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% 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.
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\begin{document}
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\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}}
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\maketitle
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 \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 $n+2\leq
j_1<\cdots<j_t\leq 2n+1$ there corresponds a term $s(T) = x_{i_1}\cdots
x_{i_s}y_{j_1-n-1}\cdots y_{j_t-n-1}$.  Then~\cite{b,k4}
 \beq
 s_{\{\la\}}(\hbox{\bf x}/\hbox{\bf y}) = \sum_T s(T)\,.
 \label{f29}
 \eeq
Using this correspondence, and performing the $x$-specialisation in the
rhs of (\ref{f29}), we obtain that
 \beq
 P_{nl}(x) = \sum_{t=0}^l\,\sum_{\mu\subset (2n)^t}\,
 \sum_{\nu\subset (2n-1)^{l-t}}\,\, x^{|\mu|+|\nu|}\,,
 \eeq
where $\mu$ is restricted to partitions with {\sl even} parts only
(contained in the rectangle $(2n)^t$) and $\nu$ is restricted to
partitions {\sl without repetitions} with {\sl odd} parts only
(contained in the rectangle $(2n-1)^{l-t}$), thus length$(\nu)=l-t$. 
The symbol $|\mu|$ denotes the weight of the partition $\mu$, which is
the sum of all its parts.  
%
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\end{document}
