Rigged configurations for all symmetrizable types

In an earlier work, the authors developed a rigged configuration model for the crystal $B(\infty)$ (which also descends to a model for irreducible highest weight crystals via a cutting procedure). However, the result obtained was only valid in finite types, affine types, and simply-laced indefinite types. In this paper, we show that the rigged configuration model proposed does indeed hold for all symmetrizable types. As an application, we give an easy combinatorial condition that gives a Littlewood-Richardson rule using rigged configurations which is valid in all symmetrizable Kac-Moody types.


Introduction
The theory of crystal bases [Kas91] has provided a natural combinatorial framework to study the representations of Kac-Moody algebras (including classical Lie algebras) and their associated quantum groups. Their applications span many areas of mathematics, and these diverse applications have compelled researchers to develop different combinatorial models for crystals which yield suitable settings to studying a particular aspect of the representation theory. See, for example, [GL05,KN94,KS97,LP08,Lit95]. The choice of using one model over the other usually depends on the underlying question at hand (and/or on the preference of the author).
We will be using rigged configurations, whose origins lie in statistical mechanics. Specifically, they correspond naturally to the eigenvalues and eigenvectors of a Hamiltonian of a statistical model via the Bethe ansatz [Bet31, KKR86,KR86]. As shown in [SS15a], the rigged configuration model for B(∞) has simple combinatorial rules for describing the crystal structure which work in all finite, affine, and all simply-laced Kac-Moody types. These combinatorial rules are only based on the nodes of the Dynkin diagram and their neighbors.
If instead we use a corner transfer matrix approach to solve the Hamiltonian, the eigenvectors become indexed by one-dimensional lattice paths [Bax89, HKK + 99, HKO + 02, NY97,SW99], which can be interpreted as highest weight vectors in a tensor product of certain crystals known as Kirillov-Reshetikhin crystals. While not mathematically rigorous, these two different approaches suggests a bijection that has been constructed in numerous special cases. See, for example, [KKR86,KR86,KSS02,OSS03a,OSSS16,SS15b,Scr16]. In [SS16], this bijection was extended to show that the rigged configuration model in [SS15a] and the marginally large tableaux model in [HL08] agree (for the appropriate types).
The purpose of this paper is to extend the crystal structure on rigged configurations B(∞) in terms of rigged configurations to all symmetrizable Kac-Moody types. There are several known models for the crystal B(∞) in finite and affine types, but only a select few which are uniformly constructed to include all symmetrizable types (e.g., modified Nakajima monomials [KKS07] and Littelmann paths [Lit95]). Having another model which works beyond finite and affine types is beneficial to studying the combinatorics of the associated representations, which, for example, has come up in the theory of automorphic forms (see [SS16] for an application of rigged configurations in finite type in this direction).
In [SS15a], our proof relied on Schilling's result [Sch06] that the crystal structure on rigged configurations satisfied the Stembridge axioms [Ste03]. While the Stembridge axioms are necessary (local) conditions for highest weight crystals, they are only sufficient conditions in simply-laced types. Then we used the technology of virtual crystals and well-known diagram foldings to extend our results to the other finite and affine types. Since rigged configurations are well-behaved under virtualization [SS15a,SS15b], the problem of showing rigged configurations model highest weight crystals and B(∞) for general symmetrizable type is reduced to determining a realization of every symmetrizable type as a diagram folding of a simply-laced type.
It is known that every Cartan type can be realized using a simple graph together with a graph automorphism [Lus93]. We can realize this graph as a Dynkin diagram of a symmetric type, where the number of edges between vertices v i and v j gives the (negative of the) (i, j)-entry of the corresponding symmetric Cartan matrix, and the automorphism as a digram folding. Therefore, we can use the corresponding embedding of root lattices and [Kas96, Thm. 5.1] to show there exists a virtualization of a crystal of any symmetrizable type into a crystal of symmetric type. An explicit virtualization map using Nakajima's and Lusztig's quiver varieties was proven in [Sav05] in this case.
In this note, we modify the construction in [Lus93] so that the resulting graph is simple, where it can be considered as a simply-laced type. We then use the aforementioned virtualization map to prove an open conjecture (see Conjecture 2.7) stated by the authors that the rigged configuration model for B(∞) and highest weight crystals B(λ) defined in [SS15a] may be extended to the case of arbitrary symmetrizable Kac-Moody algebras. Furthermore, we expect our results could to lead to a solution to the open problem of determining an analog of the Stembridge axioms for non-simplylaced types, where the only known results are for type B 2 [DKK09,Ste07]. Indeed, our results allow for a direct link between the crystal operators and a simply-laced type where the Stembridge axioms apply.
The organization of the paper goes as follows. In Section 2, we set our notation and recall basic notions about crystals, rigged configurations, and virtualization. In Section 3, we define the diagram folding required to prove our conjecture from [SS15a] in Section 4. Lastly, in Section 5, we stage the famous Littlewood-Richardson rule for decomposing tensor products of irreducible highest weight crystals in terms of the rigged configuration model.

Background
We give a background on crystals, virtual crystals, and rigged configurations.
2.1. Crystals. Let g be a symmetrizable Kac-Moody algebra with index set I, generalized Cartan matrix A = (A ij ) i,j∈I , weight lattice P , root lattice Q, fundamental weights {Λ i : i ∈ I}, simple roots {α i : i ∈ I}, and simple coroots {h i : i ∈ I}. There is a canonical pairing , : An abstract U q (g)-crystal is a nonempty set B together with maps satisfying certain conditions. The e i and f i for i ∈ I are referred to as the Kashiwara raising and Kashiwara lowering operators, respectively. See [HK02,Kas91] for details. The models used in this paper will be specific, and therefore we will give details related to those models in the subsequent sections.
We say an abstract U q (g)-crystal is simply a U q (g)-crystal if it is crystal isomorphic to the crystal basis of an integrable U q (g)-module.
Again let B 1 and B 2 be abstract U q (g)-crystals. The tensor product B 2 ⊗ B 1 is defined to be the Cartesian product B 2 × B 1 equipped with crystal operations defined by Remark 2.1. Our convention for tensor products is opposite the convention given by Kashiwara in [Kas91].
be the number of parts of length i in ν (a) . Define the vacancy numbers of ν to be p In addition, we can extend the vacancy numbers to Recall that a partition is a multiset of integers (typically sorted in weakly decreasing order). More generally, a rigged partition is a multiset of pairs of integers (i, x) such that i > 0 (typically sorted under weakly decreasing lexicographic order). Each (i, x) is called a string, while i is called the length or size of the string and x is the rigging, label , or quantum number of the string. Finally, a rigged configuration is a pair (ν, J) where J = J  i − x. For brevity, we will often denote the ath part of (ν, J) by (ν, J) (a) (as opposed to (ν (a) , J (a) )).
Definition 2.2. Let (ν ∅ , J ∅ ) be the rigged configuration with empty partition and empty riggings. Define RC(∞) to be the graph generated by (ν ∅ , J ∅ ), e a , and f a , for a ∈ I, where e a and f a acts on elements (ν, J) in RC(∞) as follows. Fix a ∈ I and let x be the smallest label of (ν, J) (a) . It is worth noting that, in this case, the definition of the vacancy numbers reduces to ∞ from the crystal structure. Example 2.3. Let g be of type A 1 , then (ν, J) ∈ RC(∞) given by (ν, J) = f k 1 (ν ∅ , J ∅ ) is the partition ν (1) = k and the rigging J Example 2.4. The top of the crystal RC(∞) in type A 2 is shown in Figure 2.1. We note that we write the rigging on the right of each row and the respective vacancy number on the left.
We can extend the crystal structure on rigged configurations to model B(λ) as follows. We consider the subcrystal RC(λ) := {(ν, J) ∈ RC(∞) : (ν, J) is λ-valid} for any λ ∈ P + . We have to modify the definition of the weight to be wt ′ (ν, J) = wt(ν, J) + λ. Thus the crystal operators become f a (ν, J) = 0 if ϕ a (ν, J) = 0, or equivalently if the result under f a above is not a λ-valid rigged configuration. This arises from the natural projection of B(∞) −→ B(λ).

Virtual crystals.
A diagram folding is a surjective map φ : I −→ I between index sets of Kac-Moody algebras and a set (γ a ∈ Z >0 : a ∈ I) of scaling factors. One may induce a map from φ on the corresponding weight lattices Ψ : P −→ P by asserting In [Bak00], it was shown that this defines a U q (g)-crystal structure on the image of v. More generally, we define a virtual crystal as follows.
Definition 2.6. Consider any symmetrizable types g and g with index sets I and I, respectively. Let φ : I −→ I be a surjection such that b is not connected to b ′ for all b, b ′ ∈ φ −1 (a) and a ∈ I. Let B be a U q ( g)-crystal and V ⊆ B. Let γ = (γ a ∈ Z >0 : a ∈ I) be the scaling factors. A virtual crystal is the quadruple (V, B, φ, γ) such that V has an abstract U q (g)-crystal structure defined using the Kashiwara operators e v a and f v a from (2.6) above, and wt := Ψ −1 • wt.
We say B virtualizes in B if there exists a U q (g)-crystal isomorphism v : B −→ V . The resulting isomorphism is called the virtualization map. We denote the quadruple (V, B, φ, γ) simply by V when there is no risk of confusion.
The virtualization map v from rigged configurations of type g to rigged configurations of type g is defined by for all b ∈ φ −1 (a). A U q (g)-crystal structure on rigged configurations is defined by using virtual crystals [OSS03a]. Moreover, we use Equation (2.8) to describe the virtual image of the type g rigged configurations into type g rigged configurations. Explicitly ( ν, J) ∈ V if and only if Next, we recall [SS15a, Conj. 5.12].
Conjecture 2.7. For all g of symmetrizable type, there exists a simply-laced type g and diagram folding φ with scaling factors (γ a ∈ Z >0 : a ∈ I) such that RC(λ) virtualizes in RC(λ v ) under the virtualization map given by Equation (2.8).

Symmetrizable types as foldings from simply-laced types
In this section, we give a modified graph construction from [Lus93, Prop. 14.1.2] which ensures that the resulting graph is simple. We identify simple graphs with simply-laced Dynkin diagrams.
Let D = (d a ) a∈I be a diagonal matrix such that DA is symmetric with d a ∈ Z >0 and gcd(d a : a ∈ I) = 1. where the indices s and s+k are taken modulo N d a and N d b , respectively. Define a map φ A : for a ∈ I and s + 1 understood modulo N d a .
Example 3.1. Let A = ( 2 −6 −4 2 ). Then D = ( 2 0 0 3 ) is a diagonal matrix such that DA = ( 4 −12 −12 6 ) is symmetric. Since d 1 = 2 and d 2 = 3 are relatively prime, set d 1 = 2, d 2 = 3. Then N = 2 and c 12 = c 21 = 2. Hence Γ A has vertices Proposition 3.2. Let A be any symmetrizable Cartan matrix and a = b ∈ I. The map φ A defined above is a Dynkin diagram automorphism of Γ A . Moreover, let E ab denote the number of edges between any fixed vertex in the φ A -orbit of a with some vertex in the φ A -orbit of b. Then −A ab = E ab .
In order to prove this proposition, we require a result from [CCC + 10], which we restate here for the reader's convenience. We also need the following technical lemma.
Lemma 3.4. With the notation as above, we have c ab ≤ N .
Proof. We have . Since N is defined as the maximum over the values given by Equation (3.1), the claim follows. Next, we consider the number of times the edge {v a,0 , v b,0 } occurs in the set above. Note that for k = 0, we have d a,b values of s such that s ≡ 0 mod N d a and s ≡ 0 mod N d b . Lemma 3.4 states that k < N , and hence, there does not exist a value k > 0 such that s ≡ 0 mod N d a such that s + k ≡ 0 mod N d b . From the construction, we can take any fixed edge and obtain the same result. Hence, where we used Proposition 3.3 and the fact that da In other words, Proposition 3.2 states that we can recover A from (Γ A , φ A ). We also have that the induced map on the weight lattice from Equation (2.5) implies 4. Proof of Conjecture 2.7 In this section we prove our main result. That is, we show that RC(λ) virtualizes in RC Ψ(λ) for any λ ∈ P + ∪ {∞}, where Ψ : P −→ P is the induced map on the weight lattices corresponding to Γ A with γ a = 1 for all a ∈ I. Indeed, by the definition of the crystal operators, we can restrict the proof to the rank two case. The fact that the crystal operators commute with the virtualization map can be made using an argument similar to [SS15b,Prop. 3.7] using Proposition 3.2 and the construction of Γ A .
We sketch the argument here. Note that m , a ∈ I, and i ∈ Z >0 . Hence e v a and f v a change ν (b) for all b ∈ φ −1 (a) in exactly the same position. Moreover, each ν (b ′ ) for b ′ ∈ φ −1 (a ′ ) has exactly −A aa ′ values of b ∈ φ −1 (a) such that A bb ′ = −1 (i.e., b and b ′ are adjacent in Γ A ), so when there is a change in vacancy numbers, and hence a change in the riggings, it is exactly A aa ′ for all a ′ ∈ I. So f v a ( ν, J) = v f a (ν, J Theorem 4.1. Let g be a Kac-Moody algebra of arbitrary symmetrizable type. Then RC(λ) ∼ = B(λ) for λ ∈ P + ∪ {∞}.

Littlewood-Richardson rule
In this section, we give a Littlewood-Richardson rule using rigged configurations, which requires a combinatorial description of ε a and ϕ a . The proof of the following Proposition follows [Sak14,SS15a].
Example 5.3. Suppose g is of type A 2 and let λ = Λ 1 + Λ 2 and µ = Λ 1 . Since B(µ) is the crystal Recall that rigged configurations in finite type can be considered as classical components of U ′ q (g)crystals, where g is of affine type, isomorphic to N i=1 B ri,1 . We note that there is an algorithm to construct all classically highest weight U ′ q (g)-rigged configurations given by Kleber in simplylaced types [Kle98] and extended to all other types by using virtualization [OSS03b]. It would be interesting to determine which nodes of the Kleber tree correspond to the highest weight elements in B(λ) ⊗ B(µ) and more generally B(λ 1 ) ⊗ · · · ⊗ B(λ ℓ ).