% Paper written in LaTeX
% ATYPICAL MODULES OF THE LIE SUPERALGEBRA $gl(m/n)$
% J. Van der Jeugt, J.W.B. Hughes, R.C. King and J. Thierry-Mieg
% in Proceedings of the XVIIIth International Colloquium on Group
% Theoretical Methods in Physics, Lecture Notes in Physics 382,
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\def\ep{\epsilon}  \def\vep{\varepsilon}
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\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}
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  }
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  {\textstyle \rm Z\kern -0.3em Z}
  {\textstyle \rm Z\kern -0.3em Z}
  {\scriptstyle \rm Z\kern -0.25em Z}
  {\scriptstyle \rm Z\kern -0.25em Z}
  }
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  {\textstyle \rm\hskip 0.2em R\kern -0.95em I\kern 0.55em}
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  }
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  {\textstyle \rm C\kern -0.55em I\kern 0.2em}
  {\textstyle \rm C\kern -0.55em I\kern 0.2em}
  {\scriptstyle \rm C\kern -0.45em I\kern 0.15em}
  {\scriptstyle \rm C\kern -0.45em I\kern 0.15em}
  }
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\def\VK{\overline V(\La)}
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\begin{document}
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\begin{center}
ATYPICAL MODULES OF THE LIE SUPERALGEBRA $gl(m/n)$
\end{center}
\vskip 2mm
\begin{center}
J.\ Van der Jeugt\footnote{Research Associate of the NFWO
(National Funds for Scientific Research of Belgium)} (University of
Ghent, Belgium), J.W.B.\ Hughes (Queen Mary and Westfield College, U.K.),
R.C.\ King (University of Southampton, U.K.) and J.~Thierry-Mieg
(University of Montpellier, France)\footnote{Talk presented by J.\
Van der Jeugt}
\end{center}
\vskip 2mm
%
Let $G = G_{\bar 0} \oplus G_{\bar 1}$ be the {\sl general linear}
Lie superalgebra
$gl(m/n)$~[2] consisting of complex matrices
$\left( {A\atop C}{B\atop D} \right)$ of size $(m+n)^2$.
The {\sl even} subspace $G_{\bar 0}$ of $G$ consists of the matrices
$\left({A\atop 0}{0 \atop D}\right)$ and the {\sl odd} subspace
$G_{\bar 1}$ consists of the matrices $\left( {0 \atop C}{B \atop 0}\right)$.
The {\sl bracket} between homogeneous elements is defined by
$[a,b] = ab - (-1)^{\al\be} ba$ for $a\in G_\al, b\in G_\be
\quad (\al,\be\in\{\bar 0, \bar 1\} = \Zah_2)$.
Thus the even
subalgebra is isomorphic to $gl(m)\oplus gl(n)$.
$G$ admits a consistent $\Zah$-grading
$G = G_{-1}\oplus G_0 \oplus G_{+1}$ where $G_0 = G_{\bar 0}$,
$G_{+1}$ is the space of matrices of the form
$\left( {0 \atop 0}{B \atop 0}\right)$ and $G_{-1}$ is the space of
matrices of the form $\left( {0 \atop C}{0 \atop 0}\right)$.
The {\sl special linear} Lie superalgebra $sl(m/n)$ is
the subalgebra of $gl(m/n)$ consisting of matrices with vanishing
supertrace.
In what follows we put $G=gl(m/n)$, but all of the
results can be reformulated for $sl(m/n)$ as well.

The {\sl Cartan subalgebra} $H$ of $G$ consists of the subspace
of diagonal matrices. The {\sl root} or {\sl weight space} $H^*$ is
the dual space of $H$ and is spanned by the forms $\ep_i$ ($i=1,\ldots,m$)
and $\de_j$ ($j=1,\ldots,n$). The inner product
on the weight space $H^*$ is given by~[5]
$\langle\ep_i|\ep_j\rangle = \de_{ij}$,
$\langle\ep_i|\de_j\rangle = 0$,
$\langle\de_i|\de_j\rangle = -\de_{ij}$,
where $\de_{ij}$ is the usual Kronecker symbol. In this $\ep\de$-basis
the even roots of $G$ are of the form $\ep_i - \ep_j$ or
$\de_i - \de_j$, and the odd roots are of the form $\pm(\ep_i - \de_j)$.
Let $\De$ denote the set of all roots, $\De_0$ the set of even roots
and $\De_1$ the set of odd roots. As a system of {\sl simple roots}
one takes the so-called {\sl distinguished set}~[3]
$\ep_1-\ep_2$, $\ep_2-\ep_3$, $\ldots$, $\ep_m-\de_1$,
$\de_1-\de_2$, $\ldots$, $\de_{n-1}-\de_n$. Then the set $\De^+$
of {\sl positive} roots consists of the elements
$\ep_i-\ep_j$ ($i<j$), $\de_i-\de_j$ ($i<j$) and $\ep_i-\de_j$.
Now the notations $\De^+_0$ and $\De^+_1$ are obvious;
in particular~:
\beq
\De^+_1 = \{ \be_{ij}=\ep_i-\de_j, \quad i=1,\dots,m,\quad j=1,\ldots,n\}.
\label{beta}
\eeq

All simple modules (i.e.\ irreducible representations) of the classical
simple Lie superalgebras were classified by Kac~[3].
Kac's result specified to $sl(m/n)$ implies that every
finite-dimensional simple $G$-module $V$
is a highest weight module $V(\La)$ specified by an {\sl integral dominant}
weight $\La$. A weight $\La\in H^*$ is said to be integral dominant if and
only if its so-called {\sl Kac-Dynkin labels}
$\La = [a_1,a_2,\ldots,a_{m-1};a_m;a_{m+1},\ldots, a_{m+n-1}]$ are such
that $a_i\in\Nat$ for $i\not= m$ whereas $a_m$ can be any complex number.
For our purpose it is sufficient to
consider only those $\La$ for which $a_m\in\Zah$.
If $\La$ is expressed in terms
of the $\ep\de$-basis as $\La = \sum\mu_i\ep_i + \sum\nu_j\de_j$, then
the Kac-Dynkin labels of $\La$ are given by
$a_i=\mu_i-\mu_{i+1}\; (i<m)$, $ a_m=\mu_m+\nu_1$,
$a_{m+j} = \nu_j-\nu_{j+1}\; (j<n)$.
Note that the coordinates in the $\ep\de$-basis represent a unique
weight of $gl(m/n)$ whereas the Kac-Dynkin labels represent a unique
weight of $sl(m/n)$ rather than $gl(m/n)$. Often it will be useful to
represent a weight $\La$ by a composite Young diagram, consisting
of the diagrams of $\{\mu\}$ and $\{\overline\nu\}$ in appropriate
positions~[5].
For example, for $gl(4/6)$ and $\La = (7,6,6,3|\bar 1,\bar 1,\bar 3,
\bar 3,\bar 5, \bar 5)$ in the $\ep\de$-basis (where $\bar k$ stands
for $-k$), the composite Young diagram is shown in~(\ref{YD-atyp}).
In this case, for example, the Kac-Dynkin labels of $\La$ are
[1,0,3;2;0,2,0,2,0].

The basic problem we are concerned with is the determination of the
weights and weight multiplicities of $V(\La)$. Such information is contained in
the so-called {\sl character} of $V(\La)$, which is by definition
equal to $\hbox{ch}V(\La) = \sum_\et(\hbox{dim}V_\et)\,e^\et$, where
$\hbox{dim}V_\et$ is the multiplicity of the weight $\et$
appearing in the weight space decomposition $V(\La) = \bigoplus_\et V_\et$.
Recall that for a (reductive) Lie algebra $G_0$ (in the present case we
can think of $G_0$ as the even part $gl(m)\oplus gl(n)$ of $gl(m/n)$)
the character formula of a $G_0$-module $V_0(\La)$ with highest weight
$\La$ is given by Weyl's character
formula~:
\beq
\hbox{ch}V_0(\La) = L_0^{-1}\,\sum_{w\in W} \vep(w)\,
 w\left(e^{\La+\rh_0}\right),\qquad
 L_0 = \prod_{\al\in\De_0^+}(e^{\al/2}-e^{-\al/2}),
\eeq
where $W$ is the Weyl group of $G_0$, $\vep(w)$ is the {\sl signature}
of $w\in W$ and $\rh_0 = {1\over 2}\sum_{\al\in\De_0^+}\al$.

A very important finite-dimensional highest weight module $\VK$, the
so-called Kac-module, was introduced in~[3].
For given integral dominant
weight $\La$, the $G_0$-module $V_0(\La)$ is uniquely determined (up to
isomorphism), and can be extended to a $G_0\oplus G_{+1}$-module by putting
$G_{+1}V_0(\La)=0$. Then one defines the induced module
\beq
\VK = \hbox{Ind}_{G_0\oplus G_{+1}}^{G} \, V_0(\La)
\cong U(G_{-1})\otimes V_0(\La).
\eeq
It follows from the structure of $U(G_{-1})$ that
\beq
\hbox{ch}\VK = \ch_K(\La) =
L_0^{-1}\,\sum_{w\in W}\vep(w)\,w\Bigl(
e^{\La+\rh_0}\,\prod_{\be\in\De_1^+}(1+e^{-\be})\Bigr).
\label{Kac-char}
\eeq

When is $\VK$ a {\sl simple} module? The answer to this question
was given by Kac~[3]~: $\VK$ is simple if and only if
$\langle\La+\rh|\be\rangle\not=0$ for all $\be$ in $\De_1^+$. Herein
$\rh=\rh_0-\rh_1$, where $\rh_0$ has been defined previously and
$\rh_1 = {1\over 2}\sum_{\be\in\De_1^+}\be$. In the case that all
$\langle\La+\rh|\be\rangle\not=0$, $\La$ and $V(\La)=\VK$ are said
to be {\sl typical}, otherwise $\La$ and $V(\La)\not=\VK$ are said
to be {\sl atypical}. If $\La$ is atypical, $\VK$ contains a unique maximal
submodule $M(\La)$ and $V(\La)\cong \VK/M(\La)$. Our main aim is to
determine characters for such atypical modules. One of the useful tools
in studying atypical modules is the so-called {\sl atypicality matrix}
$A(\La)$ consisting of the $mn$ integers $A(\La)_{ij} =
\langle\La+\rh|\be_{ij}\rangle$~[4], where $\be_{ij}$ has
been defined in~(\ref{beta}).
In terms of its components in the
$\ep\de$-basis, $A(\La)_{ij}$ is given by $\mu_i+\nu_j+m-i-j+1$.
The $m\times n$ atypicality matrix fits nicely into the compositie
Young diagram, as is illustrated here for our example,
$\La = (7,6,6,3|\bar 1,\bar 1,\bar 3,\bar 3,\bar 5, \bar 5)$ for
$gl(4/6)$~:
\beq
\vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$#$\hss}\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& &\omit& &\omit& && && &\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& &\omit& &\omit& && && &\cr
   \multispan4\norulefill&\multispan9\hrulefill\cr
   \omit& &\omit& && && && && &\cr
   \multispan4\norulefill&\multispan9\hrulefill\cr
   \omit& &\omit& && && && && &\cr
   \multispan{13}\hrulefill\cr
   & && && && && && &\cr
   \multispan{13}\hrulefill&\multispan{14}\hrulefill\cr
   \omit&9&\omit&8&\omit&5&\omit&4&\omit&1&\omit&0
     & && && && && && && && &\cr
   \multispan{12}\norulefill&\multispan{15}\hrulefill\cr
   \omit&7&\omit&6&\omit&3&\omit&2&\omit&\bar 1&\omit&\bar 2
     & && && && && && && &\cr
   \multispan{12}\norulefill&\multispan{13}\hrulefill\cr
   \omit&6&\omit&5&\omit&2&\omit&1&\omit&\bar 2&\omit&\bar 3
     & && && && && && && &\cr
   \multispan{12}\norulefill&\multispan{13}\hrulefill\cr
   \omit&2&\omit&1&\omit&\bar 2&\omit&\bar 3&\omit&\bar 6&\omit&\bar 7
     & && && && &\cr
   \multispan{12}\norulefill&\multispan7\hrulefill\cr
   }}
\label{YD-atyp}
\eeq
This $\La$ is atypical of type $\be_{1,6}$, and is {\sl singly atypical}.

Various character formulae for atypical $V(\La)$ have been proposed,
most of which are of the following form (see~[4,5] and
references therein)~:
\beq
\ch_{\De(\La)}(\La) =
L_0^{-1}\,\sum_{w\in W}\vep(w)\,w\Bigl(
e^{\La+\rh_0}\,\prod_{\be\in\De(\La)}(1+e^{-\be})\Bigr),
\label{atyp}
\eeq
where $\De(\La)$ is some subset of $\De_1^+$. In particular,
Bernstein and Leites~[1] proposed
$\De(\La)=\{ \be\in\De_1^+\,|\,\langle\La+\rh|\be\rangle\not=0\}$,
in which case $\ch_{\De(\La)}(\La)$ in~(\ref{atyp}) is
replaced by $\ch_L(\La)$. However, counterexamples were found to their
formula. Similarly, counterexamples were found to other formulae of the
type~(\ref{atyp}) [5], and in particular we were able to prove
that for $G=gl(3/4)$ and $\La=[1,1;0;0,1,0]$ no set $\De(\La)$ exists
yielding the correct character of $V(\La)$. Hence no formula of the
type~(\ref{atyp}) can give correctly the characters of all simple
modules $V(\La)$ of $gl(m/n)$.

There is, however, the important class of singly atypical modules where
the problem of finding character formulae has been solved~[4].
When there is only one $\ga$ in $\De_1^+$ with
$\langle\La+\rh|\ga\rangle =0$ (and $\langle\La+\rh|\be\rangle \not=0$
for all $\be\not=\ga$), $\La$ is singly atypical. In this case,
we proved that the maximal submodule $M(\La)$ is itself
a simple $G$-module, and that $M(\La)\cong V(\Ph)$, where
$\Ph=w\cdot(\La-k\ga) = w(\La-k\ga+\rh)-\rh$ and $\La-k\ga$ is the
first element of the sequence $\La-\ga$, $\La-2\ga$, $\ldots$ that
can be mapped into an integral dominant weight $\Ph$ by means of
a $w\cdot$ action. In terms of the composite Young diagram,
with the zero in the atypicality matrix at position $(i,j)$, we
move to the end of row $i$ in the $\mu$-part of the diagram and
to the end of column $j$ in the $\nu$-part of the diagram, and perform
a strip removal of length $k$ in both parts of the diagram,
removing one box at a time until the
composite diagram is {\sl standard}~[5].
In our example~(\ref{YD-atyp})
this leads to the following strip removal~:
\beq
\vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$#$\hss}\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& &\omit& &\omit& &&\rm X&&\rm X&\cr
   \multispan8\norulefill&\multispan5\hrulefill\cr
   \omit& &\omit& &\omit& &\omit& &&\rm X&& &\cr
   \multispan4\norulefill&\multispan9\hrulefill\cr
   \omit& &\omit& &&\rm X&&\rm X&&\rm X&& &\cr
   \multispan4\norulefill&\multispan9\hrulefill\cr
   \omit& &\omit& && && && && &\cr
   \multispan{13}\hrulefill\cr
   & && && && && && &\cr
   \multispan{13}\hrulefill&\multispan{14}\hrulefill\cr
   \omit&9&\omit&8&\omit&5&\omit&4&\omit&[1]&\omit&[0]
     && && && && && &&\rm X&&\rm X&\cr
   \multispan{12}\norulefill&\multispan{15}\hrulefill\cr
   \omit&7&\omit&6&\omit&3&\omit&2&\omit&[\bar 1]&\omit&\bar 2
     && && && && && &&\rm X&\cr
   \multispan{12}\norulefill&\multispan{13}\hrulefill\cr
   \omit&6&\omit&5&\omit&[2]&\omit&[1]&\omit&[\bar 2]&\omit&\bar 3
     && && && &&\rm X&&\rm X&&\rm X&\cr
   \multispan{12}\norulefill&\multispan{13}\hrulefill\cr
   \omit&2&\omit&1&\omit&\bar 2&\omit&\bar 3&\omit&\bar 6&\omit&\bar 7
     && && && &\cr
   \multispan{12}\norulefill&\multispan7\hrulefill\cr
   }}
\label{strip}
\eeq
Note that only after 6 box removals, the remaining Young diagrams
are standard. Thus $\Ph=w\cdot(\La-6\be_{1,6})$ for some $w\in W$, and
it follows that $\Ph = \La -(\be_{1,6}+\be_{1,5}+\be_{2,5}+\be_{3,5}+\be_{3,4}
+\be_{3,3}) = (5,5,3,3|\bar 1,\bar 1,\bar 2,\bar 2,\bar 2,\bar 4)$. Both
strip removals (indicated by X's) are necessarily of the same shape,
and the positions of the brackets $[\;]$ in the atypicality matrix (which
constitute the same shape again) determine the $\be_{ij}$ one has to
subtract from $\La$ in order to obtain $\Ph$. Also, $\Ph$ is atypical of
type $\be_{3,3}$, which corresponds to the ``tail'' of the removal strip.
Making use of these properties, of combinatorial properties of the
atypicality matrix, and of recursion, one is able to prove~[4] that
for the singly atypical case $\hbox{ch}V(\La) = \ch_L(\La)$.
Then, making a formal expansion, one can rewrite the character as
an infinite alternating series of Kac-characters $\ch_K(\la)$~:
\beq
\hbox{ch}V(\La) = \ch_L(\La) = \sum_{t=0}^{\infty}
 (-1)^t\,\ch_K(\La-t\ga).
\label{singly}
\eeq

Let us now return to the more general case of {\sl multiply atypical
modules}.
For reasons of presentation we shall illustrate here the case of
doubly atypical modules. Thus $\langle\La+\rh|\be_1\rangle=0$,
$\langle\La+\rh|\be_2\rangle=0$, and $\langle\La+\rh|\be\rangle\not=0$
for every $\be\not=\be_1,\be_2$. Similarly as
in~(\ref{singly}) one can formally expand the Bernstein-Leites formula
 as an infinite
alternating series of Kac-characters~:
\beq
\ch_L(\La) = \sum_{t_1,t_2=0}^{\infty}
 (-1)^{t_1+t_2}\,\ch_K(\La-t_1\be_1-t_2\be_2)
 =\sum_{C_\La} (-1)^{|\La-\la|}\,\ch_K(\la),
\label{doubly}
\eeq
where $C_\La=\{\la=\La-t_1\be_1-t_2\be_2\}$ is the ``cone'' with vertex
$\La$ and $(-1)^{|\La-\la|}=(-1)^{t_1+t_2}$. Let $\be_1=\ep_i-\de_j$
and $\be_2=\ep_k-\de_l$ with $i>k$ and $j<l$. Then there is a unique
$w_{12}$ in $W$ which permutes the components $i$ and $k$,
$m+j$ and $m+l$, and leaves all the other components of a weight
in the $\ep\de$-basis invariant. Let
$H_{12}=\{\et\in H^*|w_{12}\cdot(\et)=\et\}$. Clearly, such a hyperplane
splits the weight space $H^*$ into two half-spaces.
The {\sl truncated cone}
$C^+_\La$ is defined to be the set of weights of $C_\La$ that are
in the same half-space as $\La$. Then we conjecture~:
$\hbox{ch}V(\La) =
\ch_L(\La)  =\sum_{C_\La} (-1)^{|\La-\la|}\,\ch_K(\la)$
if $\La$ is not critical, and
$\hbox{ch}V(\La)  =  \sum_{C^+_\La} (-1)^{|\La-\la|}\,\ch_K(\la)$
if $\La$ is critical, where
$\La$ is {\sl critical} if and only if the entry $A(\La)_{k,j}$
in the atypicality matrix is equal to the ``hook length'' connecting
the two zeros (at positions $(i,j)$ and $(k,l)$) in the atypicality matrix,
i.e.\ equal to $i-k+l-j-1$~[5]. The ways in which this conjecture
has been tested, and how it works for atypical modules with
degree of atypicality $>2$ is described in~[5].

Let us emphasize that the given formulae are
expansions of $\hbox{ch}V(\La)$ in terms of the formal characters
$\ch_K(\la)$, which are characters of Kac-modules when $\la$ is dominant
integral.
One may also consider the inverse problem~: given the Kac-module
$\VK$, how can $\hbox{ch}\VK$ be expressed as a (necessarily finite)
sum of characters of simple modules $\hbox{ch}V(\si)$? In other words,
what are the non-zero multiplicities $n_\si$ in the expression
$\hbox{ch}\VK=\sum_\si n_\si\hbox{ch}V(\si)$? This is known
as the problem of the determination of the composition series of $\VK$. Recently,
we have made a lot of progress in solving this question. Our results
concerning the determination of the composition factors of
the Kac-module $\VK$ were presented at this Colloquium by
R.C.\ King, who reports on it elsewhere in this Volume.
%
%
\vskip 2mm
\noindent REFERENCES

\noindent
[1]
I.N.\ Bernstein and D.A.\ Leites, {\sl C.R.\ Acad.\ Bulg.\ Sci.} {\bf 33},
1049--51 (1980)

\noindent
[2]
V.G.\ Kac, {\sl Adv.\ Math.} {\bf 26}, 8--96 (1977)

\noindent
[3]
V.G.\ Kac,
{\sl Lecture Notes in Mathematics} {\bf 676}, 579--626 (1977)

\noindent
[4]
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)

\noindent
[5]
 --- , ``Character formulae for irreducible modules of the Lie
superalgebra $sl(m/n)$,'' {\sl J.\ Math.\ Phys.}, in press (1990)
%
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