Rigged configurations of type $D_4^{(3)}$ and the
filling map

Travis Scrimshaw$^{1}$†

$^{1}$University of California, Davis, CA 95616

**Abstract.** We give a statistic preserving bijection from rigged configurations to a tensor product of Kirillov–Reshetikhin crystals $\otimes_{i=1}^N B^{1,s_i}$ in type $D_4^{(3)}$ by using virtualization into type $D_4^{(1)}$. We consider a special case of this bijection with $B = B^{1,s}$, and we obtain the so-called Kirillov–Reshetikhin tableaux model for the Kirillov–Reshetikhin crystal.

**Résumé.** Nous donnons une bijection prouvant les statistiques entre les configurations gréées et les produits tensoriels de cristaux de Kirillov–Reshetikhin $\otimes_{i=1}^N B^{1,s_i}$ de type $D_4^{(3)}$, via une virtualisation en type $D_4^{(1)}$. Nous considérons un cas particulier de cette bijection pour $B = B^{1,s}$ et obtenons ainsi les modèles de tableaux appelés Kirillov–Reshetikhin pour le cristal Kirillov–Reshetikhin.

**Keywords:** rigged configuration, Kirillov–Reshetikhin crystal, bijection

# 1 Introduction

Rigged configurations were first introduced by Kerov, Kirillov, and Reshetikhin in [14, 15] as combinatorial objects that index solutions to the Bethe Ansatz for the Heisenberg spin chains. Rigged configurations were shown to be in bijection with semi-standard tableaux and classical highest weight elements of a tensor power of the vector representation in type $A_n^{(1)}$. This bijection was then extended to Littlewood–Richardson tableaux [16], to non-exceptional types [20], and to type $E_6^{(1)}$ [19]. This bijection $\Phi$ between rigged configurations and the tensor powers has been further expanded to include classically highest weight elements in a tensor product of certain Kirillov–Reshetikhin (KR) crystals [16, 21, 18, 27, 28].

Rigged configurations have been shown to display remarkable representation theoretic properties. A (classical) crystal structure was first given for simply-laced types [26], which was then extended to all finite types [27] and affine types [24]. While $\Phi$ is defined recursively, making it difficult to work with, it preserves certain natural statistics (cocharge and energy), giving a bijective proof of the $X = M$ conjecture of [5, 6]. Furthermore, the combinatorial $R$-matrix transforms into the identity map on rigged configurations under $\Phi$. Rigged configurations are also well-behaved under virtualization [21, 22, 27], a process of realizing a non-simply-laced type crystal inside of a simply-laced type, and embeddings $B(\lambda) \to B(\mu)$ where $\lambda \le \mu$ component-wise, leading to a model for $B(\infty)$ [24].

<footnote>†Partially supported by NSF grant OCI-1147247</footnote>
<figure><img src="image_1.png" /><figcaption>Fig. 2.1: Dynkin diagram of type $D_4^{(3)}$.</figcaption></figure>

KR crystals in non-exceptional types were given a combinatorial model in [4] using Kashiwara–Nakashima tableaux [13]. The bijection $\Phi$ has also lead to a new tableaux model for KR crystals, coined Kirillov–Reshetikhin (KR) tableaux, using filled rectangular tableaux [18, 25, 27]. The map between Kashiwara–Nakashima tableaux [13] and the KR tableaux is called the filling map.

The goal of this work is to extend $\Phi$ to type $D_4^{(3)}$ and describe the filling map. For this extended abstract, we will be focusing on the KR crystals $B^{1,s}$ and the rigged configurations associated with tensor products of the form $\bigotimes_{i=1}^N B^{1,s_i}$. In particular, we show $\Phi$ is a classical crystal isomorphism, and we describe the filling map for $B^{1,s}$. We do so by showing the filling map and bijection commute with the virtualization map, proving more special cases of many of the conjectures stated in [27].

This extended abstract is organized as follows. In Section 2, we give background on crystals, virtualization, and rigged configurations. In Section 3, we describe the bijection $\Phi$. In Section 4, we describe the filling map. In Section 5, we describe the virtualization map and our main results. In Section 6, we give possible extensions to $B^{2,s}$ and some open questions. We conclude in Section 7 with some examples using Sage [29].

# 2 Background

## 2.1 Crystals

For this extended abstract, let $\mathfrak{g}$ be the Kac–Moody algebra of type $D_4^{(3)}$ with index set $I = \{0, 1, 2\}$, generalized Cartan matrix $A = (A_{ij})_{i,j \in I}$, weight lattice $P$, root lattice $Q$, fundamental weights $\{\Lambda_i | i \in I\}$, simple roots $\{\alpha_i | i \in I\}$, and simple coroots $\{h_i | i \in I\}$. There is a canonical pairing $\langle \cdot, \cdot \rangle: P^\vee \times P \to \mathbb{Z}$ defined by $\langle h_i, \alpha_j \rangle = A_{ij}$, where $P^\vee$ is the dual weight lattice. Let $\mathfrak{g}_0$ denote the classical subalgebra of type $G_2$ with index set $I_0 = \{1, 2\}$, weight lattice $\bar{P}$, root lattice $\bar{Q}$, fundamental weights $\{\bar{\Lambda}_1, \bar{\Lambda}_2\}$, and simple roots $\{\bar{\alpha}_1, \bar{\alpha}_2\}$.

An abstract $U_q(\mathfrak{g})$-crystal is a nonempty set $\mathcal{B}$ together with a weight function $\mathrm{wt}: \mathcal{B} \to P$, crystal operators $e_a, f_a: \mathcal{B} \to \mathcal{B} \sqcup \{0\}$, and maps $\varepsilon_a, \varphi_a: \mathcal{B} \to \mathbb{Z} \sqcup \{-\infty\}$ for $a \in I$, subject to the conditions

1. $\varphi_a(b) = \varepsilon_a(b) + \langle h_a, \mathrm{wt}(b) \rangle$ for all $a \in I$,

2. if $e_ab \in \mathcal{B}$, then $\varepsilon_a(e_ab) = \varepsilon_a(b) - 1$, $\varphi_a(e_ab) = \varphi_a(b) + 1$, and $\mathrm{wt}(e_ab) = \mathrm{wt}(b) + \alpha_a$.

3. if $f_ab \in \mathcal{B}$, then $\varepsilon_a(f_ab) = \varepsilon_a(b) + 1$, $\varphi_a(f_ab) = \varphi_a(b) - 1$, and $\mathrm{wt}(f_ab) = \mathrm{wt}(b) - \alpha_a$.

4. $f_ab = b'$ if and only if $b = e_ab'$ for $b, b' \in \mathcal{B}$ and $a \in I$,

5. if $\varphi_a(b) = -\infty$ for $b \in \mathcal{B}$, then $e_ab = f_ab = 0$.

We define for all $b \in \mathcal{B}$

$$ \varepsilon_a(b) = \max\{k \in \mathbb{Z}_{\ge 0} \mid e_a^k b \ne 0\}, \quad \varphi_a(b) = \max\{k \in \mathbb{Z}_{\ge 0} \mid f_a^k b \ne 0\}. \quad (2.1) $$
<figure><img src="image_2.png" /><figcaption>Fig. 2.2: The KR crystal $B^{1,1}$ of type $D_4^{(3)}$ which is isomorphic to $B(\bar{\Lambda}_1) \oplus B(0)$ as $U_q(\mathfrak{g}_0)$-crystals.</figcaption></figure>

An abstract $U_q(\mathfrak{g})$-crystal with $\epsilon_a$ and $\varphi_a$ defined as above is called a *regular crystal*.

Let $\mathcal{B}_1$ and $\mathcal{B}_2$ be abstract $U_q(\mathfrak{g})$-crystals. The tensor product of crystals $\mathcal{B}_2 \otimes \mathcal{B}_1$ is defined to be the Cartesian product $\mathcal{B}_2 \times \mathcal{B}_1$ with the crystal structure

$$e_i(b_2 \otimes b_1) = \begin{cases} e_i b_2 \otimes b_1 & \text{if } \epsilon_i(b_2) > \varphi_i(b_1), \\ b_2 \otimes e_i b_1 & \text{if } \epsilon_i(b_2) \le \varphi_i(b_1), \end{cases} \quad \epsilon_i(b_2 \otimes b_1) = \max(\epsilon_i(b_2), \epsilon_i(b_1) - \langle h_i, \text{wt}(b_2) \rangle)$$

$$f_i(b_2 \otimes b_1) = \begin{cases} f_i b_2 \otimes b_1 & \text{if } \epsilon_i(b_2) \ge \varphi_i(b_1), \\ b_2 \otimes f_i b_1 & \text{if } \epsilon_i(b_2) < \varphi_i(b_1), \end{cases} \quad \varphi_i(b_2 \otimes b_1) = \max(\varphi_i(b_1), \varphi_i(b_2) + \langle h_i, \text{wt}(b_1) \rangle)$$

$$\mathrm{wt}(b_2 \otimes b_1) = \mathrm{wt}(b_2) + \mathrm{wt}(b_1).$$

**Remark 2.1** *Our tensor product convention is the opposite to that given in [12].*

Let $\mathcal{B}_1$ and $\mathcal{B}_2$ be two abstract $U_q(\mathfrak{g})$-crystals. A *crystal morphism* $\psi: \mathcal{B}_1 \to \mathcal{B}_2$ is a map $\mathcal{B}_1 \sqcup \{0\} \to \mathcal{B}_2 \sqcup \{0\}$ with $\psi(0) = 0$ such that for $b \in \mathcal{B}_1$

1. if $\psi(b) \in \mathcal{B}_2$, then $\mathrm{wt}(\psi(b)) = \mathrm{wt}(b)$, $\epsilon_i(\psi(b)) = \epsilon_i(b)$, and $\varphi_i(\psi(b)) = \varphi_i(b)$;

2. we have $\psi(e_i b) = e_i \psi(b)$ provided $\psi(e_i b) \ne 0$ and $e_i \psi(b) \ne 0$;

3. we have $\psi(f_i b) = f_i \psi(b)$ provided $\psi(f_i b) \ne 0$ and $f_i \psi(b) \ne 0$.

A crystal embedding or isomorphism is a crystal morphism such that the induced map $\mathcal{B}_1 \sqcup \{0\} \to \mathcal{B}_2 \sqcup \{0\}$ is an embedding or bijection respectively. A crystal morphism is *strict* if it commutes with all crystal operators.

If an abstract $U_q(\mathfrak{g})$-crystal $\mathcal{B}$ is isomorphic to the crystal basis of an integrable $U_q(\mathfrak{g})$-module, we simply say $\mathcal{B}$ is a $U_q(\mathfrak{g})$-crystal. In particular, an irreducible highest weight $U_q(\mathfrak{g}_0)$-module with highest weight $\lambda$ admits a crystal basis [11], which we denote by $B(\lambda)$. Moreover there is a unique element $u_\lambda \in B(\lambda)$ such that $\mathrm{wt}(u_\lambda) = \lambda$ and $e_a u_\lambda = 0$ for all $a \in I_0$. For each dominant integral weight $\lambda = k_1 \bar{\Lambda}_1 + k_2 \bar{\Lambda}_2$, we can associate a partition $(k_1 + k_2, k_2)$. We can realize $B(\lambda)$ as semistandard tableaux of shape $\lambda$ filled with entries in $B(\bar{\Lambda}_1)$ whose crystal structure is given by embedding into $B(\bar{\Lambda}_1)^{\otimes |\lambda|}$ using the reverse far-eastern reading word. The resulting tableaux were explicitly described by Kang and Misra [9].

## 2.2 Kirillov–Reshetikhin crystals

An important class of finite dimensional $U'_q(\mathfrak{g})$-representations are Kirillov–Reshetikhin (KR) modules $W^{r,s}$ indexed by $r \in I_0$ and $s \in \mathbb{Z}_{>0}$. KR modules are characterized by their Drinfeld polynomials [2, 3]
and correspond to the minimal affinization of $B(s\bar{\Lambda}_r)$ [1]. The KR modules $W^{1,s}$ admit a crystal basis called Kirillov-Reshetikhin (KR) crystals and denoted by $B^{1,s}$. As $U_q(g_0)$-crystals, we have $B^{1,s} \cong \bigoplus_{k=1}^s B(k\bar{\Lambda}_1)$, and $B^{1,s}$ is a perfect crystal [10]. This means we can use a semi-infinite tensor product of $B^{1,s}$ to realize highest weight $U_q(g)$-crystals, see [7] for details.

There is a statistic called energy defined on $B = \bigotimes_{i=1}^N B^{1,s_i}$ [5]. First we define the local energy function on $B^{1,s} \otimes B^{1,t}$ as follows. The combinatorial R-matrix is the unique $U'_q(g)$-crystal isomorphism $R: B^{1,s} \otimes B^{1,t} \to B^{1,t} \otimes B^{1,s}$ [10]. Let $c' \otimes c = R(b \otimes b')$.

$$H(e_i(b \otimes b')) = H(b \otimes b') + \begin{cases} -1 & i = 0 \text{ and } e_0(b \otimes b') = b \otimes e_0 b' \text{ and } e_0(c' \otimes c) = c' \otimes e_0 c, \\ 1 & i = 0 \text{ and } e_0(b \otimes b') = e_0 b \otimes b' \text{ and } e_0(c' \otimes c) = e_0 c' \otimes c, \\ 0 & \text{otherwise.} \end{cases} \quad (2.2)$$

The local energy function is defined up to an additive constant [8], and so we normalize $H$ by the condition $H(1^s \otimes 1^t) = 0$ where $1^k$ is row of length $k$ filled with 1. Next we define $D_{B^{1,s}}: B^{1,s} \to \mathbb{Z}$ by

$$D_{B^{1,s}}(b) = H(b \otimes b^{\sharp}) - H(1^s \otimes b^{\sharp}), \quad (2.3)$$

where $b^{\sharp}$ is the unique element such that $\varphi(b^{\sharp}) = s\Lambda_0$. Then we define

$$D(b_N \otimes \cdots \otimes b_1) = \sum_{1 \le i < j \le N} H_i R_{i+1} R_{i+2} \cdots R_{j-1} + \sum_{j=1}^{N} D_{B^{1,s_j}} R_1 R_2 \cdots R_{j-1}, \quad (2.4)$$

where $R_i$ and $H_i$ are the combinatorial R-matrix and local energy function, respectively, acting on the $i$-th and $(i+1)$-th factors and $D_{B^{1,s_j}}$ acts on the rightmost factor. We say the energy of an element $b \in B$ is $D(b)$.

## 2.3 Rigged configurations

Let $\mathcal{H}_0 = I_0 \times \mathbb{Z}_{>0}$. Consider a multiplicity array $L = (L_i^{(a)} \in \mathbb{Z}_{\ge 0} | (a, i) \in \mathcal{H}_0)$ and a dominant integral weight $\lambda$ of $\mathfrak{g}_0$. A $(L; \lambda)$-configuration is a sequence of partitions $\nu = \{\nu^{(a)} | a \in I\}$ such that

$$\sum_{(a,i) \in \mathcal{H}_0} im_i^{(a)} \bar{\alpha}_a = \sum_{(a,i) \in \mathcal{H}_0} iL_i^{(a)} \bar{\Lambda}_a - \lambda, \quad (2.5)$$

where $m_i^{(a)}$ is the number of parts of length $i$ in the partition $\nu^{(a)}$. We denote the set of $(L, \lambda)$-configurations by $C(L, \lambda)$. The vacancy numbers of $\nu \in C(L; \lambda)$ are defined as

$$p_i^{(a)} = \sum_{j \ge 1} \min(i, j) L_j^{(a)} - \sum_{(b, j) \in \mathcal{H}_0} A_{ab} \min(i, j) m_j^{(b)}. \quad (2.6)$$

A rigged configuration of classical weight $\lambda$ is a $(L; \lambda)$-configuration $\nu$, along with a sequence of multisets of integers $J = \{J_i^{(a)} | (a, i) \in \mathcal{H}_0\}$ such that $|J_i^{(a)}| = m_i^{(a)}$ (the size of $J_i^{(a)}$) and $\max J_i^{(a)} \le p_i^{(a)}$. (Often each $J_i^{(a)}$ will be sorted in weakly decreasing order.) So to each row of length $i$, we have an integer $x \in J_i^{(a)}$ and we call the pair $(i, x)$ a string. The integers $x \in J_i^{(a)}$ are called label, rigging, or
**Theorem 2.6 ([27])** Let $\mathfrak{g}_0$ be a Lie algebra of finite type. For $(\nu, J) \in \mathrm{RC}^*(L; \lambda)$, let $X_{(\nu, J)}$ be the closure of $(\nu, J)$ under $e_a$, $f_a$ for $a \in I_0$. Then $X_{(\nu, J)} \cong B(\lambda)$ as $U_q(\mathfrak{g}_0)$-crystals.

There is a statistic called cocharge on rigged configurations given by

$$cc(\nu, J) = \frac{1}{2} \sum_{a,b \in I_0} \sum_{i,j \in \mathbb{Z}_{>0}} (\alpha_a | \alpha_b) \min(i,j) m_i^{(a)} m_j^{(b)} + \sum_{(a,i) \in \mathcal{H}_0} \sum_{x \in J_i^{(a)}} x. \quad (2.8)$$

Moreover cocharge is invariant under $e_a$ and $f_a$ for $a \in I_0$ [27].

## 2.4 Virtual crystals

Let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra with index set $\hat{I}$ of type $D_4^{(1)}$ and $\hat{\mathfrak{g}}_0$ be of type $D_4$. We consider the diagram folding $\phi: \hat{I} \searrow I$ defined by $\phi(0) = 0$, $\phi(2) = 1$, and $\phi(1) = \phi(3) = \phi(4) = 2$. The folding $\phi$ restricts to a diagram folding of type $\hat{\mathfrak{g}}_0 \searrow \mathfrak{g}_0$, and by abuse of notation, we also denote this folding by $\phi$.

**Remark 2.7** To simplify our notation, for any object $X$ or $\bar{X}$ of $\mathfrak{g}_0$, we denote the corresponding object of $\hat{\mathfrak{g}}_0$ by $\hat{\bar{X}}$.

Furthermore, the folding $\phi$ induces an embedding of weight lattices $\Psi: \bar{P} \to \hat{P}$ given by

$$\bar{\Lambda}_a \mapsto \sum_{b \in \phi^{-1}(a)} \hat{\Lambda}_b, \qquad \bar{\alpha}_a \mapsto \sum_{b \in \phi^{-1}(a)} \hat{\alpha}_b. \qquad (2.9)$$

This gives an embedding of crystals as sets $v: B(\lambda) \to B(\Psi(\lambda))$, and let $V(\lambda)$ denote the image of $v$. We can define a crystal structure on $V$ which is induced from the crystal $B(\Psi(\lambda))$ by

$$e^v := \prod_{b \in \phi^{-1}(a)} \hat{e}_b, \qquad f^v := \prod_{b \in \phi^{-1}(a)} \hat{f}_b, \\ \varepsilon_a^v := \hat{\varepsilon}_x, \qquad \varphi_a^v := \hat{\varphi}_x, \\ \text{wt} := \Psi^{-1} \circ \widehat{\text{wt}}, \qquad (2.10)$$

where we fix some $x \in \phi^{-1}(a)$. We say the pair $(V(\lambda), B(\Psi(\lambda)))$ is a virtual crystal and the isomorphism $v$ is the virtualization map.

**Proposition 2.8 ([27])** Let $\mathfrak{g}_0$ be of finite type. Then we have $B(\lambda) \cong V(\lambda)$ as $U_q(\mathfrak{g}_0)$-crystals.

In particular, we can define a virtualization map on rigged configurations by

$$\hat{\nu}^{(b)} = \nu^{(a)}, \qquad (2.11a)$$

$$\hat{J}_i^{(b)} = J_i^{(a)} \qquad (2.11b)$$

for all $b \in \phi^{-1}(a)$ [27].

## 3 The bijection $\Phi$

Consider a tensor product of KR crystals $B = \bigotimes_{i=1}^N B^{r_i, s_i}$. We write $\mathrm{RC}(B)$ for $\mathrm{RC}(L)$ with $L_i^{(a)}$ equal to the number of factors $B^{a, i}$ occurring in $B$. In this section, we describe the map $\Phi: \mathrm{RC}(B) \to B$.
## 3.1 The basic algorithm $\delta$

We begin by describing the basic step $\delta$: $\text{RC}(B^{1,1} \otimes B^*) \to \text{RC}(B^*)$, where $B^*$ is some tensor product of KR crystals. Each step $\delta$ returns some element $b \in B^{1,1}$, which we use to create $B$. We note that this is the special case of the algorithm given in [17] for type $D_4^{(3)}$.

Set $\ell_0 = 1$. Do the following process for $a = 1$. Find the minimal integer $i \ge \ell_{a-1}$ such that $\nu^{(a)}$ has a singular string of length $i$. If no such $i$ exists, then set $b = a$ and $\ell_a = \infty$ and terminate. Otherwise set $\ell_a = i$ and repeat the above process for $a = 2$.

Suppose the process has not terminated. We remove the selected (singular) string of length $\ell_1$ from consideration. If there are no singular or quasi-singular strings in $\nu^{(a)}$ larger than $\ell_2$ or if $\ell_2 = \ell_1$ and there is only one string of length $\ell_1$ in $\nu^{(1)}$, then set $b = 3$ and terminate. Otherwise find the smallest $i \ge \ell_2$ that satisfies one of the following three mutually exclusive conditions:

(S) $J^{(1,i)}$ is singular and $i > 1$;

(P) $J^{(1,i)}$ is singular and $i = 1$;

(Q) $J^{(1,i)}$ is quasi-singular.

If (P) holds, set $b = \emptyset$, and $\ell_3 = i$ and terminate. If (S) holds, set $\ell_3 = i - 1$, $\bar{\ell}_3 = i$, say case (S) holds for $a = n$, and continue. If (Q) holds, find the minimal $j > i$ such that (S) holds. If no such $j$ exists, set $b = 0$ and terminate. Else set $\bar{\ell}_3 = j$ and say case (Q, S) holds and continue.

Suppose the process has not terminated, and let $a = 2$. If $\ell_a = \bar{\ell}_{a+1}$, then set $\bar{\ell}_a = \ell_a$, afterwards reset $\ell_a = \bar{\ell}_a - 1$, and say case (S2) holds for $a$. Otherwise find the minimal index $i \ge \bar{\ell}_{a+1}$ such that $\nu^{(a)}$ has a singular string of length $i$. If no such $i$ exists, set $b = \overline{a+1}$ and terminate. Otherwise set $\bar{\ell}_a = i$ and repeat this for $a = 1$ (there must exists at least two singular strings if $\bar{\ell}_3 = \bar{\ell}_1$ and case (S2) does not hold). If the process has not terminated, set $b = \overline{1}$.

Set all undefined $\ell_a$ and $\bar{\ell}_a$ for $a = 1, 2, 3$ to $\infty$.

## 3.2 Change in the rigged configuration

The rigged configurations change under $\delta$ as follows. We first remove a box from $\bar{\ell}_a$ in $\nu^{(a)}$ for $a = 1, 2$, and if case (S2) holds for $a$, we remove another box from that particular row, otherwise we remove a box from $\ell_a$. If case (S) holds, then remove two boxes from $\bar{\ell}_3$ and make the resulting string singular. If case (Q) holds, remove a box from $\ell_3$ and make the resulting string singular. If case (Q, S) holds, then we remove both boxes corresponding to $\ell_3$ and $\bar{\ell}_3$, but we make the smaller one (i.e. the row corresponding to $\ell_3$) singular and the larger one quasi-singular. Also make all the changed strings in $\nu^{(2)}$ singular.

**Remark 3.1** We can determine the inverse algorithm by roughly doing the opposite of the above; in particular, selecting largest (quasi)singular strings at most as long as before.

**Example 3.2** Using the rigged configuration $(\nu, J)$ from Example 2.2 and $B = B^{1,1} \otimes B^{1,2} \otimes B^{2,1}$. Applying the map $\delta$, we get $b = 3$ and

$$ \delta(\nu, J) = 3 \Box\Box\Box\Box 3 \quad -2 \Box\Box\Box\Box -2 $$
### 3.3 Extending to arbitrary rectangles

We now extend $\Phi$ to $B = \bigotimes_{i=1}^{N} B^{1,s_i}$ by defining the map

$$ \text{ls: } \mathrm{RC}(B^{1,s} \otimes B^*) \to \mathrm{RC}(B^{1,1} \otimes B^{1,s-1} \otimes B^*), $$

which is known as left-split. On the rigged configurations, the map ls is the identity (but perhaps increases the vacancy numbers) and a strict crystal embedding. Thus iterating ls with $\delta$, we obtain a map $\Phi: \mathrm{RC}(B) \to B$.

## 4 Filling map

We determine the highest weight rigged configurations for $B^{1,s}$ by using the virtual Kleber algorithm [22].

**Lemma 4.1** Consider the KR crystal $B^{1,s}$. We have $\mathrm{RC}(B^{1,s}) = \bigoplus_{k=0}^{s} \mathrm{RC}(B^{1,s}; k\bar{\Lambda}_1)$. Moreover the highest weight rigged configurations in $\mathrm{RC}(B^{1,s}; k\bar{\Lambda}_1)$ are given by $\nu^{(1)} = (s-k, s-k)$ and $\nu^{(2)} = (s-k)$ with all labels 0.

From Lemma 4.1 and the $U_q(\mathfrak{g}_0)$-crystal decomposition of $B^{1,s}$ is multiplicity free, there exists a natural $U_q(\mathfrak{g}_0)$-crystal isomorphism $\iota: \mathrm{RC}(B^{1,s}) \to B^{1,s}$. For type $D_4^{(3)}$, we note that $k\bar{\Lambda}_1$ can be considered as the partition $(k)$.

**Definition 4.2** Let $B^{1,s}$ be a KR crystal of type $D_4^{(3)}$ and consider the classical component $B(k\bar{\Lambda}_1) \subseteq B^{1,s}$. The filling map fill: $B^{1,s} \to (B^{1,1})^{\otimes s}$ is given by adding $\lfloor \frac{s-k}{2} \rfloor$ copies of the horizontal domino $\begin{bmatrix} 1 & 1 \end{bmatrix}$ and an additional $\emptyset$ if $s-k$ is odd.

Let $T^{1,s}$ denote the image of $B^{1,s}$ under fill written as a $1 \times s$ rectangle. We note that $T^{1,s}$ inherits a classical crystal structure from $(B^{1,1})^{\otimes s}$.

**Example 4.3** Consider the element

$$ b = \begin{bmatrix} 3 & 0 & \bar{2} & \bar{2} & \bar{1} \end{bmatrix} \in B(5\bar{\Lambda}_1) \subseteq B^{1,9}, $$

then we have

$$ \mathrm{fill}(b) = \begin{bmatrix} 3 & 0 & \bar{2} & \bar{2} & \bar{1} & \bar{1} & \bar{1} & \bar{1} \end{bmatrix}. $$

Now suppose $b \in B^{1,8}$, then we have

$$ \mathrm{fill}(b) = \begin{bmatrix} 3 & 0 & \bar{2} & \bar{2} & \bar{1} & \bar{1} & \bar{1} & \emptyset \end{bmatrix}. $$

We give a $U'_q(\mathfrak{g})$-crystal structure to $T^{1,s}$ by following [10, 30] as the conditions for $e_0$ and $f_0$ are preserved under the filling map.

**Proposition 4.4** The filling map fill: $B^{1,s} \to T^{1,s}$ given in Definition 4.2 is a $U'_q(\mathfrak{g})$-crystal isomorphism.

We also can show the following.

**Proposition 4.5** Let $B = B^{1,s}$. Then $\Phi = \mathrm{fill} \circ \iota$ with fill as in Definition 4.2 on highest weight elements.
# 5 Virtualization Map

**Lemma 5.1** The virtualization map $v: B^{1,s} \to B^{2,s}$ for types $D_4^{(3)} \to D_4^{(1)}$ is given column-by-column by

$$ \boxed{1} \mapsto \frac{\boxed{1}}{\boxed{2}} $$

$$ \boxed{2} \mapsto \frac{\boxed{1}}{\boxed{3}} $$

$$ \boxed{3} \mapsto \frac{\boxed{2}}{\boxed{3}} $$

$$ \boxed{0} \mapsto \frac{\boxed{3}}{\boxed{3}} $$

$$ \boxed{3} \mapsto \frac{\boxed{3}}{\boxed{2}} $$

$$ \boxed{2} \mapsto \frac{\boxed{3}}{\boxed{1}} $$

$$ \boxed{1} \mapsto \frac{\boxed{2}}{\boxed{1}} $$

$$ \boxed{0} \mapsto \frac{\boxed{1}}{\boxed{1}} $$

Using Lemma 5.1 and the analogue of $\Phi$ in type $D_4^{(1)}$ [18, 25], we can show the following.

**Theorem 5.2** Consider a tensor product of KR crystals $B = \bigotimes_{i=1}^N B^{1,s_i}$ of type $D_4^{(3)}$. The virtualization map $v$ commutes with the map $\Phi$.

We need to define the *complement rigging map* $\theta: \mathrm{RC}(B) \to \mathrm{RC}(B_r)$ by sending $(\nu, J) \mapsto (\nu, J')$, where $J'$ is obtained by $x' = p_i^{(a)} - x$ for all labels $x$ and $B_r$ are the factors of $B$ in reverse order. That is to say $\theta$ maps each label $x$ to its colabel. We can define $\tilde{\delta} := \theta \circ \delta \circ \theta$, and using the virtualization map, Proposition 4.5, and the results of [25], we can show the following.

**Lemma 5.3** We have $\delta \circ \tilde{\delta} = \tilde{\delta} \circ \delta$.

Using the results on the combinatorial $R$-matrix in [30], we can show the following.

**Lemma 5.4** Consider $B = B^{1,s} \otimes B^{1,1}$. We have $\Phi^{-1} \circ R \circ \Phi$ is the identity map on $\mathrm{RC}(B)$.

Then following [28, Sec. 8], the map $rs := \theta \circ ls \circ \theta$ preserves statistics using [30]. From Lemma 5.4, the $R$-matrix preserves statistics. Thus iterating $rs$ and $R$-matrices, we preserve statistics to $\bigotimes_{i=1}^{N'} B^{1,1}$. Then we use the results of [23, 25] and Theorem 5.2 to obtain our main result.

**Theorem 5.5** Let $B = \bigotimes_{i=1}^N B^{1,s_i}$ of type $D_4^{(3)}$. The map $\Phi: \mathrm{RC}(B) \to B$ is a $U_q(\mathfrak{g}_0)$-crystal isomorphism and $\Phi \circ \theta$ sends cocharge to energy.

From Proposition 4.4, Lemma 5.1, Theorem 5.5, and the filling map for type $D_n^{(1)}$ given in [18], we can show the following.

**Theorem 5.6** Let $B = B^{1,s}$. Then $\Phi = \mathrm{fill} \circ \mathrm{fill}$ with $\mathrm{fill}$ as in Definition 4.2 as $U_q(\mathfrak{g}_0)$-crystal morphisms.

Thus we can define a $U'_q(\mathfrak{g})$-crystal structure on $\mathrm{RC}(B)$ by extending $\Phi$ to be a $U'_q(\mathfrak{g})$-crystal isomorphism. Thus we have a special case in type $D_4^{(3)}$ of the conjectures given in [27].

# 6 Extensions and questions

The $U_q(\mathfrak{g}_0)$-crystal decomposition of $B^{2,s}$ and the highest weight rigged configurations will appear in the full version of this work. The author hopes to use this to determine the filling map for $B^{2,s}$.

There is a map $\mathrm{lt}: \mathrm{RC}(B^{2,1} \otimes B^*) \to \mathrm{RC}(B^{1,1} \otimes B^{1,1} \otimes B^*)$ called *left-top* which adds a singular string of length 1 to $\nu^{(1)}$. In the full version, this is used to extend the $U_q(\mathfrak{g}_0)$-crystal isomorphism $\Phi$ to tensor products also containing $B^{2,1}$.
**Example 6.1** Continuing from Example 3.2, we obtain

$$ \Phi(\nu, J) = \boxed{3} \otimes \boxed{2} \boxed{3} \otimes \boxed{\frac{1}{2}} $$

The computations for the Kleber algorithm can be modified to determine the $U_q(g_0)$-crystal decomposition of $B^{r,s}$ of type $G_2^{(1)}$. However there is a difficulty with determining what the map $\delta$ should be. This would need to be overcome to define the filling map for type $G_2^{(1)}$.

There is a conjecture [22, Conj. 3.7] that we can realize $B^{1,s}$ of type $D_4^{(3)}$ as a virtual crystal in $B^{1,s}$ of type $D_4^{(1)}$. Therefore obtaining a direct description of $e_0$ and $f_0$ on rigged configurations could lead to an answer to this conjecture using the results of [18]. The author hopes to have this description and prove this conjecture in this special case in the full version of this work.

# 7 Examples using Sage

The bijection $\Phi$ and the rigged configurations have been implemented by the author in Sage [29]. We begin by setting up the Sage environment to give a more concise printing.

```
sage: RiggedConfigurations.global_optionsess("horizontal")
```

We construct our the rigged configuration from Example 2.2 by specifying the partitions and corresponding labels.

```
sage: nu = RC(partition_list=[[4,1], [4]], rigging_list=[[3,1], [-2]]); nu
5[ ]][ ][ ][ ]3 -2[ ][ ][ ][ ]-2
1[ ]1
```

We apply the full bijection and print the output using Sage's ASCII art.

```
sage: ascii_art(nu.to par t i s o r t _ p ro d u c t _ o f _ k i r il lo v _ r es h e t ik h i n _ t a b l e a u x())
3 # 2 3 # 1
    -2
```

# Acknowledgements

The author would like to thank Masato Okado for a very valuable discussion and pointing out the reference [17]. The author would also like to thank Anne Schilling for reading a draft of this extended abstract and her mentorship.

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