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\begin{document}

\title[Supernomial coefficients \dots]{Supernomial coefficients, 
Bailey's lemma and Rogers--Ramanujan-type 
identities. \\ {\tiny A survey of results and open problems}}
 
\author[S.~O.~Warnaar]{S.~Ole Warnaar}
\address{Instituut voor Theoretische Fysica, Universiteit van Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands}
\email{warnaar@wins.uva.nl}

\keywords{Rogers--Ramanujan identities, supernomial coefficients,
Bailey's lemma} 
\subjclass{Primary 05A30, 05A19; Secondary 33D90, 33D15, 11P82}

\begin{abstract}
An elementary introduction to the recently introduced A$_2$ Bailey
lemma for supernomial coefficients is presented.
As illustration, some A$_2$ $q$-series identities are (re)de\-rived which 
are natural analogues of the classical (A$_1$) Rogers--Ramanujan identities 
and their generalizations of Andrews and Bressoud.
The intimately related, but unsolved problems of supernomial inversion,
A$_{n-1}$ and higher level extensions are also discussed.
This yields new results and conjectures involving A$_{n-1}$ basic 
hypergeometric series, string functions and cylindric partitions.
\end{abstract}
 
\maketitle

\section{Introduction}
The purpose of this paper is twofold. Firstly, it intends
to provide an easy introduction to recent
results by Andrews, Schilling and the author~\cite{ASW98}
concerning an A$_2$ Bailey lemma for supernomial coefficients.
The fact that the theorems of \cite{ASW98} have led to the discovery 
of A$_2$ analogues of the famous Rogers--Ramanujan identities,
(hopefully) justifies such an introduction.
Secondly, we hope to attract some interest in the numerous
unsolved problems directly related to the results of ref.~\cite{ASW98}.

In the first part of this paper, comprising of sections 
\ref{secA1}--\ref{secap},
we review the A$_1$ Bailey lemma and its A$_2$ supernomial generalization,
and show how this provides a natural framework for proving and deriving
identities of the Rogers--Ramanujan type. To make this part of the
paper as accessible as possible we have omitted all proofs and have
removed all the usual Bailey miscellanea. 
Also, we have chosen to cover only the simplest possible cases
that can be extracted from the general Bailey machinery 
(see e.g., \cite{Bailey49,Andrews84,Andrews85,Paule85,AAB87,Bressoud88,Paule88,ASW98}).

In the second part of the paper (sections~\ref{secRI}-\ref{secC})
we discuss various questions that have arisen in relation to
our A$_2$ Bailey lemma. Most importantly there is the problem
of generalizing the results of \cite{ASW98} to A$_{n-1}$,
but also questions concerning supernomial inversion, higher-level
Bailey lemmas and some related issues will be surveyed.

We should remark here that this paper does not in any way
discuss the A$_{n-1}$ Bailey lemma of Milne and Lilly~\cite{ML92,ML95},
nor the A$_{n-1}$ Rogers--Ramanujan identities of Milne~\cite{Milne92,Milne94}.
It is our current believe that the A$_2$ Bailey lemma for supernomials
and the A$_2$ case of Milne and Lilly's lemma are generalizations
of the classical A$_1$ Bailey lemma, which, in a sense, are orthogonal.
Also the A$_2$ Rogers--Ramanujan identity of this paper appears to be
unrelated to the A$_2$ case of Milne's A$_{n-1}$ Rogers--Ramanujan
identity.

\section{A$_1$ Rogers--Ramanujan-type identities}\label{secA1}
The Gaussian polynomial or $q$-binomial coefficient is defined as
\begin{equation*}
\qbin{n}{m}=\begin{cases}
\displaystyle
\frac{(q)_n}{(q)_m(q)_{n-m}} & \text{for } 0\leq m \leq n \\[3mm]
0 & \text{otherwise,}\end{cases}
\end{equation*}
where $(a;q)_{\infty}=(a)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k)$ and
\begin{equation*}
(a;q)_n=(a)_n=\frac{(a)_{\infty}}{(aq^n)_{\infty}}, \qquad n\in\Integer.
\end{equation*}
In particular, $(q)_0=1$, $(q)_n=(1-q)\dots(1-q^n)$ and
$1/(q)_{-n}=0$ for $n\geq 1$.
We will often use a shifted and normalized $q$-binomial coefficient,
defined as
\begin{equation}\label{SA1}
S(L,k)=\frac{1}{(q)_{2L}}\qbin{2L}{L-k}=
\begin{cases}
\displaystyle
\frac{1}{(q)_{L-k}(q)_{L+k}} & \text{for } -L\leq k\leq L \\[3mm]
0 & \text{otherwise.}\end{cases}
\end{equation}

By the $q$-Chu--Vandermonde summation (equation~(3.3.10) of \cite{Andrews76}),
it follows that the modified $q$-binomial satisfies the following 
``invariance property'' (equation~(13) of~\cite{Paule88} with 
$c_k=\delta_{k,r}$) ,
\begin{equation}\label{inv}
\sum_{L=0}^M \frac{q^{L^2}S(L,r)}{(q)_{M-L}}
=q^{r^2} S(M,r).
\end{equation}
That is, the modified $q$-binomial is (up to an overall factor) invariant 
under multiplication by $q^{L^2}/(q)_{M-L}$ followed by a sum over $L$.  
It is this property of the $q$-binomial that we shall try to
generalize to other $q$-functions.

First, however, let us demonstrate the effectiveness
of the result \eqref{inv} in deriving identities of the
Rogers--Ramanujan type. To obtain identities for odd moduli
our starting point is the following specialization of the
the $q$-binomial formula (equation~(II.4) of \cite{GR90}),
\begin{equation}\label{delta}
\sum_{r=-L}^L (-1)^r q^{\binom{r}{2}}S(L,r)=\delta_{L,0},
\end{equation}
with $\binom{r}{2}=r(r-1)/2$ for $r\in\Integer$.
Applying \eqref{inv} $k$ times (that is, multiplying \eqref{delta} by
$q^{L^2}/(q)_{M-L}$, summing over $L$ using \eqref{inv} 
and replacing $M$ by $L$, and iterating this $k$ times) yields
\begin{equation*}
\sum_{r=-L}^L (-1)^r q^{\binom{r}{2}+kr^2}S(L,r)
=\sum_{n_1,\dots,n_{k-1}\geq 0}
\frac{q^{N_1^2+\cdots+N_{k-1}^2}}
{(q)_{L-N_1}(q)_{n_1}\cdots(q)_{n_{k-1}}},
\end{equation*}
where $N_j=n_j+\cdots+n_{k-1}$.
If we let $L$ tend to infinity and use Jacobi's triple product identity
\begin{equation}\label{JTP}
\sum_{j=-\infty}^{\infty} (-z)^j q^{\binom{j}{2}}=(z,q/z,q)_{\infty},
\end{equation}
where $(a_1,\dots,a_k;q)_n=(a_1,\dots,a_k)_n=(a_1)_n\dots(a_k)_n$,
the following result is obtained.
\begin{theorem}\label{thmodd}
For $k\geq 2$, $|q|<1$, and $N_j=n_j+\cdots+n_{k-1}$,
\begin{equation}\label{AG}
\sum_{n_1,\dots,n_{k-1}\geq 0}
\frac{q^{N_1^2+\cdots+N_{k-1}^2}}
{(q)_{n_1}\cdots(q)_{n_{k-1}}}
=\frac{(q^k,q^{k+1},q^{2k+1};q^{2k+1})_{\infty}}{(q)_{\infty}}.
\end{equation}
\end{theorem}
For $k=2$ this is the (first) Rogers--Ramanujan
identity~\cite{Rogers94,Rogers17,Schur17,RR19}
\begin{equation}\label{RR}
\sum_{n\geq 0} \frac{q^{n^2}}{(q)_n}
=\prod_{n=0}^{\infty}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})}.
\end{equation}
For general $k$ equation~\eqref{AG} is (a particular case) of
Andrews' analytic counterpart~\cite{Andrews74} of
Gordon's partition theorem~\cite{Gordon61}.

To obtain a similar result for even moduli, we start with the simple identity
(equation~(40) of \cite{Paule85}),
\begin{equation}\label{even}
\sum_{r=-L}^L (-1)^r q^{r^2}S(L,r)=\frac{1}{(q^2;q^2)_L}.
\end{equation}
Applying \eqref{inv} $k-1$ times yields
\begin{equation*}
\sum_{r=-L}^L (-1)^r q^{kr^2}S(L,r)
=\sum_{n_1,\dots,n_{k-1}\geq 0}
\frac{q^{N_1^2+\cdots+N_{k-1}^2}}
{(q)_{L-N_1}(q)_{n_1}\cdots(q)_{n_{k-2}}(q^2;q^2)_{n_{k-1}}}.
\end{equation*}
When $L$ tends to infinity one can again apply the triple product~\eqref{JTP},
resulting in our next theorem.
\begin{theorem}\label{thmeven}
For $k\geq 2$, $|q|<1$, and $N_j=n_j+\cdots+n_{k-1}$,
\begin{equation}\label{Br}
\sum_{n_1,\dots,n_{k-1}\geq 0}
\frac{q^{N_1^2+\cdots+N_{k-1}^2}}
{(q)_{n_1}\cdots(q)_{n_{k-2}}(q^2;q^2)_{n_{k-1}}}
=\frac{(q^k,q^k,q^{2k};q^{2k})_{\infty}}{(q)_{\infty}}
\end{equation}
\end{theorem}
For $k=2$ this identity is due to Euler. For general
$k$ the above result was first obtained by Bressoud~\cite{Bressoud80}.

\section{Supernomial coefficients}\label{secsupern}
In this section we introduce A$_{n-1}$ generalizations of the
$q$-binomial coefficients and show how, in the case of A$_2$, the invariance
property~\eqref{inv} can be generalized. This is then used to
derive A$_2$ Rogers--Ramanujan-type identities for all moduli.
First, however, to serve as a guide for subsequent generalizations,
some of the equations of the previous section are rewritten in manifest 
A$_1$ form.
\subsection{A$_1$ again}
As is often convenient when dealing with root systems of type 
A$_{n-1}$, we introduce $n$ variables $k_1,\dots,k_n$ constrained
to the hyperplane $k_1+\cdots+k_n=0$. We denote $k=(k_1,\dots,k_n)$,
$\rho=(1,\dots,n)$, and for arbitrary $v\in\Integer^p$ we set
$|v|=\sum_{i=1}^p v_i$, so that, in particular, $|k|=0$. 
The Cartan matrix of A$_{n-1}$ will
be denoted by $C$, i.e., $C_{i,j}=2\delta_{i,j}-\delta_{|i-j|,1}$,
$i,j=1,\dots,n-1$.

Now assume $n=2$. Then $k=(k_1,k_2)=(k_1,-k_1)$, $\rho=(1,2)$ and
$C=(2)$.
We can then rewrite equation \eqref{SA1} in vector notation as
\begin{equation}\label{SA1vec}
S(L,k)=\begin{cases}
\displaystyle
\frac{1}{(q)_{L_1-k_1}(q)_{L_1-k_2}} & \text{for } k_1,k_2\leq L_1 \\[3mm]
0 & \text{otherwise.}\end{cases}
\end{equation}
where $L=(L_1)$.

Similarly, the invariance property \eqref{inv} becomes
\begin{equation}\label{inv2}
\sum_{L=0}^{M} \frac{q^{\frac{1}{2}LCL}S(L,k)}{(q)_{M-L}}
=q^{\frac{1}{2}(k_1^2+k_2^2)} S(M,k),
\end{equation}
where, generally, for $v,w\in \Integer^p$,
$vAv=\sum_{i,j=1}^p v_i A_{i,j} v_j$,
$(a)_v=(a)_{v_1}\dots (a)_{v_p}$ and
$\sum_{v=0}^w=\sum_{v_1=0}^{w_1}\dots\sum_{v_p=0}^{w_p}$.

The equations \eqref{delta} and \eqref{even} that served as input in the
derivation of the Rogers--Ramanujan-type identities become in the new
notation
\begin{equation}\label{inputodd}
\sum_{|k|=0}\sum_{\sigma\in S_2}\epsilon(\sigma)
q^{\sum_{i=1}^2(k_i-\sigma_i)k_i} S(L,2k-\sigma+\rho)=\delta_{L_1,0}.
\end{equation}
and
\begin{equation}\label{inputeven}
\sum_{|k|=0}\sum_{\sigma\in S_2}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^2(2k_i-\sigma_i+i)^2}
S(L,2k-\sigma+\rho)=\frac{1}{(q^2;q^2)_{L_1}},
\end{equation}
where $S_n$ is the permutation group on $1,2,\dots,n$ and
$\epsilon(\sigma)$ is the sign of the permutation $\sigma$.
 
\subsection{Completely antisymmetric A$_{n-1}$ supernomials}
The generalization of the $q$-binomial coefficient that is needed
here is a multivariable extension of the ``$n$-multinomial coefficient''
\begin{equation}\label{multinomial}
\frac{(q)_{\lambda_1+\cdots+\lambda_n}}{(q)_{\lambda_1}\dots (q)_{\lambda_n}},
\end{equation}
defined as follows~\cite{HKKOTY99}.
\begin{definition}\label{defsupernomial}
Let $L\in\Integer^{n-1}_+$, $\lambda\in\Integer^n_+$ and
let $\nu^{(n)}$ denote the conjugate of the partition
$(1^{L_1}\dots (n-1)^{L_{n-1}})$, i.e.,
$\nu^{(n)}_j=L_j+\cdots+L_{n-1}$.
Then, for $|\lambda|=|\nu^{(n)}|(=\sum_{a=1}^{n-1}aL_a)$,
\begin{equation}\label{supH}
\qbin{L}{\lambda}=\sum_{\nu}\prod_{a=1}^{n-1}\prod_{j=1}^a
\qbin{\nu_j^{(a+1)}-\nu_{j+1}^{(a+1)}}{\nu_j^{(a)}-\nu_{j+1}^{(a+1)}},
\end{equation}
where the sum over $\nu$ denotes a sum over sequences
$\emptyset=\nu^{(0)}\subset\nu^{(1)}\subset\dots\subset\nu^{(n)}$
of Young diagrams such that each skew 
diagram $\nu^{(a)}-\nu^{(a-1)}$ is a horizontal 
$\lambda_a$-strip\footnote{Viewing the $\nu^{(a)}$'s
as partitions, this means that $|\nu^{(a)}|-|\nu^{(a-1)}|=\lambda_a$ and
$\nu^{(a)}_i\leq \nu^{(a-1)}_{i-1}$  (i.e., the $i$th part
of $\nu^{(a)}$ does not exceed the $(i-1)$th part of $\nu^{(a-1)}$).}. 
\end{definition}
Copying the example of ref.~\cite{HKKOTY99}, we find that
for $n=3$, $L=(1,3)$ and $\lambda=(3,2,2)$,
the contributions to the above sum correspond to
$\nu=(\emptyset,(3),(3,2),(4,3))$ and
$\nu=(\emptyset,(3),(4,1),(4,3))$, yielding
$\qbins{3}{2}+\qbins{3}{2}\qbins{3}{1}=2+3q+4q^2+2q^3+q^4$.
When $L=(|\lambda|,0,\dots,0)$ the only 
term in the sum is $\nu=(\emptyset,\lambda_1,\lambda_1+\lambda_2,\dots,
|\lambda|)$, yielding
$\prod_{a=1}^n\qbins{\lambda_1+\cdots+\lambda_{a+1}}
{\lambda_1+\cdots+\lambda_a}$ which is the multinomial \eqref{multinomial}.
Perhaps not immediately evident are the symmetries~\cite{SW98b}
$$\qbin{L}{\lambda}=\qbin{L'}{|L|(1^n)-\lambda} \quad \text{and} \quad
\qbin{L}{\lambda}=\qbin{L}{\sigma(\lambda)},$$
where $L'=(L_{n-1},\dots,L_1)$ and $\sigma\in S_n$.

The (completely antisymmetric) A$_{n-1}$ supernomials have several 
interesting interpretations. In ref.~\cite{HKKOTY99} they
were defined as
\begin{equation*}
\qbin{L}{\lambda}=
\sum_{\eta\,\vdash |\lambda|}K_{\eta\lambda}K_{\eta'\mu}(q),
\end{equation*}
where $\mu=(1^{L_1}\dots (n-1)^{L_{n-1}})$,
$\lambda\in\Integer^n$ a composition such that $|\lambda|=|\mu|$,
and $K_{\lambda\mu}(q)$ and $K_{\lambda\mu}$ the Kostka polynomial and Kostka
number, respectively~\cite{Macdonald95}.
In \cite{Kirillov98} this was shown to imply that the supernomials
are connection coefficients
between the elementary symmetric functions $e_{\lambda}$ and the
Hall--Littlewood polynomials $P_{\lambda}$ in $n$ 
variables~\cite{Macdonald95}, thanks to
\begin{equation*}
e_{\lambda}(x_1,\dots,x_n)=\sum_{\mu\,\vdash |\lambda|}
\biggl(\sum_{\eta\,\vdash |\lambda|}K_{\eta\lambda}K_{\eta'\mu}(q)\biggr)
P_{\mu}(x_1,\dots,x_n;q).
\end{equation*}
In \cite{SW98b} the supernomials were introduced from a combinatorial
point of view as the generating functions of inhomogeneous
lattice paths, generalizing the fact that the multinomial coefficient
\eqref{multinomial} is the major index generating function on
words over the alphabet $\{1,\dots,n\}$.

If we now restrict \eqref{supH} to $n=3$, and set
$\nu^{(1)}=\lambda_1$, $\nu^{(2)}=(\lambda_1+m,\lambda_2-m)$ and 
$\nu^{(3)}=(L_1+L_2,L_2)$ we get
\begin{equation*}
\qbin{L}{\lambda}=\sum_m
\qbin{\lambda_1-\lambda_2+2m}{m}\qbin{L_1}{\lambda_1-L_2+m}
\qbin{L_2}{\lambda_2-m}
\end{equation*}
for $\lambda_1+\lambda_2+\lambda_3=L_1+2L_2$ and zero otherwise.
The following, more symmetric, representation may be derived 
using the $q$-Chu--Vandermonde sum,
\begin{equation*}
\qbin{L}{\lambda}=\sum_r 
\frac{q^{r_1 r_{23}}(q)_{L_1}(q)_{L_2}}
{(q)_{r_1}(q)_{r_2}(q)_{r_3}(q)_{r_{12}}(q)_{r_{13}}(q)_{r_{23}}},
\end{equation*}
where the summation over $r$ denotes a sum over $r_1,\dots,r_{23}$ such that
\begin{equation*}
r_1+r_{12}+r_{13}=\lambda_1, \quad
r_2+r_{12}+r_{23}=\lambda_2, \quad
r_3+r_{13}+r_{23}=\lambda_3
\end{equation*}
and
\begin{equation*}
r_1+r_2+r_3=L_1, \quad r_{12}+r_{13}+r_{23}=L_2.
\end{equation*}

\subsection{An A$_2$ invariance property}\label{secmain}
We now show how, in the case of A$_2$, the supernomials 
may be used the generalize the $q$-binomial invariance~\eqref{inv2}.
The first step is, of course, to again shift and normalize the 
supernomials, and for general rank we define in analogy with 
\eqref{SA1} and \eqref{SA1vec},
\begin{equation}\label{SAn}
S(L,k)=\frac{1}{(q)_{CL}}\qbin{CL}{L_{n-1}(1^n)-k}
\end{equation}
with $L\in\Integer_+^{n-1}$ and $k\in\Integer^n$ such that $|k|=0$.
Observe that $\sum_{i=1}^n (L_{n-1}-k_i)=nL_{n-1}=\sum_{a=1}^{n-1} a (CL)_a$
so that the condition $|\lambda|=|\nu^{(n)}|$
in definition \eqref{defsupernomial} is automatically satisfied.

Considering A$_2$ again, we would like to show that the following
invariance property holds (compare with \eqref{inv2})
\begin{equation}\label{invwrong}
\sum_{L=0}^M \frac{q^{\frac{1}{2}LCL}S(L,k)}{(q)_{M-L}}
=q^{\frac{1}{2}(k_1^2+k_2^2+k_3^2)} S(M,k).
\end{equation}
The analogy with A$_1$ breaks down, however, and a somewhat unexpected 
(to us at least) result arises as follows
(theorem 4.3 of~\cite{ASW98} with $a=1$).
\begin{theorem}\label{thmA2}
Let $L,M\in\Integer^2_+$, $k\in\Integer^3$, such that $|k|=0$
and let $S(L,k)$ be the A$_2$ supernomial defined in \eqref{SAn}
and $T(L,k)$ be defined in \eqref{Tdef} below. Then
\begin{equation}\label{inv3}
\sum_{L=0}^M \frac{q^{\frac{1}{2}LCL}S(L,k)}{(q)_{M-L}}
=q^{\frac{1}{2}(k_1^2+k_2^2+k_3^2)} (q)_{|M|} T(M,k)
\end{equation}
and
\begin{equation}\label{inv4}
\sum_{L=0}^M \frac{q^{\frac{1}{2}LCL}T(L,k)}{(q)_{M-L}}
=q^{\frac{1}{2}(k_1^2+k_2^2+k_3^2)} T(M,k).
\end{equation}
\end{theorem}
So we do find an invariance property, but only after summing the
supernomial $S$ to a new $q$-function $T$, given in the following definition.
\begin{definition}
For $L\in\Integer^2_+$ and $k\in\Integer^3$ such that $|k|=0$,
\begin{equation}\label{Tdef}
T(L,k)=\frac{1}{(q)^2_{L_1+L_2}}\prod_{i=1}^3 \qbin{L_1+L_2}{L_1+k_i}.
\end{equation}
\end{definition}

The fact that \eqref{invwrong} is not correct and has to be
replaced by the non-trivial theorem~\ref{thmA2} is the main
obstacle for treating the general rank case. Indeed, for arbitrary
A$_{n-1}$ we find that ($L,M\in\Integer^{n-1}_+$, $k\in\Integer^n$ such that
$|k|=0$)
\begin{equation*}
\sum_{L=0}^M \frac{q^{\frac{1}{2}LCL}S(L,k)}{(q)_{M-L}}
=q^{\frac{1}{2}(k_1^2+\cdots+k_n^2)} S(M,k)
\end{equation*}
is invalid for any $n\geq 3$. How to correct this, in a way similar
to Theorem~\ref{thmA2}, is unclear to us at present. 
A partial result on A$_{n-1}$ is given in proposition~\ref{propAn} 
of section~\ref{secAn}.

\section{A$_2$ Rogers--Ramanujan-type identities}\label{secap}
We now use the two summations of theorem~\ref{thmA2} to obtain 
A$_2$ analogues of theorems~\ref{thmodd} and \ref{thmeven}.
Were there 2 cases to consider for A$_1$, corresponding to odd
and even modulus, this time we have to consider moduli
in the residue classes of $3$.

First we need the A$_2$ generalization of \eqref{inputodd}
(proposition~5.1 of \cite{ASW98} with $\ell=0$).
\begin{proposition}
For $L\in\Integer^2$ such that $C L\in\Integer^2_+$,
\begin{equation}\label{superid}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^3(3k_i-2\sigma_i)k_i}
S(L,3k-\sigma+\rho)=\delta_{L_1,0}\delta_{L_2,0}.
\end{equation}
\end{proposition}
We now invoke theorem~\ref{thmA2}. First this gives, thanks to
\eqref{inv3}, a doubly bounded version of the A$_2$ Euler identity,
\begin{equation}\label{A2Euler}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^3(3k_i-2\sigma_i)k_i+(3k_i-\sigma_i+i)^2}
T(L,3k-\sigma+\rho)=\frac{1}{(q)_L(q)_{|L|}}.
\end{equation}
Next we can apply \eqref{inv4} and after an $(\ell-1)$-fold iteration
we arrive at 
\begin{multline*}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^3 (3k_i-2\sigma_i)k_i+\ell(3k_i-\sigma_i+i)^2}
T(L,3k-\sigma+\rho) \\
=\sum_{n_1,\dots,n_{\ell-1}\in\Integer^2_+}
\frac{q^{\frac{1}{2}\sum_{j=1}^{\ell-1} N_j C N_j}}
{(q)_{L-N_1}(q)_{n_1}\cdots (q)_{n_{\ell-1}}(q)_{|n_{\ell-1}|}},
\end{multline*}
where $N_j=n_j+\cdots+n_{\ell-1}\in \Integer^2_+$.
To transform this into identities of the
Rogers--Ramanujan type we let $L_1,L_2$ tend to infinity and apply the
A$_2$ Macdonald identity~\cite{Macdonald72}
(in a representation of \cite{Milne85})
\begin{equation}\label{MDI}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
\prod_{i=1}^3 q^{\frac{3}{2}k_i^2+\sigma_i k_i}x_i^{3k_i+\sigma_i-i}
=(q)^2_{\infty}\prod_{1\leq i<j\leq 3}(x_ix_j^{-1},q x_jx_i^{-1})_{\infty}.
\end{equation}
This leads to the following A$_2$ analogue of the identities~\eqref{AG}
(theorem~5.1 of \cite{ASW98} with $i=k$).
\begin{theorem}
For $|q|<1$, $k\geq 2$ and $N_j=n_j+\cdots+n_{k-1}$, 
\begin{multline*}
\sum_{n_1,\dots,n_{k-1}\in\Integer^2_+}
\frac{q^{\frac{1}{2}\sum_{j=1}^{k-1} N_j C N_j}}
{(q)_{n_1}\cdots (q)_{n_{k-1}}(q)_{|n_{k-1}|}} \\
=\frac{(q^k,q^k,q^{k+1},q^{2k},q^{2k+1},q^{2k+1},
q^{3k+1},q^{3k+1};q^{3k+1})_{\infty}}{(q)^3_{\infty}}.
\end{multline*}
\end{theorem}
Although this theorem is indeed very much akin to theorem~\eqref{thmodd},
there is a striking (as well as annoying) difference.
This is the fact that on the right-hand side we have a $(q)^3_{\infty}$
in the denominator
where one would have liked to see a $(q)^2_{\infty}$.
Indeed, to interpret the right-hand side combinatorially,
we have to first multiply with $(q)_{\infty}$.
Then the right-hand side becomes the generating function of pairs 
of partitions $(\lambda_1,\lambda_2)$ such that $\lambda_1$ has
no parts congruent to $0,\pm k,\pm 2k \pmod{3k+1}$ and $\lambda_2$ has
no parts congruent to $0,\pm k \pmod{3k+1}$.
Also, the right-hand side can (again after multiplication by $(q)_{\infty}$)
be identified with a character of the W$_3$ algebra.
The conclusion clearly is that the left-hand side, when multiplied with
$(q)_{\infty}$ is a series with only positive integer coefficients.
How to make this manifest is unclear to us.
Only when $k=2$ we have succeeded (section~5.4 of ~\cite{ASW98})
in rewriting the above
identity when multiplied with $(q)_{\infty}$ such that both sides
are manifestly positive series. The price for this, however, is that
the $\Integer_2$ symmetry of the summand is broken.
\begin{theorem}[A$_2$ Rogers--Ramanujan identity]
For $|q|<1$,
\begin{multline*}
(q)_{\infty}\sum_{r_1,r_2\geq 0}\frac{q^{r_1^2-r_1 r_2+r_2^2}}
{(q)_{r_1}(q)_{r_2}(q)_{r_1+r_2}}=
\sum_{r_1,r_2\geq 0}\frac{q^{r_1^2-r_1 r_2+r_2^2}}{(q)_{r_1}}
\qbin{2r_1}{r_2}\\
=\prod_{n=0}^{\infty}
\frac{1}{(1-q^{7n+1})^2(1-q^{7n+3})(1-q^{7n+4})(1-q^{7n+6})^2}.
\end{multline*}
\end{theorem}
It seems a worthwhile exercise to find a partition theoretic
interpretation for the middle term of this A$_2$ Rogers--Ramanujan
identity, or, even better,
to rewrite the left-hand side into a series
that is manifestly of A$_2$-type as well as manifestly positive,
and to then find a combinatorial interpretation.


To obtain identities of the Rogers--Ramanujan type for moduli congruent
to $2$ modulo $3$, we replace $q$ by $1/q$ in the A$_2$ Euler identity 
\eqref{A2Euler}. Using $$\qbin{m+n}{n}_{1/q}=q^{-mn}\qbin{m+n}{n}$$
and definition \eqref{Tdef} of $T$ this yields
\begin{equation*}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^3(3k_i-\sigma_i+i)^2-3(k_i-2\sigma_i)k_i}
T(L,3k-\sigma+\rho)=\frac{q^{2L_1 L_2}}{(q)_L(q)_{|L|}}.
\end{equation*}
Iterating this $\ell-1$ times using equation \eqref{inv4} leads to
\begin{multline*}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^3\ell(3k_i-\sigma_i+i)^2-3(k_i-2\sigma_i)k_i}
T(L,3k-\sigma+\rho) \\
=\sum_{n_1,\dots,n_{\ell-1}\in\Integer^2_+}
\frac{q^{\frac{1}{2}\sum_{j=1}^{\ell-2} N_j C N_j+
\frac{1}{2}N_{\ell-1} B N_{\ell-1}}}
{(q)_{L-N_1}(q)_{n_1}\cdots (q)_{n_{\ell-1}}(q)_{|n_{\ell-1}|}},
\end{multline*}
where $B_{i,j}=2\delta_{i,j}+\delta_{|i-j|,1}$, $i,j=1,2$.
Letting $L_1,L_2$ go to infinity and using the Macdonald identity
\eqref{MDI} gives the following theorem
(theorem~5.3 of \cite{ASW98} with $i=k$).
\begin{theorem}
For $|q|<1$, $k\geq 2$ and $N_j=n_j+\cdots+n_{k-1}$, 
\begin{multline*}
\sum_{n_1,\dots,n_{k-1}\in\Integer^2_+}
\frac{q^{\frac{1}{2}\sum_{j=1}^{k-2} N_j C N_j+
\frac{1}{2}N_{k-1} B N_{k-1}}}
{(q)_{n_1}\cdots (q)_{n_{k-1}}(q)_{|n_{k-1}|}} \\
=\frac{(q^{k-1},q^k,q^k,q^{2k-1},q^{2k-1},q^{2k},
q^{3k-1},q^{3k-1};q^{3k-1})_{\infty}}{(q)^3_{\infty}}.
\end{multline*}
\end{theorem}
For $k=2$ the above identity is the
first Rogers--Ramanujan identity~\eqref{RR} in disguise~\cite{ASW98}.

It remains to find identities for moduli congruent to $0\pmod{3}$.
What is needed now is the A$_2$ analogue of identity \eqref{inputeven}
provided by Gessel and Krattenthaler (equation~(6.18) of \cite{GK97}).
\begin{proposition}
For $L\in\Integer^2_+$,
\begin{equation}\label{Tid}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^3(3k_i-\sigma_i+i)^2} T(L,3k-\sigma+\rho)=
\frac{(q^3;q^3)_{|L|}}{(q^3;q^3)_L(q)_{|L|}^2}.
\end{equation}
\end{proposition}
Applying theorem~\ref{thmA2} this readily gives
\begin{multline*}
\sum_{|k|=0}\sum_{\sigma\in S_3}\epsilon(\sigma)
q^{\frac{\ell}{2}\sum_{i=1}^3(3k_i-\sigma_i+i)^2} T(L,3k-\sigma+\rho) \\
=\sum_{n_1,\dots,n_{\ell-1}\in\Integer^2_+}
\frac{q^{\frac{1}{2}\sum_{j=1}^{\ell-1}N_j C N_j}
(q^3;q^3)_{|n_{\ell-1}|}}
{(q)_{L-N_1}(q)_{n_1}\cdots (q)_{n_{\ell-2}}(q^3;q^3)_{n_{\ell-1}}
(q)^2_{|n_{\ell-1}|}}.
\end{multline*}
When $L_1,L_2$ approach infinity this yields our final Rogers--Ramanujan-type
theorem (theorem~5.4 of \cite{ASW98} with $i=k$).
\begin{theorem}
For $|q|<1$, $k\geq 2$ and $N_j=n_j+\cdots+n_{k-1}$, 
\begin{multline*}
\sum_{n_1,\dots,n_{k-1}\in\Integer^2_+}
\frac{q^{\frac{1}{2}\sum_{j=1}^{k-1}N_j C N_j}
(q^3;q^3)_{|n_{k-1}|}}
{(q)_{n_1}\cdots (q)_{n_{k-2}}(q^3;q^3)_{n_{k-1}}
(q)^2_{|n_{k-1}|}} \\
=\frac{(q^k,q^k,q^k,q^{2k},q^{2k},q^{2k},
q^{3k},q^{3k};q^{3k})_{\infty}}{(q)^3_{\infty}}.
\end{multline*}
\end{theorem}

\section{Reduction and inversion}\label{secRI}
So far, we have used the A$_1$ and A$_2$ summations~\eqref{inv}, \eqref{inv3} 
and \eqref{inv4} to derive complicated identities out of simpler ones.
Of, course, when given a complicated identity, it is of interest to know 
whether this identity is reducible to a simpler one. That is, whether,
iteration of some yet unknown simpler identity produces the complicated
identity.
The answer to this question is easily given, and in the case of A$_1$
the following result holds~\cite{Andrews86} (which actually is the $q\to 1/q$
version of \eqref{inv})
\begin{equation}\label{red}
\sum_{L=0}^M \frac{(-1)^{M-L} q^{\binom{M-L}{2}-M^2}}{(q)_{M-L}}\: S(L,r)
=q^{-r^2} S(M,r).
\end{equation}
Iterating this identity using the invariance property \eqref{inv} implies
\begin{equation*}
\sum_{M=0}^N \frac{q^{M^2}}{(q)_{N-M}} \sum_{L=0}^M 
\frac{(-1)^{M-L} q^{\binom{M-L}{2}-M^2}}{(q)_{M-L}}\: S(L,r)=S(N,r).
\end{equation*}
Interchanging the two sums on the left-hand side and then shifting
$M\to M+L$ gives
\begin{equation*}
\sum_{L=0}^N \frac{S(L,r)}{(q)_{N-L}}
\sum_{M=0}^{N-L}(-1)^M q^{\binom{M}{2}}\qbin{N-L}{M}=S(N,r),
\end{equation*}
which is indeed true since, by \eqref{delta}, 
the inner sum yields $\delta_{L,N}$. (Of course, applying \eqref{red}
to \eqref{inv}, instead of \eqref{inv} to \eqref{red} is 
consistent with the above.)

In exactly the same way one can establish that
\begin{equation*}
\sum_{L=0}^M
\Bigl(\prod_{i=1}^2 \frac{(-1)^{M_i-L_i} q^{\binom{M_i-L_i}{2}}}
{(q)_{M_i-L_i}}\Bigr) q^{-\frac{1}{2}MCM}(q)_{|L|} T(L,k)
=q^{-\frac{1}{2}(k_1^2+k_2^2+k_3^2)} S(M,k)
\end{equation*}
and
\begin{equation*}
\sum_{L=0}^M
\Bigl(\prod_{i=1}^2 \frac{(-1)^{M_i-L_i} q^{\binom{M_i-L_i}{2}}}
{(q)_{M_i-L_i}}\Bigr) q^{-\frac{1}{2}MCM} T(L,k)
=q^{-\frac{1}{2}(k_1^2+k_2^2+k_3^2)} T(M,k).
\end{equation*}
(The second equation can also be found by taking $q\to 1/q$ in \eqref{inv4}.)
Using this one can see that the two
initial condition identities \eqref{superid} and \eqref{Tid} are indeed 
``maximally reduced''. To further reduce \eqref{Tid}, for example,
one would have to find ``nice'' representations for
\begin{equation*}
\sum_{L=0}^M
\Bigl(\prod_{i=1}^2 \frac{(-1)^{M_i-L_i} q^{\binom{M_i-L_i}{2}}}
{(q)_{M_i-L_i}}\Bigr) q^{-\frac{1}{2}MCM}
\frac{(q^3;q^3)_{|L|}}
{(q^3;q^3)_L(q)_{|L|}^{2-\tau}},
\end{equation*}
with $\tau$ either $0$ or $1$. This appears not to be possible.
(Hence the analogy with A$_1$ breaks down in this case, since
\eqref{even} can be reduced to $\sum_r (-1)^r S(L,r)=(-1)^L/(q^2;q^2)_L$
using \eqref{red} and the $q$-Chu--Vandermonde sum.)

Another question of interest is that of inversion.
For A$_1$ it can be stated as follows: find a function $\bar{S}(r,L)$ such 
that
\begin{equation}\label{Sbdef}
\sum_{L\geq 0}\bar{S}(r,L)S(L,s)=\delta_{r,s} \quad\text{and}\quad
\sum_{r\geq 0}S(L,r)\bar{S}(r,M)=\delta_{L,M}.
\end{equation}
In \cite{Andrews79} Andrews provides the solution,
\begin{equation*}
\bar{S}(r,L)=(-1)^{r-L}q^{\binom{r-L}{2}}(1-q^{2r})
\frac{(q)_{L+r-1}}{(q)_{r-L}},
\end{equation*}
with $\bar{S}(0,0)=1$.
This can be used to find further identities of the type
\begin{equation}\label{fg}
\sum_{r\geq 0}\alpha_r S(L,r)=\beta_L,
\end{equation}
since \eqref{Sbdef} and \eqref{fg} imply
\begin{equation*}
\alpha_r=\sum_{L\geq 0}\bar{S}(r,L)\beta_L.
\end{equation*}
For example, taking $\beta_L=\delta_{L,0}$ immediately yields
$\alpha_r=\bar{S}(r,0)=(-1)^r q^{\binom{r}{2}}(1+q^r)$ and $\alpha_0=1$,
which gives identity \eqref{delta}.

The A$_2$ analogue of the inversion formula \eqref{Sbdef} can be 
stated as follows:
\begin{equation}\label{SbdefA2}
\sum_{L_1,L_2\geq 0}
\bar{S}(k,L)S(L,k')=\delta_{k,k'}
\quad\text{and}\quad
\sum_{\substack{k_1\geq k_2\geq k_3\\|k|=0}}
S(L,k)\bar{S}(k,L')=\delta_{L,L'}.
\end{equation}
To see that the inverse supernomial $\bar{S}(L,k)$ exists we observe
that $S(L,k)$ (for $k_1\geq k_2\geq k_3$ and $|k|=0$) is non-zero
if and only if $2L_1\geq L_2$, $2L_2\geq L_1$, $k_3\geq -L_1$ and 
$k_1\leq L_2$.
Hence, if we define $K=(k_1+k_2,k_1)$ and write $S(L,K)$ instead of $S(L,k)$
it follows that $S(L,K)$ is non-zero if and only if 
\begin{equation}\label{LT}
K_1\leq L_1,\quad K_2\leq L_2
\end{equation}
and
\begin{equation}\label{range}
2K_1\geq K_2, \quad 2K_2\geq K_1,\quad
2L_1\geq L_2,\quad 2L_2\geq L_1.
\end{equation} 
Consequently, if we view $S(L,K)$ as an entry of an infinite-dimensional 
matrix with $L$ and $K$ in the ranges given by \eqref{range}, then
equation \eqref{LT} implies that $S$ is invertible.
We have computed $\bar{S}(K,L)$ for many different $L$ and $K$, and despite
the fact that we identified $\bar{S}(K,L)$ for all $K=(r,2r-p)$ with
$p=0,\dots,3$ we failed to observe enough regularity to guess (and then
prove, of course) a formula for arbitrary $K$. 
For those in for a challenge, here are the cases $p=0$ and $1$,
($\bar{S}((0,0),(0,0))=1$)
\begin{equation*}
\bar{S}((r,2r),L)=(-1)^{L_2}q^{\binom{r-L_1}{2}+\binom{r+L_1-L_2}{2}}
(1-q^{3r})\frac{(q)_{L_2+r-1}}{(q)_{r-L_1}} 
\end{equation*}
and
\begin{align*}
\bar{S}((r_,2r-1),L)&=(-1)^{L_2+1}
q^{\binom{r-L_1}{2}+\binom{r+L_1-L_2-1}{2}}
(1-q^{3r-2})\frac{(q)_{L_2+r-2}(q)_{L_1-L_2+r}}
{(q)_{r-L_1}(q)_{L_1-L_2+r-1}}\\
&-(-1)^{L_2+1}
q^{\binom{r-L_1-1}{2}+\binom{r+L_1-L_2-1}{2}+3r-3}
\frac{(q)_{L_2+r-2}(q)_1}{(q)_{r-L_1-1}},
\end{align*}
both for $r\geq 1$.
(To correctly get $\bar{S}((1,1),(0,0))$, first set $L_1=L_2=0$
and simplify to $\bar{S}((2,2r-1),(0,0))=-q^{r(r-2)}(q+q^r+q^{2r})$.
Then set $r=1$.)
 
The situation for $T(L,k)$ is much simpler, in that an inverse does 
not exists. Indeed, assuming again that
$k_1\geq k_2\geq k_3$ (and, of course, $|k|=0$), and writing
$T(L,K)$, we find that $T(L,K)$ is non-zero if and only if
\eqref{LT} and 
\begin{equation}\label{range2}
2K_1\geq K_2, \quad 2K_2\geq K_1,\quad
L_1\geq 0,\quad L_2\geq 0
\end{equation} 
hold.
Hence, viewing $T$ as an infinite-dimensional matrix 
with rows indexed by $L$ and columns by $K$, with ranges
given by \eqref{range2}, $T$ is no longer invertible,
its rows and columns ranging over different (infinite) sets.
(A right and/or a left inverse might of course exist, but the highly
overdetermined set of equations does not admit a solution.)

\section{Higher rank and level}\label{secAn}
Perhaps the two most important open questions are those of generalizing 
theorem~\eqref{thmA2} to A$_{n-1}$ and to higher level.

First addressing the A$_{n-1}$ problem, we would like
to establish the arbitrary rank version of the A$_1$ sum
\eqref{inv} and the A$_2$ summations \eqref{inv3} and \eqref{inv4}.
As mentioned in section~\ref{secmain}, the non-trivial
nature of theorem~\ref{thmA2}, has so-far prevented us from
making much progress in this direction. The only general result that
we have established can be stated as follows. 
\begin{proposition}\label{propAn}
Let $L\in\Integer^{n-1}_+$ and $k\in\Integer^n$ such that $|k|=0$ and 
let $S(L,k)$ be the A$_{n-1}$ supernomial of equation~\eqref{SAn}.
Then, for $M_1,M_{n-1}\in\Integer_+$,
\begin{equation}\label{eqprop}
\sum_{L\in\Integer^{n-1}_+}
\frac{q^{\frac{1}{2} LCL}S(L,k)}
{(q)_{M_1-L_1}(q)_{M_{n-1}-L_{n-1}}}
=\frac{q^{\frac{1}{2}\sum_{i=1}^n k_i^2}
(q)^{n-1}_{M_1+M_{n-1}}}
{\prod_{i=1}^n(q)_{M_1+k_i}(q)_{M_{n-1}-k_i}}.
\end{equation}
\end{proposition}
The proof of this only requires the $q$-Chu--Vandermonde sum
and will be presented in the appendix.
Letting $M_1,M_{n-1}$ tend to infinity,
this result yields an A$_{n-1}$ version of
what is referred to in ref.~\cite{Andrews92} as the
weak form of Bailey's lemma,
\begin{equation}\label{Aninf}
\sum_{L\in\Integer_+}
q^{\frac{1}{2} LCL}S(L,k)
=\frac{q^{\frac{1}{2}\sum_{i=1}^n k_i^2}}
{(q)_{\infty}^{n-1}}.
\end{equation}
Thus, given a supernomial identity one may derive a new $q$-series identity by
the above summation, but one cannot iterate ad infinitum.

For A$_3$ we further have the following isolated result.
\begin{proposition}\label{propA3}
Let $L\in\Integer^3_+$ and $k\in\Integer^4$ such that $|k|=0$ and 
let $S(L,k)$ be the A$_3$ supernomial of equation~\eqref{SAn}.
Then, for $M_2\in\Integer_+$,
\begin{equation}\label{A3}
\sum_{L\in\Integer^3_+}
\frac{q^{\frac{1}{2} LCL}S(L,k)}
{(q)_{M_2-L_2}}
=\frac{q^{\frac{1}{2}\sum_{i=1}^4 k_i^2}(q)^2_{2M_2}}
{\prod_{1\leq i<j\leq 4}(q)_{M_2+k_i+k_j}}.
\end{equation}
\end{proposition}
But even for A$_3$ we have not found a sufficiently simple expression for 
the more general
\begin{equation*}
\sum_{L\in\Integer^3_+}
\frac{q^{\frac{1}{2} LCL}S(L,k)}{(q)_{M-L}}
\end{equation*}
(with $M\in\Integer^3_+$) to suggest how (after
taking out possible factors) to iterate further.

Another important problem is to generalize \eqref{Aninf}
to higher levels. Here the observation is that 
$1/(q)_{\infty}^{n-1}$ (divided by $q^{(n-1)/24}$)
is the level-$1$ A$_{n-1}^{(1)}$ string function.
It is thus natural to ask for a generalization of \eqref{Aninf}
involving level-$N$ A$_{n-1}$ string functions~\cite{KP84}.
The simplest such functions admit the following representation
\begin{equation*}
C_k(q)=\frac{1}{(q)_{\infty}^{n-1}}\sum_{\eta}
\frac{q^{\frac{1}{2}\eta(C\otimes C^{-1})\eta}}{(q)_{\eta}},  
\end{equation*}
for $k\in\Integer^n$ such that $|k|=0$.
Here $\eta$ is a vector in the tensor-product space $\Integer^{n-1}\otimes
\Integer^{N-1}$ with entries $\eta_j^{(a)}$, $a=1,\dots,n-1$, 
$j=1,\dots,N-1$, $C\otimes C^{-1}$ denotes the tensor product of the
A$_{n-1}$ Cartan matrix and the inverse A$_{N-1}$ Cartan matrix, and
the sum is over $\eta$ such that
\begin{equation*}
\frac{\sum_{i=a}^{n-1}k_i}{N}
-\sum_{j=1}^{N-1}C_{1,j}^{-1}\eta_j^{(a)}\in\Integer,\qquad a=1,\dots,n-1. 
\end{equation*}
Multiplied by $q^{-(n^2-1)N/24(N+n)}$, $C_k$ is the string function
$C^{\Lambda}_{\mu}$ in the representation of Georgiev~\cite{Georgiev95},
with $\Lambda=N\Lambda_0$ and
$\mu=\sum_{a=1}^{n-1} (k_a-k_{a-1})\bar{\Lambda}_a$ ($k_0=k_n$).
It was conjectured in~\cite{SW98b} (equation~(9.9) with
$\ell=\lambda=\sigma=0$) that the following identity holds.
\begin{conjecture}\label{conj1}
For $n\geq 2$, $N\geq 1$ and $k\in\Integer^n$ such that $|k|=0$,
\begin{equation}\label{eqconj1}
\sum_{L\in\Integer_+^{n-1}}
q^{\frac{1}{2N} LCL}S(L,k)\sum_{\eta}
q^{\frac{1}{2}\eta(C\otimes C^{-1})\eta}\qbin{\mu+\eta}{\eta}
=q^{\frac{1}{2N}(k_1^2+\cdots+k_n^2)}C_k(q).
\end{equation}
\end{conjecture}
Here the following notation is employed on the left-hand side.
The sum over $\eta\in\Integer^{n-1}\otimes\Integer^{N-1}$ denotes a
sum such that
\begin{equation*}
\frac{L_a}{N}
-\sum_{j=1}^{N-1}C_{1,j}^{-1}\eta_j^{(a)}\in\Integer ,\qquad a=1,\dots,n-1.
\end{equation*}
The vector $\mu$ is fixed by $\eta$ through the equation
\begin{equation*}
(C\otimes I)\eta+(I\otimes C)\mu=CL\otimes e_{N-1},
\end{equation*}
where $I$ is the identity matrix and $e_j$ is the $j$th standard unit vector.
For $v,w\in\Integer^p$, $\qbins{v+w}{v}=\prod_{i=1}^p\qbins{v_i+w_i}{v_i}$.
A proof of conjecture~\ref{conj1} for $n=2$ has been given in \cite{SW98a}.

The simplest application of the previous propositions and conjecture
requires the A$_{n-1}$ form of equations \eqref{inputodd} and \eqref{superid}.
\begin{proposition}\label{propSS}
For $L\in\Integer^{n-1}$ such that $C L\in\Integer^{n-1}_+$,
\begin{equation}\label{eqconj2}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^n(nk_i-2\sigma_i)k_i}
S(L,nk-\sigma+\rho)=\delta_{L_1,0}\dots\delta_{L_{n-1},0}.
\end{equation}
\end{proposition}
A proof using crystal base theory has recently been obtained
by Schilling and Shimozono (equation.~(6.6) of~\cite{SS98}).
If we apply proposition~\ref{propAn} to this identity we find
a doubly bounded A$_{n-1}$ Euler identity,
\begin{equation*}
\sum_{|k|=0}
q^{\binom{n+1}{2}\sum_{i=1}^n k_i^2+(n+1)\sum_{i=1}^n ik_i}
\det_{1\leq i,j\leq n}\Bigl(q^{j(j-i-nk_i)}
\qbins{M+M'}{M+i-j+nk_i}\Bigr)=\qbins{M+M'}{M},
\end{equation*}
where we have traded the sum over $S_n$ for a determinant.
Using some results of \cite{GK97} this can be rewritten
in $q$-hypergeometric notation as
\begin{multline}\label{hyper}
\sum_{|k|=S} 
q^{\binom{n+1}{2}\sum_{i=1}^n k_i^2+\sum_{i=1}^n ik_i}
\prod_{1\leq i<j\leq n} (1-q^{nk_j-nk_i+j-i}) \\
\times
\prod_{i=1}^n \frac{(q)_{M+M'+i-1}}
{(q)_{M+nk_i+i-1}(q)_{M'-nk_i-i+n}}
=(-1)^{(n-1)S}q^{(n+1)\binom{S+1}{2}}\qbins{M+M'}{M+S}.
\end{multline}
This generalizes Milne's theorem~1.9 of \cite{Milne92} (or theorem~6.1 
of \cite{Milne94}), which is recovered when $M=0$ and $M'\to\infty$.
It also generalizes theorem~22 of Gessel and Krattenthaler~\cite{GK97}
which corresponds to \eqref{hyper} with $M'\to\infty$.
A proof of \eqref{hyper} based on Milne and Lilly's A$_{n-1}$ analogue
of Watson's $q$-Whipple transform~\cite{ML95} has recently been found by 
Krattenthaler~\cite{Krattenthaler98}.

If instead of \eqref{eqprop} we use \eqref{A3} it follows that
\begin{multline}\label{A3hyper}
\sum_{|k|=S} q^{10\sum_{i=1}^4 k_i^2+\sum_{i=1}^4 ik_i}
\prod_{1\leq i<j\leq 4} \frac{1-q^{4k_j-4k_i+j-i}}
{(q)_{M+4k_i+4k_j+i+j-3}} \\
=\frac{(-1)^S q^{5\binom{S+1}{2}}}{(q)_{2M+4S+4}(q)_{2M+4S+2}(q)_{M+2S}}
\end{multline}
which we have failed to recognize as a (generalization of a) known
A$_3$ $q$-hyper\-geo\-metric identity. In fact, \eqref{A3hyper}
is very misleading in that it incorrectly hints at the possibility to
sum
\begin{multline*}
\sum_{|k|=S} q^{\binom{n+1}{2}\sum_{i=1}^n k_i^2+\sum_{i=1}^n ik_i}
\prod_{1\leq i<j\leq n} (1-q^{nk_j-nk_i+j-i}) \\
\times
\prod_{1\leq i_1<\dots<i_p\leq n}\frac{1}
{(q)_{M+n k_{i_1}+\cdots+n k_{i_p}+i_1+\cdots+i_p-\binom{p+1}{2}}}.
\end{multline*}
Computer experimentations reveal that only 
for $p=1$ and $p=n-1$ (equation \eqref{hyper} with $M'\to\infty$ and
\eqref{hyper} with $M\to\infty$) and for $n=3$, $p=2$ this is the case.

Finally we note that a level-$N$ A$_{n-1}$ Euler identity is 
obtained if we sum \eqref{eqconj2} by application of \eqref{eqconj1},
\begin{equation}\label{EulernN}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^n(nk_i-2\sigma_i)k_i+\frac{1}{N}(nk_i-\sigma_i+i)^2}
C_{nk-\sigma+\rho}(q)=1.
\end{equation}

\section{Supernomial identities and beyond?}\label{secC}
So far we have given two applications of theorem~\ref{thmA2},
based on the initial condition identities \eqref{superid} and \eqref{Tid}.
Many more identities can however be derived. In ref.~\cite{SW98b}
an infinite hierarchy of A$_2$ supernomial identities was conjectured, 
of which \eqref{superid} is the first instance.
Taking this conjectured hierarchy as input to theorem~\ref{thmA2}
leads to a doubly-infinite family of A$_2$ $q$-series identities.
We will not carry out this programme in full here, but shall instead
make some intriguing observations concerning some of the
identities that may be derived. In fact, since all will be conjectural,
we shall present our speculations in a more general A$_{n-1}$ setting.

First we need the conjecture of \cite{SW98b}
(equation (9.2) with $q\to 1/q$ and $N=1$). Using the tensor-product 
notation of the previous section we define for all integers $p\geq n$,
\begin{equation*}
F_{p,L}(q)=\frac{q^{\frac{LCL}{2(p-n)}}}
{(q)_{CL}}
\sum_{\mu} q^{\frac{1}{2}\eta(C\otimes C^{-1})\eta} 
\qbin{\mu+\eta}{\mu}.
\end{equation*}
Here $L\in\Integer^{n-1}_+$, $C\otimes C^{-1}$ is the tensor product of
the A$_{n-1}$ and inverse A$_{p-n-1}$ Cartan matrices, and the sum is over 
$\mu\in \Integer^{n-1}\otimes \Integer^{p-n-1}$, with entries $\mu_j^{(a)}$,
$a=1,\dots,n-1$, $j=1,\dots,p-n-1$, such that
\begin{equation}\label{mures}
(C^{-1}\otimes I)\mu\in\Integer^{p-n-1}.
\end{equation}
The vector $\eta$ is determined by $L$ and $\mu$ through the relation
\begin{equation}\label{etamu}
\eta=L\otimes e_1 - (C^{-1}\otimes C)\mu.
\end{equation}
As special cases we have 
$$F_{n,L}(q)=\delta_{L_1,0}\dots\delta_{L_{n-1},0}$$
and 
$$F_{n+1,L}(q)=\frac{q^{\frac{1}{2}LCL}}{(q)_{CL}}.$$
\begin{conjecture}
For $n\geq 2$, $p\geq n$, $L\in\Integer^{n-1}_+$ and $k\in\Integer^n$ 
such that $|k|=0$,
\begin{equation}\label{superidp}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\frac{1}{2}\sum_{i=1}^n(pk_i-2\sigma_i)k_i}
S(L,pk-\sigma+\rho)=F_{p,L}(q).
\end{equation}
\end{conjecture}
For $p=n$ the conjecture becomes proposition~\ref{propSS} and for
$p=n+1$ a proof has been found by Schilling and Shimozono
(equation.~(6.5) of~\cite{SS98}).

Now we apply proposition~\ref{propAn} to this conjecture
and replace $p$ by $p-1$. 
Extending definition \eqref{Tdef} of $T$ to A$_{n-1}$ by
\begin{equation*}
T(L,k)=\frac{1}{(q)^2_{L_1+L_2}}\prod_{i=1}^n \qbin{L_1+L_2}{L_1+k_i}
\end{equation*}
for $|k|=0$, this gives
\begin{multline}\label{Tc}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\phi_{p,k,\sigma}}
T(M,(p-1)k-\sigma+\rho) \\
= \frac{1}{(q)_{|M|}}
\sum_{\mu} \frac{q^{\frac{1}{2}\mu(C^{-1}\otimes C)\mu}}
{(q)_{M_1-\sum_bC^{-1}_{1,b}{\mu}_1^{(b)}}
(q)_{M_2-\sum_bC^{-1}_{n-1,b}{\mu}_1^{(b)}}}\qbin{\mu+\eta}{\mu},
\end{multline}
where the sum over $\mu$ is again restricted by \eqref{mures}, $\eta$ is given
by \eqref{etamu} with $L\to(\infty^{n-1})$ and
$$
\phi_{p,k,\sigma}
=\frac{1}{2}\sum_{i=1}^n((p-1)k_i-2\sigma_i)k_i+((p-1)k_i-\sigma_i+i)^2.
$$

Next we observe that a very similar identity can be derived by 
replacing $q\to 1/q$ in \eqref{superidp}. Defining the reciprocal
A$_{n-1}$ supernomial $T'$ as
\begin{equation*}
T'(L,k) \sim S(L,k;1/q)
\end{equation*}
such that $T'(L,k)=\sum_{j\geq 0} a_j q^j$ with $a_0>0$, we find
\begin{equation}\label{Tpc}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\phi_{p,k,\sigma}}
T'(L,pk-\sigma+\rho)
=\sum_{\mu} \frac{q^{\frac{1}{2}\mu(C^{-1}\otimes C)\mu}}{(q)_{CL}} 
\qbin{\mu+\eta}{\mu}, 
\end{equation}
where \eqref{mures} and \eqref{etamu} again apply.

Still this is not the end of the story. In \cite{GK97} Gessel and
Krattenthaler generalized plane partitions to what they termed cylindric
partitions. For a special class of these cylindric partitions they derived
an expression for the generating function close to the left-hand sides
of \eqref{Tc} and \eqref{Tpc}. Supported by computer checks, we turn
this into the following conjecture. 
For $L,M\in\Integer_+$ and $k,k'\in\Integer^n$ such that $|k|=|k'|=0$, set
\begin{equation*}
T''(L,M,k,k')=\prod_{i=1}^n \qbin{M+L-k_i+k'_i}{L-k_i}.
\end{equation*}
Then, for $L,M\in\Integer_+$,
\begin{multline}\label{Tppc}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\phi_{p,k,\sigma}}
T''(L,M,pk-\sigma+\rho,(p-1)k-\sigma+\rho) \\
=\sum_{\mu} q^{\frac{1}{2}\mu(C^{-1}\otimes C)\mu} 
\qbin{M+nL-\sum_b C^{-1}_{1,b}\mu_1^{(b)}}{nL}
\qbin{\mu+\eta}{\mu}
\end{multline}
with sum over $\eta$ as in \eqref{mures} and $\eta$ given by
\begin{equation*}
\eta=nL(C^{-1}e_1\otimes e_1)-(C^{-1}\otimes C)\mu.
\end{equation*}
The left-hand side of this identity coincides with the generating
function in theorem~3 of \cite{GK97} with
$r\to n$, ${\boldsymbol \lambda}\to(L^n)$, ${\boldsymbol \mu}\to(0^n)$,
$d\to p-n$, ${\boldsymbol \alpha}\to(0^{p-n})$, ${\boldsymbol \beta}\to
(0,\dots,0,n-p+1)$, $a_i\to M$ and $b_i\to 0$.
For $p=n+1$ the above is identity (6.2) of \cite{GK97}.

We note that \eqref{Tc}, \eqref{Tpc} and \eqref{Tppc} are consistent.
Specifically, \eqref{Tc} with $M_2\to\infty$ and \eqref{Tppc} with
$L\to\infty$ coincide (identifying $M_1$ with $M$),
and \eqref{Tpc} with $L=((n-1)L_1,\dots,2L_1,L_1)$ and
\eqref{Tppc} with $M\to\infty$ coincide (identifying $L_1$ and $L$).


The three above conjectures strongly suggest the existence of a 
unifying identity of the form
\begin{multline}\label{Hcon}
\sum_{|k|=0}\sum_{\sigma\in S_n}\epsilon(\sigma)
q^{\phi_{p,k,\sigma}}
{\mathcal T}
(L,M,pk-\sigma+\rho,(p-1)k-\sigma+\rho) \\
=\sum_{\mu} q^{\frac{1}{2}\mu(C^{-1}\otimes C)\mu} 
\prod_{a=1}^{n-1}
\Bigl(\qbin{M_a+(CL)_a-\sum_b C^{-1}_{a,b}\mu_1^{(b)}}{(CL)_a}\Bigr)
\qbin{\mu+\eta}{\mu}
\end{multline}
with sum over $\mu$ such that \eqref{mures} holds,
$\eta$ given by \eqref{etamu} and $L,M\in\Integer^{n-1}_+$.
The generalized supernomial ${\mathcal T}(L,M,k,k')$ (where
$L,M\in\Integer^{n-1}$, $k,k'\in\Integer^n$ and $|k|=|k'|=0$) 
must satisfy the following
consistency conditions:
\begin{align*}
\lim_{CL\to(\infty^{n-1})} {\mathcal T}(L,(M_1,0^{n-3},M_2),k,k')&=
T((M_1,M_2),k)(q)_{M_1+M_2}\\
\lim_{M\to(\infty^{n-1})} {\mathcal T}(L,M,k,k')&=T'(L,k)  \\
{\mathcal T}(((n-1)L_1,\dots,2L_1,L_1),(M_1,0^{n-2}),k,k')&=T''(L_1,M_1,k,k').
\end{align*}
(The first condition applies when $n\geq 3$ only.)
A further restriction on the possible form of ${\mathcal T}$ is obtained by
observing that the right-hand side is, up to a factor $q^{MCL}$,
invariant under the change $q\to 1/q$, so that
\begin{equation*}
\frac{{\mathcal T}(L,M,k,k';1/q)}{{\mathcal T}(L,M,k,k';q)}
=q^{-MCL+\sum_{i=1}^n k_i k'_i}.
\end{equation*}
Despite these strong restrictions on ${\mathcal T}$ (especially when $n=3$)
we have not succeeded in
finding a closed form expression when $n\geq 3$.
For $n=2$ the third condition specifies ${\mathcal T}$ 
and \eqref{Hcon} has been proven in \cite{FLW97} using the Burge 
transform~\cite{Burge93}.

\subsection*{Acknowledgements}
I thank George Andrews and Anne Schilling for collaboration on
\cite{ASW98}, from which all of the material of sections~\ref{secsupern}
and \ref{secap} is taken. The organizers of the 42th S\'eminaire 
Lotharingien de Combinatoire are greatfully acknowledged for
a wonderful conference and for providing the opportunity to
present this work.
This work is supported by a fellowship of the Royal 
Netherlands Academy of Arts and Sciences.

\appendix

\section{Proof of proposition~\ref{propAn}}

To prove the proposition we recall the definitions \eqref{defsupernomial}
and \eqref{SAn} of the A$_{n-1}$ supernomials.
Inserting these, the left-hand side of \eqref{eqprop} corresponds to
the following multiple sum
\begin{equation}\label{multsum}
\sum_{L\in\Integer^{n-1}_+} \sum_{\nu}
\frac{q^{\frac{1}{2} LCL}}
{(q)_{M_1-L_1}(q)_{M_{n-1}-L_{n-1}}(q)_{CL}}
\prod_{a=1}^{n-1}\prod_{j=1}^a
\qbin{\nu_j^{(a+1)}-\nu_{j+1}^{(a+1)}}{\nu_j^{(a)}-\nu_{j+1}^{(a+1)}},
\end{equation}
where the sum over $\nu$ denotes a sum over the variables
$\nu_j^{(a)}$ for $1\leq a\leq n$ and $1\leq j\leq a$
such that
\begin{equation*}
\nu_j^{(n)}=L_{n-1}+L_j-L_{j-1}, \qquad j=1,\dots,n
\end{equation*}
(with $L_0=L_n=0$) and
\begin{equation*}
\sum_{j=1}^a \nu_j^{(a)}-\sum_{j=1}^{a-1}\nu_j^{(a-1)}=L_{n-1}-k_a
\end{equation*}

Instead of working with the variables $L$ and $\nu_j^{(a)}$ we find it more
convenient to introduce new variables
$\mu_j^{(a)}=\nu_j^{(a+1)}-\nu_j^{(a)}$
for $1\leq a\leq n-1$ and $1\leq j\leq a$.
If we also define the quantities
\begin{equation*}
A_j^{(a)}=k_{a+1}-k_j+\sum_{i=1}^{a-j}\mu_j^{(a-i)}
-\sum_{i=1}^{j-1}\mu_i^{(j-1)},
\end{equation*}
(again for $1\leq a\leq n-1$ and $1\leq j\leq a$) then
it is elementary to show the following string of relations:
\begin{equation*}
\nu_j^{(a)}=\begin{cases}
L_{n-1}-k_a+\mu_j^{(a-1)}+A_j^{(a-1)} & \text{for $j=1,\dots a-1$,} \\[1mm]
L_{n-1}-k_a-\sum_{i=1}^{a-1}\mu_i^{(a-1)} & \text{for $j=a$,}
\end{cases}
\end{equation*}
\begin{equation*}
\sum_{j=1}^a A_j^{(a)}=a k_{a+1}-\sum_{j=1}^a k_j,
\end{equation*}
\begin{equation*}
L_a=\sum_{j=1}^a(\mu_j^{(n-1)}+A_j^{(n-1)}-k_n),
\end{equation*}
and
\begin{equation*}
\nu_j^{(n)}-\nu_{j+1}^{(n)}=(CL)_j=\begin{cases}
\mu_j^{(n-1)}-\mu_{j+1}^{(n-1)}+A_j^{(n-1)}-A_{j+1}^{(n-1)}& j=1,\dots,n-2, \\
\mu_{n-1}^{(n-1)}+\sum_{i=1}^{n-1}\mu_i^{(n-1)}+A_{n-1}^{(n-1)}& j=n-1.
\end{cases}
\end{equation*}
Using all of these, the expression \eqref{multsum} can be rewritten as
\begin{multline*}
q^{\frac{1}{2}\sum_{i=1}^n k_i^2}
\sum_{\mu}
\frac{q^{\sum_{a=1}^{n-1}\sum_{j=1}^a\mu_j^{(a)}
(A_j^{(a)}+\sum_{i=1}^j\mu_i^{(a)})}}
{(q)_{M_1+k_n-A_1^{(n-1)}-\mu_1^{(n-1)}}
(q)_{M_{n-1}-k_n-\sum_{j=1}^{n-1}\mu_j^{(n-1)}}} \\
\times (q)^{-1}_{\mu_{n-1}^{(n-1)}
+\sum_{j=1}^{n-1}\mu_j^{(n-1)}+A_{n-1}^{(n-1)}}
\prod_{j=1}^{n-2}
(q)^{-1}_{\mu_j^{(n-1)}-\mu_{j+1}^{(n-1)}+A_j^{(n-1)}-A_{j+1}^{(n-1)}} \\
\times
\prod_{a=1}^{n-1}\Biggl(
\qbin{\mu_a^{(a)}+\sum_{j=1}^a\mu_j^{(a)}+A_a^{(a)}}{\mu_a^{(a)}}
\prod_{j=1}^{a-1}
\qbin{\mu_j^{(a)}-\mu_{j+1}^{(a)}
+A_j^{(a)}-A_{j+1}^{(a)}}{\mu_j^{(a)}}\Biggr).
\end{multline*}
Observing that $A_j^{(a)}$ depends on $\mu_k^{(b)}$ with
$1\leq b\leq a-1$ only, we can now successively sum over 
$\mu^{(n-1)},\dots,\mu^{(1)}$ by repeatedly applying
\begin{multline}\label{key}
\sum_{\mu^{(a)}}
\frac{q^{\sum_{j=1}^a\mu_j^{(a)}(A_j^{(a)}+\sum_{i=1}^j\mu_i^{(a)})}}
{(q)_{M_1+k_{a+1}-A_1^{(a)}-\mu_1^{(a)}}
(q)_{M_{n-1}-k_{a+1}-\sum_{j=1}^a\mu_j^{(a)}}} \\
\times \frac{1}{(q)_{\mu_a^{(a)}}(q)_{A_a^{(a)}+\sum_{j=1}^a\mu_j^{(a)}}
\prod_{j=1}^{a-1}(q)_{\mu_j^{(a)}}
(q)_{A_j^{(a)}-A_{j+1}^{(a)}-\mu_{j+1}^{(a)}}} \\
=\frac{(q)_{M_1+M_{n-1}}}{(q)_{M_1+k_{a+1}}(q)_{M_{n-1}-k_{a+1}}
(q)_{M_1+k_{a+1}-A_1^{(a)}}(q)_{M_{n-1}-k_{a+1}+A_a^{(a)}}
\prod_{j=1}^{a-1}(q)_{A_j^{(a)}-A_{j+1}^{(a)}}}
\end{multline}
and by rewriting the right-hand side as
\begin{multline}\label{rew}
\frac{(q)_{M_1+M_{n-1}}}{(q)_{M_1+k_{a+1}}(q)_{M_{n-1}-k_{a+1}}
(q)_{M_1+k_a-A_1^{(a-1)}-\mu_1^{(a-1)}}
(q)_{M_{n-1}-k_a-\sum_{j=1}^{a-1}\mu_j^{(a-1)}}} \\
\times
\frac{1}{(q)_{A_{a-1}^{(a-1)}+\mu_{a-1}^{(a-1)}+\sum_{j=1}^{a-1}\mu_j^{(a-1)}}}
\prod_{j=1}^{a-2}\frac{1}{(q)_{
A_j^{(a-1)}-A_{j+1}^{(a-1)}+\mu_j^{(a-1)}-\mu_{j+1}^{(a-1)}}}
\end{multline}
using
\begin{equation*}
A_j^{(a)}=\begin{cases}
k_{a+1}-k_a+A_j^{(a-1)}+\mu_j^{(a-1)} & \text{for $j=1,\dots,a-1$}, \\
k_{a+1}-k_a-\sum_{i=1}^{a-1}\mu_i^{(a-1)} & \text{for $j=a$}.
\end{cases}
\end{equation*}
The final sum over $\mu_1$ then yields the right-hand side of
\eqref{eqprop}.
The proof of the key identity \eqref{key} follows by successively
summing over $\mu^{(a)}_a,\dots,\mu^{(a)}_1$ using the 
$q$-Chu--Vandermonde sum (equation~(3.3.10) of \cite{Andrews76}),
\begin{equation*}
\sum_{j\geq 0}q^{j(j+a)}\qbin{b}{j}\qbin{a+c}{j+a}=\qbin{a+b+c}{c}.
\end{equation*}

Of course, in a complete proof, the statements that one 
can successively sum over the $\mu^{(n-1)},\dots,\mu^{(1)}$ 
using \eqref{key} and \eqref{rew}, and that
\eqref{key} follows from successively summing over $\mu^{(a)}_a,
\dots \mu^{(a)}_1$ require a proof by induction.
This takes (a lot of) space, but requires
no intellectual effort other than that of avoiding typos.
It is therefore omitted here.

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\end{thebibliography}

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