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%% define your title in the usual way
\title{On the \texorpdfstring{$\mathrm{A}_2$}{A2} Andrews--Schilling--Warnaar identities}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author %[Optional shorter names]
{Shashank Kanade\thanks{
		\href{mailto:shashank.kanade@du.edu}{shashank.kanade@du.edu}. 
		S.\ Kanade is supported by Collaboration Grant for Mathematicians \#636937 awarded by the Simons Foundation.}\addressmark{1}
%	\and 
%	you\addressmark{2}
}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Denver, USA 
%	\addressmark{2}School of Something, University of Something Else, Somewhere
}

%% put the date of submission here
\received{11/14/2022}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
	In a groundbreaking work, Andrews--Schilling--Warnaar invented an $\mathrm{A}_2$ generalization of the $\mathrm{A}_1$ Bailey machinery and discovered many identities related to the principal characters of standard modules for $\widehat{\mathfrak{sl}}_3$, or equivalently, for the vertex operator algebra $\mathscr{W}_3(3,p')$. Jointly with Russell, we have given conjectures for completing this set of identities and proved these conjectures for small values of $p'$. In another direction, character of $\mathscr{W}_r(p,p')$ has been related to an appropriate limit of certain $\mathfrak{sl}_r$ coloured Jones polynomials of torus knots $\mathrm{T}(p,p')$ under some restrictions on $r,p,p'$. This note summarizes these developments.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

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\keywords{Rogers--Ramanujan-type identities, Affine Lie algebras, Vertex operator algebras, Jones polynomials}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above



\section{Introduction}

\subsection{Rogers--Ramanujan identities and their origins}
Let us begin by recalling the celebrated pair of Rogers--Ramanujan identities:
\begin{align}
	\sum_{n\geq 0}\dfrac{q^{n^2}}{(q)_n} &= \dfrac{1}{\theta(q;q^5)},\quad 
	\sum_{n\geq 0}\dfrac{q^{n^2+n}}{(q)_n} = \dfrac{1}{\theta(q^2;q^5)}. \label{eqn:RR}
\end{align}
where we use the standard notations for the Pochhammer symbols $(q)_n$ (and more generally $(a;q)_n$) 
for $n\in\ZZ_{\geq 0}\cup\{\infty\}$ as found in \cite{And-book} and we use the notation:
\begin{align}
	\theta(a_1,a_2,\cdots,a_k;q)=\prod_{1\leq i\leq k} (a_i;q)_\infty(q/a_i;q)_\infty.
\end{align}

Rogers--Ramanujan identities, their various generalizations, and analogues appear in many branches of mathematics. Most importantly for us, they appear in the study of affine Lie algebras and vertex operator algebras (henceforth, VOAs). Secondly, they also appear in relation to the coloured Jones polynomials of knots and links.

The identities \eqref{eqn:RR} were first proved completely representation-theoretically by Lepowsky and Wilson \cite{LepWil-1, LepWil-2, LepWil-3, LepWil-4} by considering 
standard (i.e., integrable, highest-weight) level $3$ modules for the affine Lie algebra $\widehat{\la{sl}}_2$.
Identities \eqref{eqn:RR} also arise by considering the representation theory of Virasoro VOA $\vir(2,5)$.
% In both these cases, the identities are proved by calculating characters of relevant modules in two very different ways. The sums arise by explicitly constructing bases, and the products arise by utilizing character formulas. 
% The construction of explicit bases is usually achieved by using vertex algebraic relations to cut down a hugely redundant spanning set. 
% On the other hand, the character formulas are in turn a manifestation of various resolutions of the relevant irreducible modules in terms of (generalized) Verma modules.
In general, representation theory of VOAs is an enormously rich source of Rogers--Ramanujan-type identities having the ``sum=product'' shape of \eqref{eqn:RR}.

On the knot-theoretical side, the second Rogers--Ramanujan identity in \eqref{eqn:RR} seems to have been proved first by Armond and Dasbach \cite{ArmDas-RR} by considering tails of Jones polynomials for torus knots $\torus(2,5)$ coloured with finite dimensional irreducible $\la{sl}_2$ modules. Here, the product side is a result of using the Rosso--Jones formula \cite{RosJon-torus} as explained by Morton, \cite{Mor-jones}. The sum side is calculated using a combinatorial walk model \cite{Arm-walks}. More generally, Armond and Dasbach proved one Andrews--Gordon identity at each modulus $2k+1$ by considering the torus knot $\torus(2,2k+1)$.


\subsection{Bailey machinery}

In the world of $q$-series, Bailey machinery provides a highly productive mechanism to prove Rogers--Ramanujan-type identities. 
By Bailey machinery, we mean the framework that takes as input a Bailey pair (as defined in \cite{And-qserbook}, for instance) and modifies this pair iteratively (using the Bailey lemma, the Bailey lattice, or any other suitable transformations). The modified pairs then lead to requisite identities by taking appropriate limits. 

From the point of view of representation theory and knot theory, this mechanism is especially well suited to handle cases that are closely related to the Lie algebra $\la{sl}_2$. 
In particular, it produces identities related to principal characters (see \eqref{eqn:principal} below) for standard modules for affine Lie algebras $\widehat{\la{sl}_2}=\typeA_1^{(1)}$ and $\typeA_2^{(2)}$, rational Virasoro algebras $\vir(p,q)$, etc. 
This machinery has also been successfully used (see \cite{HikLov-torus}, for instance) in understanding Jones polynomials
of links coloured with irreducible, finite-dimensional $\la{sl}_2$ modules.

For these and some more reasons explained in \cite{ASW}, we refer to the classical Bailey machinery by the designation $\la{sl}_2$ or $\typeA_1$.

\subsection{Andrews--Schilling--Warnaar's \texorpdfstring{$\typeA_2$}{A\_2} Bailey machinery}
In \cite{ASW}, Andrews--Schilling--Warnaar invented  an $\typeA_2$  generalization of the $\typeA_1$ Bailey machinery.
Specifically, they defined a notion of the $\typeA_2$ Bailey pair, found the unit $\typeA_2$ Bailey pair, and gave a transformation law for $\typeA_2$ Bailey pairs. 
Note that this transformation law is not yet as general as the one for $\typeA_1$ Bailey pairs.

In \cite{ASW}, the authors then used this mechanism to discover ``sum=product'' identities related to 
principal characters of certain standard modules at every level $\ell\in\ZZ_{>0}$ for the affine Lie algebra $\widehat{\la{sl}_3}$. At levels $\ell\in\ZZ_{>0}$ coprime to $3$, these principal characters also arise by considering characters of modules for the rational VOA $\sW_3(3,\ell+3)$ (see \cite{ASW, FodWel}). 

The actual number of possible identities at level $\ell\in\ZZ_{>0}$ grows as a quadratic in $\ell$, but the number of identities found in \cite{ASW} grows linearly in $\ell$. Consequently, a major subset of identities were yet to be found. 


Our recent work \cite{KanRus-cylindric} gives precise conjectures for all these missing identities. We are further able to prove our conjectures at levels $3$ and $7$. 
A further family of results valid for all levels divisible by $3$ is detailed in \cref{sec:completing} below.

% Let us now summarize the set of identities found in \cite{ASW}. Highest weights for standard modules of $\la{sl}_3$ at level $\ell\in\ZZ_{>0}$ are given by $\lambda=c_0\Lambda_0+c_1\Lambda_1+c_2\Lambda_2$ with $c_i\in\ZZ_{>0}$ and $c_0+c_1+c_2=\ell$. Denote the standard module with highest weight $\lambda$ by $L(\lambda)$.
% In \cite{ASW}, ``sum=product'' identities related to principal characters of \sk{add}.
% This means that except at small levels such as \sk{$\ell=2$}, the set of identities given in \cite{ASW} is incomplete. 


\subsection{Certain \texorpdfstring{$\la{sl}_r$}{sl\_r} coloured Jones polynomials of torus knots}

Morton's work \cite{Mor-jones} effectively relates characters of $\vir(p,q)$ VOA to 
$\la{sl}_2$ coloured Jones polynomials of torus knots $\torus(p,q)$ where $2\leq p<q$ are coprime. Now, $\vir(p,q)$ VOA is the $r=2$ member in the family of principal $\sW_r$ algebras of type $\typeA$. It is thus natural to investigate whether characters of $\sW_r(p,q)$ algebras are analogously related to torus knots. In \cite{Kan-slrtorus} we show that limits of certain $\la{sl}_r$ coloured invariants of torus knots $\torus(p,q)$ similarly give rise to characters of $\sW_r(p,q)$ VOAs. For precise conditions and statements, see in \cref{sec:torus}. 
Importantly, our hope is that generalizing the work of Armond and Dasbach \cite{ArmDas-RR} to $\la{sl}_3$ coloured invariants of torus knots $\torus(3,p+3)$ will put the Andrews--Schilling--Warnaar identities in a very different light.


\subsection{Recent and related developments}
Recently, Andrews--Schilling--Warnaar's $\typeA_2$ identities have gained much attention. Especially,
the interested reader may look at \cite{CorDouUnc}, \cite{War-A2further}, \cite{Tsu}, \cite{Unc-1113} and \cite{War-A2proof}.
%Corteel--Dousse--Uncu's work for level $5$ \cite{CorDouUnc}, Warnaar's work on proving new identities at all levels \cite{War}, and Tsuchioka's work for level $3$ \cite{Tsu}. 
%Uncu has proved our conjectures in levels $8$ and $10$ in \cite{Unc-1113}.
%Regrettably, we shall not discuss these here due to the limitations of space.

\section{Foundations}
In this section, we begin by collecting the basic setup required for this note.

\subsection{Notation regarding the Lie algebras \texorpdfstring{$\la{sl}_r$}{sl\_r}} 
% The Lie algebra $\la{sl}_r$ is defined to be the usual Lie algebra of trace-zero $r\times r$ matrices with
% complex entries. 
% We let $\la{h}$ be the Cartan subalgebra of diagonal matrices and 
As usual, we let the simple roots of $\la{sl}_r$ be $\alpha_1,\dots,\alpha_{r-1}$.
The set of positive roots will be denoted by $\Phi_r^+$ and the set of all roots will be $\Phi_r$.
The fundamental weights will be denoted by $\Lambda_1,\dots,\Lambda_{r-1}$.
$Q_r$ and $P_r$ will denote the root and the weight lattices, respectively. The Weyl group will be denoted by $\mathfrak{S}_r$. The lattices $Q_r\subsetneq P_r$ come equipped with a symmetric, positive definite and 
$\mathfrak{S}_r$-invariant bilinear form $(\cdot,\cdot)$ such that $(\alpha_i,\alpha_i)=2$ for all $0\leq i\leq r-1$. We will let $\|\cdot\|$ be the norm corresponding to $(\cdot,\cdot)$.
The Weyl vector will be denoted by $\delta$. 
Recall that $w\mapsto (-1)^{\ell(w)}$ is the sign representation of $\mathfrak{S}_r$ where $\ell(\cdot)$ is the standard length function on $\mathfrak{S}_r$.

\subsection{Affine Lie algebras \texorpdfstring{$\widehat{\la{sl}_r}$}{sl\_r\^{}} }
\label{sec:aff}
For $\widehat{\la{sl}_r}$, we will reuse some notation from $\la{sl}_r$ case.
We shall provide adequate clarification in the cases of potential confusion.

For $0\leq i\leq r-1$, let $\alpha_i$ be the simple roots and let $\Lambda_i$ be the fundamental weights of the affine Lie algebra $\widehat{\la{sl}_r}$.
Fix $\ell\in \ZZ_{\geq 0}$ (called the level) and let $\lambda=c_0\Lambda_0+\cdots+c_{r-1}\Lambda_{r-1}$, where $c_i\in\ZZ_{\geq 0}$
and $c_0+\cdots+c_{r-1}=\ell$.
By $L(\lambda)$ we denote the irreducible highest-weight $\widehat{\la{sl}_r}$ module with highest-weight $\lambda$.
The character of this module is denoted by $\ch(L(\lambda))$ and it belongs to:
\begin{align}
\ch(L(\lambda)) \in  e^{\lambda}\ZZ[[e^{-\alpha_0},\cdots,e^{-\alpha_{r-1}}]].
\end{align}
Here, $e^\lambda$ and $e^{\alpha_i}$ are formal symbols.
We define the \emph{principally specialized character} and the \emph{principal character}\footnote{Note that the first equality in \eqref{eqn:principal} as a definition of principal characters is only valid for the affine Lie algebras of type $X_r^{(t)}$ with $X=\mathrm{A}, \mathrm{D}, \mathrm{E}$ and $t\in \{1,2,3\}$.}
of $L(\lambda)$ to be respectively --
\begin{align}
\chi(L(\lambda))&= \left(e^{-\lambda}\ch(L(\lambda))\right)\vert_{e^{-\alpha_0}\mapsto q,\dots,e^{-\alpha_{r-1}}\mapsto q },\\
\chi(\Omega(\lambda))&= \dfrac{\chi(L(\lambda))}{\chi(L(\Lambda_0))}
=\dfrac{(q)_\infty}{(q^{r};q^{r})_\infty} \chi(L(\lambda))
%\,\,
%\mathrm{for}
%\,\,\widehat{\la{sl}_r}
%\,\,\mathrm{modules}
\label{eqn:principal}
\end{align}
Due to the Dynkin diagram symmetries in the $\widehat{\la{sl}_r}$ case, we also have that:
\begin{align}
	\chi(\Omega(\lambda))=\chi(\Omega(\sigma\lambda)),
	\label{eqn:charsym}
\end{align}
where $\sigma$ is any dihedral permutation of $c_0,c_1,\dots,c_{r-1}.$
In the case of $\widehat{\la{sl}_3}$, we have the following for $c_0,c_1,c_2\in\ZZ_{\geq 0}$, $c_0+c_1+c_2=\ell$, $m=\ell+3$ (see \cite{FodWel, War}, etc.):
\begin{align}
\chi(\Omega(c_0\Lambda_0+c_1\Lambda_1+c_2\Lambda_2))
=\dfrac{(q^{m};q^{m})^2_\infty}{(q)_\infty^2}
\theta(q^{c_0+1};q^{m})
\theta(q^{c_1+1};q^{m})
\theta(q^{c_2+1};q^{m}).
\label{eqn:sl3principal}
\end{align}

\subsection{Principal \texorpdfstring{$\sW_r$}{W\_r} algebras}
Let $r\leq p,p'$ be a pair of coprime integers.
Now, principal $\sW_r(p,p')$ algebras are certain simple vertex operator algebras (VOAs)
that can be obtained as the quantum Hamiltonian reductions of affine VOAs based on
$\widehat{\la{sl}_r}$ at levels {$\frac{p}{p'}-r$}. For deep and foundational properties of these VOAs, see \cite{Ara-c2cofw, Ara-prinwrat}.

%Due to fundamental results of Arakawa
%\cite{Ara-c2cofw, Ara-prinwrat} these VOAs satisfy important properties such as
%$C_2$-cofiniteness and rationality. Thus, the characters of these VOAs and their modules satisfy modular invariance properties \cite{Zhu-modinv, DonLinNg}.

From \cite{Miz} and \cite{FodWel}, we recall characters of $\sW_r(p,p')$ VOAs and their modules.

\begin{enumerate}
	\item In the notation of \cite{FodWel}, the character of the VOA $\sW_r(p,p')$ is
	$\chi^{r,p,p'}_{0,0}$ and up to a pure power of $q$ it equals the following normalized character:
	\begin{align}
		\overline{\chi}^{r,p,p'}_{0,0}=(q)_\infty^{1-r}
		\sum_{\alpha\in Q_r}\sum_{\sigma\in\mathfrak{S}_r}
		(-1)^{\ell(\sigma)}q^{\frac{1}{2}pp'\|\alpha\|^2
		-p'(\alpha,\delta)+p(\alpha,\sigma(\delta))-(\delta,\sigma(\delta)-\delta)}.
	\end{align}
	\item When $p=r$, inequivalent irreducible modules of $\sW_r(r,p')$ are enumerated
	by $\la{sl}_r$ weights $\zeta=c_1\Lambda_1+\cdots+c_{r-1}\Lambda_{r-1}\in P_r$ such that 
	$c_1+\cdots+c_{r-1}\leq p'-r$.
%	by compositions (see \cref{sec:cylindric}) $c=(c_0,c_1,\cdots, c_{r-1})$ of $p'-r$. In this case, we let 
%	$\zeta=c_1\Lambda_1+\cdots +c_{r-1}\Lambda_{r-1}$ and consider $\zeta$ to be a weight of $\la{sl}_r$. 
	The character of the corresponding irreducible $\sW_r(r,p')$ module
	is denoted by $\chi^{r,r,p'}_{0,\zeta}$ \cite{FodWel} and up to a pure power of $q$ it equals the normalized character:
	\begin{align}
		\overline{\chi}^{r,r,p'}_{0,\zeta}
		=(q)_\infty^{1-r}
		\sum_{\alpha\in Q_r}\sum_{\sigma\in\mathfrak{S}_r}
		(-1)^{\ell(\sigma)}q^{\frac{1}{2}rp'\|\alpha\|^2
			-p'(\alpha,\delta)+r(\alpha,\sigma(\zeta+\delta))-(\delta,\sigma(\zeta+\delta)-\zeta-\delta)}.
	\end{align}
\end{enumerate}

\subsection{Cylindric partitions}
\label{sec:cylindric}
We let $r\geq 1$ and $\ell\geq 1$.

A \emph{composition} of $\ell$ of length $r$ is an (ordered) sequence of non-negative integers $c=(c_0,c_1,\dots,c_{r-1})$ that adds up to $\ell$. 
%It would have been natural to denote the length of $c$ by $\ell$, however, 
Note crucially that $\ell$ stands for the level and $r$ for rank as in \cref{sec:aff}.

A \emph{cylindric partition} of profile $c$ is defined to be an (ordered) sequence of partitions $\Lambda=(\lambda^{(0)},\dots,\lambda^{(r-1)})$ such that:
\begin{enumerate}
\item Each $\lambda^{(j)}=\lambda_1^{(j)}+\lambda_2^{(j)}+\cdots$ is written in a non-increasing order and is assumed to continue indefinitely by appending an infinite string of zeroes. 
\item For all $0\leq i\leq r-2$ and all $j$, $\lambda^{(i)}_j\geq \lambda^{(i+1)}_{j+c_{i+1}}$ and 
$\lambda^{(r-1)}_j\geq \lambda^{(0)}_{j+c_{0}}$.
\end{enumerate}

Given a composition $c=(c_0,\dots,c_{r-1})$ of $\ell$, define its set of non-negative indices by
$$I_c=\{0\leq i\leq r-1\,\mid\,c_i>0\}.$$
For $\emptyset \subsetneq J\subseteq I_c$, define $c(J)=(c_0(J),\dots,c_{r-1}(J))$ as 
\begin{align}
c_i(J)=
\begin{cases}
c_i-1 & i\in J, i-1\not\in J\\
c_i+1 & i\not\in J, i-1\in J\\
c_i &\mathrm{otherwise},
\end{cases}
\end{align}
where we think of the indices $0,\dots, r-1$ (and consequently elements of the sets $I_c$ and $J$) as elements of the cyclic group $\ZZ_r$.
Note that if $c$ is a composition of $\ell$, then so is $c(J)$.

Let $\cC_c$ be the collection of all cylindric partitions of profile $c$ and define:
\begin{align}
H_c(z,q)=\dfrac{(zq)_\infty}{(q)_\infty}\sum_{\Lambda\in\cC_c} z^{\mathrm{max}(\Lambda)}q^{\mathrm{wt}(\Lambda)}.
\end{align}

Now, the following properties of cylindric partitions and the functions $H_c$ are crucial.

\begin{enumerate}
\item The functions $H_c$ are governed by the Corteel--Welsh recurrence \cite{CorWel}.
For a fixed $r$ and $\ell$, the functions $H_c$ as $c$ varies over length $r$ compositions of $\ell$ are the unique solutions in $\ZZ[[z,q]]$ to the following finite system of recurrences and initial conditions \footnote{The referee has kindly pointed out that $H_c(z,0)=1$ from \eqref{eqn:CWinit} is redundant.}:
\begin{align}
H_c(z,q)&=\sum_{\emptyset\subsetneq J\subseteq I_c} (-1)^{|J|-1}(zq)_{|J|-1}H_{c(J)}(zq^{|J|},q),
\label{eqn:CWrec}\\
\quad H_c(0,q)&=(q)_\infty^{-1},\quad H_c(z,0)=1\label{eqn:CWinit}.
\end{align}
\item $H_c$ appear as characters as follows (\cite{Bor, Tin-crystal, GesKra, FodWel}):
\begin{align}
{(q)_\infty} H_{(c_0,\dots,c_{r-1})}(1,q)=\chi(\Omega(c_0\Lambda_0+\cdots+c_{r-1}\Lambda_{r-1}))
=\overline{\chi}^{r,r,\ell+r}_{0,\,c_1\Lambda_1+\cdots+c_{r-1}\Lambda_{r-1}}.
\label{eqn:Hchar}
\end{align}
\item $H_c$ functions have certain symmetries. The first follows from the definition of cylindric partitions.
The second follows from the previous relation to principal characters, but can also be deduced from the properties of cylindric partitions.
\begin{align}
H_{(c_0,\cdots,c_{r-1})}(z,q)&=H_{(c_1,c_2,\dots,c_{r-1},c_0)}(z,q), \label{eqn:H_cyclic}\\
H_{(c_0,\cdots,c_{r-1})}(1,q)&=H_{(c_{r-1},c_{r-2},\dots,c_1,c_0)}(1,q). \label{eqn:H_reflect}
\end{align}
Due to \eqref{eqn:H_cyclic}, we cyclically permute $c$ and assume that $c_0$ is the largest part.
\end{enumerate}


\section{Completing the \texorpdfstring{$\typeA_2$}{A\_2} identities from \cite{ASW}}
\label{sec:completing}
In this section, we will now solely focus on the $r=3$ case. 
That is, we will only consider principal characters of standard $\widehat{\la{sl}_3}$ modules of level $\ell\in\ZZ_{>0}$. 
Equivalently, these are the characters $\overline{\chi}^{3,3,3+\ell}_{0,\zeta}$ for irreducible modules for the VOA $\sW_3(3,3+\ell)$ whenever $\ell$ is coprime to $3$, however, we will stick with the former description.



In \cite{ASW}, using an appropriate generalization of the usual Bailey machinery to the $\typeA_2$ root system, ``sum=product'' identities involving certain principal characters were found. 
For identities found in \cite{ASW}, the triple $(c_0,c_1,c_2)$ appearing in \eqref{eqn:sl3principal} satisfies:
\begin{align}
&(c_{\sigma(0)}+1,c_{\sigma(1)}+1,c_{\sigma(2)}+1) \notag\\
&\in \{ 
(i,i,3k-2i+s), 
% (i,i,3k-2i\pm 1), 
(1,k-t,2k+t-1\pm 1)
% (1,k,2k-1\pm 1), (1,k-1,2k\pm 1)
\,\mid\,
k\geq 2;\, 1\leq i\leq k;\, t=0,1;\, s=-1,0,1
 \}.
\end{align}
% % (recall \eqref{eqn:charsym}).
% \begin{align*}
% % \chi(\Omega((\ell-i)\Lambda_{\sigma(0)}+i\Lambda_{\sigma(1)}+i\Lambda_{\sigma(2)})),&\quad  \ell>0,\,\, i\geq 0,\,\, \ell-i\geq 0,\,\, \ell-i+1\geq i,\\
% \chi(\Omega(  i\Lambda_{\sigma(0)}+i\Lambda_{\sigma(1)}+ (3k-3\pm 1 -2i)\Lambda_{\sigma(2)})),
% &\quad 
% k\geq 2,\,\,0\leq i\leq k-1,
% \\   
% % \ell-i\geq i\geq 0, \ell>0,\\
% \chi(\Omega((2k-1\pm 1)\Lambda_{\sigma(0)}+(k-2)\Lambda_{\sigma(1)})),&\quad k\geq 2,\\
% \chi(\Omega((2k-2\pm 1)\Lambda_{\sigma(0)}+(k-1)\Lambda_{\sigma(1)})),&\quad 2k-2\pm 1\geq 0,
% \end{align*}
 where $\sigma$ is some permutation of the indices $0,1,2$ (recall \eqref{eqn:charsym}). 
This implies that a majority of identities at every level were yet to be found. In \cite{KanRus-cylindric}, jointly with Russell, we were able to conjecture all of the missing identities and prove our conjectures at levels $3$ and $7$.
In fact, our conjectures pertain to the two-variable generating functions $H_c(z,q)$. Upon setting $z=1$, one gets conjectures for the principal characters of standard modules (see \eqref{eqn:Hchar}).

\subsection{The arrangement of modules}
Fix $\ell>0$ and let 
$m=\ell+3$ and let $k$ be such that
$m=\ell+3\in 3k+\{-1,0,1\}.$

We now arrange length $3$ compositions of $\ell$ in a triangular array.
Suppose that $c=(c_0,c_1,c_2)$ is a composition of $\ell$. 
Due to the cyclic symmetry \eqref{eqn:H_cyclic}, we assume that $c_0$ is the largest part of $c$.
Let $(\lambda_0, \lambda_1, \lambda_2)$ be the sequence obtained by permuting $(c_0,c_1,c_2)$ in a weakly decreasing order. Now place the composition $c$ in row $\lambda_1$ and column $\lambda_2$. This gives rise to a triangular arrangement of compositions.  Lastly, we draw a horizontal line after row $k-1$. 
For example, here is the arrangement for $\ell=11,m=14,k=5$:
\begin{align}
	{\arraycolsep=0.3cm
			\def\arraystretch{1.2}
			\begin{array}{llllll}
				\mathrm{row}\, 0: & 11,0,0 		\\
				\mathrm{row}\, 1: &	10,1,0/10,0,1 & 9,1,1    \\
				\mathrm{row}\, 2: &	9,2,0/9,0,2  & 8,2,1/8,1,2  & 7,2,2\\
				\mathrm{row}\, 3: &	8,3,0/8,0,3 & 7,3,1/7,1,3  & 6,3,2/6,2,3 & 5,3,3\\
				\mathrm{row}\, 4: &	7,4,0/7,0,4 & 6,4,1/6,1,4    & 5,4,2/5,2,4 & 4,4,3\\
				\hline
				\mathrm{row}\, 5: &	6,5,0/6,0,5  & 5,5,1
				\end{array}
		}
	\label{eqn:level11}
\end{align}
%For example, here is the arrangement for $\ell=14,m=17,k=6$:
%\begin{align}
%	{\arraycolsep=0.3cm
%		\def\arraystretch{1.2}
%		\begin{array}{llllll}
%		\mathrm{row}\, 0: & 14,0,0 		\\
%		\mathrm{row}\, 1: &	13,1,0/14,0,1 & 12,1,1    \\
%		\mathrm{row}\, 2: &	12,2,0/14,0,2 & 11,2,1/11,1,2  & 10,2,2\\
%		\mathrm{row}\, 3: &	11,3,0/13,0,3 & 10,3,1/10,1,3  & 9,3,2/9,2,3 & 8,3,3\\
%		\mathrm{row}\, 4: &	10,4,0/12,0,4 & 9,4,1/9,1,4    & 8,4,2/8,2,4 & 7,4,3/7,3,4 & 6,4,4\\
%		\mathrm{row}\, 5: &	9,5,0/11,0,5  & 8,5,1/8,1,5    & 7,5,2/7,2,5 & 6,5,3/6,3,5 & 5,5,4\\
%			\hline
%		\mathrm{row}\, 6: &	8,6,0/8,0,6   & 7,6,1/7,1,6    & 6,6,2\\
%		\mathrm{row}\, 7: &	7,7,0
%		\end{array}
%	}
%	\label{eqn:level14}
%\end{align}

\begin{thm} \cite[Thm 5.1]{KanRus-cylindric}
	\label{thm:belowtheline}
	Fix $\ell>0$. Suppose that expressions for all $H_c(z,q)$ are known where $c$ runs over length $3$ compositions of $\ell$ that lie above the line, i.e., $\lambda_1\leq k-1$. Then, the expressions for
	$H_c(z,q)$ where $c$ is a length $3$ composition of $\ell$ that lies below the line, i.e., $\lambda_1\geq k$ are determined by the Corteel--Welsh recursions 
	\eqref{eqn:CWrec}.
\end{thm}

This theorem implies that it is enough to produce conjectures for the compositions that lie above the line.

\subsection{Conjectures for compositions above the line}
Let us continue to let $\ell, m, k$ as in the previous subsection. 
For $0\leq j\leq k-1$, define:
{\small
\begin{align}
	\mathrm{e_j}&=[\overbrace{0,\dots,0}^{j},1,\dots,1]\in\ZZ^{k-1},\quad
	\mathrm{e_{-1}}=[2,1,\dots,1]\in \ZZ^{k-1}.
\end{align}
For $\rho, \sigma\in \ZZ^{k-1}$, define (with the convention that $(q)_n^{-1}=0$ if $n<0$):
\begin{align}
	&S_{3k-1}(\rho\mid \sigma)
	= \sum_{r,s\in\ZZ^{k-1}_{\geq 0}}
	\dfrac{z^{r_1}q^{\left(\sum_{1\leq i \leq k-1} r_i^2-r_is_i+s_i^2\right) + \rho\cdot r + \sigma\cdot s}}
	{\prod_{1\leq i\leq k-2}(q)_{r_i-r_{i+1}}(q)_{s_i-s_{i+1}}}
	\cdot\dfrac{q^{2r_{k-1}s_{k-1}}}{(q)_{r_{k-1}}(q)_{s_{k-1}}(q)_{r_{k-1}+s_{k-1}+1}}
	%\label{eqn:S3k-1}
	\nonumber
	\\
	&S_{3k+1}(\rho\mid \sigma)
	= \sum_{r,s\in\ZZ^{k-1}_{\geq 0}}
	\dfrac{z^{r_1}q^{\left(\sum_{1\leq i \leq k-1} r_i^2-r_is_i+s_i^2\right)+\rho\cdot r + \sigma\cdot s}}
	{\prod_{1\leq i\leq k-2}(q)_{r_i-r_{i+1}}(q)_{s_i-s_{i+1}}}
	\cdot\dfrac{1}{(q)_{r_{k-1}}(q)_{s_{k-1}}(q)_{r_{k-1}+s_{k-1}+1}}
	\label{eqn:S}
	\\
	&S_{3k}(\rho\mid \sigma)
	= \sum_{r,s\in\ZZ^{k-1}_{\geq 0}}
	\dfrac{z^{r_1}q^{\left(\sum_{1\leq i \leq k-1} r_i^2-r_is_i+s_i^2\right)+\rho\cdot r + \sigma\cdot s}}
	{\prod_{1\leq i\leq k-2}(q)_{r_i-r_{i+1}}(q)_{s_i-s_{i+1}}}
	\cdot\dfrac{1}{(q)_{r_{k-1}+s_{k-1}}(q)_{r_{k-1}+s_{k-1}+1}}\qbin{r_{k-1}+s_{k-1}}{r_{k-1}}_{q^3}
	\nonumber
	%\label{eqn:S3k+1}
\end{align}
}

\begin{conj} \cite[Conj.\ 5.1]{KanRus-cylindric}
	\label{conj:KRASWconj}
	Fix $\ell$ and let $c=(c_0,c_1,c_2)$ be a composition above the line (i.e., $\lambda_1\leq k-1$).
	We then have:
	\begin{align}
		H_{(c_0,c_1,c_2)}(z,q)
		&=
		\begin{cases}
			S_{\ell+3}(\mathrm{e}_{c_1}\mid \mathrm{e}_{c_2})
			- qS_{\ell+3}(\mathrm{e}_{c_1-1}\mid \mathrm{e}_{c_2-1}) & c_1,c_2>0\\
			S_{\ell+3}(\mathrm{e}_{c_1}\mid \mathrm{e}_0) & c_2=0\\
			S_{\ell+3}(\mathrm{e}_0\mid \mathrm{e}_{c_2})
			-q(1-z)S_{\ell+3}(\mathrm{e}_{-1}\mid \mathrm{e}_{c_2-1}) & c_1=0, c_2\neq 0
		\end{cases}
		\label{eqn:aboveline}
	\end{align}
\end{conj}	
\begin{thm} \cite[Thm 5.3]{KanRus-cylindric}
The conjectures above satisfy the initial conditions \eqref{eqn:CWinit}, i.e., 
$H_c(0,q)=(q)_\infty^{-1}$ and $H_c(z,0)=1$.
\end{thm}	
\subsection{Proving the conjectures at low levels and other results}

We were able to prove our conjectures in some specific (small) levels:
\begin{thm} \cite{KanRus-cylindric}
	\label{thm:lowlevels}
	\cref{conj:KRASWconj} is true for $\ell\in\{2,3,4,5,7\}$ (the cases $\ell\in\{3, 7\}$ are new).
\end{thm}
\begin{proof}[Idea of proof]
Broadly, the proof has two steps. First we find certain fundamental (recurrence) relations satisfied by the $S$ functions defined in \eqref{eqn:S}. 
This step is straight-forward (see also \cite{Che}).
The next step is to show that the Corteel--Welsh recursions \eqref{eqn:CWrec} are a consequence of these fundamental relations. This step requires a non-trivial amount of computer assistance, especially in levels $5$ and $7$.
\end{proof}

Recently, Uncu has extended the theorem above to include two further cases.
\begin{thm} \cite{Unc-1113}
	\label{thm:lowlevels-uncu}
	\cref{conj:KRASWconj} is true for $\ell\in\{8, 10\}$.
\end{thm}

In an important development, S.\ O.\ Warnaar has given a far reaching enhancement of the $\mathrm{A}_2$ Bailey machinery of \cite{ASW} and proved the following.
\begin{thm} \cite{War-A2proof}
	\cref{conj:KRASWconj} is true with $z=1$.
\end{thm}


In the case of $3\mid \ell$ and with $z=1$, we are in fact able to improve \cref{thm:belowtheline}. 
%Due to \eqref{eqn:Hsym1} and \eqref{eqn:Hsymz}, we have that $H_c(1,q)=H_{\widetilde{c}}(1,q)$ where $\widetilde{c}$ is any permutation of $c$.
In this case, 
%conditional upon the truth of our \cref{conj:KRASWconj}, 
we prove concrete formulas for $H_c(1,q)$ when $c$ is below the line.
This process crucially uses an identity of Weierstra\ss{} (see \cite[Lem 6.1]{KanRus-cylindric}). 
Note that the original formulation in \cite{KanRus-cylindric} of the theorem below  was conditional upon the validity of \cref{conj:KRASWconj} with $z=1$, which in turn has been proved in \cite{War-A2proof}.
\begin{thm} \cite[Thm. 6.4]{KanRus-cylindric}, \cite{War-A2proof}
	\label{thm:mod3k}
	\footnote{Here we correct two minor errors in \cite{KanRus-cylindric}: In equations (6.10) and (6.11) of \cite{KanRus-cylindric}, $S_{3m}$ are supposed to be $S_{3k}$. Additionally, in equations (6.7) and (6.10) of \cite{KanRus-cylindric}, the second case never occurs.}
	\label{thm:mod3krel}
	Let $k\geq 3$. 
	Let $c=(c_0,c_1,c_2)$ with $c_0\geq c_1\geq c_2\geq 0$, $c_0+c_1+c_2=\ell=3k-3$, and $c_1\geq k$.
%	If the formulas in \cref{conj:KRASWconj} are true for $z=1$, then: 
%	we must have:
	Then:
	\begin{align}
		H_c(1,q)=
		X(1,q)-q^{c_2+1}Y(1,q),
	\end{align}
	where 
	\begin{align}
		X(z,q)&=
		S_{3k}(\mathrm{e}_{2k-c_1-2}\mid\mathrm{e}_{2k-c_0-2})-qS_{3k}(\mathrm{e}_{2k-c_1-3}\mid\mathrm{e}_{2k-c_0-3}),\\
		Y(z,q)&=
		\begin{cases}
			S_{3k}(\mathrm{e}_{c_0-k}\mid\mathrm{e}_{c_1-k})-qS_{3k}(\mathrm{e}_{c_0-k-1}\mid\mathrm{e}_{c_1-k-1}), & c_1>k\\
			S_{3k}(\mathrm{e}_{c_0-k}\mid\mathrm{e}_0) & c_1=k.
		\end{cases}
	\end{align}
\end{thm}
%Combined with the special cases of \cref{conj:KRASWconj}, with $z\mapsto 1$, already proved in \cite{ASW}, this gives us the following (unconditional) theorem.
%\begin{thm}
%	\label{thm:mod3k}
%	Let $k\geq 3$, and let $i\geq k$ such that $3k-2i-3\geq 0$.
%	Then, \cref{thm:mod3krel} holds unconditionally for
%	$c=(i,i,3k-2i-3)$.
%%	\begin{align}
%%		H_{(i,i,3k-3-2i)}(1,q)
%%		= 
%%		% \dfrac{(q^{3k};\,q^{3k})^2_\infty}{(q)^3_\infty}
%%		% \theta(q^{i+1},q^{i+1},q^{3k-2-2i};\,q^{3k})
%%		% =
%%		X(1,q)-q^{3k-2-2i}Y(1,q),
%%	\end{align}
%%	where:
%%	\begin{align}
%%		X(z,q)=
%%		\begin{cases}
%%			S_{3m}(\mathrm{e}_{2k-i-2}\mid\mathrm{e}_{2k-i-2})-qS_{3m}(\mathrm{e}_{2k-i-3}\mid\mathrm{e}_{2k-i-3}),
%%			& i<2k-2\\
%%			S_{3m}(\mathrm{e}_{2k-i-2}\mid\mathrm{e}_0),
%%			& i=2k-2
%%		\end{cases}
%%	\end{align}
%%	and 
%%	\begin{align}
%%		Y(z,q)=
%%		\begin{cases}
%%			S_{3m}(\mathrm{e}_{i-k}\mid\mathrm{e}_{i-k})-qS_{3m}(\mathrm{e}_{i-k-1}\mid\mathrm{e}_{i-k-1}), & i>k\\
%%			S_{3m}(\mathrm{e}_{i-k}\mid\mathrm{e}_0) & i=k.
%%		\end{cases}
%%	\end{align}
%\end{thm}

\subsection{Examples}
As examples, we single out two new identities that emerge from the theorems above.
In both, $\sum$ stands for a sum over $r_1,r_2,s_1,s_2\geq 0$.
\begin{align}
	\sum&
	\dfrac{q^{r_1^2-r_1s_1+s_1^2+r_2^2-r_2s_2+s_2^2+r_2+s_2}(1-2q^{1+r_1+s_1})}
	{(q)_{r_1-r_2}(q)_{s_1-s_2}(q)_{r_2+s_2}(q)_{r_2+s_2+1}}
	\qbin{r_2+s_2}{r_2}_{q^3}
	\notag\\
	&=\dfrac{1}{(q)_\infty}\dfrac{1}{\theta(q,q^2,q^2,q^3,q^3;\,q^9)_\infty}
	=\dfrac{\chi(\Omega(3\Lambda_0+3\Lambda_1))}{(q)_\infty},
	\label{eqn:330}
\end{align}
%\begin{align}
%	\sum_{r_1,r_2,s_1,s_2\geq 0}
%	&\dfrac{q^{r_1^2-r_1s_1+s_1^2+r_2^2-r_2s_2+s_2^2}
%		(
%		q^{-r_1+s_2}-q^{r_2}+q^{1+r_1+r_2+s_2}-q^{1-r_1+r_2+s_1+s_2}-q^{s_1+s_2}
%		)
%	}{(q)_{r_1-r_2}(q)_{s_1-s_2}(q)_{r_2}(q)_{s_2}(q)_{r_2+s_2+1}}
%	\notag\\
%	&=
%	\dfrac{1}{(q)_\infty}\dfrac{1}{\theta(q,q,q^2,q^3,q^3,q^5;\,q^{10})}
%	=\dfrac{1}{(q)_\infty}\chi(\Omega(3\Lambda_0+3\Lambda_1+\Lambda_2)).
%\end{align}
\begin{align}
	\sum&\dfrac{q^{r_1^2-r_1s_1+s_1^2+r_2^2-r_2s_2+s_2^2}(q^{-r_1+s_1+s_2}-q^{r_2+s_2}+q^{1+r_1+r_2+s_1+s_2})}
	{(q)_{r_1-r_2}(q)_{s_1-s_2}(q)_{r_2}(q)_{s_2}(q)_{r_2+s_2+1}}\notag\\
%	&=
%	\sum
%	%_{r_1,r_2,s_1,s_2\geq 0}
%	\dfrac{q^{r_1^2-r_1s_1+s_1^2+r_2^2-r_2s_2+s_2^2}
%	}{(q)_{r_1-r_2}(q)_{s_1-s_2}(q)_{r_2}(q)_{s_2}(q)_{r_2+s_2+1}}\notag\\
%	&\qquad\times
%	(
%	q^{s_2}-q^{1+r_2+s_1+s_2}-q^{1+r_1+s_1+s_2}-q^{r_1+r_2}
%	+(1-q)q^{2r_1}+q^{2+2r_1+r_2+s_2}+q^{1+2r_1+r_2}
%	)\notag\\
	&=
	\dfrac{1}{(q)_\infty}\dfrac{1}{\theta(q,q^2,q^2,q^3,q^3,q^4;\,q^{10})}
	=\dfrac{\chi(\Omega(4\Lambda_0+3\Lambda_1))}{(q)_\infty}.
	\label{eqn:430}
\end{align}
Identity \eqref{eqn:330} arises from \cref{thm:mod3k} with $(c_0,c_1,c2)=(3,3,0)$.
\eqref{eqn:430} is one of the identities in \cref{thm:lowlevels} with $\ell=7$.


\section{Coloured \texorpdfstring{$\la{sl}_r$}{sl\_r} invariants of torus knots}
\label{sec:torus}
Let us begin by collecting a few further facts about the representation theory of $\la{sl}_r$.

Recall that the Weyl denominator associated with $\la{sl}_r$ is:
\begin{align*}
	\Delta_r = \sum_{w\in \mathfrak{S}_r}(-1)^{\ell(w)}e^{w(\delta)},
\end{align*}
where $e^{w(\delta)}$ is a formal exponential, to be thought of as an element of the group algebra of
the weight lattice $P_r$. We denote:
\begin{align*}
	\qdim(\Delta_r)=\sum_{w\in \mathfrak{S}_r}(-1)^{\ell(w)}q^{(w(\delta),\delta)}.
\end{align*}
Weyl denominator formula implies that:
\begin{align*}
	\qdim(\Delta_r)=\prod_{\alpha\in\Phi_r^+} (q^{\frac{1}{2}(\delta,\alpha)}-q^{-\frac{1}{2}(\delta,\alpha)}).
\end{align*}
The function $\qdim$ can be defined more generally by $\qdim(e^\mu)=q^{(\mu,\delta)}$ for $\mu\in P_r$.
In particular, for a finite-dimensional irreducible module $L$, we may define $\qdim(L)$ as simply
$\qdim(\ch(L))$ where the character $\ch(L)$ belongs to the group algebra of $P_r$.

Let $L_r(n\Lambda_1)$ denote the (finite-dimensional) irreducible $\la{sl}_r$ module of highest weight $n\Lambda_1$ and let $\Pi_{r,n}$ be the set of weights of $L_r(n\Lambda_1)$. 
%The following hold:
%\begin{enumerate}
%	\item We have $\lambda\in \Pi_{r,n}$ iff $\lambda=n\Lambda_1 - a_1\alpha_1-\cdots a_{r-1}\alpha_{r-1}$ with $n\geq a_1\geq\cdots a_{r-1}\geq 0$.
%	\item Every weight of $L_r(n\Lambda_1)$ appears with multiplicity one. 
%	\item $\Pi_{r,n}\subseteq \Pi_{r,n+r}$. 
%	\item $\bigcup_{j\geq 0}\Pi_{r,n+jr}=n\Lambda_1+Q_r$.
%\end{enumerate}

Given a knot $K$ with a fixed framing $f$, a finite-dimensional simple Lie algebra $\la{g}$ along with a finite-dimensional irreducible module $L$, let $J^f_K(L)$ denote the framing-dependent, un-normalized, $L$ coloured Jones invariant of $K$. Here, un-normalized means that for the unknot with zero framing this invariant equals $\qdim(L)$.


Let $p, p'$ be a pair of coprime positive integers, and let $\torus(p,p')$ be the torus knot obtained by 
the $(p,p')$ cabling of the unknot of zero framing, see \cite{Mor-jones} for details. In this case,  
$\torus(p,p')$ has writhe $pp'$. We will fix (and then forget) this framing of $\torus(p,p')$.
Now, the Rosso--Jones formula \cite{RosJon-torus} (as explained by \cite{Mor-jones})
in conjunction with some elementary properties of $\Pi_{r,n}$ leads to the following theorem.
\begin{thm} Let $p,p'$ be a pair of positive coprime integers. 
	\begin{enumerate}
		\item \cite[Thm.\ 5.1]{Kan-slrtorus} The framing-dependent, un-normalized invariants of torus knots satisfy:
		\begin{align*}
			J_{\torus(p,p')}(L_r(n\Lambda_1))
			= \dfrac{q^{-\frac{p'}{2p}\|\delta\|^2}}{\qdim(\Delta_r)}
			\sum_{\lambda\in\Pi_{r,n},w\in\mathfrak{S}_r}(-1)^{\ell(w)}q^{(p\lambda+w\delta,\delta)+\frac{p'}{2p}\|p\lambda+w\delta\|^2}.
		\end{align*}
		\item \cite[Thm.\ 6.1]{Kan-slrtorus} We have:
			\begin{align}
			\lim_{n\rightarrow \infty} J_{\torus(p,p')}(L_r(nr\Lambda_1)) = 
			\dfrac{(q)_\infty^{r-1}}{\prod_{\alpha\in\Phi_r^+}(1-q^{(\alpha,\delta)})}\overline{\chi}^{r,p,p'}_{0,0}.
			\label{eqn:toruslim}
			\end{align}
		\item \cite[Prop.\ 3.4]{Kan-slrtorus} If $p<p'$ and $1\leq p\leq r-1$, then, $\overline{\chi}^{r,p,p'}_{0,0}=0$ and the limit in \eqref{eqn:toruslim} is $0$.
	\end{enumerate}
\end{thm}

In part 3 above, the reason the limit equals $0$ is that the lowest degree of the corresponding Jones invariant approaches $\infty$ as $n\rightarrow\infty$.
Thus, it is interesting to understand limits of the shifted invariants $\widehat{J}^f_K(L)$ 
which equal $J^f_K(L)$ divided by its trailing monomial.
We have the following conjecture:
\begin{conj} \cite[Conj.\ 6.6]{Kan-slrtorus}
	Let $2\leq p<p'$ be a pair of coprime positive integers such that $p<r$. Letting $\Phi_1^+=\emptyset$ and denoting the Weyl vector of $\la{sl}_r$ by $\delta_r$, we have:
	\begin{align*}
		\lim_{n\rightarrow \infty} \widehat{J}_{\torus(p,p')}(L_r(n\Lambda_1)) = 
		\dfrac{\prod_{\alpha\in\Phi_{r-p}^+}(1-q^{(\alpha,\delta_{r-p})})}
			{\prod_{\alpha\in\Phi_{r}^+}(1-q^{(\alpha,\delta_{r})})}
			(q)_\infty^{p-1}\,
			\overline{\chi}^{p,p,p'}_{0,0}.
	\end{align*}
Note that unlike \eqref{eqn:toruslim} which involves $L_r(nr\Lambda_1)$, this conjecture involves  $L_r(n\Lambda_1)$.
\end{conj}	

The $r=3$ case of this conjecture holds due to the work \cite{GarVuo} (see also \cite{Kan-slrtorus}).

Our hope is that this connection of $\sW_3(3,3+\ell)$ algebra characters to knot theory will provide useful insights on the corresponding Andrews--Schilling--Warnaar identities. 

\acknowledgements{
A major part of this extended abstract is based on joint work \cite{KanRus-cylindric} with Matthew C.\ Russell. The work \cite{KanRus-cylindric} benefitted enormously from insights generously shared by Ole Warnaar. His inputs are gratefully acknowledged.}


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