Adjoint crystals and Young walls for Up(sl2)

We develop the combinatorics of Young walls associated with higher level adjoint crystals for the quantum affine algebra Uq(c sl2). The irreducible highest weight crystal B(λ) of arbitrary level is realized as the affine crystal consisting of reduced Young walls on λ. We also give a Young wall realization of the crystal B(∞) for U− q (c sl2). Introduction The theory of perfect crystals was developed by Kang et al. in an effort to understand the theory of vertex models in terms of representation theory of quantum affine algebras [7]. A perfect crystal of level l is a finite classical crystal whose minimal vectors are in 1-1 correspondence with dominant integral weights of level l. If B is a perfect crystal of level l and B(λ) is an irreducible highest weight crystal of level l, then there exists a crystal isomorphism B(λ) ∼ −→ B(ε(bλ)) ⊗B, where bλ is the unique minimal vector such that φ(bλ) = λ. Using this crystal isomorphism, they obtained the path realization of irreducible highest weight crystals for quantum affine algebras. In [8] and [6], Kang et al. constructed at least one coherent family of perfect crystals for all classical quantum affine algebras. Thus the next natural problem would be to construct perfect crystals for exceptional quantum affine algebras. In [1], Benkart et al. gave a uniform construction of level 1 perfect crystals B = B(0) ⊕B(θ) for all quantum affine algebras (including exceptional cases), where θ denotes the (shortest) maximal root of the corresponding finite dimensional simple Lie algebras. Therefore, one can naturally conjecture that the crystal B = B(0) ⊕ B(θ) ⊕ · · · ⊕ B(lθ) is a perfect crystal of level l for all quantum affine algebras. For quantum affine algebras of type A (1) n , C (1) n , A (2) 2n and D (2) n+1, this conjecture was proved in [8] and [6]. In [18], Schilling and Sternberg proved this conjecture for quantum affine algebras of type D (1) n , which yields This research was supported by KRF Grant # 2005-070-C00004. This research was supported by KRF Grant # 2007-341-C00001. This research was supported by BK21 Mathematical Sciences Division. This research was partially supported by KIAS, RIMS, and Yonsei BK21. 1 2 JI HYE JUNG ET AL. the proof for the cases A (2) 2n−1 and B (1) n . We will call the perfect crystal B = B(0) ⊕B(θ) ⊕ · · · ⊕B(lθ) the adjoint crystal of level l. On the other hand, in [4], Kang introduced the combinatorics of Young walls for classical quantum affine algebras. The Young walls consist of colored blocks with various shapes that are built on a given ground-state wall and they can be viewed as generalizations of Young diagrams. The rules and patterns for building Young walls and the action of Kashiwara operators are given explicitly using combinatorics of Young walls. Now the crystal graphs for basic representations are characterized as the affine crystals consisting of reduced Young walls and the characters of basic representations can be computed easily by counting the number of colored blocks in reduced Young walls that have been added to the ground-state wall. In [9], Kang and Lee extended the combinatorics of Young walls to the higher level cases for all classical quantum affine algebras. (See [3] for A (1) n case.) As is shown in [5], it turned out that the combinatorics of Young walls and the theory of perfect crystals are very closely related. The Young walls known so far can be understood as the combinatorial models for the paths arising from the perfect crystals corresponding to the fundamental weight l̟1. Thus it is a very natural and interesting problem to construct Young wall models associated with various perfect crystals. In particular, the Young wall models for adjoint crystals would yield an explicit combinatorial realization of irreducible highest weight crystals for exceptional quantum affine algebras. In this paper, as the first step toward this goal, we develop the combinatorics of Young walls associated with higher level adjoint crystals for the quantum affine algebra Uq(ŝl2). The irreducible highest weight crystal B(λ) of arbitrary level is realized as the affine crystal consisting of reduced Young walls on λ. We would like to emphasize that our Young walls are different from the ones given in [3] and [9]. The main difference lies in the following: The previously known higher level Young walls are l-tuples or l-layers of level 1 Young walls. Thus they are of thickness l. Actually, they are made up of level l perfect crystals corresponding to the fundamental weight ̟1. But the Young walls introduced in this paper are made up from the level l adjoint crystal and are of thickness ≤ 1. The perfect crystals corresponding to the fundamental weight l̟1 have not yet been constructed for exceptional affine algebras. But level l adjoint crystals were constructed for some exceptional cases including D (3) 4 , and there is a good conjecture that level l adjoint crystals are perfect for all quantum affine algebras. (For classical quantum affine algebras, this conjecture was recently proved in [16].) Thus we believe our paper will give us a new insight how to construct level l Young walls of thickness 1, which are not l layers of level 1 Young walls for all quantum affine algebras. Another advantage of Young walls in this paper is in the connection with Littelmann’s path model. Up to now, there was no explicit combinatorial ADJOINT CRYSTALS AND YOUNG WALLS 3 explanation between Young walls and Littelmann’s path model. The Young walls constructed in this paper are found to be in 1-1 correspondence with Littelmann’s path model, which is explicitly explained in combinatorial way for level 1 case in [10]. We expect this result can be generalized for higher level cases. This paper is organized as follows. In Section 1, we briefly review the basic theory of perfect crystals for the quantum affine algebra Uq(ŝl2). In Section 2, we give a detailed description of higher level adjoint crystals, (affine) energy functions and (affine) path realization of irreducible highest weight crystals. In Section 3, we explain the combinatorics of Young walls associated with adjoint crystals. In Section 4, we define the crystal structure on the set of Young walls and show that the irreducible highest weight crystal B(λ) of arbitrary level is isomorphic to the affine crystal consisting of reduced Young walls on λ. The level 1 case was treated in [19]. Finally, in Section 5, we give a Young wall realization of the crystal B(∞) for U q (ŝl2). 1. The algebra Uq(ŝl2) and perfect crystals Recall that the affine Cartan datum (A,Π,Π, P, P) of type A (1) 1 consists of : (i) the affine generalized Cartan matrix A = (aij)i,j∈I = ( 2 −2 −2 2 ) , (ii) the dual weight lattice P = Zh0 ⊕ Zh1 ⊕ Zd, (iii) the set of simple coroots Π = {h0, h1}, (iv) the set of simple roots Π = {α0, α1} ⊂ h ∗ satisfying αi(hj) = aij , αi(d) = δi,0 (i, j ∈ I), where h = C ⊗ Z P, (v) the weight lattice P = ZΛ0 ⊕ ZΛ1 ⊕ Zδ, where Λi(hj) = δij , Λi(d) = 0 (i, j ∈ I), and δ = α0 + α1 is the null root. Let q be an indeterminate and let [n] = q − q q − q−1 . The quantum affine algebra Uq(ŝl2) is the quantum group associated with the affine Cartan datum (A,Π,Π, P, P) of type A (1) 1 . That is, it is the associative algebra over C(q) generated by the elements ei, fi, K ±1 i = q ±hi , (i = 0, 1) and q with the following defining relations: (1.1) q = 1, qq ′ = q ′ for all h, h ∈ P, qeiq −h = qiei, q fiq −h = qifi for h ∈ P , eifj − fjei = δij Ki −K −1 i q − q−1 , ei ej − [3]e 2 i ejei + [3]eieje 2 i − eje 3 i = 0 (i 6= j), f i fj − [3]f 2 i fjfi + [3]fifjf 2 i − fjf 3 i = 0 (i 6= j). 4 JI HYE JUNG ET AL. We denote by U ′ q(ŝl2) the algebra generated by ei, fi,K ±1 i (i ∈ I). It can be regarded as the quantum group associated with the classical Cartan datum (A,Π,Π, P̄ , P̄), where P̄ = ZΛ0 ⊕ ZΛ1, P̄ ∨ = Zh0 ⊕ Zh1, and αi,Λi (i ∈ I) are considered as linear functionals on h̄ = C ⊗ Z P̄. The elements of P (respectively, P̄ ) are called the affine weights (respectively, classical weights), and the elements of P+ = {λ ∈ P |λ(hi) ∈ Z≥0 for all i ∈ I} (respectively, P̄+ = {λ ∈ P̄ | λ(hi) ∈ Z≥0 for all i ∈ I}) are called the affine dominant integral weights (respectively, classical dominant integral weights). The level of a dominant integral weight λ = a0Λ0 + a1Λ1 is defined to be value λ(c) = a0 + a1, where c = h0 + h1 is the canonical central element. Definition 1.1. An affine crystal or a Uq(ŝl2)-crystal (respectively, a classical crystal or a U ′ q(ŝl2)-crystal) is a set B together with the maps wt : B → P (respectively, wt : B → P̄ ), ẽi, f̃i : B → B ⋃ {0} and εi, φi : B → Z ⋃ {−∞} (i ∈ I) satisfying the following properties: (i) φi(b) = εi(b) + 〈hi,wt(b)〉 for all i ∈ I (respectively, φi(b) = εi(b) + 〈hi,wt(b)〉 for all i ∈ I), (ii) wt(ẽib) = wt(b) + αi if ẽib ∈ B (respectively, wt(ẽib) = wt(b) + αi if ẽib ∈ B), (iii) wt(f̃ib) = wt(b) − αi if f̃ib ∈ B (respectively, wt(f̃ib) = wt(b) − αi if f̃ib ∈ B), (iv) εi(ẽib) = εi(b) − 1, φi(ẽib) = φi(b) + 1 if ẽib ∈ B, (v) εi(f̃ib) = εi(b) + 1, φi(f̃ib) = φi(b) − 1 if f̃ib ∈ B, (vi) f̃ib = b ′ if and only if b = ẽib ′ for b, b ′ ∈ B, i ∈ I, (vii) If φi(b) = −∞ for b ∈ B , then ẽib = f̃ib = 0. For a classical crystal B, we define ε(b) = ε0(b)Λ0 + ε1(b)Λ1, φ(b) = φ0(b)Λ0 + φ1(b)Λ1 for b ∈ B. Example 1.2. (a) Let V (λ) be the irreducible highest weight Uq(ŝl2)module with the highest weight λ ∈ P+. The crystal graph B(λ) of V (λ) is an affine crystal. (b) The crystal graph B of a finite dimensional U ′ q(ŝl2)-module V is a classical crystal. The crystals have a very nice and simple behavior with respect to taking tensor product. If B1 and B2 are crystals, then the set B1 ⊗ B2 := B1 × B2 ADJOINT CRYSTALS AND YOUNG WALLS 5 is given a crystal structure as follows: ẽi(b1 ⊗ b2) = { ẽib1 ⊗ b2 if φi(b1) ≥ εi(b2), b1 ⊗ ẽib2 if φi(b1) < εi(b2), f̃i(b1 ⊗ b2) = { f̃ib1 ⊗ b2 if φi(b1) > εi(b2), b1 ⊗ f̃ib2 if φi(b1) ≤ εi(b2), wt(b1 ⊗ b2) = wt(b1) + wt(b2), εi(b1 ⊗ b2) = max(εi(b1), εi(b2) − 〈hi,wt(b1)〉), φi(b1 ⊗ b2) = max(φi(b2), φi(b1) + 〈hi,wt(b2)〉). Definition 1.3. Let l be a positive integer. A finite classical crystal B is called a perfect crystal of level l if (i) there exists a finite dimensional U ′ q(ŝl2)-module with a crystal basis whose crystal graph is isomorphic to B, (ii) B ⊗ B is connected, (iii) there exists a classical weight λ0 ∈ P̄ such that wt(B) ⊂ λ0 + ∑ i6=0 Z≤0 αi, #(Bλ0) = 1, where Bλ0 = {b ∈ B | wt(b) = λ0}, (iv) for any b ∈ B, 〈c, ε(b)〉 ≥ l, (v) for any λ ∈ P̄+ with λ(c) = l, there exist unique b, bλ ∈ B such that ε(b) = λ = φ(bλ). The heart of the theory of perfect crystals is the following crystal isomorphism theorem: Theorem 1.4. ([7]) Let B be a perfect crystal of level l. For any λ ∈ P̄+ with λ(c) = l, there exists a unique crystal isomorphism Ψ : B(λ) ∼ −→ B(ε(bλ)) ⊗ B uλ 7−→ uε(bλ) ⊗ bλ, where uλ is the highest weight vector in B(λ) and bλ is the unique vector in B such that φ(bλ) = λ. By applying the crystal isomorphism Ψ repeatedly, we get a sequence of crystal isomorphisms B(λ) ∼ −→ B(λ1) ⊗ B ∼ −→ B(λ2) ⊗ B ⊗ B ∼ −→ · · · uλ 7−→ uλ1 ⊗ b0 7−→ uλ2 ⊗ b1 ⊗ b0 7−→ · · · , where λ0 = λ, b0 = bλ0 , λk+1 = ε(bλk), bk+1 = bλk+1 . The infinite sequence pλ = (bk) ∞ k=0 ∈ B ⊗∞ is called the ground-state path of weight λ and the elements of P(λ) := {p = (pk) ∞ k=0 ∈ B ⊗∞ | pk ∈ B, pk = bk for all k ≫ 0} 6 JI HYE JUNG ET AL. are called the λ-paths. One of the main results of [7] is the path realization of B(λ) given below. Theorem 1.5. ([7]) The set P(λ) can be given a classical crystal structure, and there exists a classical crystal isomorphism Ψλ : B(λ) ∼ −→ P(λ) uλ 7−→ pλ, where uλ is the highest weight vector in B(λ). 2. Adjoint crystals Fix a positive integer l and let B = {(x, y) ∈ Z≥0 × Z≥0 | x + y = 2k, 0 ≤ k ≤ l}. It is straightforward to verify that B is given a classical crystal structure with the following maps: (2.1) wt(x, y) = (y − x)Λ0 + (x− y)Λ1, ε1(x, y) = y, φ1(x, y) = x, ε0(x, y) = l − y + 1 2 |x− y|, φ0(x, y) = l − x + 1 2 |x− y|, ẽ1(x, y) = (x + 1, y − 1), f̃1(x, y) = (x− 1, y + 1), ẽ0(x, y) = { (x− 2, y) if x > y, (x, y + 2) if x ≤ y, f̃0(x, y) = { (x + 2, y) if x ≥ y, (x, y − 2) if x < y. The U ′ q(ŝl2)-crystal B is called the adjoint crystal of level l. It was shown in [8] that B is a perfect crystal of level l with minimal vectors b = bλ = (a, a) for λ = (l − a)Λ0 + aΛ1. Hence we have the crystal isomorphism Ψ : B(λ) ∼ −→ B(λ) ⊗ B, uλ 7→ uλ ⊗ (a, a), which yields the path realization B(λ) ∼= P(λ) = {p = ((xk, yk))k≥0 | (xk, yk) ∈ B for all k ≥ 0, (xk, yk) = (a, a) for all k ≫ 0}. ADJOINT CRYSTALS AND YOUNG WALLS 7 Recall that an energy function on a (finite) classical crystal B is a Z-valued function h : B ⊗ B → Z such that h(ẽ0(b1 ⊗ b2)) = { h(b1 ⊗ b2) + 1 if φ0(b1) ≥ ε0(b2), h(b1 ⊗ b2) − 1 if φ0(b1) < ε0(b2), h(ẽ1(b1 ⊗ b2)) = h(b1 ⊗ b2), whenever ẽ0(b1 ⊗ b2), ẽ1(b1 ⊗ b2) ∈ B ⊗ B (see, for example, [2]). Proposition 2.1. Let B be the adjoint crystal of level l. Define a function h : B ⊗ B → Z by h((x1, y1) ⊗ (x2, y2)) =    |a− b| if x1 > y2 ≥ a or y1 ≤ x1 ≤ y2, |a− y2| + |b− y2| if x1 > y2, a > y2, |a− x1| + |b− x1| if x1 ≤ y2, x1 < y1, where a = x1 + y1 2 and b = x2 + y2 2 . Then h is an energy function on B satisfying h((0, 0) ⊗ (0, 0)) = 0. Proof. Suppose x1 > y2 ≥ a = x1 + y1 2 . In this case, we have x1 > y1, φ0(x1, y1) = l − a, ε0(x2, y2) = l − y2 + 1 2 |x2 − y2| and h((x1, y1) ⊗ (x2, y2)) = |a− b|. Hence ẽ1((x1, y1) ⊗ (x2, y2)) = (x1 + 1, y1 − 1) ⊗ (x2, y2) and h((x1 + 1, y1 − 1) ⊗ (x2, y2)) = |a− b|. If φ0(x1, y1) ≥ ε0(x2, y2), then y2 − a ≥ 1 2 |x2 − y2|, which yields a ≤ b. Hence by definition, we have ẽ0((x1, y1) ⊗ (x2, y2)) = (x1 − 2, y1) ⊗ (x2, y2) and h((x1 − 2, y1) ⊗ (x2, y2)) = |a− b− 1| = |a− b| + 1. If φ0(x1, y1) < ε0(x2, y2) and x2 > y2, then we have b = x2 + y2 2 > y2 ≥ a. Hence we obtain ẽ0((x1, y1) ⊗ (x2, y2)) = (x1, y1) ⊗ (x2 − 2, y2) and h((x1, y1) ⊗ (x2 − 2, y2)) = |a− (b + 1)| = |a− b| − 1. If φ0(x1, y1) < ε0(x2, y2) and x2 ≤ y2, then we have y2 − a < y2 − x2 2 , which yields a > b. Hence we obtain ẽ0((x1, y1) ⊗ (x2, y2)) = (x1, y1) ⊗ (x2, y2 + 2) 8 JI HYE JUNG ET AL. and h((x1, y1) ⊗ (x2, y2 + 2)) = |a− b− 1| = |a− b| − 1. The other cases can be verified in a similar manner. ¤ Let B be the adjoint crystal of level l and B̂ = {b(m) | b ∈ B,m ∈ Z} be the affinization of B (see, for example, [2, 7]). The affine crystal structure on B̂ is given as follows: (2.2) wt(b(m)) = wt(b) + mδ, ẽ0(b(m)) = (ẽ0b)(m + 1), ẽ1(b(m)) = (ẽ1b)(m), f̃0(b(m)) = (f̃0b)(m− 1), f̃1(b(m)) = (f̃1b)(m), εi(b(m)) = εi(b), φi(b(m)) = φi(b) for b ∈ B, i = 0, 1, and m ∈ Z. Let h : B ⊗ B → Z be an energy function on B. We define the affine energy function H : B̂ ⊗ B̂ → Z associated with h : B ⊗ B → Z to be (2.3) H(b1(m) ⊗ b2(n)) = m− n− h(b1 ⊗ b2). Note that (2.4) H(ẽi(b1(m) ⊗ b2(n))) = H(f̃i(b1(m) ⊗ b2(n))) = H(b1(m) ⊗ b2(n)) for i = 0, 1. From the crystal isomorphism Ψ : B(λ) ∼ −→ B(λ) ⊗ B, uλ 7→ uλ ⊗ (a, a), we obtain a crystal embedding Ψ̂ : B(λ) →֒ B(λ) ⊗ B̂, uλ 7→ uλ ⊗ (a, a)(0). By applying the crystal embedding Ψ̂ repeatedly, we obtain a sequence of embeddings B(λ) →֒ B(λ) ⊗ B̂ →֒ B(λ) ⊗ B̂ ⊗ B̂ →֒ · · · uλ 7→ uλ ⊗ (a, a)(0) 7→ uλ ⊗ (a, a)(0) ⊗ (a, a)(0) 7→ · · · . Hence B(λ) is isomorphic to the connected component P̂(λ) of B̂ containing the affine ground-state path p̂λ = (· · · (a, a)(0), · · · (a, a)(0), (a, a)(0)). Using the affine energy function H, the connected component P̂(λ) is characterized as follows. Proposition 2.2. ([2, 7, 11]) For a dominant integral weight λ = (l−a)Λ0+ aΛ1, there exists an affine crystal isomorphism B(λ) ∼ −→ P̂(λ), uλ 7→ p̂λ, ADJOINT CRYSTALS AND YOUNG WALLS 9