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\title[Principal subspaces and $q$-series multisums]{Principal subspaces of basic modules for twisted affine Lie algebras, $q$-series multisums, and Nandi's identities}
\author[\initial{K.} \lastname{Baker}]{\firstname{Katherine} \lastname{Baker}}
\address{
Department of Mathematics, Computer Science, and Statistics\\
Ursinus College\\
Collegeville\\
PA 19426}
\email{kabaker2@ursinus.edu}
\author[\initial{S.} \lastname{Kanade}]{\firstname{Shashank} \lastname{Kanade}}
\address{Department of Mathematics\\
University of Denver\\
Denver\\
CO 80208}
\email{shashank.kanade@du.edu}
\thanks{S.K. acknowledges support from the Collaboration Grant for Mathematicians \#636937 awarded by the Simons Foundation.}
\author[\initial{M.} \middlename{C.} \lastname{Russell}]{\firstname{Matthew} \middlename{C.} \lastname{Russell}}
\address{Department of Mathematics\\
University of Illinois Urbana-Champaign\\
Urbana\\
IL 61801}
\email{mcr39@illinois.edu}
\author[\initial{C.} \lastname{Sadowski}]{\firstname{Christopher} \lastname{Sadowski}}
\address{Department of Mathematics, Computer Science, and Statistics\\
Ursinus College\\
Collegeville\\
PA 19426}
\email{csadowski@ursinus.edu}
\keywords{Principal subspaces, vertex operator algebras, affine Lie algebras, Nandi's identities}
\subjclass{05A15, 05A17, 11P84, 17B69}
\datepublished{2024-01-08}
\begin{document}
\begin{abstract}
We provide an observation relating several known and conjectured $q$-series
identities to the theory of principal subspaces of basic modules for
twisted affine Lie algebras. We also state and prove two new families of
$q$-series identities. The first family provides quadruple sum
representations for Nandi's identities, including a manifestly positive
representation for the first identity. The second is a family of new mod 10
identities connected with principal characters of integrable, level 4,
highest-weight modules of $\mathrm{D}_4^{(3)}$.
% \SuppMaterial
\end{abstract}
\maketitle
\section{Introduction}
Principal subspaces of standard (i.e., highest-weight and integrable) modules for untwisted affine Lie algebras were introduced and studied by Feigin and Stoyanovsky~\cite{FS1,FS2}, and their study from a vertex-algebraic point of view has been developed by Calinescu, Capparelli, Lepowsky, and Milas~\cite{CLM1,CLM2,CalLM1,CalLM2,CalLM3}, and many others. In particular, the graded dimensions of principal subspaces are interesting due to their connection to various partition identities and recursions they satisfy. The study of principal subspaces for standard modules of twisted affine Lie algebras was initiated by Calinescu, Lepowsky, and Milas~\cite{CalLM4}, and further developed in works by Calinescu, Milas, Penn, and the fourth author~\cite{CalMPe,PS1,PS2,CPS}. The multigraded dimensions for principal subspaces of basic (i.e., the standard module with highest weight $\Lambda_0$, see, for example, Carter's book \cite[P.\ 508]{Car-book}) modules for twisted affine Lie algebras are well-known, and have been studied in several of those papers~\cite{CalMPe,PS1,PS2}. In particular, they take the form
\begin{equation}\label{twistedsum}
\sum_{{\bf m} \in (\mathbb{Z}_{\ge 0})^d}\frac{q^{\frac{{\bf m}^t A[\nu] {\bf m}}{2}}}{(q^{\frac{k}{l_1}};q^{\frac{k}{l_1}})_{m_1}\cdots (q^{\frac{k}{l_d}};q^{\frac{k}{l_d}})_{m_d}}x_1^{m_1}\cdots x_d^{m_d}
\end{equation}
where $A[\nu]$ is a matrix obtained by ``folding" a Cartan matrix $A$ of type $\mathrm{A}$, $\mathrm{D}$,
or $\mathrm{E}$ by a Dynkin Diagram automorphism $\nu$ of order $k$, and $l_1,\dots,l_d$ are the sizes of the orbits of various simple roots (this folding is defined generally by Penn, Webb, and the fourth author~\cite{PSW}). The matrices $A[\nu]$ are symmetrized Cartan matrices of types $\mathrm{B}$, $\mathrm{C}$, $\mathrm{F}$, $\mathrm{G}$, and, in the case $\mathrm{A}_{2n}^{(2)}$, of the tadpole Dynkin diagram.
Another recently active field of research is finding and proving Rogers--Ramanujan-type (multi)sum-to-product identities corresponding mainly to the principal characters of various affine Lie algebras.
Here, the principal characters refer to principally specialized characters divided by a certain factor depending on the affine Lie algebra in question.
For more on this terminology, see the works of the second and third authors~\cite{KanRus-a22}, \cite{KanRus-cylindric} or Sills' textbook~\cite{Sil-book}.
This use of ``principal'' is not to be confused with the ``principal'' subspaces mentioned above; these correspond to completely different notions.
The second and third authors conjectured identities~\cite{KanRus-stair} regarding principal characters of standard modules with level 2 of $\mathrm{A}_9^{(2)}$, which were later proved by Bringmann, Jennings-Shaffer, and Mahlburg~\cite{BriJenMah} and Rosengren~\cite{Ros}; Takigiku and Tsuchioka~\cite{TakTsu-a22} provided various results on levels $5$ and $7$ of $\mathrm{A}_2^{(2)}$ and some conjectures on standard modules with level $2$ of $\mathrm{A}_{13}^{(2)}$;
the authors~\cite{KanRus-a22} proved identities for all standard modules of $\mathrm{A}_2^{(2)}$. Andrews, Schilling, and Warnaar~\cite{ASW}, Corteel, Dousse, and Uncu~\cite{CorDouUnc}, Warnaar~\cite{War2}, the second and third authors~\cite{KanRus-cylindric}, and Tsuchioka~\cite{Tsu-a21level3} all provided conjectures and/or proved results on identities related to the standard modules of $\mathrm{A}_2^{(1)}$.
Finally, Griffin, Ono, and Warnaar~\cite{GriOnoWar} demonstrated many identities for (not necessarily principal) characters for a variety of affine Lie algebra modules. For an excellent overview of Rogers--Ramanujan-type identities, we refer the reader to the textbook of Sills~\cite{Sil-book}.
This work grew out of the following observations:
Calinescu, Milas, and Penn~\cite{CalMPe} studied
the graded dimension of the principal subspace of the basic $\mathrm{A}_{2n}^{(2)}$ module and showed that it is given by
\begin{equation}\label{A2n-sum}
\sum_{{\bf m} \in (\mathbb{Z}_{\ge 0})^n}\frac{q^{\frac{{\bf m}^t T_n {\bf m}}{2}}}{(q;q)_{m_1}\cdots (q;q)_{m_n}}
\end{equation}
where $T_n$ is the Cartan matrix of the tadpole Dynkin diagram. Meanwhile, Calinescu, Penn, and the fourth author~\cite{CPS} conjectured that the graded dimension of the principal subspace of the
standard $\mathrm{A}_2^{(2)}$ module having the highest weight $n\Lambda_0$ is given by
\begin{equation}\label{stem-sum}
\sum_{{\bf m} \in (\mathbb{Z}_{\ge 0})^n}\frac{q^{\frac{{\bf m}^t 2T_n^{-1} {\bf m}}{2}}}{(q^2;q^2)_{m_1}\cdots (q^2;q^2)_{m_n}},
\end{equation}
which is the sum side of Stembridge's variant of the Andrews--Gordon and G\"ollnitz--Gordon--Andrews identities~\cite{S} (see also ~\cite{BIS} and ~\cite{War1} for more general sums of this form). This conjectured graded dimension has been proved by Takenaka~\cite{Ta}. Similarly, Penn and the fourth author~\cite{PS1} showed that the graded dimension of the basic $\mathrm{D}_4^{(3)}$-module is given by
\begin{equation}
\label{eqn:d43}
\sum_{{\bf m} \in (\mathbb{Z}_{\ge 0})^2}\frac{q^{\frac{{\bf m}^t A[\nu] {\bf m}}{2}}}{(q^3;q^3)_{m_1}(q;q)_{m_2}},
\end{equation}
where
\begin{equation}
A[\nu] = \begin{bmatrix} 6 & -3\\ -3 & 2 \end{bmatrix}.
\end{equation}
Meanwhile, Penn, Webb, and the fourth author~\cite{PSW} constructed a principal subspace of a twisted module for a lattice vertex operator algebra whose graded dimension is
\begin{equation}
\label{eqn:d43inv}
\sum_{{\bf m} \in (\mathbb{Z}_{\ge 0})^2}\frac{q^{\frac{{\bf m}^t 3A[\nu]^{-1} {\bf m}}{2}}}{(q;q)_{m_1}(q^3;q^3)_{m_2}},
\end{equation}
which is precisely the sum side of one of the mod $9$ conjectures of the second and third authors \cite{KanRus-idf}
as found by Kur\c{s}ung\"{o}z~\cite{Kur-KR}.
The present work is the result of a discussion at the AMS Fall Sectional Meeting at Binghamton University involving Alejandro Ginory, the second author, and the fourth author. Namely, generalizing the shapes of \eqref{A2n-sum} and \eqref{stem-sum} (or \eqref{eqn:d43} and \eqref{eqn:d43inv}), do the sum sides of any other identities emerge when the matrix used in the exponent of the numerator is replaced by a multiple of its inverse and the $q$-Pochhammer symbols in the denominator are modified in some systematic way?
In this work, we show that, for matrices $A$ of type $\mathrm{A}$, $\mathrm{D}$, or $\mathrm{E}_6$, we obtain sum-sides for certain $q$-series identities, and give several new identities. In particular, we replace $A[\nu]$ with a suitable multiple (large enough to clear fractional entries) of $A[\nu]^{-1}$ and manipulate the denominator of each sum of the form~\eqref{twistedsum} in the following way: if the diagram automorphism has order $k=2$ or $k=3$, we replace instances of $(q;q)_n$ with $(q^k;q^k)_n$ and vice-versa. In all cases except when $A$ is of type $\mathrm{A}_{2n-1}$ for $n \ge 2$ these identities come in pairs, producing two families of identities.
We emphasize that at present, we do not have a clear understanding of why this remarkable ``duality'' among the families of identities holds nor do we have an algebraic explanation for why manipulating the graded dimensions of these principal subspaces causes these identities to arise. It would surely be a worthy goal to understand representation-theoretic, number-theoretic, and combinatorial underpinnings of this phenomenon.
After experimentation by the first and fourth authors using Garvan's \texttt{qseries} Maple package~\cite{Ga}, new identities were found using a matrix $A$ of type $\mathrm{E}_6^{(2)}$:
\begin{align}
\sum_{i,j,k,\ell \ge 0}
\frac{q^{4i^2 + 12 ij + 8ik + 4i\ell + 12j^2 + 16jk + 8j\ell + 6k^2+6k\ell+2\ell^2}}
{\left(q^2;q^2\right)_i\left(q^2;q^2\right)_j\left(q;q\right)_k\left(q;q\right)_\ell}
&= \left(q^2,q^3,q^4,q^{10},q^{11},q^{12};q^{14}\right)^{-1}_\infty \label{eqn:Nandi_intro}, \\
\sum_{i,j,k,\ell \ge 0}
\frac{q^{2i^2 + 6 ij + 4ik + 2i\ell + 6j^2 + 8 jk + 4 j\ell + 3k^2+3k\ell+\ell^2}}
{\left(q^2;q^2\right)_i\left(q^2;q^2\right)_j\left(q;q\right)_k\left(q;q\right)_\ell}
&=\left(q,q^2,q^4,q^6,q^8,q^9;q^{10}\right)^{-1}_\infty\label{eqn:mod10_intro}.
\end{align}
Notably, the product side of~\eqref{eqn:Nandi_intro} matches one of Nandi's identities. These were first conjectured by Nandi in his thesis~\cite{Nan-thesis} and later proved by Takigiku and Tsuchioka~\cite{TakTsu-nandi}. Remarkably, the (new) expression on the left side of~\eqref{eqn:Nandi_intro} above is a manifestly positive quadruple sum. (The sums used in Takigiku and Tsuchoika's proof are double sums, but are not manifestly positive.) Nandi's identities are connected to principal characters of standard modules of $\mathrm{A}_2^{(2)}$ of level $4$ (and also level 2 of $\mathrm{A}_{11}^{(2)}$).
In~\eqref{eqn:mod10_intro}, the left side is again a manifestly positive quadruple sum. In fact, the exponent of $q$ in the terms on the left side of~\eqref{eqn:Nandi_intro} is exactly twice the exponent of $q$ in the terms on the left side of~\eqref{eqn:mod10_intro}. The product side here is connected to level 4 of $\mathrm{D}_4^{(3)}$. The same relationship (of doubling the quadratic form) holds between Capparelli's identities \cite{Cap-1,KanRus-stair,Kur-Cap} which reside at level $3$ of $\mathrm{A}_2^{(2)}$ and Kur\c{s}ung\"oz's (multi)sum-to-product companions \cite{Kur-KR} to the conjectures of the second and third authors~\cite{KanRus-idf} related to level $3$ of $\mathrm{D}_4^{(3)}$.
Sections 5 and 6 are dedicated to proofs of these identities and others in their families. We now outline our proof strategy.
We begin by deducing $x,q$-relations (or, in the second case, $x,y,q$-relations) that sum sides equalling each respective product side are known to satisfy. After appropriately generalizing the quadruple sums, we then demonstrate relations that these generalized quadruple sums must satisfy. We finish our proofs by showing that that the desired relations follow from these known relations. In the case of the Nandi identities, this proof requires the use of a computer. This technique is similar to one that the second and third authors used in a previous paper~ \cite{KanRus-cylindric} (see also the work of Chern~\cite{Che-linked}). The Maple code for the computations used in our research can be found on the journal's website under \printDOI.
\section{Notation and preliminaries}
\label{sec:notations}
\subsection{Partitions}
We will write partitions in a weakly decreasing order. If $\lambda=\lambda_1+\lambda_2+\cdots +\lambda_j$ is a partition of $n$,
we will say that the weight of $\lambda$ is $|\lambda|=n$ and
we will let $j=\ell(\lambda)$ be the length, or the number of parts, of $\lambda$.
Each $\lambda_i$ will be called a part of $\lambda$.
Given a positive integer $i$,
we denote by $m_i(\lambda)$ the multiplicity of $i$ in~$\lambda$.
By a (contiguous) sub-partition of $\lambda$, we mean the subsequence
$\lambda_s+\cdots+\lambda_t$ of $\lambda$.
We say that $\lambda$ satisfies the difference condition $[d_1,d_2,\cdots,d_{j-1}]$ if $\lambda_i-\lambda_{i+1}=d_i$ for all~$1\leq i\leq j-1$.
Let $\sC$ be any set of partitions. In the usual way, the two-variable generating function of $\sC$ is defined as
\begin{align}
f_{\sC}(x,q)=\sum_{\lambda\in\sC}x^{\ell(\lambda)}q^{|\lambda|}.
\end{align}
Then, the $q$-generating function of $\sC$ is simply $f_{\sC}(1,q)$ (sometimes denoted just by~$f_{\sC}(q)$).
As usual, we will let $\ZZ[[x,q]]$ be the ring of power series in variables $x,q$ with coefficients in $\ZZ$. However, note that all of our series will actually be in the sub-ring $\ZZ[x][[q]]$.
We will require a subset $\sS\subset \ZZ[[x,q]]$:
\begin{align}
\sS = \{ f\in \ZZ[[x,q]]\,\mid\, f(0,q)=f(x,0)=1\}
\label{eqn:S}.
\end{align}
\subsection{\texorpdfstring{$q$-Series}{q-Series}}
We shall use standard notation regarding $q$-series.
All of our series are formal, and issues of analytic convergence are disregarded.
For $n\in\ZZ_{\geq 0}\cup\{\infty\}$ we define:
\begin{align}
(a;q)_n=\prod_{0\leq t S(A,B,C,D)
-x^2*q^(4+A)*(1+q^2)*S(A+8,B+12,C+8,D+4)
-S(A+4,B,C,D)
+x^4*q^(18+2*A)*S(A+16,B+24,C+16,D+8):
N2 := (A,B,C,D) -> S(A,B,C,D)
-S(A,B+2,C,D)
-x^3*q^(12+B)*S(A+12,B+24,C+16,D+8):
N3 := (A,B,C,D) -> S(A,B,C,D)
-x^2*q^(6+C)*S(A+8,B+16,C+12,D+6)
-S(A,B,C+2,D)
-x^2*q^(7+C)*S(A+8,B+16,C+12,D+6)
+x^4*q^(25+2*C)*S(A+16,B+32,C+24,D+12):
N4 := (A,B,C,D) -> S(A,B,C,D)
-x*q^(2+D)*S(A+4,B+8,C+6,D+4)-S(A,B,C,D+2)
-x*q^(3+D)*S(A+4,B+8,C+6,D+4)
+x^2*q^(9+2*D)*S(A+8,B+16,C+12,D+8):
\end{verbatim}
\end{quote}
We now read the file that contains a linear combination of
\eqref{eqn:reln1}--\eqref{eqn:reln4} which when expanded is supposed
to yield \eqref{eqn:recS_F1}.
\begin{quote}
\begin{verbatim}
F1rel := parse(FileTools[Text][ReadFile]("F1.txt")):
\end{verbatim}
\end{quote}
Simply simplifying this entire expression takes too long.
Thus, we collect all like $S$ terms together and simplify the coefficients.
\begin{quote}
\begin{verbatim}
collect(F1rel, S, simplify);
\end{verbatim}
\end{quote}
Naturally, most coefficients are $0$ and get dropped from the expression.
The output is:
\begin{quote}
\begin{verbatim}
-x^3*q^16*(q^11*x+q^9*x+q^8*x-q^3-q-1)*S(18, 26, 17, 8)
+x*q^3*(q^8*x+q^6*x+q^2+q-1)*S(10, 14, 9, 4)
+x^3*q^19*(q^18*x^2-q^10*x-q^8*x+1)*S(22, 32, 21, 10)
+S(2, 2, 1, 0)+(-q^5*x-q^4*x-q^2*x-1)*S(6, 8, 5, 2)
+x^2*q^8*(q^8*x+q^6*x-q^3+q-1)*S(14, 20, 13, 6)
\end{verbatim}
\end{quote}
which is exactly \eqref{eqn:recS_F1}.
We repeat the process for \eqref{eqn:recS_F5}:
\begin{quote}
\begin{verbatim}
F5rel := parse(FileTools[Text][ReadFile]("F5.txt")):
collect(F5rel, S, simplify);
\end{verbatim}
\end{quote}
and the answer matches \eqref{eqn:recS_F5}:
\begin{quote}
\begin{verbatim}
S(0, -2, -2, -1)
-x^2*q^4*(q^11*x-q^8*x-q^7*x-q^6*x+q^5-q^3+1)*S(12, 16, 10, 5)
-(q^10*x+q^9*x+q^8*x-q^2-q-1)*q^11*x^3*S(16, 22, 14, 7)
+(q^8*x+q^7*x+q^6*x-q^3*x+q^3+q^2-1)*q*x*S(8, 10, 6, 3)
+(q^18*x^2-q^10*x-q^8*x+1)*q^13*x^3*S(20, 28, 18, 9)
+(-q^4*x-q^3*x-q^2*x-1)*S(4, 4, 2, 1)
\end{verbatim}
\end{quote}
For \eqref{eqn:recS_F7}:
\begin{quote}
\begin{verbatim}
F7rel := parse(FileTools[Text][ReadFile]("F7.txt")):
collect(F7rel, S, simplify);
\end{verbatim}
\end{quote}
and the answer matches \eqref{eqn:recS_F7}:
\begin{quote}
\begin{verbatim}
S(0, 0, 0, 0)
+(q^5*x+q^4*x+q^3*x-x+1)*q^4*x*S(8, 12, 8, 4)
+x^3*q^17*(q^14*x^2-q^8*x-q^6*x+1)*S(20, 30, 20, 10)
-x^2*q^6*(q^9*x-q^6*x-q^5*x-q^4*x+1)*S(12, 18, 12, 6)
+(-q^4*x-q^3*x-q^2*x-1)*S(4, 6, 4, 2)
-x^3*q^13*(q^8*x+q^7*x+q^6*x-q^2-q-1)*S(16, 24, 16, 8)
\end{verbatim}
\end{quote}
\longthanks{
We are indebted to S.\ Ole Warnaar for his most valuable comments on an earlier draft of this manuscript.
We also thank S.\ Tsuchioka for helpful correspondence.
We are thankful to the anonymous referee for carefully reading the manuscript, suggesting many valuable changes, and for asking insightful questions.}
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\end{document}