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Uniform description of the rigged configuration bijection

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Abstract

We give a uniform description of the bijection \(\Phi \) from rigged configurations to tensor products of Kirillov–Reshetikhin crystals of the form \(\bigotimes _{i=1}^N B^{r_i,1}\) in dual untwisted types: simply-laced types and types \(A_{2n-1}^{(2)}\), \(D_{n+1}^{(2)}\), \(E_6^{(2)}\), and \(D_4^{(3)}\). We give a uniform proof that \(\Phi \) is a bijection and preserves statistics. We describe \(\Phi \) uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that \(\Phi \) is a bijection for \(\bigotimes _{i=1}^N B^{r_i,s_i}\) when \(r_i\), for all i, map to 0 under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov–Reshetikhin crystals \(B^{r,1}\) using tableaux of a fixed height \(k_r\) depending on r in all affine types. Additionally, we are able to describe crystals \(B^{r,s}\) using \(k_r \times s\) shaped tableaux that are conjecturally the crystal basis for Kirillov–Reshetikhin modules for various nodes r.

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Notes

  1. Our results also include type \(A_n^{(1)}\), where we instead have \(B^{1,1} \otimes B^{n,1}\) as the atomic object.

  2. To obtain the formulas of [20, 22], we need to substitute \(q = q^{-1}\).

  3. A map for when the left factor is \(B^{2,1}\) of type \(E_6^{(1)}\) was conjectured in [4].

  4. Note that if \(\ell _{j-1} = 1\) and there exists a singular row of length 1, then we would not be in this case as \(i_a = 1 < \max \{\ell _{j-1},2\} = 2\).

  5. Recall that \(\delta _r^{\star }\) in [89], where it was denoted by \({\text {rh}}\), was defined by removing the rightmost factor (and then going to the highest weight element) in contrast to our definition of conjugating the left factor removal by \(\star \). However, these definitions are equivalent by [89, Prop. 5.9(5)].

  6. From the definition of \({{\,\mathrm{lb}\,}}\) and that \({{\,\mathrm{lb}\,}}\) preserves vacancy numbers, we can consider \(\widetilde{\delta } \circ {{\,\mathrm{lb}\,}}\) as one operation that is the same as \(\widetilde{\delta }\) except that it begins at \(\nu ^{(r)}\) instead of \(\nu ^{(1)}\).

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Acknowledgements

The author would like to thank Masato Okado for the reference [4] and useful discussions. The author would like to thank Anne Schilling for useful discussions. The author would like to thank Ben Salisbury for comments on an early draft of this paper. The author would like to thank the referee for their useful comments. This work benefited from computations using SageMath [15, 79].

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  1. School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia

    Travis Scrimshaw

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The author was partially supported by the National Science Foundation RTG grant NSF/DMS-1148634.

Appendices

Appendix A: KR tableaux for fundamental weights

In this section, we list the classically highest weight KR tableaux for \(B^{r,1}\) for types \(E_{6,7,8}^{(1)}\), \(E_6^{(2)}\), \(F_4^{(1)}\), and \(G_2^{(1)}\).

Recall that highest weight rigged configurations must have \(0 \le p_i^{(a)}\) for all \((a,i) \in \mathcal {H}_0\). Moreover, all rigged configurations in this section will have \(p_i^{(a)} = 0\) except for possibly one \((b, j) \in \mathcal {H}_0\) where \(p_j^{(b)} = 1\). Therefore we describe the rigged configuration simply by its configuration \(\nu \) and if \(p_i^{(a)} = 1\), then we write the corresponding rigging x as the subscript (x). We also write the column tableau \( \begin{aligned} \hline x_1 &{} x_2 &{} x_3 &{} \cdots &{} x_r \\ \hline \end{aligned}^{t} \) as \([x_1, x_2, x_3, \dotsc , x_r]\).

Example A.1

In type \(E_7^{(1)}\) for \(B^{4,1}\), we denote the rigged configuration

figure f

by \((1, 11, 1_{(1)} 1_{(0)}, 1111, 111, 11, 1)\) or more compactly \((1, 1^2, 1^2_{(1,0)}, 1^4, 1^3, 1^2, 1)\).

In the remaining part of this section, we give the KR tableaux for \(B^{r,1}\) for the exceptional types (except for \(r = 4, 5\) in type \(E_8^{(1)}\), which can be generated using SageMath [15]).

1.1 A.1. Type \(E_6^{(1)}\)

\(r=1\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1] \end{aligned}$$

\(r=2\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}6, 2 \overline{6}] \\ (1, 1^2, 1^2, 1^3, 1^2, 1)\mapsto & {} [1, \overline{1}6, \overline{6}] \end{aligned}$$

\(r=3\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}3] \\ (1, 1, 1^2, 1^2, 1, \emptyset )\mapsto & {} [1, \overline{1}6] \end{aligned}$$

\(r=4\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}3, \overline{3}4] \\ (\emptyset , 1, 1, 1^2, 1, \emptyset )\mapsto & {} [1, \overline{1}3, 1 \overline{3}6] \\ (1, 1_{(0)}, 1, 1^3, 1^2, 1)\mapsto & {} [1, \overline{1}6, 2\overline{6}] \\ (1, 1_{(1)}, 1^2, 1^3, 1^2, 1)\mapsto & {} [1, \overline{1}3, 2 \overline{3}] \\ (1^2, 1^3, 1^4, 1^6, 1^4, 1^2)\mapsto & {} [1, \overline{1}6, \overline{6}] \end{aligned}$$

\(r=5\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}6, 2 \overline{6}, \overline{2}5] \\ (\emptyset , 1, 1, 1^2, 1^2, 1)\mapsto & {} [1, \overline{1}6, 2 \overline{6}, 1 \overline{2}] \end{aligned}$$

\(r=6\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}6] \end{aligned}$$

1.2 A.2. Type \(E_7^{(1)}\)

\(r=1\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7, 1 \overline{7}] \\ (1^2, 1^2, 1^3, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, \overline{7}] \end{aligned}$$

\(r=2\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7, 1 \overline{7}, \overline{1}2] \\ (1, 1^2, 1^2, 1^3, 1^2, 1, \emptyset )\mapsto & {} [7, 1 \overline{7}, \overline{1}7] \end{aligned}$$

\(r=3\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7, 1 \overline{7}, \overline{1}2, \overline{2}3] \\ (1, 1, 1^2, 1^2, 1, \emptyset , \emptyset )\mapsto & {} [7, 1 \overline{7}, \overline{1}2, \overline{2}6] \\ (1_{(0)}, 1^2, 1^3, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, 1 \overline{7}, \overline{1}7, 1 \overline{7}] \\ (1_{(1)}, 1^2, 1^3, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, 1 \overline{7}, \overline{1}2, 1 \overline{2}] \\ (1^3, 1^4, 1^6, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [7, 1 \overline{7}, \overline{1}7, \overline{7}] \end{aligned}$$

\(r=4\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7, 6 \overline{7}, 5 \overline{6}, 4 \overline{5}] \\ (\emptyset , 1, 1, 1^2, 1, \emptyset , \emptyset )\mapsto & {} [7, 6 \overline{7}, 5 \overline{6}, 1 \overline{5}6] \\ (\emptyset , 1^2, 1^2, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, 6 \overline{7}, 1 \overline{6}7, 1 \overline{7}] \\ (1, 1_{(0)}, 1^2, 1^3, 1^2, 1, \emptyset )\mapsto & {} [7, 6 \overline{7}, 1 \overline{6}7, \overline{1}2] \\ (1, 1_{(1)}, 1^2, 1^3, 1^2, 1, \emptyset )\mapsto & {} [7, 6 \overline{7}, 5 \overline{6}, 2 \overline{5}7] \\ (1, 1^2, 1^2_{(0,0)}, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, 1 \overline{7}, \overline{1}2, \overline{2}3] \\ (1, 1^2, 1^2_{(1,0)}, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, 6 \overline{7}, 1 \overline{6}7, \overline{1}3 \overline{7}] \\ (1, 1^2, 1^2_{(1,1)}, 1^4, 1^3, 1^2, 1)\mapsto & {} [7, 6 \overline{7}, 5 \overline{6}, 3 \overline{5}] \\ (1^2, 1^3, 1^4, 1^6, 1^4, 1^2, \emptyset )\mapsto & {} [7, 6 \overline{7}, 1\overline{6}7, \overline{1}7] \\ (1^2, 1^3, 1^4, 1^6, 1^4, 1^2_{(0,0)}, 1)\mapsto & {} [7, 1 \overline{7}, \overline{1}2, \overline{2}6] \\ (1^2, 1^3, 1^4, 1^6, 1^4, 1^2_{(1,0)}, 1)\mapsto & {} [7, 6 \overline{7}, 1 \overline{6}7, \overline{1}6 \overline{7}] \\ (1^2, 1^3, 1^4, 1^6, 1^4, 1^2_{(1,1)}, 1)\mapsto & {} [7, 6 \overline{7}, 5 \overline{6}, \overline{5}6] \\ (1^2_{(0,0)}, 1^4, 1^5, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [7, 1 \overline{7}, \overline{1}7, 1 \overline{7}] \\ (1^2_{(1,0)}, 1^4, 1^5, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [7, 1 \overline{7}, \overline{1}2, 1 \overline{2}] \\ (1^2_{(1,1)}, 1^4, 1^5, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [7, 6 \overline{7}, 1 \overline{6}7, \overline{7}] \\ (1^4, 1^6, 1^8, 1^{12}, 1^9, 1^6, 1^3)\mapsto & {} [7, 1 \overline{7}, \overline{1}7, \overline{7}] \\ (2, 21, 2^2, 2^2 1^2, 2 1^2, 1^2, 1)\mapsto & {} [7, 6 \overline{7}, 2 \overline{6}, \overline{2}6] \\ (1^2, 2 1^2, 2 1^3, 2^2 1^4, 2^2 1^2, 2^2, 2)\mapsto & {} [7, 6 \overline{7}, 2 \overline{6}, \overline{2}1] \\ (2^2, 2^2 1^2, 2^3 1^2, 2^4 1^4, 2^3 1^3, 2^2 1^2, 21)\mapsto & {} [7, 6 \overline{7}, \overline{6}7, \overline{7}] \end{aligned}$$

\(r=5\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7, 6 \overline{7}, 5\overline{6}] \\ (\emptyset , 1, 1, 1^2, 1^2, 1, \emptyset )\mapsto & {} [7, 6 \overline{7}, 1 \overline{6}7] \\ (1, 1_{(0)}, 1^2, 1^3, 1^3, 1^2, 1)\mapsto & {} [7, 1 \overline{7}, \overline{1}2] \\ (1, 1_{(1)}, 1^2, 1^3, 1^3, 1^2, 1)\mapsto & {} [7, 6 \overline{7}, 2\overline{6}] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^3, 1_{(0)})\mapsto & {} [7, 1 \overline{7}, \overline{1}7] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^3, 1_{(1)})\mapsto & {} [7, 6 \overline{7}, \overline{6}7] \end{aligned}$$

\(r=6\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7, 6 \overline{7}] \\ (\emptyset , 1, 1, 1^2, 1^2, 1^2, 1)\mapsto & {} [7, 1 \overline{7}] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^2)\mapsto & {} [7, \overline{7}] \end{aligned}$$

\(r=7\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [7] \end{aligned}$$

1.3 A.3. Type \(E_8^{(1)}\)

\(r=1\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [8, 1\overline{8}] \\ (1^2, 1^2, 1^3, 1^4, 1^3, 1^2, 1, \emptyset )\mapsto & {} [8, \emptyset ] \\ (1^4, 1^5, 1^7, 1^{10}, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [\emptyset , \emptyset ] \end{aligned}$$

\(r=2\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}2] \\ (1, 1^2, 1^2, 1^3, 1^2, 1, \emptyset , \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}7] \\ (1_{(0)}, 1^3, 1^3, 1^5, 1^4, 1^3, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}1] \\ (1_{(1)}, 1^3, 1^3, 1^5, 1^4, 1^3, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \emptyset ] \\ (1^3, 1^5, 1^6, 1^9, 1^7, 1^5, 1^3, 1_{(0)})\mapsto & {} [8, \emptyset , 8\overline{8}] \\ (1^3, 1^5, 1^6, 1^9, 1^7, 1^5, 1^3, 1_{(1)})\mapsto & {} [8, \emptyset , \emptyset ] \\ (1^5, 1^8, 1^{10}, 1^{15}, 1^{12}, 1^9, 1^6, 1^3)\mapsto & {} [\emptyset , \emptyset , \emptyset ] \end{aligned}$$

\(r=3\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}2, \overline{2}3] \\ (1, 1, 1^2, 1^2, 1, \emptyset , \emptyset , \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}2, \overline{2}6] \\ (1_{(0)}, 1^2, 1^3, 1^4, 1^3, 1^2, 1, \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}7, 1\overline{7}8] \\ (1_{(1)}, 1^2, 1^3, 1^4, 1^3, 1^2, 1, \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}2, 1\overline{2}8] \\ (1^2, 1^2_{(0,0)}, 1^4, 1^5, 1^4, 1^3, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}7, 2 \overline{7}] \\ (1^2, 1^2_{(1,0)}, 1^4, 1^5, 1^4, 1^3, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}2, \overline{2}2] \\ (1^2, 1^2_{(1,1)}, 1^4, 1^5, 1^4, 1^3, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}2, \emptyset ] \\ (1^3, 1^4, 1^6, 1^8, 1^6, 1^4, 1^2, \emptyset )\mapsto & {} [8, 1\overline{8}, \overline{1}7, \overline{7}8 8] \\ (1^3, 1^4, 1^6, 1^8, 1^6, 1^4, 1^2_{(0,0)}, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}1, \overline{1}7] \\ (1^3, 1^4, 1^6, 1^8, 1^6, 1^4, 1^2_{(1,0)}, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}7, \overline{7}7] \\ (1^3, 1^4, 1^6, 1^8, 1^6, 1^4, 1^2_{(1,1)}, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}7, \emptyset ] \\ (1^3_{(0,0,0)}, 1^5, 1^7, 1^{10}, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [8, \emptyset , \overline{8}8, 1 \overline{8}] \\ (1^3_{(1,0,0)}, 1^5, 1^7, 1^{10}, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [8, 1\overline{8}, \overline{1}1, \overline{1}1] \\ (1^3_{(1,1,0)}, 1^5, 1^7, 1^{10}, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [8, 1\overline{8}, \overline{1}1, \emptyset ] \\ (1^3_{(1,1,1)}, 1^5, 1^7, 1^{10}, 1^8, 1^6, 1^4, 1^2)\mapsto & {} [8, 1\overline{8}, \emptyset , \emptyset ] \\ (1^5, 1^7, 1^{10}, 1^{14}, 1^{11}, 1^8, 1^5, 1^2_{(0,0)})\mapsto & {} [8, \emptyset , \overline{8}8, \overline{8}8] \\ (1^5, 1^7, 1^{10}, 1^{14}, 1^{11}, 1^8, 1^5, 1^2_{(1,0)})\mapsto & {} [8, \emptyset , \overline{8}8, \emptyset ] \\ (1^5, 1^7, 1^{10}, 1^{14}, 1^{11}, 1^8, 1^5, 1^2_{(1,1)})\mapsto & {} [8, \emptyset , \emptyset , \emptyset ] \\ (1^7, 1^{10}, 1^{14}, 1^{20}, 1^{16}, 1^{12}, 1^8, 1^4)\mapsto & {} [\emptyset , \emptyset , \emptyset , \emptyset ] \\ (21, 2^2, 2^2 1^2, 2^3 1^2, 2^2 1^2, 21^2, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}2, \overline{2}7] \\ (1^3, 21^3, 21^5, 2^2 1^6, 2^2 1^4, 2^2 1^2, 2^2, 2)\mapsto & {} [8, 1\overline{8}, \overline{1}7, 1 \overline{7}] \\ (2^2 1_{(0)}, 2^2 1^3, 2^3 1^4, 2^4 1^6, 2^3 1^5, 2^2 1^4, 2 1^3, 1^2)\mapsto & {} [8, 1\overline{8}, \overline{1}1, \overline{1}8] \\ (2^2 1_{(1)}, 2^2 1^3, 2^3 1^4, 2^4 1^6, 2^3 1^5, 2^2 1^4, 2 1^3, 1^2)\mapsto & {} [8, 1\overline{8}, \emptyset , \overline{1}8] \\ (2^2 1^3, 2^3 1^4, 2^4 1^6, 2^6 1^8, 2^5 1^6, 2^4 1^4, 2^3 1^2, 2^2)\mapsto & {} [8, \emptyset , \emptyset , \overline{8}] \end{aligned}$$

\(r=6\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [8, 7\overline{8}, 6\overline{7}] \\ (\emptyset , 1, 1, 1^2, 1^2, 1^1, 1, \emptyset )\mapsto & {} [8, 7\overline{8}, 1\overline{7}8] \\ (1, 1_{(0)}, 1^2, 1^3, 1^3, 1^3, 1^2, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}2] \\ (1, 1_{(1)}, 1^2, 1^3, 1^3, 1^3, 1^2, 1)\mapsto & {} [8, 7\overline{8}, 2 \overline{7}] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^2, \emptyset )\mapsto & {} [8, 7\overline{8}, \overline{7}8 8] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^2_{(0,0)}, 1)\mapsto & {} [8, 1\overline{8}, \overline{1}7] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^2_{(1,0)}, 1)\mapsto & {} [8, 7\overline{8}, \overline{7}7] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^2_{(1,1)}, 1)\mapsto & {} [8, 7\overline{8}, \emptyset ] \\ (1^2_{(0,0)}, 1^4, 1^5, 1^8, 1^7, 1^6, 1^4, 1^2)\mapsto & {} [8, \overline{8}8, 1 \overline{8}] \\ (1^2_{(1,0)}, 1^4, 1^5, 1^8, 1^7, 1^6, 1^4, 1^2)\mapsto & {} [8, 1 \overline{8}, \overline{1}1] \\ (1^2_{(1,1)}, 1^4, 1^5, 1^8, 1^7, 1^6, 1^4, 1^2)\mapsto & {} [8, 1\overline{8}8, \emptyset ] \\ (1^4, 1^6, 1^8, 1^{12}, 1^{10}, 1^8, 1^5, 1^2_{(0,0)})\mapsto & {} [8, \overline{8}8, \overline{8}8] \\ (1^4, 1^6, 1^8, 1^{12}, 1^{10}, 1^8, 1^5, 1^2_{(1,0)})\mapsto & {} [8, \overline{8}8, \emptyset ] \\ (1^4, 1^6, 1^8, 1^{12}, 1^{10}, 1^8, 1^5, 1^2_{(1,1)})\mapsto & {} [8, \emptyset , \emptyset ] \\ (1^6, 1^9, 1^{12}, 1^{18}, 1^{15}, 1^{12}, 1^8, 1^4)\mapsto & {} [\emptyset , \emptyset , \emptyset ] \\ (1^2, 21^2, 21^3, 2^2 1^4, 2^2 1^3, 2^2 1^2, 2^2, 2)\mapsto & {} [8, 7\overline{8}, 1 \overline{7}] \\ (2^2, 2^2 1^2, 2^3 1^2, 2^4 1^4, 2^3 1^4, 2^2 1^4, 21^3, 1^2)\mapsto & {} [8, 1\overline{8}, \overline{1}8] \\ (2^2 1^2, 2^3 1^3, 2^4 1^4, 2^6 1^6, 2^5 1^5, 2^4 1^4, 2^3 1^2, 2^2)\mapsto & {} [8, \emptyset , \overline{8}] \end{aligned}$$

\(r=7\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [8, 7\overline{8}] \\ (\emptyset , 1, 1, 1^2, 1^2, 1^2, 1^2, 1)\mapsto & {} [8, 1\overline{8}] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^3, 1_{(0)})\mapsto & {} [8, 8\overline{8}] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^3, 1_{(1)})\mapsto & {} [8, \emptyset ] \\ (1^4, 1^6, 1^8, 1^{12}, 1^{10}, 1^8, 1^6, 1^3)\mapsto & {} [\emptyset , \emptyset ] \end{aligned}$$

\(r=8\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [8] \\ (1^2, 1^3, 1^4, 1^6, 1^5, 1^4, 1^3, 1^2)\mapsto & {} [\emptyset ] \end{aligned}$$

1.4 A.4. Type \(E_6^{(2)}\)

\(r=1\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1] \\ (1^2, 1^3, 1^2, 1)\mapsto & {} [\emptyset ] \end{aligned}$$

\(r=2\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}2] \\ (1, 1^2, 1, \emptyset )\mapsto & {} [1, \overline{1}4] \\ (1_{(0)}, 1^3, 1^2, 1)\mapsto & {} [1, \overline{1}1] \\ (1_{(1)}, 1^3, 1^2, 1)\mapsto & {} [1, \emptyset ] \\ (1^3, 1^6, 1^4, 1^2)\mapsto & {} [\emptyset , \emptyset ] \end{aligned}$$

\(r=3\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}2, \overline{2}3] \\ (\emptyset , 1, 1, \emptyset )\mapsto & {} [1, \overline{1}2, 1 \overline{2}4] \\ (\emptyset , 1^2, 1^2, 1)\mapsto & {} [1, \overline{1}2, 11\overline{2}] \\ (1, 1^2_{(0,0)}, 1^2, 1)\mapsto & {} [1, \overline{1}4, 2\overline{4}] \\ (1, 1^2_{(1,0)}, 1^2, 1)\mapsto & {} [1, \overline{1}2, \overline{2}2] \\ (1, 1^2_{(1,1)}, 1^2, 1)\mapsto & {} [1, \overline{1}2, \emptyset ] \\ (1^2, 1^4, 1^3, 1_{(0)})\mapsto & {} [1, \emptyset , \overline{1}4] \\ (1^2, 1^4, 1^3, 1_{(1)})\mapsto & {} [1, \overline{1}4,\emptyset ] \\ (1^2_{(0,0)}, 1^5, 1^4, 1^2)\mapsto & {} [1, \overline{1}1, \overline{1}1] \\ (1^2_{(1,0)}, 1^5, 1^4, 1^2)\mapsto & {} [1, \overline{1}1, \emptyset ] \\ (1^2_{(1,1)}, 1^5, 1^4, 1^2)\mapsto & {} [1, \emptyset ,\emptyset ] \\ (1^4, 1^8, 1^6, 1^3)\mapsto & {} [\emptyset , \emptyset , \emptyset ] \\ (2, 2^2, 21, 1)\mapsto & {} [1, \overline{1}2, \overline{2}4] \\ (1^2, 2 1^3, 2 1^2, 2)\mapsto & {} [1, \overline{1}4, 1\overline{4}] \\ (2^2, 2^3 1^2, 2^2 1^2, 21)\mapsto & {} [1, \emptyset , \overline{1}] \end{aligned}$$

\(r=4\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [1, \overline{1}4] \\ (\emptyset , 1, 1, 1)\mapsto & {} [1, \emptyset ] \\ (1^2, 1^4, 1^3, 1^2)\mapsto & {} [\emptyset , \emptyset ] \end{aligned}$$

1.5 A.5. Type \(F_4^{(1)}\)

We follow Proposition 4.3 to describe the elements of \(B(\overline{\Lambda }_4)\).

\(r=1\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [4, 1\overline{4}] \\ (1^2, 1^3, 2^2, 2)\mapsto & {} [4, \overline{4}] \end{aligned}$$

\(r=4\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [4, 3\overline{4}, 2\overline{3}] \\ (1, 1^2, 2, \emptyset )\mapsto & {} [4, 3\overline{4}, \overline{3}44] \\ (1_{(0)}, 1^3, 2^2, 2)\mapsto & {} [4, 4\overline{4}, 1\overline{4}] \\ (1_{(1)}, 1^3, 2^2, 2)\mapsto & {} [4, 3\overline{4}, 1\overline{3}] \\ (1^3, 1^6, 2^4, 2^2)\mapsto & {} [4, 4\overline{4}, \overline{4}] \end{aligned}$$

\(r=3\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [4, 3\overline{4}] \\ (1, 1^2, 21, 1)\mapsto & {} [4, 4\overline{4}] \end{aligned}$$

\(r=4\)

$$\begin{aligned} (\emptyset , \emptyset , \emptyset , \emptyset )\mapsto & {} [4] \end{aligned}$$

1.6 A.6. Type \(G_2^{(1)}\)

\(r=1\)

$$\begin{aligned} (\emptyset , \emptyset )\mapsto & {} [1] \end{aligned}$$

\(r=2\)

$$\begin{aligned} (\emptyset , \emptyset )\mapsto & {} [1,2] \\ (3, 1^2)\mapsto & {} [1, \overline{1}] \end{aligned}$$

Appendix B: Examples with SageMath

We give some examples using SageMath [15], where rigged configurations, KR tableaux, and the bijection \(\Phi \) has been implemented by the author.

We first construct Example 4.4.

figure g

Next, we construct Example 5.3.

figure h

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Scrimshaw, T. Uniform description of the rigged configuration bijection. Sel. Math. New Ser. 26, 42 (2020). https://doi.org/10.1007/s00029-020-00564-8

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