% Plain tex file % Character formulae for the Lie superalgebra C(n) % Joris Van der Jeugt % Commun. Algebra 19 (1991) 199-222. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % 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} % % the end-of-proof box % \def\mybox{\setbox1=\vbox{\hrule height 4pt width 4pt} \hfill\llap{\box1}} % % some fonts % \font\titlefont=cmbx10 scaled \magstep2 \font\sectionfont=cmbx10 scaled \magstep1 \def\titlepaper#1{\vskip 1cm \centerline{\titlefont #1} \vskip 1cm} \outer\def\section#1\par{\vskip0pt plus.05\vsize\penalty-250 \vskip0pt plus-.05\vsize\bigskip\vskip\parskip \message{#1}\leftline{\sectionfont#1}\nobreak\smallskip\noindent} \def\subsection#1{\vskip 1cm \noindent\line{\bf #1\hfil}\vskip 5mm \nobreak} % % some references % \def\refp#1#2#3#4#5#6#7 {\item{[#1]} #2, ``#3,'' {\sl#4~\bf#5} (#6), #7.\hfill} \def\refpb#1#2#3#4#5#6#7#8 {\item{[#1]} #2, ``#3,'' in {\sl#4}, #5 (#6), ed.~#7, pp.~#8.\hfill} \def\refb#1#2#3#4#5 {\item{[#1]} #2, {\sl#3}, #4 (#5).\hfill} \def\ss{\vskip 2mm \noindent} \def\br{\hfill\break\noindent} \def\refa#1#2#3{\vskip 2mm \noindent #1, {\sl #2}, #3.} % % the maths. symbols for natural, integer, real and complex numbers % \def\Nat{\mathchoice {\textstyle \rm\hskip 0.2em N\kern -0.95em I\kern 0.55em} {\textstyle \rm\hskip 0.2em N\kern -0.95em I\kern 0.55em} {\scriptstyle \rm\hskip 0.15em N\kern -0.7em I\kern 0.3em} {\scriptstyle \rm\hskip 0.15em N\kern -0.7em I\kern 0.3em} } \def\Zah{\mathchoice {\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} } \def\Real{\mathchoice {\textstyle \rm\hskip 0.2em R\kern -0.95em I\kern 0.55em} {\textstyle \rm\hskip 0.2em R\kern -0.95em I\kern 0.55em} {\scriptstyle \rm\hskip 0.15em R\kern -0.7em I\kern 0.3em} {\scriptstyle \rm\hskip 0.15em R\kern -0.7em I\kern 0.3em} } \def\Q{\mathchoice {\textstyle \rm Q\kern -0.60em I\kern 0.25em} {\textstyle \rm Q\kern -0.60em I\kern 0.25em} {\scriptstyle \rm Q\kern -0.5em I\kern 0.2em} {\scriptstyle \rm Q\kern -0.5em I\kern 0.2em} } \def\C{\mathchoice {\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} } % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Finally the twelve point text stuff \font\twelverm=cmr12 \font\twelvei=cmmi12 %\font\twelvesy=cmsy12 \font\twelvesy=cmsy10 at 12 true pt \font\twelvebf=cmbx12 \font\twelvett=cmtt12 \font\twelveit=cmti12 \font\twelvesl=cmsl12 % The skew chars for these new fonts \skewchar\twelvei='177 \skewchar\twelvesy='177 \hyphenchar\twelvett=-1 \font\drtwelverm=cmr10 at 12 true pt \font\drtwelvei=cmmi10 at 12 true pt \font\drtwelvesy=cmsy10 at 12 true pt \font\drtwelvebf=cmbx10 at 12 true pt \font\drtwelvett=cmtt10 at 12 true pt \font\drtwelveit=cmti10 at 12 true pt \font\drtwelvesl=cmsl10 at 12 true pt \skewchar\drtwelvei='177 \skewchar\drtwelvesy='177 \hyphenchar\drtwelvett=-1 % Keep the same sizes for scripts and scriptscripts as plain TeX % since this is a GR thesis with loads of indices \def\twelvepoint{\def\rm{\fam=0\twelverm}% \textfont0=\twelverm \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm \textfont1=\twelvei \scriptfont1=\teni \scriptscriptfont1=\seveni \textfont2=\twelvesy \scriptfont2=\tensy \scriptscriptfont2=\sevensy \textfont\itfam=\twelveit \def\it{\fam\itfam\twelveit}% \textfont\slfam=\twelvesl \def\sl{\fam\slfam\twelvesl}% \textfont\ttfam=\twelvett \def\tt{\fam\ttfam\twelvett}% \textfont\bffam=\twelvebf \def\bf{\fam\bffam\twelvebf}% \scriptfont\bffam=\tenbf \scriptscriptfont\bffam=\sevenbf \lineskip=1pt \baselineskip=17pt\lineskiplimit=0pt\rm} %NB ten pt copied from 12 pt therefore sub- super- etc might not work. \def\tenpoint{\def\rm{\fam=0\tenrm}% \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont\itfam=\tenit \def\it{\fam\itfam\tenit}% \textfont\slfam=\tensl \def\sl{\fam\slfam\tensl}% \textfont\ttfam=\tentt \def\tt{\fam\ttfam\tentt}% \textfont\bffam=\tenbf \def\bf{\fam\bffam\tenbf}% \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \normalbaselines\rm} \def\twelvepointdraft{\def\rm{\fam=0\drtwelverm}% \message{***DRAFT FONTS IN USE***} \textfont0=\drtwelverm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \textfont1=\drtwelvei \scriptfont1=\seveni \scriptscriptfont1=\fivei \textfont2=\drtwelvesy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont\itfam=\drtwelveit \def\it{\fam\itfam\drtwelveit}% \textfont\slfam=\drtwelvesl \def\sl{\fam\slfam\drtwelvesl}% \textfont\ttfam=\drtwelvett \def\tt{\fam\ttfam\drtwelvett}% \textfont\bffam=\drtwelvebf \def\bf{\fam\bffam\drtwelvebf}% \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \normalbaselines\rm} \font\sc=cmr10 %small caps font %\font\sc=amr7 %small caps font %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\VK{\overline V(\La)} \def\cha{\hbox{ch}} \twelvepoint \hyphenation{Krijgs-laan} \centerline{\titlefont Character formulae for the Lie superalgebra C(n)} \vskip 2cm \centerline{by} \centerline{Joris Van der Jeugt\footnote*{Aangesteld Navorser N.F.W.O. (National Fund for Scientific Research of Belgium)}} \vskip 1cm \noindent Laboratorium voor Numerieke Wiskunde en Informatica, Rijksuniversiteit Gent, Krijgslaan 281--S9, 9000~Gent, BELGIUM and \noindent Faculty of Mathematical Studies, University of Southampton, Southampton SO9 5NH, U.K. \vskip 3cm \noindent {\bf Abstract.} \noindent {\sl Using a recent result on the structure of the maximal submodule of the Kac-module for type~I Lie superalgebras, we derive two character formulae for all finite-dimensional simple modules (i.e.~all irreducible representations) of the Lie superalgebra $C(n)=osp(2,2n-2)$. This is the first time a character formula is proven for a series of simple Lie superalgebras covering both the typical and atypical modules.} \vfill \section{1. Introduction} Lie superalgebras and their representations play an important role in the understanding and exploitation of supersymmetry in physical systems in the areas of particle physics, nuclear physics and condensed matter physics. An early review of their use was given by Corwin {\sl et al} (1975); a comprehensive review with the emphasis on their applications in these fields is given in a paper by Bars (1985). A complete classification of the finite-dimensional simple Lie superalgebras over $\C$ has been given by Kac (1975, 1977a) and by Scheunert {\sl et al} (1976, 1979). Kac showed that the basic classical Lie superalgebras are closely analogous to the finite-dimensional simple Lie algebras over $\C$. In particular they can also be constructed from a (super) Cartan matrix or, equivalently, from a Kac-Dynkin diagram (Kac 1977a, Van de Leur 1986). In his seminal paper on the representations of the basic classical Lie superalgebras (Kac 1977b), Kac proved that all inequivalent finite-dimensional irreducible representations (or, in the mathematicians terminology, all {\sl simple modules}) may be labelled by means of Kac-Dynkin labels which serve to specify the highest weight, $\La$, of the corresponding simple module $V(\La)$. He showed that the simple modules of a basic classical Lie superalgebra $G=G_{\bar 0}\oplus G_{\bar 1}$ fall into two classes, which were called {\sl typical} and {\sl atypical}. If the even, resp.~odd, positive roots of $G$ are denoted by $\De_0^+$, resp.~$\De_1^+$, and ${\overline\De}_1^+=\{\be\in\De_1^+| 2\be\notin\De_0^+\}$, then the highest weight $\La$ or the module $V(\La)$ is said to be typical if and only if $$ \langle\La+\rh\mid\be\rangle\not=0,\quad\forall\be\in{\overline\De}_1^+, \eqno(1.1) $$ where $\langle\;\mid\;\rangle$ is a non-degenerate bilinear form, and $\rh=\rh_0-\rh_1$, with $\rh_0$ and $\rh_1$ half the sum of the even and odd positive roots, respectively, of $G$. Conversely, if $$ \langle\La+\rh\mid\be\rangle =0,\quad\hbox{for any }\be\in{\overline\De}_1^+, \eqno(1.2) $$ then $\La$ and the module $V(\La)$ are said to be atypical. The weight structure of a $G$ module is completely determined by its character. For typical modules, Kac gave a character formula for all basic classical Lie superalgebras (Kac 1977b). For atypical modules no general character formula has been given so far, although many partial results have been obtained. The Lie superalgebra that has been studied most extensively is $sl(m/n)$, or $A(m-1,n-1)$ in Kac's notation. Some progress has been made there in understanding the properties of atypical modules through the introduction of supertableaux methods (Balantekin 1984, Berele and Regev 1987, Cummins and King 1987, Dondi and Jarvis 1981, Hurni 1987), and the use of shift operators and weight space techniques (Hurni and Morel 1983, Palev 1986, Van der Jeugt 1987), but to date these methods have not provided character formulae for all atypical modules of $sl(m/n)$. Bernstein and Leites (1980, Leites 1980) published a generalisation of Kac's character formula, but it was soon afterwards observed (Thierry-Mieg 1983) that their formula was not correct for all atypical modules (for example, in general it failed to give the correct character for the identity representation) although it seemed to work for many particular cases. Realizing the failure of the Bernstein--Leites formula, Hughes and King (1987) conjectured another character formula for $sl(m/n)$, and yet another formula is found in the work of Serganova and Serge'ev (1987). In a recent paper, it is shown that none of the formulae published so far covers all atypical modules of $sl(m/n)$ (Hughes {\sl et al} 1989a). However, an important step forward was recently obtained by Hughes {\sl et al} (1989b) in the case of {\sl singly atypical} modules of Type~I Lie superalgebras (i.e.~for the series $A(m,n)$ and $C(n)$). The module $V(\La)$ or the weight $\La$ is called singly atypical if the condition (1.2) is satisfied for just one $\be\in\De_1^+$. In that paper, a unique characterisation was given of the maximal proper submodule of an indecomposable module introduced by Kac, the so-called {\sl Kac-module} $\VK$, when $\La$ is singly atypical. Using this and various combinatorial properties of the so-called {\sl atypicality matrix}, a character formula for all singly atypical simple modules of $sl(m/n)$ was deduced (Hughes {\sl et al} 1989b). In the present paper we use the theorem proved by Hughes {\sl et al} (1989b) for singly atypical modules of type~I Lie superalgebras (Theorem 3.8 in this paper) and apply it to the case of $G=C(n)$. The series $C(n)$ contains only typical or singly atypical modules, and hence our analysis covers {\sl all} simple modules of $C(n)$. In Section~2, we introduce the Lie superalgebra $C(n)$, and fix some notation. In Section~3, we give the definition and some properties of the Kac-module, including the theorem on the maximal proper submodule of a singly atypical Kac-module. The particular case of a $C(n)$ module which is singly atypical of type $\al_1$ (where $\al_1$ is the unique odd simple root of $C(n)$) is considered in Section~4, and it turns out to be rather easy to obtain a character formula in this situation. Then we study some properties of a matrix consisting of the numbers $\langle\La+\rh\mid\be\rangle$, with $\be\in\De_1^+$; this is the atypicality matrix (Section~5). Using these properties, we obtain a character formula for all atypical modules of $C(n)$ in Section~6 (Theorem~6.2 or Corollary~6.3). Then, exploiting Weyl symmetries of the character formula obtained, we prove that it is equivalent to yet another character formula (Theorem~6.6) which is superior to the previous one from the computational point of view. Finally, some examples are given to illustrate our results. \section{2. The Lie superalgebra C(n)} A complex Lie superalgebra $G$ is a $\Zah_2$-graded linear vector space, $G=G_{\bar 0}\oplus G_{\bar 1}$ over $\C$ with a bracket $[\;,\;]$ such that $\forall a\in G_\al$, $\forall b\in G_\be$ and $\forall\al,\be\in\Zah_2$ (Kac 1977a, Scheunert 1979) $$ \eqalign{ [a,b]&\in G_{\al+\be},\cr [a,b]&=-(-1)^{\al\be}[b,a],\cr [a,[b,c]]&=[[a,b],c]+(-1)^{\al\be}[b,[a,c]].\cr}\eqno(2.1) $$ Note that the {\sl even} part $G_{\bar 0}$ is a complex Lie algebra, and that the {\sl odd} part $G_{\bar 1}$ is a $G_{\bar 0}$ module under the adjoint action. The simplest example of a Lie superalgebra is given by $gl(m/n)$ with $m,n\in\Nat$. Its natural matrix realisation takes the form: $$ gl(m/n)=\left\lbrace x=\left(\matrix{A&B\cr C&D\cr}\right) \mid A\in M_{m\times m}, B\in M_{m\times n}, C\in M_{n\times m}, D\in M_{n\times n} \right\rbrace , \eqno(2.2) $$ 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 $D=0$. In the case of $G=gl(m/n)$, the bracket is determined in the natural matrix representation by $$ [a,b]=ab-(-1)^{\al\be}ba,\qquad\forall a\in G_\al\hbox{ and } \forall b\in G_\be. \eqno(2.3) $$ One defines the {\sl supertrace} $\hbox{str}(x)$ and the {\sl supertranspose} $x^T$ of $x=\bigl({A\atop C}{B\atop D}\bigr)$ as (Kac 1977a, Scheunert 1979) $$ \eqalign{ \hbox{str}(x)&=\hbox{tr}(A)-\hbox{tr}(D),\cr x^T&=\left(\matrix{A^t&-C^t\cr B^t&D^t\cr}\right),\cr}\eqno(2.4) $$ where {\sl tr} is the ordinary trace and $A^t$ is the transpose of $A$. Using the supertrace and the supertranspose, one can define various subalgebras of $gl(m/n)$. The {\sl special linear Lie superalgebra} $sl(m/n)$ is: $$ sl(m/n)=\lbrace x\in gl(m/n)\mid \hbox{str}(x)=0\rbrace. \eqno(2.5) $$ In Kac's notation, $A(m-1,n-1)=sl(m/n)$ for $m\not=n$. If $m=2p$ and $n=2q$ are both even, one can define the following subalgebra of $gl(m/n)$ (Kac 1977a, Scheunert 1979): $$ \eqalign{ \hbox{osp}(2p,2q)=\Bigl\lbrace x&=\left(\matrix{A&B\cr C&D\cr}\right)\in sl(2p/2q) \mid x_\al^TJ+(-1)^\al J x_\al=0,\cr &\forall x_\al\in sl(2p/2q)_\al\;(\al=\bar 0,\bar 1)\Bigr\rbrace,\cr} \eqno(2.6) $$ where $$ J=\left(\matrix{0&I_m& & \cr I_m&0& & \cr & &0&I_n\cr & &-I_n&0\cr}\right) \eqno(2.7) $$ and $I_r$ is the $r\times r$ identity matrix. In the notation of Kac, $D(p,q)=\hbox{osp}(2p,2q)$ for $p\not=1$ and $C(q+1)=\hbox{osp}(2,2q)$. In what follows, we put $G=C(n)$ ($n>1$). Kac (1977a) showed that $C(n)$ is a {\sl simple} Lie superalgebra of {\sl type~I}. Note that $C(n)_{\bar 0}=\hbox{so}(2)\oplus\hbox{sp}(2n-2) \cong\C\oplus\hbox{sp}(2n-2)$, where $\hbox{sp}(2n-2)$ is the simple symplectic Lie algebra in $2n-2$ dimensions, is a reductive Lie algebra, the simple modules of which are well known. A Cartan subalgebra $H$ of $G$ has dimension $n$ and is the space of diagonal matrices in (2.6). The dual space $H^*$ is described in the basis of forms $\ep$ and $\de_j$ ($j=1,\ldots,n-1$), where $\ep\colon x\rightarrow A_{1,1}$ and $\de_j\colon x\rightarrow D_{j,j}$ for $x=\bigl({A\atop C}{B\atop D}\bigr)$. Using the notation $E_{ij}$ for the $2n\times 2n$ matrix with entry 1 at position $(i,j)$ and 0 elsewhere, the roots and corresponding root vectors of $C(n)$ are given by $$ \eqalign{\hbox{even }&\hbox{roots:}\cr \de_j-\de_k\;&\leftrightarrow\; E_{2+j,2+k}-E_{n+1+k,n+1+j}\;(1\leq j,k\leq n-1;j\not=k);\cr \de_j+\de_k\;&\leftrightarrow\; E_{2+j,n+1+k}+E_{2+k,n+1+j}\;(1\leq j,k\leq n-1)\cr -\de_j-\de_k\;&\leftrightarrow\; E_{n+1+j,2+k} +E_{n+1+k,2+j}\;(1\leq j,k\leq n-1)\cr} \eqno(2.8) $$ $$ \eqalign{\hbox{ odd }&\hbox{roots:}\cr \ep-\de_k\;&\leftrightarrow\;E_{1,2+k}-E_{n+1+k,2}\cr \ep+\de_k\;&\leftrightarrow\;E_{1,n+1+k}+E_{2+k,2}\cr -\ep-\de_k\;&\leftrightarrow\;E_{2,2+k}-E_{n+1+k,1}\cr -\ep+\de_k\;&\leftrightarrow\;E_{2,n+1+k}+E_{k+2,1}\;(1\leq k\leq n-1).\cr} \eqno(2.9) $$ Denote by $\De$ the set of all roots, by $\De_0$ the set of even roots, by $\De_1$ the set of odd roots, and by $e(\al)$ the root vector (2.8)--(2.9) corresponding to the root $\al\in\De$. $G$ has the root space decomposition $$ G=H\oplus\left(\bigoplus_{\al\in\De}\C e(\al)\right). \eqno(2.10) $$ A set of simple roots $\{\al_1,\al_2,\ldots,\al_n\}$ of $\De$ may be chosen as follows (Kac 1977b): $$ \al_1=\ep-\de_1,\;\al_i=\de_{i-1}-\de_i\;(2\leq i\leq n-1),\; \al_n=2\de_{n-1}. \eqno(2.11) $$ Note that $\al_1$ is the only odd simple root. Let $e_i$ (resp. $f_i$) be root vectors for $\al_i$ (resp. $-\al_i$), and $h_i=[e_i,f_i]$: $$ \eqalign{ h_1&=E_{1,1}-E_{2,2}+E_{3,3}-E_{n+2,n+2},\cr h_i&=E_{i+1,i+1}-E_{i+2,i+2}-E_{n+i,n+i}+E_{n+i+1,n+i+1},\;(2\leq i\leq n-1)\cr h_n&=E_{n+1,n+1}-E_{2n,2n}.\cr}\eqno(2.12) $$ These elements form a basis for $H$. With the choice of a simple root system (2.11), the elements of $H^*$ are partially ordered by $$ \la,\mu\in H^*\;\colon\quad\la\geq\mu\quad\Leftrightarrow\quad\la-\mu =\sum_{i=1}^{n}k_i\al_i \hbox{ with }k_i\geq 0. \eqno(2.13) $$ This partial ordering $\geq$ can be extended to a total ordering $\succeq$ compatible with $\geq$, i.e. $$ \la\geq\mu\quad\Rightarrow\quad\la\succeq\mu; \eqno(2.14) $$ the most natural example of such a total ordering is lexicographical ordering with respect to the simple roots. The even and odd {\sl positive} roots of $C(n)$ are given by $$ \eqalign{ \De_0^+&=\{\de_i-\de_j\;(i\be_1 >\cdots>\be_q$ from $\De_1^+$ such that the sequence of weights $\nu_0=\La, \nu_1,\ldots,\nu_q,\nu_{q+1}$ with $\nu_{t+1}=\nu_t-\be_t$ satisfies $$ \eqalignno{ &\langle\nu_t+\rh|\be_t\rangle=0,\quad 0\leq t\leq q;&(5.6a)\cr &\nu_t\hbox{ is vanishing for }0A(\La)_i+1$ then $q=0$ and all statements of the lemma are satisfied. If $A(\La)_{i-1}=A(\La)_i+1$, then $A(\La-\be_0)$ has two zeros at positions $i-1$ and $i$; then $\be_1=\ep-\de_{i-1}$. The sequence is now constructed by iteration: if $A(\La)_{i-2}>A(\La)_{i-1}+1$ then $q=1$, and if $A(\La)_{i-2}=A(\La)_{i-1}+1$ then $\be_2=\ep-\de_{i-2}$, etc. From this construction the existence and uniqueness of the sequence follows, and the matrices ${\cal A}_0=A(\La)$, ${\cal A}_1=A(\La-\be_0)$, ${\cal A}_2=A(\La-\be_0-\be_1)$, $\ldots$ satisfy $$ \eqalignno{ &{\cal A}_t\hbox{ has two zeros, at positions }i-t\hbox{ and }i-t+1\hbox{ for }1\leq t\leq q;&(5.8a)\cr &{\cal A}_{q+1}\hbox{ has one zero at position }i-q+1;&(5.8b)\cr &{\cal A}_t\hbox{ is obtained from }A(\La-t\be)\hbox{ by }t-1\hbox{ transpositions in the}\cr &\hbox{ left part of }A(\La)\hbox{ (and with (5.3b) also in the right part).}&(5.8c)\cr} $$ Now (5.6a) follows from (5.8a), and (5.6b) follows from Lemma~5.5 and (5.8a). By constuction, ${\cal A}_{q+1}$ satisfies the conditions of Lemma~5.2b, so using this and (5.8b) leads to (5.6c). Finally, (5.8c) implies (5.6d). Now consider the case where the zero of $A(\La)$ appears in the right part of $A(\La)$. Then the counterpart of (5.7) is given by \tenpoint $$ \matrix{ A(\la)&=&\Bigl(A_1&\cdots&A_i&\cdots&A_{n-1}&\parallel&A_n&\cdots &A_{2n-1-i}&\cdots&A_{2n-2}\Bigr)\cr A(\la-\ep-\de_i)&=&\Bigl(A_1-1&\cdots&A_i-2&\cdots&A_{n-1}-1&\parallel&A_n-1&\cdots &A_{2n-1-i}&\cdots&A_{2n-2}-1\Bigr)\cr} \eqno(5.9) $$ \twelvepoint Hence, if $a_n\not=0$, the analysis of the sequence is completely analogous to the previous case, and the lemma follows. There is, however, one situation that needs more attention. This is the case with $\be_0=\be=\ep+\de_i$ and $a_{i+1}=a_{i+2}=\cdots=a_n=0$. In this case $\be_1=\ep+\de_{i+1}$, $\ldots$, $\be_{n-i-1}=\ep+\de_{n-1}$, and since $a_n=0$, $\be_{n-i}=\ep-\de_{n-1}$. In other words, the positions of the zeros in ${\cal A}_t$ appear in the left part of the atypicality matrix as $t\geq n-i$. Making use of (5.9) and (5.7), one can verify that in this case $\be_q=\ep-\de_i$. All the properties (5.8) are still true, but for (5.8c) one of the transpositions is the interchanging of the $(n-1)^{th}$ and $n^{th}$ element in the atypicality matrix if $t\geq n-i$; this corresponds to the dot action of $r_{\de_{i-1}}$ discussed previously. The lemma is then proven similarly.\mybox \noindent {\bf Example.} The following three weights illustrate the three distinct cases in the proof of Lemma~5.6. Let $G=C(5)$; the sequence $\be_t$ of Lemma~5.6 is denoted by underlining the corresponding elements $\langle\La+\rh|\be_t\rangle$ in the atypicality matrix in (5.10): $$ \eqalign{ \La=[2;0,0,0,0]\quad&A(\La)=\bigl(\underline 2,\underline 1, \underline 0,-1\parallel -3,-4,-5,-6\bigr),\cr \La=[10;0,0,1,0]\quad&A(\La)=\bigl(10,9,8,6\parallel 4, \underline 2,\underline 1,\underline 0\bigr),\cr \La=[6;0,0,0,0]\quad&A(\La)=\bigl(6,5,\underline 4,\underline 3\parallel \underline 1,\underline 0,-1,-2\bigr).\cr} \eqno(5.10) $$ \section{6. The character formulas} Using the lemmas of Section~5, we are now able to prove a character formula for $V(\La)$, where $\La$ is a singly atypical integral dominant weight. The proofs given here are similar to the proofs given for $sl(m/n)$ in a previous paper (Hughes {\sl et al} 1989b). \proclaim {Lemma 6.1}. Let $\La$ be singly atypical of type $\ga$ and let $\be_0=\ga,\be_1,\ldots,\be_q=\ga'$ be the sequence of Lemma~5.6. Let $q(\La)=\La-\be_0-\cdots-\be_q$ be the dominant weight defined in Lemma~5.6, which is singly atypical of type $\ga'$. Then, using the notation (4.9): $$ \ch_\ga(\La-\ga)=\ch_{\ga'}(q(\La)). \eqno(6.1) $$ \noindent {\sl Proof.} As in the proof of Theorem 4.3, we can expand $\ch_\ga(\La-\ga)$ in a series of $\ch_K(\la)$-terms: $$ \ch_\ga(\La-\ga)=\ch_K(\La-\ga)-\ch_K(\La-2\ga)+\ch_K(\La-3\ga)-\cdots +(-1)^q\ch_K(\La-(q+1)\ga)+\cdots\eqno(6.2) $$ But for $0\al_1$. Let $\La_0=\La$ and $\ga_0=\ga$, and using the notation of Lemma~6.1 we define a sequence of dominant weights and elements of $\De_1^+$ by $$ \La_{k+1}=q(\La_k),\qquad \ga_{k+1}=\ga_k',\qquad(k\geq 0). \eqno(6.6) $$ Clearly, every $\ga_{k+1}\leq\ga_k$, with equality if and only if $q(\La_k)=\La_k-\ga_k$, i.e. if and only if $\La_k-\ga_k$ is dominant. But $\ga_{k+1}=\ga_k$ can happen only a finite number of times if $\ga_k>\al_1$. Therefore, there exists an $s$ such that $\ga_{s-1}>\ga_s=\al_1$, $\al_1$ being the smallest element of $\De_1^+$ according to the partial ordering (2.13). Since every $\La_k$ is dominant, we find, using (3.18)--(3.19) and (4.11): $$ \cha\overline V(\La_k)=\ch_{\ga_k}(\La_k)+\ch_{\ga_k}(\La_k-\ga_k). \eqno(6.7) $$ Using Lemma 6.1, this becomes: $$ \cha\overline V(\La_k)=\ch_{\ga_k}(\La_k)+\ch_{\ga_{k+1}}(\La_{k+1}). \eqno(6.8) $$ On the other hand, since every $\La_k$ is singly atypical, Theorem~3.8 implies: $$ \cha\overline V(\La_k)=\cha V(\La_k)+\cha X(\La_k), \eqno(6.9) $$ where both $V(\La_k)$ and $X(\La_k)$ are simple $G$ modules. Applying (6.8) and (6.9) for $k=s-1$, using Theorem~4.3, leads to $$ \eqalignno{ \cha\overline V(\La_{s-1})&=\ch_{\ga_{s-1}}(\La_{s-1})+\cha V(\La_s),&(6.10a)\cr \cha\overline V(\La_{s-1})&=\cha V(\La_{s-1})+\cha X(\La_{s-1}).&(6.10b)\cr} $$ From Remark 4.4, (6.7) and (6.10a), we see that $V(\La_s)$ has $w_0(\La_{s-1}-2\rh_1)$ as lowest weight. Since a simple $G$ module is characterised by its lowest weight, it follows that $V(\La_s)$ is isomorphic to $X(\La_{s-1})$, and (6.10) implies: $$ \cha V(\La_{s-1})=\ch_{\ga_{s-1}}(\La_{s-1}). \eqno(6.11) $$ By iteration one finds: $$ \cha V(\La_k)=\ch_{\ga_k}(\La_k),\qquad k=s,s-1,\ldots,1,0, \eqno(6.12) $$ proving the theorem.\mybox From Theorem~3.7, (3.19), Theorem~6.2, and the fact that $C(n)$ has only typical or singly atypical modules, one obtains the following result: \proclaim {Corollary 6.3}. Let $G=C(n)$ and $\La$ be integral dominant. Then $$ \cha V(\La)=L_0^{-1}\sum_{w\in W}\vep(w)w\Bigl\lbrace e^{\La+\rh_0} \prod_{{\scriptstyle\be\in\De_1^+}\atop{\scriptstyle\langle\La+\rh|\be\rangle \not=0}}(1+e^{-\be})\Bigr\rbrace. \eqno(6.13) $$ Note that this formula is in fact the Bernstein--Leites formula (1980, Leites 1980), wrongly claimed to be true for all simple Lie superalgebras but proven here to be correct for the series $C(n)$. Finally, we shall derive an alternative character formula for the atypical case. The validity of this alternative formula is proven by means of the following lemma: \proclaim {Lemma 6.4}. Let $\La$ be atypical of type $\be$. Then there exists a unique sequence of distinct elements $\be_0=\be<\be_1 <\cdots<\be_p$ from $\De_1^+$ such that the sequence of weights $\nu_0=\La, \nu_1,\ldots,\nu_p,\nu_{p+1}$ with $\nu_{t+1}=\nu_t+\be_t$ satisfies $$ \eqalignno{ &\langle\nu_t+\rh|\be_t\rangle=0,\quad 0\leq t\leq p;&(6.14a)\cr &\nu_t\hbox{ is vanishing for }0\la_{i-p}\hbox{ if }i-p>1. \eqno(6.18a) $$ \item{2.} The zero in $A(\La)$ appears in the left part of $A(\La)$, and $a_n\not=0$. Then $\be=\be_0=\ep-\de_i$, $\be_1=\ep-\de_{i+1}$, $\ldots$, $\be_p=\ep-\de_{i+p}$, with $1\leq i\leq i+p\leq n-1$, and $\La$ satisfies $$ \et\equiv\la_i=\la_{i+1}=\cdots=\la_{i+p}\;\hbox{ and } \la_{i+p}>\la_{i+p+1}\hbox{ if }i+p