Categorifying the tensor product of the Kirillov-Reshetikhin crystal $B^{1,1}$ and a fundamental crystal

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of"trivial"modules. The nodes of the fundamental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.


Introduction
Kang-Kashiwara [9] and Webster [23] show the cyclotomic Khovanov-Lauda-Rouquier (KLR) algebra R Λ categorifies the highest weight representation V (Λ) in arbitrary symmetrizable type. (KLR algebras are also known in the literature as quiver Hecke algebras.) By a slight abuse of language, we will say the combinatorial version of this statement is that R Λ categorifies the crystal B(Λ), where simple modules correspond to nodes, and functors that take socle of restriction correspond to arrows, i.e. the Kashiwara crystal operators [16]. Webster [23] and Losev-Webster [18] categorify the tensor product of highest weight modules, and hence the tensor product of highest weight crystals. However, one can consider a tensor product of crystals (1) † Email: hkvinge@math.ucdavis.edu ‡ Email: vazirani@math.ucdavis.edu Partially supported by NSA grant H98230-12-1-0232, ICERM, and the Simons Foundation.
where Λ, Λ ∈ P + are of level k and B is a perfect crystal of level k. We (combinatorially) categorify the crystal isomorphism (1) in the case Λ = Λ i is a fundamental weight and B = B 1,1 is a Kirillov-Reshetikhin crystal. In other words, our main theorems give a purely module-theoretic construction of this crystal isomorphism. (One must modify the form of the crystal isomorphism in the case B 1,1 is not perfect or when Λ i is not of level 1.) Each node of B 1,1 corresponds to an infinite family of "trivial" modules, but note this does not give a categorification of B. These "trivial" modules T p;k are the KLR analogues of the nodes in highest weight crystals studied in [21].
If we apply Theorem 6.3 to iterating (1), this corresponds to constructing a simple module as a quotient of Ind T pa;ka · · · T pz;kz . In type A, this is somewhat intermediate between the crystal operator construction and the Specht module construction. See [22] for details. This paper also describes how socle of restriction interacts with the construction. One can also recover the paper's results for finite type whose Dynkin diagram is a subdiagram of that of type X studied here. For a construction of simple modules related to the crystal B(∞) for finite type KLR algebras see [2]. This paper generalizes the theorems and constructions from [22] for type A affine.

Cartan datum
Fix an integer ≥ 2. X will be one of the following types: 2 −1 . I = {0, 1, . . . , } will denote the indexing set. Let [a ij ] i,j∈I denote the associated Cartan matrix. We direct the reader to [7] for the explicit matrices. Following [7] we let h be a Cartan subalgebra, = {α 0 , . . . , α } its system of simple roots, ∨ = {h 0 , . . . , h } its simple coroots, and Q the root lattice. Then set Q + = i∈I Z ≥0 α i . For an element ν ∈ Q + , we define its height |ν| to be the sum of the coefficients, i.e. if ν = i∈I ν i α i then There is a canonical pairing , : h × h * → C with a ij = h i , α j . Using this pairing we define the fundamental weights {Λ i | i ∈ I} via h i , Λ j = δ ij . The weight lattice is i∈I ZΛ i and the integral dominant weights are P + = i∈I Z ≥0 Λ i . We also have a symmetric bilinear form ( , ) : h * × h * → C, For the Dynkin diagrams of the types under consideration we direct the reader to [7].

The tensor product of two crystals
We refer the reader to [11] for the definition of a crystal. Let B 1 and B 2 be two crystals with nodes b 1 ∈ B 1 and b 2 ∈ B 2 . We recall that the crystal structure on the tensor product B 1 ⊗ B 2 is given by Given a crystal B, we can draw its associated crystal graph with nodes (or vertices) B and I-colored arrows (directed edges) as follows. When e i b = a (so b = f i a) we draw an i-colored arrow a i − → b. We also say b has an incoming i-arrow and a has an outgoing i-arrow.

Type A
In type A (1) , the highest weight crystal B(Λ i ) has a model with nodes the ( + 1)-restricted partitions, i.e. λ = (λ 1 , . . . , λ t ) such that λ r ∈ Z ≥0 , 0 ≤ λ r − λ r+1 < + 1 for all r. Let B 1,1 be the crystal graph is an example of a perfect crystal (see [10] for the definition and important properties). One key property this level 1 perfect crystal has is that tensoring it with a fundamental (or highest weight level 1) crystal yields an isomorphism to another level 1 highest weight crystal. In particular, for i ∈ I there exists an isomorphism of crystals, The isomorphism is pictured in Figure 1 for i = 2 and = 2. Note the underlying graph of B(Λ i ) is identical to that of B(Λ 0 ), but the colors of the arrows are obtained from those of B(Λ 0 ) by adding i mod ( +1).

General type
The perfect crystal B 1,1 in type A (1) is also an example of a Kirillov-Reshetikhin (KR) crystal. For a quantized affine algebra U q (g), the KR crystals B r,s correspond to a special family of finite dimensional modules W r,s indexed by a positive integer s and a Dynkin node r from the classical subalgebra g 0 of g [14], [20]. In all of the types we consider, with the exception of C (1) , the crystal B 1,1 is perfect of level 1 [5]. When B 1,1 is perfect and Λ i is a level 1 fundamental weight for i ∈ I, B 1,1 has a unique node b i such that ε(b i ) = Λ i and ϕ(b i ) = Λ σ(i) for some σ(i) ∈ I. There then exists a crystal isomorphism [10] T : In the interest of space we do not provide details of the case where either B 1,1 is not perfect or Λ i is not a level one weight. However, with modification the theorems and proofs in this paper hold in these cases also.

Key definitions: class A, B, D nodes and cyclotomic paths
The analogues of trivial modules that we study below are constructed from crystal data given by walks on B 1,1 . Below are definitions used to describe these walks.
Definition 3.1 Let X be one of the affine types listed in Section 2.1 and let I be the indexing set of its Dynkin nodes. A type X path p of length k, is a function p : {0, 1, . . . , k − 1} → I such that there is a directed walk in the type X crystal B 1,1 whose ith step corresponds to a p(i)-arrow.
When k > 1, a path p of length k corresponds to a unique directed walk in B 1,1 . In this case we use the terms "type X path" and "walk in B 1,1 " interchangeably. Definition 3.2 For a path p : {0, 1, . . . , k − 1} → I we call the arrow corresponding to p(0) the tail of p, and the arrow corresponding to p(k − 1) the head of p. An extension to the tail of p by a j-arrow is a path p : {0, 1, . . . , k} → I such that p (t) = p(t − 1) for 1 ≤ t ≤ k and p (0) = j. An extension to the head of p by a j-arrow, is a path p : {0, 1, . . . , k} → I such that p (t) = p(t) for 0 ≤ t ≤ k − 1 and p (k) = j.
Let π(j) be the length 1 path π(j) : {0} → I, π(j)(0) = j. For a path p, we denote the extension of its tail by a j-arrow by π(j) p and the extension of its head by a j-arrow by p π(j). We can think of extension as concatenation of paths. If the tail (respectively head) of p cannot be extended by a j-arrow then we set π(j) p = 0 (respectively p π(j) = 0).
The set of colors of arrows that can extend the tail of a path p of length k > 1 is denoted ext − p , The set of colors of arrows that can extend the head of p is denoted ext + p , Type class A class B class D pairs When ext − p (respectively ext + p ) contains a single element i, we allow ourselves the convenience of writing p(−1) = i (respectively p(k) = i).
In Table 1 we classify the elements of I into three classes: A, B, and D.
Below is one of our key definitions. 3. ext − p = {i 1 }, i.e. p has a unique extension by an i 1 -arrow to its tail.
The following proposition follows from inspection of the B 1,1 crystal graphs.
Proposition 3.4 Let X be any type not equal to D 3 . Then for all i ∈ I there exists some j ∈ I such that there is a cyclotomic path p of tail weight (Λ j , Λ i ). When these paths exist they can be of any length k ∈ Z ≥1 . Definition 3.5 We refer to 1 ∈ I in type D  While paths are about i ∈ I and hence the arrows of B 1,1 , informally, a cyclotomic path corresponds to a walk that starts at a "level 1 node," that is a node b with ε(b) = Λ i1 and ϕ(b) = Λ i2 .
For ν ∈ Q + with |ν| = m, the KLR algebra R(ν) associated with Cartan matrix [a ij ] i,j∈I is the associative, graded, unital C-algebra generated by subject to relations which can be found in [12] or [16]. We set R = ν∈Q + R(ν) (which is not unital).
The elements 1 i are orthogonal idempotents since they satisfy 1 i 1 j = δ i,j 1 i and the identity element of R(ν) is R(α i ) has a unique (up to grading shift) simple module L(i). It is 1-dimensional and x 1 1 i acts as zero. R(nα i ) has a unique simple module L(i n ).

Induction, co-induction, and restriction
It was shown in [12] and [13] that for ν, µ ∈ Q + there is a non-unital embedding Using this embedding one can define induction and restriction functors, In the future we will write Ind ν+µ ν,µ = Ind and Res ν+µ ν,µ = Res when the algebras are understood from the context. More generally we can extend this embedding to finite tensor products We refer to the image of this embedding as a parabolic subalgebra and denote it by R(ν) ⊂ R(ν (1) + · · · + ν (k) ).

Crystal operators
Define the functor e i : R(ν)mod → R(ν − α i )mod as the restriction, • Res ν ν−αi,αi M. When M is simple we can further refine this functor by setting e i M := soc e i M . Recall for a module N that soc N is its largest semisimple submodule while cosoc N is its largest semisimple quotient. We measure how many times we can apply e i to M by Let (still assuming M is simple). We refer the reader to [12] for the most important facts about e i , e i , f i .
It is important to note that by the exactness of restriction, e i , e ∨ i are exact functors, while e i and e ∨ i are only left exact, and f i and f ∨ i are only right exact.

Rep Λ and the functor pr Λ
For Λ = i∈I λ i Λ i ∈ P + define I Λ ν to be the two-sided ideal of R(ν) generated by the elements x λi 1 1 1 i for all i ∈ Seq(ν). When ν is clear from the context we write, I Λ ν = I Λ . The cyclotomic KLR algebra of weight Λ is then defined as The algebra R Λ (ν) is finite dimensional [3,16]. The category of finite dimensional R-modules on which I Λ vanishes is denoted Rep Λ . We construct a right-exact functor, pr Λ : and extend it to pr Λ : Rmod → Rmod. We will primarily consider Λ = Λ i in which case I Λi ν is generated by x 1 1 ii2...im and 1 ji2...im , j = i ranging over i ∈ Seq(ν).

Module-theoretic model of B(Λ)
Let M be a simple R(ν)-module. Set Let Irr R be the set of isomorphism classes of simple R-modules and Irr R Λ be the set of isomorphism classes of simple modules in In [16] it was shown that the tuple (Irr R, ε i , ϕ i , e i , f i , wt) defines a crystal isomorphic to B(∞) and (Irr R Λ , ε i , ϕ Λ i , e i , f i , wt) defines a crystal isomorphic to the highest weight crystal B(Λ). Proposition 4.2 [16] Let M be a simple R(ν)-module with pr Λ M = 0. Then We mimic the conventions usually used in the theory of crystals and define 5 The family of modules T p;k Denote by 1 the trivial R(0)-module.
Definition 5.1 For any type X and for a fixed path p of length k in B 1,1 , define . . . f p(0) 1 ∼ = T p;k (16) and γ p;k := k−1 i=0 α p(i) so that T p;k is an R(γ p;k )-module. When T p;k / ∈ Rep Λ p(0) then T p;k ∈ Rep Λ p(0) +Λ p (1) . However, such p will not ever arise in our main theorems. If p is a cyclotomic path of tail weight (Λ i1 , Λ i2 ) then the modules T p;k belong to Rep Λi 2 for any k ≥ 0. This is part of the motivation for Definition 3.3 of cyclotomic path.
Note that this definition implies that if j ∈ ext + p then f j T p;k ∼ = T p π(j);k+1 . Observe T p;0 = 1. For k > 1, we define ϕ − j (p; k) and ϕ + j (p; k) as in Table 2, so that We also define, Remark 5.2 Recall that if p is a cyclotomic path of tail weight (Λ i1 , Λ i2 ) then p has a unique extension to its tail by an i 1 -arrow, i.e. ext − p = {i 1 }.
While the proof is too technical to give in full here, an outline of it follows. Set R 0 (A) := A, and We show inductively that there exist surjections For each of these surjections Lemma 6.2 implies that R t (A) ∈ Rep Λ p t (−1) +ϕ + (pt;t) . The induction will end at the smallest k such that R k (A) ∈ Rep Λ p(−1) . Then we set p = p k , r(A) = k, and R(A) = R k (A). We further conjecture the following.
Remark 6.5 In some types, having chosen p(0) = i, the choice of p(1) and consequently p t , t < r(A) is forced upon us. In other types, such as C (1) , there can be 2 choices for p(1) (and hence p(−1)). This choice is mirrored by the combinatorial structure of B 1,1 ⊗ B(Λ i ).
Note that by Section 4.5, the crystal-theoretic consequence of Theorem 6.3 is a map and where j runs over all possibilities for p(−1). (See Remark 6.5.) By abuse of notation, we let p(k−1) be the node in B 1,1 that the path p ends at. In this way, each node of B 1,1 corresponds to the infinite collection of paths p that end at that node, and in turn to the collection of modules T p;k . (As remarked in the introduction, this is not a categorification, but it is a useful correspondence.) If we choose to specify B(Λ i ) as above, that further specifies that we consider the p with p(0) = i from that collection. To recover the crystal isomorphism (3) from Theorem 6.3 one must actually fix p(−1) and let i vary (whereas the theorem fixes i). In many types X , i ∈ I, specifying p(0) = i determines p(−1) (in particular when Λ i is of level 1 and B 1,1 is perfect). In type A the relationship between (3) and (2) is transparent. [22] discusses the crystal isomorphism in type A in more depth. At the moment, the above theorem just gives us a map of nodes. Our second main theorem in Section 6.1 below will show that it is a morphism of crystals.

The action of the crystal operators
Next we study the action of the crystal operators e j and f j on (2) to show that the map in part 2 of Theorem 6.3 categorifies our crystal isomorphism T .
Compare the theorems below with the crystal-theoretic statement (2). As in [16] simple modules in Rep Λi correspond to nodes in the highest weight crystal B(Λ i ). Each node b of the KR crystal B 1,1 corresponds to an infinite family of R(γ p;k )-modules T p;k , k ∈ Z ≥0 that satisfy ε(T p;k ) = ε(b). It is in this manner that the main theorems of this paper give a categorification of the crystal isomorphism T .