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\title{Multi-Grounded Partitions and Character Formulas}

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\author[J.~Dousse and I.~Konan]{Jehanne Dousse\thanks{Partially supported by the ANR COMBIN\'e ANR-19-CE48-0011 and the project IMPULSION of IDEXLYON.}\addressmark{1} \and Isaac Konan\thanks{\href{mailto:konan@math.univ-lyon1.fr}{konan@math.univ-lyon1.fr}. Supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit\'e de Lyon, within the program ''Investissements d'Avenir'' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).}\addressmark{1}}

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\address{\addressmark{1}Univ Lyon, CNRS, Universit\'e Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, F-69622 Villeurbanne, France}

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\abstract{We introduce a new generalisation of partitions, multi-grounded partitions,
related to ground state paths indexed by dominant weights of Lie algebras.
We use these to express characters of irreducible highest weight
modules of Kac--Moody algebras of affine type as generating functions for
multi-grounded partitions, and obtain new non-specialised character formulas.}

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\keywords{integer partitions, coloured partitions, representations of affine Lie algebras, character formulas, crystal bases}

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\begin{document}

\maketitle

\section{Introduction and statement of results}
Let $\gf$ be a Kac--Moody affine Lie algebra and let $\h^*$ be the dual of its Cartan subalgebra. Let $P^+$ be the set of dominant integral weights, and $L(\lambda)$ an irreducible highest weight $\gf$-module of highest weight $\lambda \in P^+$.
Then the character of $L(\lambda)$ is defined as
$$
\mathrm{ch} (L(\lambda)) = \sum_{\mu\in \h^*} \dim L(\lambda)_{\mu} \cdot e^{\mu},
$$
where $e$ is a formal exponential, and $\dim L(\lambda)_{\mu}$ is the dimension of the weight space $L(\lambda)_{\mu}$.
Character formulas have been widely studied, starting with the famous Weyl--Kac character formula \cite{Kac}:
\begin{equation}
\label{eq:weylkac}
\mathrm{ch} (L(\lambda)) = \frac {\sum _{ w \in W } \mathrm{sgn}(w) e ^ { w ( \lambda + \rho ) - \rho } } { \prod _ { \alpha \in \Delta ^ { + }} ( 1 - e ^ { -\alpha } ) ^ { \text{dim} \mathfrak{g} _ { \alpha }  } },
\end{equation}
where $W$ is the Weyl group of $\gf$, $\Delta^{+}$ the set of positive roots of $\gf$, $\mathrm{sgn}(w)$ the signature of $w$, $\rho \in \h^*$ the Weyl vector, and  
$\mathfrak{g} _ { \alpha }$ the $\alpha$ root space of $\gf$.

Equation \eqref{eq:weylkac} is beautiful, but it is not so well suited to compute characters in practice. Moreover, even though by definition $e^{-\lambda}\mathrm{ch} (L(\lambda))$ is a series with positive coefficients in the $e^{-\alpha_i}$'s, this positivity is not explicit from the formulation in \eqref{eq:weylkac}. We now briefly explain what solutions have been given to work around these issues, and present a new method which allows us to give simple non-specialised character formulas using perfect crystals and a new generalisation of integer partitions.

\medskip
The first solution to obtain simple character formulas is to perform certain specialisations, \textit{i.e.}\ for each of the simple roots $\alpha_i$, applying the transformations $e^{-\alpha_i} \rightarrow q^{s_i}$ for some integer $s_i$. Using this method, it is possible to transform the Weyl--Kac character formula into infinite products. In particular, the principal specialisation, where $e^{-\alpha_i} \rightarrow q$ for all $i$, has been widely exploited in the theory of partition identities related to representations of affine Lie algebras, see for example \cite{Capparelli,GOW16,Meurman,Nandi}. Lepowsky and Milne~\cite{Le-Mi1} were the first to expose the connection by noting that up to a $(q;q^2)_{\infty}$ factor, the principal specialisation of the Weyl--Kac character formula for level $3$ standard modules of the affine Lie algebra $A_1^{(1)}$ is the product side of the Rogers--Ramanujan identities:
$$
\sum_{n\geq 0}\frac{q^{n(n+i)}}{(q;q)_n}=\frac{1}{(q^{1+i},q^{4-i};q^5)_{\infty}}, \quad i \in \{0,1\}.
$$
Here and in the whole paper, we use the standard $q$-series notation: for $n \in \mathbb{N} \cup \{\infty\}$ and $j \in \mathbb{N}$,
$
(a;q)_n := \prod_{k=0}^{n-1} (1-aq^k)$ and $(a_1, \dotsc, a_j ; q)_n := (a_1;q)\cdots(a_j;q)$.

Lepowsky and Wilson \cite{Lepowsky} later gave an interpretation of the sum side by constructing a basis of these standard modules using vertex operators. Their method has then led to the discovery of many new $q$-series and partition identities, see, \textit{e.g.},~\cite{Capparelli,Nandi,PrimcSikic,Siladic}. However, without performing a specialisation, it is in general difficult to reduce the Weyl--Kac character formula to obtain a combinatorially simple character formula.

On the other hand, Bartlett and Warnaar \cite{BW15} gave non-specialised formulas with explicitly positive coefficients for the characters of certain highest weight modules of the affine Lie algebras $C_n^{(1)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$ as sums using Hall--Littlewood polynomials. This led them to generalisations for the Macdonald identities for $B_n^{(1)},$ $C_n^{(1)},$ $A_{2n-1}^{(2)},$ $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}.$ Using Macdonald--Koornwinder theory, Rains and Warnaar \cite{WarnaarRains} found additional character formulas for these Lie algebras, together with new Rogers--Ramanujan type identities.


\medskip
In a different direction, Kang, Kashiwara, Misra, Miwa, Nakashima, and Nakaya\-shi\-ki~\cite{KMN2a,KMN2b} introduced the theory of perfect crystals to study the irreducible highest weight modules over quantum affine algebras. The proved the so-called ``(KMN)$^2$ crystal base character formula'' \cite{KMN2a},
$$\mathrm{ch}(L(\lambda)) = \sum_{\p \in \mathcal P(\lambda)} e^{\mathrm{wt}\, \p},$$
which expresses the character $\mathrm{ch}(L(\lambda))$ as a series indexed by $\lambda$-paths (see Section \ref{sec:ground}). Here the weight $\mathrm{wt} \p$ is computed using the energy function of a perfect crystal.

Primc \cite{Primc} was the first to use this character formula to give new Rogers--Ramanujan type identities related to the level $1$ standard modules of $A_1^{(1)}$ and $A_2^{(1)}$. The product side came from the principally specialised Weyl--Kac character formula again, while the sum side came from the principally specialised (KMN)$^2$ crystal base character formula.
In a couple of previous papers \cite{DK19,DK19-2}, we generalised Primc's identities to $A_{n-1}^{(1)}$ for all $n$, and managed to avoid doing a specialisation. To do this, we established a bijection between $\lambda$-paths and a new generalisation of partitions, in the case where the ground state path is constant.

But the case of standard modules whose ground state paths are not constant was still open, and the goal of \cite{DKmulti}, of which this article is an extended abstract, was to extend our method to treat this case as well. To do so, we introduced so-called ``multi-grounded partitions''. We explain their connection with crystals and characters in Section \ref{sec:ground}. Our main result is a character formula using generating functions for multi-grounded partitions (see Theorem \ref{theo:formchar2}). We proved it by transforming the (KMN)$^2$ crystal base character formula using a bijection with $\lambda$-paths. One big advantage of this formula is that one does not need to perform a specialisation, and that, being generating functions for combinatorial objects, the series always have obviously positive coefficients. Moreover, as we shall see on an example (Theorem \ref{theo:chara2n1}) in Section \ref{sec:examples}, these generating functions are relatively easy to compute in practice.

In \cite{DKmulti}, we used multi-grounded partitions to compute new character formulas for irreducible highest weight level one modules of classical Lie algebras.
All these formulas are non-specialised, with obviously positive coefficients, and are either infinite products or sums of two infinite products. 


Let $G=G(x_1, \dotsc, x_n)$ be a power series in several variables $x_1, \dotsc, x_n.$ For $k \leq n$, we denote by $\E_{x_1, \dotsc, x_k} (G)$ the sub-series of $G$ where we only keep the terms in which the sum of the powers of $x_1, \dotsc, x_k$ is even.
Note that if $G$ has only positive coefficients, then the same is true for $\E_{x_1, \dotsc, x_k} (G)$ for all $k$. There is a simple formula to obtain $\E_{x_1, \dotsc, x_k} (G)$ from $G$:
\begin{equation}
\label{eq:evennumberparts}
\E_{x_1, \dotsc, x_k} (G)= \frac{1}{2} \Big(G(x_1, \dotsc, x_k, x_{k+1}, \dotsc, x_n) + G(-x_1, \dotsc, -x_k, x_{k+1}, \dotsc, x_n)\Big).
\end{equation}

Our character formulas are the following.

\begin{theo}\label{theo:chara2n1}
Let $n\geq 3$, and let $\Lambda_0, \dotsc, \Lambda_{n}$ be the fundamental weights  and $\alpha_0, \dotsc, \alpha_{n}$ be the simple roots of $A_{2n-1}^{(2)}$.
Let $\delta= \alpha_0+\alpha_1+2\alpha_2+\cdots+2\alpha_{n-1}+\alpha_n$ be the null root.
Set
\begin{gather*}
q=e^{-\delta/2}, \quad c_i = e^{\alpha_i + \cdots + \alpha_{n-1} +\alpha_n/2} \text{ for all } i \in \{1,\dots , n \},
\\ \text{and} \quad
\Pi_{a,b} = \prod_{k=a}^b (-c_k q^2;q^2)_{\infty} (-c_k^{-1};q^2)_{\infty}.
\end{gather*}
We have
\begin{align}
e^{-\Ll_0}\mathrm{ch}(L(\Ll_0))&= \E_{c_1, \dotsc, c_n} \left( (q^2;q^4)_{\infty} \Pi_{1,n} \right), \label{eq:a210}
\\e^{-\Ll_1}\mathrm{ch}(L(\Ll_1))&= \E_{c_1, \dotsc, c_n} \left( (q^2;q^4)_{\infty} (-c_1 q^3;q^2)_{\infty} (-c_1^{-1}q^{-1};q^2)_{\infty} \Pi_{2,n} \right). \nonumber
\end{align}
\end{theo}

The next theorem concerns the Lie algebra $B_{n}^{(1)}$. Note that the second author proved a character formula for $L(\Ll_n)$, another level $1$ module, in \cite{Konan_ww2}.

\begin{theo}\label{theo:charbn1}
Let $n\geq 3$, and let $\Lambda_0, \dotsc, \Lambda_{n}$ be the fundamental weights  and $\alpha_0, \dotsc, \alpha_{n}$ be the simple roots of $B_{n}^{(1)}$.
Let $\delta= \alpha_0+\alpha_1+2\alpha_2+\cdots+2\alpha_n$ be the null root.
Set
\begin{gather*}
q=e^{-\delta/2}, \quad c_0=1, \quad c_i = e^{\alpha_i + \cdots + \alpha_{n-1} +\alpha_n} \text{ for all } i \in \{1,\dotsc, n \},
\\ \text{and} \quad
\Pi_{a,b} = \prod_{k=a}^b (-c_k q^2;q^2)_{\infty} (-c_k^{-1};q^2)_{\infty}.
\end{gather*}
We have
\begin{align*}
e^{-\Ll_0}\mathrm{ch}(L(\Ll_0))&= \E_{c_0,c_1, \dotsc, c_n}  \left((-c_0q;q^2)_{\infty} \Pi_{1,n} \right),
\\e^{-\Ll_1}\mathrm{ch}(L(\Ll_1))&= \E_{c_0,c_1, \dotsc, c_n}  \left((-c_0q;q^2)_{\infty} (-c_1 q^3;q^2)_{\infty} (-c_1^{-1}q^{-1};q^2)_{\infty} \Pi_{2,n} \right).
\end{align*}
\end{theo}

We conclude with the four level $1$ standard modules of $D_{n}^{(1)}$.
\begin{theo}\label{theo:chardn1}
Let $n\geq 4$, and let $\Lambda_0, \dotsc, \Lambda_{n}$ be the fundamental weights  and $\alpha_0, \dotsc, \alpha_{n}$ be the simple roots of $D_{n}^{(1)}$.
Let $\delta= \alpha_0+\alpha_1+2\alpha_2+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_n$ is the null root.
Set
\begin{gather*}
q=e^{-\delta/2},
\quad
c_i = e^{\alpha_i + \cdots + \alpha_{n-2} +\alpha_{n-1}/2 +\alpha_n/2} \; \text{ for all } i \in \{1,\dotsc, n \},
\\ \text{and} \quad
\Pi_{a,b} = \prod_{k=a}^b (-c_k q^2;q^2)_{\infty} (-c_k^{-1};q^2)_{\infty}.
\end{gather*}
We have
\begin{align*}
e^{-\Ll_0}\mathrm{ch}(L(\Ll_0))&= \E_{c_1, \dotsc, c_n} \Pi_{1,n}, 
\\e^{-\Ll_1}\mathrm{ch}(L(\Ll_1))&= \E_{c_1, \dotsc, c_n} \left((-c_1 q^3;q^2)_{\infty} (-c_1^{-1}q^{-1};q^2)_{\infty} \Pi_{2,n} \right), 
\\e^{-\Ll_{n-1}}\mathrm{ch}(L(\Ll_{n-1})) &= \E_{c_1, \dotsc, c_n} \left((-c_n;q^2)_{\infty} (-c_n^{-1}q^2;q^2)_{\infty} \Pi_{1,n-1} \right), 
\\e^{-\Ll_n}\mathrm{ch}(L(\Ll_n)) &= \E_{c_1, \dotsc, c_n} \left((-c_n q^2;q^2)_{\infty} (-c_n^{-1};q^2)_{\infty} \Pi_{1,n-1} \right).
\end{align*}
\end{theo}

 Some more formulas of the same kind, but involving only products (not sums of products), can be found in Wakimoto's book \cite[pages 50--58]{WakimotoBook}.




\section{Perfect crystals and multi-grounded partitions}
\label{sec:ground}
In this section, we define multi-grounded partitions and make the connection with characters of Lie algebra modules.
We start by briefly recalling the theory of perfect crystals~\cite{HK}. Let $\gf$ be an affine Kac--Moody algebra with simple positive roots $\alpha_{0},  \cdots,  \alpha_{n}$ and with null root $\delta = d_{0}\alpha_{0} + \cdots + d_{n}\alpha_{n} $. For $\lambda \in \overline P^{+}$, let $\B(\lambda)$ be the crystal graph of a crystal basis of $L(\lambda)$. For an integer level $\ell \geq 1$ and a weight $\lambda \in \overline P^+_\ell$, Kashiwara \textit{et al.}\ ~\cite[Section 1.4]{KMN2a} define the notion of a \emph{perfect crystal $\B$ of level $\ell$}, an \emph{energy function} $H\colon \B \otimes \B \to \mathbb{Z}$, and a particular element 
$$
  {\p}_\lambda = \bigl(g_k)_{k=0}^\infty =  \cdots \ot g_{k+1} \ot g_k \ot \cdots \ot g_1 \ot g_0 \in \B^{\infty},
$$
called the \emph{ground state path of weight} $\lambda$. From this they consider all elements of the form
$$\p = (p_k)_{k=0}^\infty =  \cdots \ot p_{k+1} \ot p_k \ot \cdots \ot p_1 \ot p_0 \in \B^{\infty},$$
which satisfy $p_k = g_k$ for large enough $k$. Such elements are called $\lambda$-paths; their collective set is denoted $\mathcal{P}(\lambda)$. The crystal $\B(\lambda)$ can then be realised on the set of $\lambda$-paths, in particular the affine weight function is given by the following theorem.
\begin{theo}[(KMN)$^2$ crystal base character formula \cite{KMN2a}]
\label{theorem:wtchar}
Let $\lambda \in \overline P^+_{\ell}$, let $H$ be an energy function on $\B \ot \B$, and let  $\p = (p_k)_{k=0}^\infty \in \mathcal P(\lambda)$.
Then the weight of $\p$ and the character of the irreducible highest weight $U_q(\widehat \gf)$-module
$L(\lambda)$ are given by the following expressions:
\begin{align*}
\mathrm{wt} \p &= \lambda + \sum_{k=0}^\infty \left(\left(\wt p_k -\wt g_k\right)  -  \frac{\delta}{d_0}\sum_{j=k}^\infty (H(p_{j+1} \ot p_j) - H(g_{j+1}\ot g_j))\right),\\
\mathrm{ch}(L(\lambda)) &= \sum_{\p \in \mathcal P(\lambda)}  e^{\mathrm{wt} \p}.
\end{align*}
\end{theo}

To make the connection between character formulas and partitions, we introduce the concept of \textit{multi-grounded partitions}.

\begin{deff}
Let $\C$ be a set of colours, and let $\Z_{\C} = \{k_c : k \in \Z, c \in \C\}$ be the set of integers coloured with the colours of $\C$.
Let $\succ$ be a binary relation defined on $\Z_{\C}$.
A \textit{generalised coloured partition} with relation $\succ$ is a finite sequence $(\pi_0,\ldots,\pi_s)$ of coloured integers, such that for all $i \in \{0, \dotsc, s-1\},$ $\pi_i\succ \pi_{i+1}.$
\end{deff}

In the following, if $\pi=(\pi_0,\ldots,\pi_s)$ is a generalised coloured partition, then $c(\pi_i) \in \C$ denotes the colour of the part $\pi_i$. The quantity $|\pi|=\pi_0+\cdots+\pi_s$ is the weight of $\pi$, and $C(\pi) = c(\pi_0)\cdots c(\pi_s)$ is its colour sequence. 

\begin{deff}\label{deff:multiground}
 Let $\Co$ be a set of colors, $\Z_{\Co}$ the set of integers coloured with colours in $\C$, and $\succ$ a binary relation defined on $\Z_{\Co}$. Suppose that there exist some colors $c_{g_0},\ldots,c_{g_{t-1}}$ in $\Co$ 
 and \textbf{unique} coloured integers $u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)}$ such that
 \begin{align}
  &u^{(0)}+\cdots+u^{(t-1)}=0,  \label{eq:sumu} \\
  &u_{c_{g_0}}^{(0)}\succ u_{c_{g_1}}^{(1)}\succ \cdots\succ u_{c_{g_{t-1}}}^{(t-1)}\succ u_{c_{g_0}}^{(0)}.   \label{eq:cyclgrounds}
 \end{align}
Then a \textit{multi-grounded partition} with ground $c_{g_0},\ldots,c_{g_{t-1}}$ and relation $\succ$ is a non-empty generalised coloured partition $\pi = (\pi_0,\dotsc,\pi_{s-1},u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)})$ with relation $\succ$, such that
$(\pi_{s-t},\cdots,\pi_{s-1})\neq (u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)})$ in terms of coloured integers. Let $\Psucm$ denote the set of multi-grounded partitions with ground $g_0,\ldots,g_{t-1}$ and relation $\succ$. 
\end{deff}


\begin{ex}
\label{ex}
 Let us consider the set of colours $\Co=\{c_1,c_2,c_3\}$, the matrix 
 $$
 M = 
 \begin{pmatrix}
   2&2&2\\
   0&0&2\\
   -2&0&2\\
 \end{pmatrix},$$
 and define the relation $\succ$ on $\Z_{\C}$ by $k_{c_b}\succ k'_{c_{b'}} \text{ if and only if } k-k' \geq M_{b,b'}$.
 
If we choose $(g_0,g_1) = (1,3)$, the pair $(u^{(0)},u^{(1)})=(1,-1)$ is the unique pair satisfying \eqref{eq:sumu} and \eqref{eq:cyclgrounds}. The generalised coloured partitions
$(3_{c_3},3_{c_2},3_{c_1},-1_{c_3},1_{c_1},-1_{c_3})$  and $(1_{c_3},3_{c_1},1_{c_3},3_{c_1},-1_{c_3},1_{c_1},-1_{c_3})$
are examples of multi-grounded partitions with ground $c_1,c_3$ and relation $\succ$, while $(1_{c_1},-1_{c_3},1_{c_1},-1_{c_3})$ and $(2_{c_1},1_{c_1},-1_{c_3})$
are not.
\end{ex}

Let $\B$ be a perfect crystal of level $\ell$, and let $\lambda \in \overline P_{\ell}^+$ be a level $\ell$ dominant classical weight with ground state path $\p_{\lambda}=(g_k)_{k\geq 0}$.
The fact that the set $P_{\ell}$ is finite implies the periodicity of the sequence $(g_i)_{i\geq 0}$. Let us set $t$ to be the \textit{period} of the ground state path, \textit{i.e.}\ the smallest non-negative integer $k$ such that $g_k=g_0$.
Let $H$ be an energy function on $\B\ot\B$, which is uniquely determined by fixing its value on a particular $b_0 \ot b_0' \in \B \ot \B$. 

We now define the function $H_{\lambda}$, for all $b, b' \in \B\ot\B$, by
  \begin{equation}\label{eq:Hlamb}
   H_{\lambda} (b\ot b') := H(b\ot b')-\frac{1}{t}\sum_{k=0}^{t-1} H(g_{k+1}\ot g_k)\,.
  \end{equation}
Thus we have
$$\sum_{k=0}^{t-1} H_{\lambda}(g_{k+1}\ot g_k)=0.$$
The function $H_{\lambda}$ satisfies all the properties of energy functions, except that it only has integer values when $t$ divides $\sum_{k=0}^{t-1} H(g_{k+1}\ot g_k).$

  Note that for any energy function $H$, we always have  
\[\sum_{k=0}^{t-1} (k+1) H_{\lambda}(g_{k+1}\ot g_k)= \sum_{k=0}^{t-1} (k+1) H(g_{k+1}\ot g_k) -\frac{t+1}{2}\sum_{k=0}^{t-1} H(g_{k+1}\ot g_k) \in \frac{1}{2}\Z  .\]
The quantity above is an integer as soon as $t$ is odd, and is equal to $0$ when $t=1$. Thus we can choose a suitable divisor $D$ of $2t$ such that 
$DH_{\lambda}(\B\ot\B)\subset \Z$ and $\frac{1}{t}\sum_{k=0}^{t-1} (k+1) DH_{\lambda}(g_{k+1}\ot g_k) \in \Z$. In the whole paper, $D$ always denotes such an integer.

Let us now consider the set of colours $\Co_{\B}$ indexed by $\B$, and let us define, for the remainder of this paper, the relations $\gtrdot$  and $\gg$ on $\Z_{\Co_{\B}}$ by 
\begin{align*}
 k_{c_b}\gtrdot k'_{c_{b'}} & \text{ if and only if } k-k'= DH_\lambda(b'\ot b), \\
 k_{c_b}\gg k'_{c_{b'}} & \text{ if and only if } k-k'\geq DH_\lambda(b'\ot b).
\end{align*}
We can define multi-grounded partitions associated with these relations, as can be seen in the next proposition.

\begin{prop}
\label{prop:multigroundedparts}
The set $\Pppm$ (resp.\ $\Ppm$) of multi-grounded partitions with ground $c_{g_0},\ldots,c_{g_{t-1}}$ and relation $\gtrdot$ (resp.\ $\gg$) is the set of non-empty generalised coloured partitions $\pi = (\pi_0,\cdots,\pi_{s-1},u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)})$ with relation $\gtrdot$ (resp.\ $\gg$), such that
$(\pi_{s-t},\cdots,\pi_{s-1})\neq (u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)})$, and for all $k \in \{0, \dotsc, t-1\},$
\begin{equation}\label{eq:choiceofu}
 u^{(k)} = -\frac{1}{t} \sum_{j=0}^{t-1} (j+1) DH_{\lambda}(g_{j+1}\ot g_j) + \sum_{j=k}^{t-1}DH_{\lambda}(g_{j+1}\ot g_j).
\end{equation}
\end{prop}

Moreover, for any positive integer $d$, we denote by 
$^d\Ppm$ the set of multi-grounded partitions $\pi=(\pi_0,\cdots,\pi_{s-1},u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)})$ of $\Ppm$ such that for all $k\in\{0,\ldots,s-1\}$, $\pi_k-\pi_{k+1} - DH_{\lambda}(p_{k+1}\ot p_{k}) \in d\Z_{\geq 0}$,
where $c(\pi_k)=c_{p_k}$ and $\pi_s=u_{c_{g_0}}^{(0)}$. 
Finally, let $_t^d\Ppm$ (resp.\ $_t\Ppm$, $_t\Pppm$) denote the set of multi-grounded partitions of $^d\Ppm$ (resp.\ $\Ppm$, $\Pppm$) with number of parts divisible by $t$.

\begin{ex}
Assume that for all $b,b' \in \B$, $DH_{\lambda}(b' \otimes b)=M_{b,b'}$ given in Example \ref{ex}. Then $^2\Pp^{\gg}_{c_1,c_3}$ is the set of multi-grounded partitions $\pi=(\pi_0,\cdots,\pi_{s-1},1_{c_1},-1_{c_3})$ of $\Pp^{\gg}_{c_1,c_3}$ such that for all $k\in\{0,\ldots,s-1\}$,
$ \pi_k-\pi_{k+1} - DH_{\lambda}(p_{k+1}\ot p_{k}) \in 2\Z_{\geq 0}$.
Given that all the values in the matrix $M$ are even and that these multi-grounded partitions always end with $(1_{c_1},-1_{c_3})$, the multi-grounded partitions in $^2\Pp^{\gg}_{c_1,c_3}$ only have $odd$ parts.
For example, the multi-grounded partition $(7_{c_1},5_{c_2},3_{c_3},1_{c_2},1_{c_1},-1c_{c_3})$ belongs to $^2\Pp^{\gg}_{c_1,c_3}$. It also belongs to $_2^2\Pp^{\gg}_{c_1,c_3}$ as its number of parts is divisible by $2$.
\end{ex}

Now that we have introduced all the relevant notation, we can state our main theorem, which makes the connection between perfect crystals and multi-grounded partitions.

\begin{theo}\label{theo:formchar2}
Setting $q=e^{-\delta/(d_0D)}$ and $c_b=e^{\wt b}$ for all $b\in \B$, we have $c_{g_0}\cdots c_{g_{t-1}}=1$, and the character of the irreducible highest weight $U_q(\gf)$-module
$L(\lambda)$ is given by the following expressions:
\begin{align*}
\sum_{\mu\in _t\Pppm} C(\pi)q^{|\pi|} &= e^{-\lambda}\mathrm{ch}(L(\lambda)),\\
 \sum_{\pi\in \, _t^d\Ppm} C(\pi)q^{|\pi|} &= \frac{e^{-\lambda}\mathrm{ch}(L(\lambda))}{(q^d;q^d)_{\infty}}.
\end{align*}
\end{theo}

Theorem \ref{theo:formchar2} is proved in detail in \cite{DKmulti}, using the (KMN)$^2$ crystal base character formula and the two following bijections.

\begin{prop}
\label{prop:flat2}
Define a map $\phi \colon \Pp(\lambda) \to \Pppm$ by $\phi(\p_{\lambda}) = (u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)})$ and for all  $\p=(p_k)_{k\geq 0} \in \Pp(\lambda)$ different from $\p_{\lambda}$,
\[ \phi(\p) = (\pi_0,\cdots,\pi_{mt-1},u_{c_{g_0}}^{(0)},\ldots, u_{c_{g_{t-1}}}^{(t-1)}),\]
where for all $k\in \{0,\cdots,mt-1\}$, $c(\pi_k) = c_{p_k}$,
\[
 \pi_k = -\frac{1}{t}\sum_{j=0}^{t-1} (j+1) DH_{\lambda}(g_{j+1}\ot g_j)+\sum_{j=k}^{mt-1} DH_{\lambda}(p_{j+1} \ot p_j),
\]
and $m$ is the unique positive integer such that $(p_{(m-1)t},\ldots,p_{mt-1}) \neq (g_0,\ldots,g_{t-1})$ and $(p_{m't},\ldots,p_{(m'+1)t-1}) = (g_0,\ldots,g_{t-1})$ for all  $m'\geq m$.

Then $\phi$ defines a bijection between $\Pp(\lambda)$ and the set $_t\Pppm$ of partitions of $\Pppm$ whose number of parts is divisible by $t$.
\end{prop}


\begin{prop}\label{prop:diff2}
Let $^d\Pp$ be the set of classical partitions where all parts are divisible by $d$. There is a bijection $\Phi_d$ between $_t^d\Ppm$ and $_t\Pppm\times \, ^d\Pp$, such that if $\Phi_d(\pi) =(\mu,\nu)$, then $|\pi|=|\mu|+|\nu|$, and by setting $c_{g_0}\cdots c_{g_{t-1}}=1$, we have $C(\pi)=C(\mu)$.
\end{prop}


The additional parameter $d$ allows us to have a refined equality and to simplify some calculations. It is 
particularly useful when
$DH_{\lambda}(\B\ot\B)\in d\Z$, in which case the parts of our partitions all belong to the same congruence class modulo $d$.


\section{Example of application: character formulas for the standard level $1$ modules of the Lie algebra $A_{2n-1}^{(2)}(n\geq 3)$}
\label{sec:examples}

The crystal $\B$ of the vector representation of $A_{2n-1}^{(2)}(n\geq 3)$ is given by the crystal graph in Figure \ref{fig5} with, for all $u\in \{1,\ldots,n\}$, the weights
$\wt (u) = -\wt (\overline{u}) = \frac{1}{2}\alpha_n+\sum_{i=u}^{n-1} \alpha_i .$
Here, the null root is $\delta = \alpha_0+\alpha_1+\alpha_n+2\sum_{i=2}^{n-1} \alpha_i$.

\begin{figure}[h]
\[
\begin{tikzpicture}[scale=0.8, every node/.style={scale=0.8}]
%%crystal graph B
\draw (-1,-0.75) node{$\B$ :};
\draw (-6,-0.75) node{$b^{\Ll_0}=b_{\Ll_1}=1\qquad b^{\Ll_1}=b_{\Ll_0}=\overline{1}$};
\draw (-6,-1.5) node{$\p_{\Ll_0}=(\cdots\,\overline{1}\,1\,\overline{1}\,1\,\overline{1})\qquad \p_{\Ll_1}=(\cdots\,1\,\overline{1}\,1\,\overline{1}\,1)$};
\draw (0,0) node {$1$};
\draw (1.5,0) node {$2$};
\draw (5,0) node {$n-1$};
\draw (6.75,0) node {$n$};
\draw (3.15,0) node {$\cdots$};
\draw (0,-1.5) node {$\overline{1}$};
\draw (1.5,-1.5) node {$\overline{2}$};
\draw (5,-1.5) node {$\overline{n-1}$};
\draw (6.75,-1.5) node {$\overline{n}$};
\draw (3.15,-1.5) node {$\cdots$};

\foreach \x in {0,1,4.5} 
\foreach \y in {0,1}
\draw (-0.2+1.5*\x,0.2-1.5*\y)--(0.2+1.5*\x,0.2-1.5*\y)--(0.2+1.5*\x,-0.2-1.5*\y)--(-0.2+1.5*\x,-0.2-1.5*\y)--(-0.2+1.5*\x,0.2-1.5*\y);
\foreach \x in {0} 
\foreach \y  in {0,1}
\draw (-0.45+5+2*\x,0.2-1.5*\y)--(0.45+5+2*\x,0.2-1.5*\y)--(0.45+5+2*\x,-0.2-1.5*\y)--(-0.45+5+2*\x,-0.2-1.5*\y)--(-0.45+5+2*\x,0.2-1.5*\y);

\draw [thick, <-] (0.05,-0.25)--(1.45,-1.25);
\draw [thick, <-] (1.45,-0.25)--(0.05,-1.25);
\draw (0.3,-0.9) node {\footnotesize{$0$}};
\draw (1.2,-0.9) node {\footnotesize{$0$}};


\draw [thick, ->] (0.3,0)--(1.2,0);
\draw [thick, <-] (0.3,-1.5)--(1.2,-1.5);
\draw (0.75,0.15) node {\footnotesize{$1$}};
\draw (0.75,-1.67) node {\footnotesize{$1$}};
\draw [thick, ->] (1.8,0)--(2.7,0);
\draw [thick, <-] (1.8,-1.5)--(2.7,-1.5);
\draw (2.25,0.15) node {\footnotesize{$2$}};
\draw (2.25,-1.67) node {\footnotesize{$2$}};
\draw [thick, ->] (3.6,0)--(4.5,0);
\draw [thick, <-] (3.6,-1.5)--(4.5,-1.5);
\draw (4.05,0.15) node {\footnotesize{$n-2$}};
\draw (4.05,-1.67) node {\footnotesize{$n-2$}};
\draw [thick, ->] (5.6,0)--(6.4,0);
\draw [thick, <-] (5.6,-1.5)--(6.4,-1.5);
\draw (6,0.15) node {\footnotesize{$n-1$}};
\draw (6,-1.67) node {\footnotesize{$n-1$}};

\draw [thick, ->] (6.75,-0.25)--(6.75,-1.25);
\draw (6.9,-0.75) node {\footnotesize{$n$}};

\end{tikzpicture}
\]
\caption{Crystal graph $\B$  of the vector representation for the Lie algebra $A_{2n-1}^{(2)}(n\geq 3)$.}
\label{fig5}
\end{figure}



%We use the crystal graph for $\B\ot \B$ given in Figure \ref{fig6} to compute the energy function $H$ on $\B\ot \B$ such that  $H(1\ot \overline{1})=-1$.
%
%\begin{figure}[H]
%\begin{center}
%\begin{tikzpicture}[scale=0.5, every node/.style={scale=0.5}]
%\draw [thick,->] (-10,-1-0.5)--(-9,-1-0.5);
%\draw (-8.8,-1-0.5) node[right] {: $0$-arrow};
%\draw [->>] (-10,-2-0.5)--(-9,-2-0.5);
%\draw (-8.8,-2) node[right] {: $1$-arrow};
%\draw [->] (-10,-3-0.5)--(-9,-3-0.5);
%\draw (-8.8,-3-0.5) node[right] {: $n$-arrow};
%\draw [dashed,->] (-10,-4-0.5)--(-9,-4-0.5);
%\draw (-8.8,-4-0.5) node[right] {: paths of $i$-arrows, for consecutive $i\neq 0,1,n$};
%\draw [blue] (-10,-4.6-0.5)--(-9,-4.6-0.5)--(-9,-5.4-0.5)--(-10,-5.4-0.5)--(-10,-4.6-0.5);
%\draw (-8.8,-5-0.5) node[right] {: connected component of $1\ot \overline{1}$};
%\draw [dashed,red] (-10,-5.6-0.5)--(-9,-5.6-0.5)--(-9,-6.4-0.5)--(-10,-6.4-0.5)--(-10,-5.6-0.5);
%\draw (-8.8,-6-0.5) node[right] {: connected component of $1\ot 2$};
%\draw [thick,foge] (-10,-6.6-0.5)--(-9,-6.6-0.5)--(-9,-7.4-0.5)--(-10,-7.4-0.5)--(-10,-6.6-0.5);
%\draw (-8.8,-7-0.5) node[right] {: connected component of $1\ot 1$};
%%%crystal graph B
%\foreach \x in {1,...,7} 
%\foreach \y in {1,...,7}
%\draw (-0.2-0.4+3*\x,0.2-1.5*\y)--(0.2-0.4+3*\x,0.2-1.5*\y)--(0.2-0.4+3*\x,-0.2-1.5*\y)--(-0.2-0.4+3*\x,-0.2-1.5*\y)--(-0.2-0.4+3*\x,0.2-1.5*\y);
%\foreach \x in {1,...,7} 
%\foreach \y in {1,...,7}
%\draw (-0.2+0.4+3*\x,0.2-1.5*\y)--(0.2+0.4+3*\x,0.2-1.5*\y)--(0.2+0.4+3*\x,-0.2-1.5*\y)--(-0.2+0.4+3*\x,-0.2-1.5*\y)--(-0.2+0.4+3*\x,0.2-1.5*\y);
%
%\foreach \x in {1,...,7} 
%\foreach \y in {1,...,7}
%\draw (3*\x,-1.5*\y) node {$\ot$};
%
%\foreach \x in {1,...,7} 
%\draw (0.4+3*\x,-10.5) node {$\overline{n}$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+21,-1.5*\y) node {$\overline{n}$};
%
%\foreach \x in {1,...,7}
%\draw (0.4+3*\x,-1.5) node {$\overline{n}$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+3,-1.5*\y) node {$\overline{n}$};
%
%\foreach \x in {1,...,7}
%\draw (0.4+3*\x,-3) node {$\overline{2}$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+6,-1.5*\y) node {$\overline{2}$};
%
%\foreach \x in {1,...,7}
%\draw (0.4+3*\x,-4.5) node {$\overline{1}$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+9,-1.5*\y) node {$\overline{1}$};
%
%\foreach \x in {1,...,7}
%\draw (0.4+3*\x,-6) node {$1$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+12,-1.5*\y) node {$1$};
%
%\foreach \x in {1,...,7}
%\draw (0.4+3*\x,-7.5) node {$2$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+15,-1.5*\y) node {$2$};
%
%\foreach \x in {1,...,7}
%\draw (0.4+3*\x,-9) node {$n$};
%\foreach \y in {1,...,7} 
%\draw (-0.4+18,-1.5*\y) node {$n$};
%
%\foreach \x in {1,...,5,7} 
%\draw [->] (3*\x,-9.25)--(3*\x,-10.25);
%
%\foreach \x in {2,3,4,6,7} 
%\draw [dashed,->] (3*\x,-1.75)--(3*\x,-2.75);
%\foreach \x in {2,3,4,6,7} 
%\draw [dashed,->] (3*\x,-7.75)--(3*\x,-8.75);
%
%\foreach \x in {2,4}
%\foreach \y in {1,2,4,6,7}
%\draw [->>] (0.9+3*\x,-1.5*\y)--(2.1+3*\x,-1.5*\y);
%
%\foreach \x in {2,3}
%\foreach \y in {1,2,3,6,7}
%\draw [thick,->] (0.65+3*\x,0.25-1.5*\y) arc (110:70:7);
%
%\foreach \y in {2,4}
%\foreach \x in {1,3,5,6,7}
%\draw [->>] (3*\x,-0.25-1.5*\y)--(3*\x,-1.25-1.5*\y);
%
%\foreach \y in {2,3}
%\foreach \x in {1,4,5,6,7}
%\draw [thick,->] (0.65+3*\x,-0.25-1.5*\y) arc (20:-20:3.7);
%
%\foreach \y in {2,...,6} 
%\draw [->] (18.9,-1.5*\y)--(20.1,-1.5*\y);
%
%\foreach \y in {1,3,4,5,7} 
%\draw [dashed,->] (3.9,-1.5*\y)--(5.1,-1.5*\y);
%\foreach \y in {1,3,4,5,7} 
%\draw [dashed,->] (15.9,-1.5*\y)--(17.1,-1.5*\y);
%
%\draw [blue] (-0.7+12,0.25-4.5)--(0.7+12,0.25-4.5)--(0.7+12,-0.25-4.5)--(-0.7+12,-0.25-4.5)--(-0.7+12,0.25-4.5);
%
%\draw [dashed,red] (-0.7+12,0.25-1.5)--(0.7+18,0.25-1.5)--(0.7+18,-0.25-4.5)--(-0.7+15,-0.25-4.5)--(-0.7+12,-0.25-3)--(-0.7+12,0.25-1.5);
%\draw [dashed,red] (-0.7+3,1-3)--(0.7+6,0.25-4.5)--(0.7+6,-0.25-4.5)--(-0.7+3,-0.25-4.5)--(-0.7+3,1-3);
%\draw [dashed,red] (-0.7+12,0.25-7.5)--(0.7+12,0.25-7.5)--(-1.2+18,-0.25-9)--(-0.7+12,-0.25-9)--(-0.7+12,0.25-7.5);
%
%\draw [thick,foge] (-0.7+3,0.25-6)--(0.7+9,0.25-6)--(0.7+9,-0.25-9)--(-0.7+3,-0.25-9)--(-0.7+3,0.25-6);
%\draw [thick,foge] (-0.7+3,0.25-1.5)--(0.7+9,0.25-1.5)--(0.7+9,-0.25-4.5)--(-0.7+9,-0.25-4.5)--(-0.7+3,-0.25-1.5)--(-0.7+3,0.25-1.5);
%\draw [thick,foge] (-0.7+12,0.25-6)--(0.7+18,0.25-6)--(0.7+18,-0.25-9)--(-0.7+18,-0.25-9)--(-0.7+12,-0.25-6)--(-0.7+12,0.25-6);
%\end{tikzpicture}
%\end{center}
%\caption{Crystal graph of $\B\ot \B$ for the Lie algebra $A_{2n-1}^{(2)} (n\geq 3)$}
%\label{fig6}
%\end{figure}
The energy function such that $H(1\ot \overline{1})=-1$ is given by the following matrix, where $H(b_1\ot b_2)$ is the entry in column $b_1$ and row $b_2$:
$$
H=
\bordermatrix{
\text{}&1& 2 &\cdots &n & \overline{n} & \cdots & \overline{2} &\overline{1}
\cr 1&1&\cdots&\cdots&\cdots&\cdots& \cdots& \cdots&1 
\cr 2&0&\ddots&&&&&&\vdots 
\cr \vdots&\vdots &\ddots&\ddots&&&1^*&&\vdots
\cr n&\vdots&&\ddots&\ddots & &&&\vdots
\cr \overline{n}&\vdots&&&\ddots&\ddots&& &\vdots
\cr \vdots&\vdots&&0^*&&\ddots&\ddots&&\vdots
\cr \overline{2}&0&0&&& &\ddots &\ddots &\vdots
\cr \overline{1}&-1&0&\cdots&\cdots&\cdots&\cdots& 0&1
} .
$$

We start by computing the character of $L(\Lambda_0)$.
Recall that the ground state path of $\Lambda_0$ is $\p_{\Lambda_0}=(g_k)_{k=0}^\infty$ with $g_{2k}=\overline{1}$ and $g_{2k+1}=1$ for all $k\geq 0$. Here, the period of the ground state path is $t=2$, and our choice of particular value $H(1\ot \overline{1})=-1$ for the energy function gives
$H(g_{2k+2}\ot g_{2k+1})=-H(g_{2k+1}\ot g_{2k})=1$.
Thus we have $H(1\ot \overline{1})+H(\overline{1} \ot 1)=0$, and by \eqref{eq:Hlamb},  $H_{\Lambda_0}=H$. 
By \eqref{eq:choiceofu}, we obtain that $u^{(0)}=-1$ and $u^{(1)}=1$.

We apply \Thm{theo:formchar2} with $d=2$ and $D=2$, which is allowed because $H(g_{1}\ot g_0)+2H(g_2\ot g_1)=1$. We obtain
\begin{equation}
\label{eq:tha210}
\sum_{\pi\in \, _2^2\Pp_{c_{\overline{1}}c_1}^{\gg}} C(\pi)q^{|\pi|} = \frac{e^{-\Ll_0}\mathrm{ch}(L(\Lambda_0))}{(q^2;q^2)_{\infty}},
\end{equation}
where $q=e^{-\delta/2}$ and $c_b=e^{\wt b}$ for all $b\in \B$.
 Recall that $_2^2\Pp_{c_{\overline{1}}c_1}^{\gg}$ is the set of multi-grounded partitions $\pi=(\pi_0,\ldots,\pi_{2s-1},-1_{c_{\overline{1}}},1_{c_1})$ with relation $\gg$ and ground $c_{\overline{1}},c_1$, \textbf{having an even number of parts}, such that for all $k \in \{0, \dotsc, 2s-1\}$,
\begin{equation}
\label{eq:a21parity}
\pi_k-\pi_{k+1} - 2H(p_{k+1}\ot p_{k}) \in 2 \Z_{\geq0},
\end{equation}
where $c(\pi_k)=c_{p_k}$ and $\pi_{2s}=-1_{c_{\overline{1}}}$.

We observe that, by \eqref{eq:a21parity} and the fact that $u^{(0)}=-1$, the multi-grounded partitions of $_2^2\Pp_{c_{\overline{1}}c_1}^{\gg}$ have parts with odd sizes, as the differences between consecutive parts are even and the grounds' sizes are odd (indeed, we always have the fixed tail $((-1)_{c_{\overline{1}}},1_{c_1})$).
Besides, computing the generating function for partitions in $_2^2\Pp_{c_{\overline{1}}c_1}^{\gg}$ is not too difficult. It suffices to notice that, combined with \eqref{eq:a21parity}, $\gg$ is the following partial order on the set of coloured odd integers:
\[\begin{array}{c} (-1)_{c_{\overline{1}}} \\ 1_{c_1}\end{array}\ll 1_{c_2}\ll\cdots \ll 1_{c_n}\ll 1_{c_{\overline{n}}}\ll \cdots \ll 1_{c_{\overline{2}}}\ll\begin{array}{c} 1_{c_{\overline{1}}} \\ 3_{c_1}\end{array}\ll 3_{c_2}\ll\cdots .\]
 We also note that, since $H(b\ot b)=1$ for all $b\in \B$, only parts coloured $c_1$ and $c_{\overline{1}}$ can appear several times, in sequences of the form
 \begin{equation*}\label{eq:twistedseq}
 \cdots \ll (2k-1)_{c_{\overline{1}}}\ll (2k+1)_{c_1}\ll(2k-1)_{c_{\overline{1}}}\ll \cdots \ll(2k-1)_{c_{\overline{1}}}\ll (2k+1)_{c_1}\ll\cdots .
 \end{equation*}
The generating function of these sequences for a fixed integer $k \geq 1$ is given by
$$ \frac{(1+c_{\overline{1}}q^{2k-1})(1+c_1q^{2k+1})}{(1-c_{\overline{1}}c_1q^{4k})}\, ,$$
where the denominator generates pairs $((2k-1)_{c_{\overline{1}}},(2k+1)_{c_1})$ that can repeat arbitrarily many times, and the numerator accounts for the possibility of having an isolated $(2k+1)_{c_1}$ on the left end of the sequence, or an isolated $(2k-1)_{c_{\overline{1}}}$ on the right end of the sequence.

% To compute the generating function of such sequences  for a fixed $k$, we distinguish $4$ cases:
% \begin{enumerate}
%  \item When the sequences begin and end with $(2k-1)_{c_{\overline{1}}}$, we then have an odd number of parts, and by gathering the pairs $(2k+1)_{c_1}\ll(2k-1)_{c_{\overline{1}}}$ after the first element (eventually no pairs),
%  we obtain the series
%  \[\frac{c_{\overline{1}}q^{2k-1}}{(1-c_{\overline{1}}c_1q^{4k})} .\]
%  \item In the same way, when the sequences begin and end with $(2k+1)_{c_1}$, we have 
%  \[\frac{c_{1}q^{2k+1}}{(1-c_{\overline{1}}c_1q^{4k})} .\]
%  \item When the sequences have an even non zero number of parts, by taking pairwise and considering whether the sequences begin by either $(2k-1)_{c_{\overline{1}}}$ or $(2k+1)_{c_{1}}$, we obtain
%  \[\frac{2c_{\overline{1}}c_1q^{4k}}{(1-c_{\overline{1}}c_1q^{4k})} .\]
%  \item Finally, when we have neither $(2k-1)_{c_{\overline{1}}}$ nor $(2k+1)_{c_{1}}$, the generating function is $1$.
% \end{enumerate}
%Gathering these $4$ cases, the generating function of such sequences (possibly empty or having one element) for a fixed positive integer $k$ is 
%\begin{equation}\label{eq:twisted}
% \frac{(1+c_{\overline{1}}q^{2k-1})(1+c_1q^{2k+1})}{(1-c_{\overline{1}}c_1q^{4k})} .
%\end{equation}

Note that for $k=0$, only the sequence $(1_{c_1},(-1)_{c_{\overline{1}}},1_{c_1})$ can occur at the tail of the partitions grounded in $c_{\overline{1}},c_1$, but not the sequence $((-1)_{c_{\overline{1}}},1_{c_1},(-1)_{c_{\overline{1}}},1_{c_1})$, as this would violate the definition of multi-grounded partitions.
So, if we temporarily forget the condition on the even number of parts in $_2^2\Pp_{c_{\overline{1}}c_1}^{\gg}$, the generation function would be 
\[(1+c_1q)\cdot\frac{(-c_1q^3,-c_{\overline{1}}q,\ldots,-c_nq,-c_{\overline{n}}q;q^2)_{\infty}}{(c_{\overline{1}}c_1q^4;q^4)_{\infty}} =\frac{(-c_1q,-c_{\overline{1}}q,\ldots,-c_nq,-c_{\overline{n}}q;q^2)_{\infty}}{(c_{\overline{1}}c_1q^4;q^4)_{\infty}} .\]
Now to take into account the fact that there are an even number of parts, we use \eqref{eq:evennumberparts}.
Then the final expression \eqref{eq:a210} follows by using \eqref{eq:tha210}.

\bi

We now turn to $L(\Lambda_1)$ with a similar reasoning. 
Recall that the ground state path of $\Lambda_1$ is $(g_k)_{k=0}^\infty$ with $g_{2k+1}=\overline{1}$ and $g_{2k}=1$ for all $k\geq 0$. We still have $H_{\Lambda_1}=H$, and by setting $D=2$, we have
by \eqref{eq:choiceofu} that $u^{0}=1$ and $u^{(1)}=-1$.
\Thm{theo:formchar2} gives
\begin{equation*}
\label{eq:tha211}
\sum_{\pi\in \, _2^2\Pp_{c_1c_{\overline{1}}}^{\gg}} C(\pi)q^{|\pi|} = \frac{e^{-\Ll_1}\mathrm{ch}(L(\Lambda_1))}{(q^2;q^2)_{\infty}},
\end{equation*}
where $q=e^{-\delta/2}$ and $c_b=e^{\wt b}$ for all $b\in \B$.

So we need to study the set of multi-grounded partitions with ground $c_1,c_{\overline{1}}$ corresponding to $_2^2\Pp_{c_1c_{\overline{1}}}^{\gg}$.
We have almost the same set of partitions as in $_2^2\Pp_{c_{\overline{1}}c_1}^{\gg}$, except that now the tail is always $(1_{c_1},(-1)_{\overline{1}})$, and we can end with the sequence $((-1)_{\overline{1}},1_{c_1},(-1)_{\overline{1}})$, but 
not with $(1_{c_1},(-1)_{\overline{1}},1_{c_1},(-1)_{\overline{1}})$.

Thus the generating function where we temporarily omit the condition on the parity of the number of parts is given by
%\begin{footnotesize}
\[(1+c_{\overline{1}}q^{-1})\cdot\frac{(-c_1q^3,-c_{\overline{1}}q,\ldots,-c_nq,-c_{\overline{n}}q;q^2)_{\infty}}{(c_{\overline{1}}c_1q^4;q^4)_{\infty}} =\frac{(-c_1q^3,-c_{\overline{1}}q^{-1},\ldots,-c_nq,-c_{\overline{n}}q;q^2)_{\infty}}{(c_{\overline{1}}c_1q^4;q^4)_{\infty}},\]
%\end{footnotesize}
and the proof ends similarly as for $L(\Lambda_0)$.

\acknowledgements{We thank Leonard Hardiman and Ole Warnaar for their comments on earlier versions of this paper and their helpful suggestions to improve it.}

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