% Latex file % Unimodal polynomials associated with Lie algebras and superalgebras % J.W.B. Hughes and J. Van der Jeugt % J. Comp. Appl. Math. 37 (1991), 81-88. % \documentstyle[12pt]{article} % \setlength{\topmargin}{0cm} \setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm} \setlength{\textheight}{225mm} \setlength{\textwidth}{150mm} % % SOME OF THE MACROS AND DEFINITIONS % % 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} % % LaTeX begin and end of numbered equation % \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\nn{\nonumber} % % 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} } % % a definition for my Young diagrams: % \def\mystrut{\hbox{\vrule height9.6pt depth2pt width0pt}} % % \begin{document} % \title{Unimodal polynomials associated with Lie algebras and superalgebras} \author{J.W.B.~Hughes\\ {\normalsize Department of Mathematics, Queen Mary and Westfield College,}\\ {\normalsize Mile End Road, London E1 4NS, U.K.}\\ and \\ J.\ Van der Jeugt\thanks{Senior Research Assistant of the NFWO (National Funds for Scientific Research of Belgium)} \\ {\normalsize Laboratorium voor Numerieke Wiskunde en Informatica,}\\ {\normalsize Rijksuniversiteit Gent, Krijgslaan 281-S9, B9000 Gent, Belgium}} % \maketitle % \begin{abstract} It is well known that the theory of Lie algebras can be used to prove the unimodality of certain polynomials. Recently, it was shown that also Lie superalgebras can give rise to unimodal polynomials. In this paper, we present a unified approach to the question. We do not go into the technical details of Lie algebras and superalgebras, but give a lot of old and new examples of unimodal polynomials associated with these structures. \end{abstract} \vskip 5mm \noindent {\sl Keywords~:} unimodal polynomials, sl(2), Lie algebras. \section{Introduction.} A polynomial in $q$ of the form \beq f(q) = \sum_{k=0}^N a_k q^k\,,\qquad a_k\in\Zah^+ \label{f1} \eeq is said to be {\sl symmetric} if $a_k = a_{N-k}$ for all $k=0,1,\ldots,N$, or equivalently if $q^Nf(q^{-1}) = f(q)$. The polynomial~(\ref{f1}) is said to be {\sl unimodal} if there exists a number $m$ such that the coefficients $a_k$ satisfy \beq a_0\leq a_1\leq a_2\leq\cdots\leq a_m\geq a_{m+1}\geq\cdots\geq a_N . \label{f2} \eeq For a symmetric unimodal polynomial, we have that $m=\lfloor(N+1)/2\rfloor$, where $\lfloor x\rfloor$ denotes the integer part of a positive real number $x$. Similarly, a sequence of positive integers $(a_i)_{i=0}^N$ is unimodal if~(\ref{f2}) is satisfied. If $f(q)$ is given in the form~(\ref{f1}), it is a straightforward matter to verify whether $f$ is unimodal or not. However, in many situations $f$ is given in a completely different form, and then it often becomes a non-trivial task to prove or disprove its unimodality. We shall give two examples in this introduction. The first is~\cite{h1} \beq f(q) = (1+q)(1+q^2)\cdots(1+q^n) = \prod_{i=1}^n (1+q^i) = \sum_k p(k;n,n)\,q^k\,, \label{f3} \eeq where $p(k;n,n)$ is the number of partitions of $k$ into at most $n$ strictly decreasing integers none of which exceeds $n$. Obviously, (\ref{f3}) is symmetric, but a proof of the unimodality of (\ref{f3}) using only elementary algebra and combinatorics does not exist. Nevertheless, the unimodality of (\ref{f3}), which can be proven by means of the Lie algebraic method explained here, has a simple combinatorial interpretation, namely~: $p(k-1;n,n)\leq p(k;n,n)$ for $k\leq \lfloor {1\over 4}n(n+1)\rfloor$. The second example is the Gaussian polynomial~\cite{a2,h1} \beq f(q) = \left[ m\atop n \right]_q ={(1-q^m)(1-q^{m-1})\cdots(1-q^{m-n+1}) \over (1-q^n)(1-q^{n-1})\cdots(1-q) } = \sum_{k=0}^{n(m-n)} (k;m-n,n)\,q^k\,, \label{f4} \eeq where $(k;m-n,n)$ is the number of partitions of $k$ into at most $m-n$ integers none of which exceeds $n$. The first proof of the unimodality of (\ref{f4}) was by means of invariant theory~\cite{s6,s2}, but the shortest proof is undoubtedly the Lie algebraic proof~\cite{h1}. It is remarkable that a proof using just (advanced) combinatorics was given only very recently~\cite{o2}. Note that (\ref{f4}) has several combinatorial interpretations, the most popular being~: $(k-1;m-n,n)\leq(k;m-n,n)$ for $k\leq\lfloor n(m-n)/2\rfloor$. It is the purpose of this paper to explain the Lie algebraic method in order to prove the unimodality of certain polynomials without going into the details of the technicalities involved. Essentially, the method is based upon two fundamental facts~\cite{d,h1,s3,s4,a1}~: the first is the unimodal weight distribution of sl(2) representations, the second is the embedding of a principal sl(2) subalgebra in every given simple Lie algebra~\cite{k5}. Then, we point out how this argument can be extended to Lie superalgebras, where osp(1,2) representations play a fundamental role~\cite{s5,v2}. Several examples are given of unimodal polynomials arising from Lie superalgebras. Finally, we show how a combinatorial property can be deduced from the unimodality of such a polynomial. \section{Representations of sl(2) and osp(1,2).} The Lie algebra sl(2) is spanned by the three matrices~\cite{h3,j} \beq h=\left(\begin{array}{cc} 1/2 & 0 \\ 0 & -1/2 \end{array}\right), \qquad e_+=\left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right), \qquad e_-=\left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right), \eeq satisfying the commutation relations \beq [h,e_\pm] = \pm e_\pm \,,\qquad [e_+,e_-] = 2h\,. \label{f6} \eeq Every mathematical physicist will immediately recognize the angular momentum algebra in (\ref{f6}). The finite-dimensional irreducible representations of sl(2) are familiar to both physicists and mathematicians~: they are labelled by a number $j$ ($j\in\{0,{1\over 2},1,{3\over 2},2,\ldots\} = {1\over 2}\Nat$), and the $2j+1$ basis vectors are labelled by $v(j,m)$ where $m=-j,-j+1,\ldots,j$ is called the weight of the vector $v(j,m)$. In general, an sl(2) representations space $V$ can be written as a direct sum $V=\oplus_\mu V_\mu$, where $V_\mu$ is the weight space of weight $\mu$. Then the $q$-character of such a representation is \beq \hbox{ch}_q\,V = \sum_\mu \hbox{dim}\,(V_\mu) q^\mu\,. \eeq So in particular for the irreducible representation $V^{(j)}$ labelled by the number $j$, the $q$-character is given by \beq \hbox{ch}_q\,V^{(j)} = q^{-j}+q^{-j+1}+\cdots +q^j = q^{-j} (1+q+q^2+\cdots q^{2j})\,. \label{f8} \eeq An arbitrary finite-dimensional representation $V$ can always be decomposed into a direct sum of irreducible representations~\cite{h3,j}, hence the $q$-character of $V$ can be written as \beq \hbox{ch}_q\,V = b_0\,\hbox{ch}_q\,V^{(0)} + b_{1/2}\,\hbox{ch}_q\,V^{(1/2)} + b_{1}\,\hbox{ch}_q\,V^{(1)} + \cdots + b_{l}\,\hbox{ch}_q\,V^{(l)} = \sum_{i=-l}^l a_i q^i\,, \label{f9} \eeq where every $b_j\geq 0$, $l\in{1\over 2}\Nat$ and $i=-l,-l+{1\over 2}, \ldots,l$. Hence, we obtain from (\ref{f8}) \proclaim Theorem 1. If $\hbox{ch}_q\,V = \sum_{i=-l}^l a_i q^i$ then the two sequences $\ldots,a_{-2}$,$a_{-1}$,$a_0$,$a_1$,$a_2,\ldots$ and $\ldots,a_{-3/2},a_{-1/2},a_{1/2},a_{3/2},\ldots$ are unimodal and symmetric about 0. In particular, if only integer $j$'s appear in the decomposition of $V$ (or only half-integers), then the polynomial $q^{-l}\hbox{ch}_q\,V$ is a symmetric and unimodal polynomial. \vskip 5mm The Lie superalgebra osp(1,2) is spanned by the five matrices~\cite{h2,k1} \begin{eqnarray} h&=&\left(\begin{array}{ccc} 1/2&0&0\\0&-1/2&0\\0&0&0\end{array}\right)\,, \qquad e_+=\left(\begin{array}{ccc} 0&1&0\\0&0&0\\0&0&0\end{array}\right)\,, \qquad e_-=\left(\begin{array}{ccc} 0&0&0\\1&0&0\\0&0&0\end{array}\right)\,, \nonumber\\ v_+&=&\left(\begin{array}{ccc} 0&0&1\\0&0&0\\0&-1&0\end{array}\right)\,, \qquad\qquad v_-=\left(\begin{array}{ccc} 0&0&0\\0&0&-1\\1&0&0\end{array}\right)\,, \end{eqnarray} satisfying the commutation and anti-commutation relations \begin{eqnarray} [h,e_\pm] &=& \pm e_\pm\,,\quad [e_+,e_-]=2h\,, \nn\\ {}[h,v_\pm] &=& \pm{1\over 2} v_\pm\,, \quad [e_\pm,v_\mp] = - v_\pm\,, \\ \{v_+,v_-\} &=& 2h\,, \quad \{v_\pm, v_\pm\} = 2e_\pm\,. \nn \end{eqnarray} Clearly, the Lie algebra sl(2) is a subalgebra of the Lie superalgebra osp(1,2). The finite-dimensional irreducible representations of osp(1,2) have been classified~\cite{h2,s1}, and are again labelled by a number $j\in{1\over 2}\Nat$. An irreducible representation $W^{(j)}$ has dimension $4j+1$, and the basis vectors $w(j,m)$ have weight $m=-1,-1+{1\over 2},\ldots,j$. Thus the $q$-character of $W^{(j)}$ is \beq \hbox{ch}_q\,W^{(j)} = q^{-j}+q^{-j+{1\over 2}}+\cdots+q^j\,. \eeq In order to deal with polynomials, we put $x=q^{1/2}$, such that the $x$-character becomes \beq \hbox{ch}_x\,W^{(j)} = x^{-2j}+x^{-2j+1}+\cdots+x^{2j} = x^{-2j}(1+x+x^2+\cdots+x^{4j})\,. \eeq Although it is not a general property of Lie superalgebras that any finite-dimen\-sion\-al representation $W$ can be decomposed into a direct sum of irreducible representations~\cite{k1,k2}, it certainly is true for the case of osp(1,2). Hence, just as in (\ref{f9}), the $x$-character of $W$ is of the form \beq \hbox{ch}_x\,W =b_0\,\hbox{ch}_x\,W^{(0)}+b_{1/2}\,\hbox{ch}_x\,W^{(1/2)}+ b_1\,\hbox{ch}_x\,W^{(1)}+\cdots+b_l\,\hbox{ch}_x\,W^{(l)} = \sum_{i=-2l}^{2l} a_i x^i\,, \label{f14}) \eeq where every $b_j\geq 0$, $l\in{1\over 2}\Nat$ and $i=-2l,-2l+1,\ldots,2l$. This time, (\ref{f14}) implies that the sequence $a_{-2l}, a_{-2l+1},\ldots,a_{2l}$ is unimodal and symmetric about 0, hence \proclaim Theorem 2. The polynomial $x^{-2l}\hbox{ch}_x\,W$ is a symmetric and unimodal polynomial. \section{Some unimodal polynomials arising from Lie algebras.} We begin this section with some technical definitions concerning Lie algebras and representations~\cite{h3,j}. Let $G$ be a simple Lie algebra of rank $l$. $G$ contains an $l$-dimensional diagonalisable abelian subalgebra $H$, the so-called Cartan subalgebra. Denote the dual space of $H$ by $H^*$ and let $\langle\,,\,\rangle$ be the pairing of $H$ and $H^*$. For $\al\in H^*\backslash\{0\}$, we put $G_\al=\{ g\in G | [h,g]=\langle\al,h\rangle g,\;\forall h\in H\}$. If $G_\al\not= \{0\}$, $\al$ is called a root of $G$, and $G_\al$ is the root space. Let $\De$ be the set of all roots of $G$. There exists a subset $\Ph=\{\al_1,\ldots,\al_l\}$ of $\De$ such that $\Ph$ is a basis of $H^*$ and such that every $\al\in\De$ can be written as $\al=\sum_{i=1}^l k_i\al_i$ with either all $k_i\in\Zah^+ = \{0,1,2,\ldots\}$ (in which case we call $\al$ positive) or else all $k_i\in\Zah^- =\{0,-1,-2,\ldots\}$ (in which case $\al$ is negative). Thus $\De = \De_-\cup\De_+$ with $\De_+$ (resp.\ $\De_-$) the set of positive (resp.\ negative) roots of $G$. As usual, we put $\rh = {1\over 2}\sum_{\al\in\De_+}\al$, and for $\al\in H^*$ we denote its dual element (with respect to the basis $\al_1,\ldots,\al_l$) in $H$ by $\al^\wedge$. Let $V$ be a representation of $G$, i.e.\ a $G$-module. For $\mu\in H^*$, we put $V_\mu = \{ v\in V | h\cdot v=\langle\mu,h\rangle v,\quad \forall h \in H\}$. If $V_\mu\not=\{0\}$, $\mu$ is a weight of $V$, and $V_\mu$ is the weight space. If $V$ is an irreducible representation (or a simple $G$-module), $V$ is characterized by its highest weight $\la$, and $V$ is denoted by $V(\la)$. It is well known that every simple Lie algebra $G$ contains a so-called principal sl(2) subalgebra $G\supset\hbox{sl(2)}$~\cite{k5}. If $V$ is a representation of $G$, it becomes automatically a representation of sl(2) by restriction to the principal sl(2) subalgebra of $G$. The $q$-specialised character of $V$ is then the $q$-character of $V$ as an sl(2) representation. Thus by Theorem~1 it is, apart from a factor $q^{-l}$, automatically a unimodal polynomial provided only integer powers of $q$ appear. For $V$ an irreducible representation, $V=V(\la)$, there exists an explicit formula for the $q$-specialised character~\cite{k3}~: \beq \hbox{ch}_q\,V(\la) = q^{-\langle\la,\rh^\wedge\rangle} \prod_{\al\in\De_+} {1-q^{\langle\la+\rh,\al^\wedge\rangle} \over 1-q^{\langle\rh,\al^\wedge\rangle} } \,. \label{f15} \eeq This formula can be deduced from Weyl's character formula~\cite{w}. The simple form given in (\ref{f15}) allows one to calculate the $q$-specialised character in some interesting cases, and to deduce the unimodality of certain polynomials. For the case of the special linear Lie algebra $G=A_n=\hbox{sl}(n+1)$, the highest weight $\la$ of an irreducible representation $V(\la)$ is characterized by a partition $\{\la\}$ with length($\la$) $\leq n$~\cite{l,m}; we shall use reference~\cite{m} for the terminology concerning partitions. Using (\ref{f15}), one can calculate the $q$-specialised character for three interesting cases~: (1) $\{\la\} = \{k\}$ ($k\in\Nat$), which corresponds to the totally symmetric part of the $k$-th tensor product of the natural representation; (2) $\{\la\} = \{1^k\} =\{1,1,\ldots,1\}$ ($k\leq n$), which corresponds to the totally anti-symmetric part of the $k$-th tensor product of the natural representation; (3) $\{\la\}=\{\rh\} =\{n,n-1,\ldots,1,0\}$. One finds~: \begin{eqnarray} \hbox{ch}_q\,V\{k\} & = & q^{-nk/2} \left[ n+k \atop n \right]_q\,, \label{f16}\\ \hbox{ch}_q\,V\{1^k\} & = & q^{-k(n-k+1)/2} \left[ n+1 \atop n+1-k \right]_q\,, \label{f17}\\ \hbox{ch}_q\,V\{\rh\} & = & q^{-n(n+1)(n+2)/12} \prod_{i=1}^n \prod_{j=i+1}^{n+1} (1+q^{j-i}) \nn\\ & & \quad = q^{-n(n+1)(n+2)/12} \prod_{i=1}^n (1+q)(1+q^2)\cdots(1+q^i)\,. \label{f18} \end{eqnarray} Then it follows from Theorem~1 and (\ref{f16}) or (\ref{f17}) that the Gaussian polynomials (\ref{f4}) are symmetric and unimodal. The second case we shall consider here is $G=B_n=\hbox{so}(2n+1)$, the orthogonal Lie algebra in $2n+1$ dimensions. It is well known that $B_n$ has a so-called spin representation of dimension $2^n$. For this spin representation, the highest weight of which we denote by $\la_s$, (\ref{f15}) leads to~\cite{h1} \beq \hbox{ch}_q\,V(\la_s) = q^{-n(n+1)/4} \prod_{j=1}^n (1+q^j)\,, \label{f19} \eeq thus proving the unimodality of (\ref{f3}). It is remarkable that until recently the Lie algebraic method presented here was the only way to prove the unimodality of the polynomial appearing in the rhs of (\ref{f19}); a second proof, using an advanced analytic method, was given in 1982~\cite{o1}. Note that the unimodality of the polynomial in (\ref{f19}) implies the unimodality of the polynomial in (\ref{f18}), since the product of two symmetric and unimodal polynomials is again symmetric and unimodal~\cite{a2}. \section{Some unimodal polynomials arising from Lie superalgebras.} Let $G$ be a simple classical Lie superalgebra, and $W$ a simple $G$-module (or an irreducible representation of $G$). The situation here is rather different from the case of Lie algebras for two reasons. First, not all Lie superalgebras contain a principal osp(1,2) subalgebra. The classical Lie superalgebras which do contain a principal five-dimensional subalgebra have been classified~\cite{v1}, and are~: sl$(n+1,n)=A(n,n-1)$, $B(n-1,n)$, $B(n,n)$, $D(n,n-1)$ and $D(n,n)$. Here, Kac's notation for Lie superalgebras has been used~\cite{k1}. The second difference is that no character formula exists which is valid for all irreducible representations of all classical simple Lie superalgebras (like Weyl's character formula for Lie algebras). The only classes of irreducible representations where a character formula exists are (1) all {\sl typical} representations of classical Lie superalgebras~\cite{k2}, (2) all {\sl covariant tensor} representations of sl$(m/n)=A(m-1,n-1)$~\cite{b}, (3) all {\sl singly atypical} representations of type~I Lie superalgebras~\cite{v3}. It is for such classes that we shall be able to calculate the $x$-specialised character of a representation $W$ of $G$, which is the $x$-character of $W$ when restricted to the osp(1,2) subalgebra of $G$. First, let $W$ be a covariant tensor representation of sl($n+1,n$), which is specified by a partition $\{\la\}$~\cite{b}. Consider the situation where $\{\la\} =\{l\}$ ($l\in\Nat$). Then the theory of characters for covariant tensors implies that \beq \hbox{ch}\,W\{l\} = \sum_{k=0}^l \hbox{ch}\,V^{\hbox{\scriptsize sl}(n+1)}\{k\}\, \cdot\hbox{ch}\,V^{\hbox{\scriptsize sl}(n)}\{1^{l-k}\}\,, \eeq where we have denoted by a superscript the Lie algebra of which $V$ is a representation. Then, with $q=x^2$, \beq \hbox{ch}_x\,W\{l\} = \sum_{k=0}^l \hbox{ch}_{x^2}\,V^{\hbox{\scriptsize sl}(n+1)}\{k\}\, \cdot\hbox{ch}_{x^2}\,V^{\hbox{\scriptsize sl}(n)}\{1^{l-k}\}\,. \eeq Using (\ref{f16}) for sl($n+1$) and (\ref{f17}) for sl($n$), we find~\cite{s5} \begin{eqnarray} \hbox{ch}_x\,W\{l\} & = & x^{-nl}\,P_{nl}(x),\qquad\hbox{with}\nn\\ P_{nl}(x) & = & \sum_{k=0}^l x^{(l-k)^2}\, \left[ n+k\atop n\right]_{x^2}\,\left[ n\atop n-l+k\right]_{x^2}\,. \label{f22} \end{eqnarray} Subsequently, we shall give some examples of unimodal polynomials arising from Kac's character formula for typical representations. For $G=\hbox{sl}(n+1/n)$ and $\{\la\}=\{n^{n+1}\}$, this implies the unimodality of~\cite{v2} \beq d(x)=\prod_{j=1}^{n+1}\prod_{k=1}^n\,(1+x^{|2k-2j+1|}) =\prod_{i=1}^n\,(1+x^{2i-1})^{2(n-i+1)}\,. \label{f23} \eeq A generalisation of (\ref{f23}) can be obtained for $\{\la\} = \{(n+1)^s,n^{n+1-s},t\}$, in which case we find that the $x$-specialised character is, apart from a factor $q^{-l}$, equal to \beq \prod_{i=1}^n\,(1+x^{2i-1})^{2(n-i+1)}\, \left[n+1\atop s\right]_{x^2}\,\left[n\atop t\right]_{x^2}\,, \quad (s\leq n+1;\,t\leq n)\,. \eeq Note that $\left[n+1\atop s\right]_{x^2}\,\left[n\atop t\right]_{x^2}$ on its own is not unimodal (since the odd powers of $x$ are missing), but the product with (\ref{f23}) is. As a second example, consider the orthosymplectic Lie superalgebra $G=B(m,n)=\hbox{osp}(2m+1,2n)$. Let $\la=(0,\ldots,0;m+{1\over 2}; 0,\ldots,0,1)$ in terms of the Kac-Dynkin labels~\cite{k1}. Then, the $x$-specialised character of $W(\la)$ is proportional to~\cite{v2} \beq \prod_{j=1}^n\,\prod_{k=1}^m\,(1+x^{|2j-2k-1|}) (1+x^{2j+2k-1})(1+x^{2k})\,. \label{f25} \eeq Consequently, as only $B(n,n)$ and $B(n-1,n)$ contain a principal osp(1,2) subalgebra, the polynomial (\ref{f25}) is {\sl unimodal} when $m=n,n-1$. Next, in view of the new theory~\cite{v3} established for singly atypical modules of sl($m/n$), it is interesting to study an example in that situation. Let $G=\hbox{sl}(n+1/n)$, and take $\{\la\} =\{n^n\}$. Then $W\{\la\}$ is singly atypical of type $\be$, where $\be$ is the unique odd simple root of $G$. The theory of singly atypical characters then implies that~\cite{v3} \beq \hbox{ch}\,W\{\la\} = \ch_K(\la)-\ch_K(\la-\be)+\ch_k(\la-2\be)-\ldots\,, \label{f26} \eeq where $\ch_K(\mu)$ stands for the so-called Kac-character, that is the character of the {\sl Kac module} with highest weight $\mu$~\cite{k2}. The Kac-character is well known, and it can be easily specialised to $\ch_{K,x}(\la-k\be)$~: \beq \ch_{K,x}(\la-k\be) = x^{-l_{nk}}\, d(x)\,\left[2n+k+1\atop n+1\right]_{x^2}\, \left[n+k\atop n\right]_{x^2}\,, \eeq where $d(x)$ has been given in (\ref{f23}) and $l_{nk} = {1\over 6}n(n+1)(2n+1)+nk+(n+1)(n+k)$. Then it follows from (\ref{f26}) that \beq \ch_x\,W(\la) = d(x)\,\sum_{k=0}^{\infty}\,(-1)^k\,x^{-l_{nk}}\, \left[2n+k+1\atop n+1\right]_{x^2}\, \left[n+k\atop n\right]_{x^2}\,. \label{f28} \eeq Although the summation in the rhs of (\ref{f28}) is a formal infinite alternating series, it follows from the finite-dimensionality of $W\{\la\}$ that, after multiplication by $d(x)$, it is a {\sl finite} expression of the form $x^{-n^2(n-1)/2}\,p(x)$, with $p(x)$ a polynomial. Moreover, by virtue of Theorem~2, $p(x)$ is a {\sl unimodal} polynomial. Finally, we shall give a combinatorial expression for the polynomial $P_{nl}(x)$ in (\ref{f22}). This arises from the fact that the character of $W\{l\}$ is the supersymmetric function $s_{\{\la\}} (x_1,\ldots,x_{n+1}/y_1,\ldots,y_n)$ in $n+1$ variables $x_i$ and $n$ variables $y_j$~\cite{b,k4}. The $x$-specialised character is then equivalent to making the specialisation $x_i\rightarrow x^{2i-2}$, $y_j\rightarrow x^{2j-1}$ in $s_{\{\la\}}(\hbox{\bf x}/\hbox{\bf y})$. On the other hand $s_{\{\la\}}(\hbox{\bf x}/\hbox{\bf y})$ has an expression in terms of Young tableaux. For the case $\{l\}$ of sl($n+1/n$), a Young tableau $T$ of shape $\{l\}$ is a labelling of $l$ boxes $ \vcenter {\offinterlineskip \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$#$\hss}\cr \multispan{19}\hrulefill\cr & && && && && && &\omit&\cdots&\omit& && &\cr \multispan{19}\hrulefill\cr }} $ with numbers $\{1,2,\ldots,2n+1\}$ such that the numbers appear in non-decreasing order from left to right, with repititions allowed for the numbers $\{1,2,\ldots,n+1\}$ only. To every such tableau $ T = \vcenter {\offinterlineskip \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$#$\hss}\cr \multispan{10}\hrulefill&\multispan{13}\hrulefill\cr &i_1&&i_2&& &\omit&\cdots&\omit& &&i_s&&j_1&& &\omit&\cdots&\omit& &&j_t&\cr \multispan{10}\hrulefill&\multispan{13}\hrulefill\cr }} $ with $1\leq i_1\leq i_2\leq\cdots\leq i_s\leq n+1$ and $n+2\leq j_1<\cdots