}+\be_{22}$. The five composition factors
have highest weights $(32|\bar 2\bar 3)=[1;0;1]$, $(31|\bar 1\bar 3)=
[2;0;2]$,
$(22|\bar 2\bar 2)=[0;0;0]$, $(21|\bar 1\bar 2)=[1;0;1]$ and
$(11|\bar 1\bar 1)=[0;0;0]$ corresponding to the strip removals:
$$
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan2\norulefill&\multispan3\hrulefill\cr
   \omit& && &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill&\multispan6\hrulefill\cr
   \omit&0&\omit&0
     && && && &\cr
   \multispan4\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&0
     && && &\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   }}
\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan2\norulefill&\multispan3\hrulefill\cr
   \omit& && &\cr
   \multispan5\hrulefill\cr
   &1&& &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill&\multispan6\hrulefill\cr
   \omit&0&\omit&0
     && && && &\cr
   \multispan4\norulefill&\multispan7\hrulefill\cr
   \omit&1&\omit&0
     && &&1&\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   }}
\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan2\norulefill&\multispan3\hrulefill\cr
   \omit& &&2&\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill&\multispan6\hrulefill\cr
   \omit&0&\omit&2
     && && &&2&\cr
   \multispan4\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&0
     && && &\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   }}
\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan2\norulefill&\multispan3\hrulefill\cr
   \omit& &&2&\cr
   \multispan5\hrulefill\cr
   &1&& &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill&\multispan6\hrulefill\cr
   \omit&0&\omit&2
     && && &&2&\cr
   \multispan4\norulefill&\multispan7\hrulefill\cr
   \omit&1&\omit&0
     && &&1&\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
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\quad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan2\norulefill&\multispan3\hrulefill\cr
   \omit& &&2&\cr
   \multispan5\hrulefill\cr
   &2&&2&\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill&\multispan6\hrulefill\cr
   \omit&2&\omit&2
     && &&2&&2&\cr
   \multispan4\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&2
     && &&2&\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   }}
$$

(iii) Finally, the $sl(3/4)$ case $\La=(432|\bar 2\bar 2\bar 3\bar 3)=
[11;0;010]$
is doubly atypical of type $\be_1=\be_{31}$ and $\be_2=\be_{14}$:
$$\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& && &&2&\cr
   \multispan9\hrulefill\cr
   &1&& && && &\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&4&\omit&3&\omit&1&\omit&0
     && && && &&2&\cr
   \multispan8\norulefill&\multispan9\hrulefill\cr
   \omit&2&\omit&1&\omit&\bar 1&\omit&\bar 2
     && && && &\cr
   \multispan8\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&\bar 1&\omit&\bar 3&\omit&\bar 4
     && &&1&\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   }}
$$

$\La$ is critical since $x=4$ and $h=4$. This time $S(\be_1)=\be_{31}$
and the usual method of determining $S(\be_2)$ leads to a strip which reaches
and wraps around that associated with $\be_1$. There are now only
three composition factors with highest weights $(432|\bar 2\bar 2\bar 3\bar 3)=
[11;0;010]$, $(431|\bar 1\bar 2\bar 3\bar 3)=[12;0;110]$ and
$(432|\bar 2\bar 2\bar 3\bar 3)=[11;0;010]$ obtained by means of the strip
removals:
$$\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& && && &\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&0&\omit&0
     && && && && &\cr
   \multispan8\norulefill&\multispan9\hrulefill\cr
   \omit&0&\omit&0&\omit&0&\omit&0
     && && && &\cr
   \multispan8\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&0&\omit&0&\omit&0
     && && &\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   }}
\qquad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& && && &\cr
   \multispan9\hrulefill\cr
   &1&& && && &\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&0&\omit&0
     && && && && &\cr
   \multispan8\norulefill&\multispan9\hrulefill\cr
   \omit&0&\omit&0&\omit&0&\omit&0
     && && && &\cr
   \multispan8\norulefill&\multispan7\hrulefill\cr
   \omit&1&\omit&0&\omit&0&\omit&0
     && &&1&\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   }}
\qquad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& &&2&&2&\cr
   \multispan9\hrulefill\cr
   &1&&2&&2&& &\cr
   \multispan9\hrulefill\cr
   &2&&2&& && &\cr
   \multispan{17}\hrulefill\cr
   \omit&0&\omit&0&\omit&2&\omit&2
     && && &&2&&2&\cr
   \multispan8\norulefill&\multispan9\hrulefill\cr
   \omit&2&\omit&2&\omit&2&\omit&0
     &&2&&2&&2&\cr
   \multispan8\norulefill&\multispan7\hrulefill\cr
   \omit&1&\omit&2&\omit&0&\omit&0
     &&2&&1&\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   }}
$$

To deal with cases for which the degree of atypicality is greater than two
an algorithm has been developed, based on the notion of a strip removal
scheme in which the question of linking and wrapping is determined from a
criticality matrix. The whole process is codified, and an algorithm has
been constructed and implemented on a computer. Many checks have been
carried out covering cases of atypicality degree as large as five, for which
the number of composition factors rises as high as 132.

The converse problem of determing all those Kac-modules $\VK$ which
contain a specific irreducible module $V(\Si)$ as a composition factor
turns out to have a simpler solution. The algorithm for its solution starts
from the atypicality matrix $A(\Si)$. The first step is to determine
those $\be$'s which belong to a set $\De_S(\Si)\subseteq\De_1�+$. This may
be done in several ways [4] but for our purposes here a diagramatic way is
preferable. It is illustrated in the following diagram in which the entries
$*$ specify the $\be$'s belonging to $\De_S(\Si)$ for $sl(3/4)$ with
$\Si=(432|\bar 2\bar 2\bar 3\bar 3)=[11;010]$.

$$\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan4\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& && && &\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan9\hrulefill&\multispan8\hrulefill\cr
   \omit&4&\omit&3&\omit&1&\omit&0
     & && && && && &\cr
   \multispan8\norulefill&\multispan9\hrulefill\cr
   \omit&2&\omit&1&\omit&\bar 1&\omit&\bar 2
     & && && && &\cr
   \multispan8\norulefill&\multispan7\hrulefill\cr
   \omit&0&\omit&\bar 1&\omit&\bar 3&\omit&\bar 4
     & && && &\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   }}
\qquad
\vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 9.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan{13}\hrulefill\cr
   & && &&*&&*&&0&\omit&0&\cr
   \multispan9\hrulefill\cr
   & && &&*&&0&\omit&0&\omit&0&\cr
   \multispan7\hrulefill&\multispan1\norulefill&\multispan5\hrulefill\cr
   & && &&0&\omit&0&&*&&*&\cr
   \multispan{13}\hrulefill\cr
   \omit& &\omit& && && && && &\cr
   \multispan4\norulefill&\multispan9\hrulefill\cr
   \omit& &\omit& && && && && &\cr
   \multispan4\norulefill&\multispan9\hrulefill\cr
   }}
$$
The entries $*$ are those covered by boxes of
$F�{\mu}$ and $F�{\bar{\ka}�\prime}$
positioned so that the $i$th row of $F�{\mu}$ and the $j$th
column of $F�{\bar{\ka}�\prime}$ terminate just to the left and just below
the position of the leftmost $0$ in $A(\Si)$.
If overlapping had occurred it would have
been necessary to truncate the diagram and reposition a new portion
around the position of the next zero in the atypicality matrix.
In the case of no overlap, as above,
the top boundary of $F�{\mu}$ and the right hand boundary of
$F�{\bar{\ka}�\prime}$ are extended until they meet.

The algorithm is then as follows. Each connected set of zeros
in the matrix of $*$'s and $0$'s constructed as above is renumbered
consecutively step by step in a shifting process whereby at every stage
$F�{\mu}$ and $F�{\bar{\ka}�\prime}$ either slide one step south and one
step west, respectively, or one step east and one step north, respectively.
In these two cases the zeros covered in this way are all to be renumbered
either $1$ or $1�\prime$, appropriately. The process is then repeated
until all zeros are covered. At each stage there is a choice of a
south-west (SW) or north-east (NE) slide leading to new unprimed and primed
entries. The process terminates after precisely $r$ steps, leading to a total
of $2�r$ distinctly labelled matrices $CF(\Si)$. The significance of this
labelling lies in the following:

{\bf Conjecture 6.} Let $\Si$ be multiply atypical of degree $r$. Each of the
$2�r$ matrices $CF(\Si)$ defines $\La$ such that $V(\Si)$ is a composition
factor of $\VK$. $\La$ is found by adding to $\Si$ those $\be$'s associated
with the positions of the unprimed numbers.

The procedure is exemplified as follows in the doubly atypical $sl(3/4)$
case $\Si=(432|\bar2\bar2\bar3\bar3)=[11;0;010]$.
$${\offinterlineskip
    \matrix{ & & & & & & & & & & & & & & & &*&*&1�\prime&2�\prime\cr
             & & & & & & & & & & & & & &NE& &*&1�\prime&1�\prime&1�\prime\cr
             & & & & & & & &*&*&1�\prime&0& &\nearrow& &
 &1�\prime&1�\prime&*&*\cr
             & & & & & &NE& &*&1�\prime&1&1& & & & & & & & \cr
             & & & & &\nearrow & & &1�\prime&1�\prime&*&*& &\searrow& &
 &*&*&1�\prime&2\cr
             & & & & & & & & & & & & & &SW& &*&1�\prime&1�\prime&1�\prime\cr
             &*&*&0&0& & & & & & & & & & & &1�\prime&1�\prime&*&*\cr
             &*&0&0&0& & & & & & & & & & & & & & & \cr
             &0&0&*&*& & & & & & & & & & & &*&*&1&1\cr
             & & & & & & & & & & & & & &NE& &*&1&2�\prime&1\cr
             & & & & &\searrow& & &*&*&1&1& &\nearrow& & &1&1&*&*\cr
             & & & & & &SW& &*&1&0&1& & & & & & & & \cr
             & & & & & & & &1&1&*&*& &\searrow & & &*&*&1&1\cr
             & & & & & & & & & & & & & &SW& &*&1&2&1\cr
             & & & & & & & & & & & & & & & &1&1&*&*\cr}}
$$

It can be inferred from the final four diagrams that $V(\Si)$, with
$\Si= (432|\bar 2\bar 2\bar 3\bar 3) = [11;0;010]$, is a
composition factor of four Kac-modules $\VK$ having highest weights:
$$
\eqalign{(432|\bar 2\bar 2\bar 3\bar 3)+(000|0000)
&= (432|\bar 2\bar 2\bar 3\bar 3)=[11;0;010];\cr
(432|\bar 2\bar 2\bar 3\bar 3)+(100|000\bar 1)
&= (532|\bar 2\bar 2\bar 3\bar 4)=[21;0;011];\cr
(432|\bar 2\bar 2\bar 3\bar 3)+(222|\bar 1\bar 2\bar 1\bar 2)
&= (654|\bar 3\bar 4\bar 4\bar 5)=[11;1;101];\cr
(432|\bar 2\bar 2\bar 3\bar 3)+(232|\bar 1\bar 2\bar 2\bar 2)
&= (664|\bar 3\bar 4\bar 5\bar 5)=[02;1;110].\cr}
$$

This procedure is very easy to program and all our checks to date
indicate that the results are entirely consistent with the determination
of composition factors of Kac-modules by means of the algorithm based on
criticality and strip removals. Indeed it was the nice combinatorial
features of this algorithm in its identification of $\be$'s to be
subtracted from $\La$ to give $\Si$ that led to the discovery of the very
simple converse procedure just described for obtaining all $2�r$
possible $\La$ from a knowledge of the $r$-fold atypical $\Si$.

{\bf References}
\item{[1]} J.\ Van der Jeugt, J.W.B.\ Hughes and R.C.\ King
{\sl Atypical modules of the Lie superalgebra $gl(m/n)$} (to be
published elsewhere in this volume).
\item{[2]} V.G.\ Kac,
{\sl Lecture Notes in Mathematics} {\bf 676}, 579--626 (1977)
\item{[3]} M.D.\ Gould, {\sl J.\ Phys.} {\bf A22}, 1209--1221 (1989)
\item{[4]} J.\ Van der Jeugt, J.W.B.\ Hughes, R.C.\ King and J.\ Thierry-Mieg,
``Character formulae for irreducible modules of the Lie
superalgebra $sl(m/n)$'', {\sl J.\ Math.\ Phys.}, in press (1990)
\item{[5]} J.\ Van der Jeugt, J.W.B.\ Hughes, R.C.\ King and J.\ Thierry-Mieg,
``A character formula for singly atypical modules of the Lie superalgebra
$sl(m/n)$'', {\sl Commun. Algebra}, in press (1990)
\end

 Notice that there
is no question of the strip associated with $\be_2$ linking with or
wrapping that associated with $\be_1$ since any extension of the former
leads to a regular diagram before contact is made with the latter.

