Quiver Varieties and Path Realizations arising from Adjoint Crystals of type $A_n^{(1)}$

Let $B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(\Lambda_0)$ of $B(\Lambda_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(\Lambda_0)$ of $B(\Lambda_0)$ arising from the adjoint crystal $\adjoint$.


Introduction
The theory of perfect crystals developed in [7] has a lot of important and interesting applications to the representation theory of quantum affine algebras and the theory of vertex models in mathematical physics. In particular, the crystal B(λ) of an integrable highest weight module over a quantum affine algebra can be realized as the crystal P B (λ) consisting of λ-paths arising from a given perfect crystal B. In [1], Benkart, Frenkel, Kang and Lee gave a uniform construction of level 1 perfect crystals for all quantum affine algebras. These perfect crystals are called the adjoint crystals because, when forgetting 0-arrows, they coincide with the direct sums of the trivial crystals and the crystals of adjoint or little adjoint representations of finite dimensional simple Lie algebras.
On the other hand, for a symmetric Kac-Moody algebra g, Lusztig gave a geometric construction of U − q (g) in terms of perverse sheaves on quiver varieties and introduced the notion of canonical basis which yields natural bases for all integrable highest weight modules as well [13,14]. In [12], Kashiwara and Saito defined a crystal structure on the set B(∞) of irreducible components of Lusztig's quiver varieties and showed that B(∞) is isomorphic to the crystal B(∞) of U − q (g). Moreover, in [16,17], Nakajima defined a new family of quiver varieties associated with a dominant integral weight λ and gave a geometric realization of the integrable highest weight g-module V (λ). In [18], Saito defined a crystal structure on the set B(λ) of irreducible components of certain Lagrangian subvarieties of Nakajima's quiver varieties, and showed that B(λ) is isomorphic to the crystal B(λ) of V (λ).
Therefore, for quantum affine algebras, it is natural to investigate the crystal isomorphism between the geometric realization B(λ) and the path realization P B (λ) for various perfect crystals B. In this paper, we will focus on the level 1 highest weight crystal B(Λ 0 ) of the quantum affine algebra U q (A the path realization arising from the adjoint crystal B ad . We will also give a geometric interpretation of the fundamental isomorphism theorem for perfect crystals: B(Λ 0 ) ∼ = B(Λ 0 ) ⊗ B ad . One of the key ingredients of our construction is the explicit 1-1 correspondence between B(Λ 0 ) and Y(Λ 0 ) discovered in [3,19], where Y(Λ 0 ) is the crystal consisting of (n + 1)-reduced Young diagrams. We hope our construction will provide a new insight toward the understanding of the connection between B(Λ 0 ) and P ad (Λ 0 ) for all quantum affine algebras.
The free abelian group Q = n i=0 Zα i is called the root lattice and the semigroup Q + = n i=0 Z ≥0 α i is called the positive root lattice. For α = i∈I k i α i ∈ Q + , the number ht(α) = i∈I k i is called the height of α. For λ, µ ∈ h * , we define λ ≥ µ if and only if λ − µ ∈ Q + . The elements in P + = {λ ∈ P | λ(h i ) ≥ 0, i ∈ I} are called the dominant integral weights. Note that the minimal imaginary root is given by δ = α 0 + α 1 + · · · + α n ∈ Q + . The element c = h 0 + h 1 + · · · + h n ∈ P ∨ is called the canonical central element.
Given n ∈ Z and any symbol q, we define [n] q = q n − q −n q − q −1 and set [0] q ! = 1, [n] q ! = [n] q [n − 1] q · · · [1] q . For m ≥ n ≥ 0, let Definition 1.1. The quantum affine algebra U q (g) = U q (A (1) n ) is an associative algebra over C(q) with 1 generated by e i , f i (i ∈ I) and q h (h ∈ P ∨ ) satisfying the defining relations: The definition of category O q int , Kashiwara operators and crystal bases can be found in [10,4], etc. It was shown in [10] that every U q (g)-module in the category O q int has a crystal basis. The notion of abstract crystals was introduced in [11]. For convenience, we recall some of the basic definitions and properties of abstract crystals. Definition 1.2. An abstract crystal associated with the Cartan datum (A, Π, Π ∨ , P, P ∨ ) is a set B together with the maps wt : B → P,ẽ i ,f i : B → B ⊔{0}, and ε i , ϕ i : B → Z∪{−∞} (i ∈ I) satisfying the following properties: We often say that B is a U q (g)-crystal. We denote B λ = {b ∈ B| wt(b) = λ} so that B = λ∈P B λ . (1) Let (L, B) be a crystal basis of M ∈ O q int . Then B has a crystal structure, where the maps ε i , ϕ i are given by In particular, we denote by B(λ) the crystal of the irreducible highest weight module V (λ) with highest weight λ ∈ P + . (2) Let (L(∞), B(∞)) be a crystal basis of U − q (g). Then B(∞) has a crystal structure, where the maps ε i , ϕ i are given by (3) For λ ∈ P , let us consider T λ = {t λ } with the maps: Then T λ is a crystal. (4) Let C = {c}. We define the maps Then C is a crystal. (1) A map ψ : B 1 → B 2 is a crystal morphism if it satisfies the following properties: Here, we understand ψ(0) = 0.
(3) ψ is called an embedding if the underlying map ψ : B 1 → B 2 is injective.
Let B 1 and B 2 be crystals. The tensor product B 1 ⊗ B 2 is defined to be the set B 1 × B 2 together with the following maps: It was shown in [11] that there is a unique strict crystal embedding Here, u λ is the highest weight element of B(λ). We denote by ι λ the composition of the strict embedding and the natural projection: Note that ι λ is injective, but not a crystal morphism.

Path realization
Let U ′ q (g) be the subalgebra of U q (g) generated by e i , f i , q ±hi (i ∈ I) and we set P ∨ = n i=0 Zh i , h = C ⊗ Z P ∨ , P = n i=0 ZΛ i and P + = n i=0 Z ≥0 Λ i . Denote by cl : P → P the natural projection from P to P . An abstract crystal B associated with U ′ q (g) is called a classical crystal. For b ∈ B, we define Definition 2.1. A perfect crystal of level ℓ is a finite classical crystal B satisfying the following conditions: (a) there exists a finite dimensional U ′ q (g)-module with a crystal basis whose crystal graph is isomorphic to B, Given a dominant integral weight λ with λ(c) = ℓ and a perfect crystal B of level ℓ, it was shown in [7] that there exists a unique crystal isomorphism, called the fundamental isomorphism theorem for perfect crystals, sending u λ to u ε(b λ ) ⊗ b λ . By applying this crystal isomorphism repeatedly, we get a sequence of crystal isomorphisms k=0 is called the ground-state path of weight λ and a sequence p = (p k ) ∞ k=0 of elements p k ∈ B is called a λ-path in B if p k = b k for all k ≫ 0. We denote by P B (λ) the set of λ-paths in B, which gives rise to the path realization of B(λ). We list some examples of perfect crystals of level 1 and the corresponding ground-state paths (see [1,8], etc).

Example 2.3.
(1) The crystal B 1 and its ground-state p 1 Λi of weight Λ i (i ∈ I) are given as We denote by P 1 (Λ i ) the set of all Λ i -paths in B 1 . (2) The crystal B n and its ground-state p n Λi of weight Λ i (i ∈ I) are given as We denote by P n (Λ i ) the set of all Λ i -paths in B n . (3) The adjoint crystal B ad is given as follows. Let where α ij := α i + α i+1 + · · · + α j for 1 ≤ i ≤ j ≤ n. We define the i-arrows (i ∈ I) by where θ = α 1 + · · · + α n . The crystal B ad is a perfect crystal of level 1 with the ground-state path of weight Λ 0 p ad Λ0 = (· · · , ∅, ∅, · · · , ∅, ∅).
There is a crystal isomorphism p ad : B n ⊗ B 1 → B ad given by We denote by P ad (Λ 0 ) the set of all Λ 0 -paths in B ad .

Combinatorics of Young Walls
In [4,6], Kang gave a combinatorial realization of crystal graphs for basic representations of quantum affine algebras of type A (1) 2n , D (2) n+1 (n ≥ 2), and B (1) n (n ≥ 3) by using new combinatorial objects called Young walls, which are a generalization of colored Young diagrams used in [2,5,15]. In this work, we focus on the quantum affine algebra The Young wall Y 1 (resp. Y n ) is a wall consisting of colored blocks stacked by the following rules: (a) the colored blocks should be stacked in the pattern P 1 (resp. P n ) of weight Λ k given below, (b) except for the right-most column, there should be no free space to the right of any block.
The patterns are given as follows. Note that the heights of the columns of a Young wall Y are weakly decreasing from right to left, so we denote it by Y = (y i ) i≥0 , where y i is the i-th column of Y . Definition 3.1. Let Y be a Young wall corresponding to the pattern P 1 (resp. P n ).
(1) An i-block in Y is called a removable i-block if Y remains a Young wall after removing the block. (2) A place in Y is called an admissible i-slot if one may add an i-block to obtain another Young wall. (3) A column in Y is said to be i-removable (resp. i-admissible) if the column has a removable i-block (resp. an admissible i-slot).
Now we define the action of Kashiwara operatorsẽ i ,f i (i ∈ I) on Young walls. Let Y = (y k ) k≥0 be a Young wall corresponding to the pattern P 1 (resp. P n ).
(a) To each column y k of Y , we assign (b) From this sequence of +'s and −'s, cancel out all (+, −) pairs to obtain a finite sequence of −'s followed by +'s. This sequence (−, · · · , −, +, · · · , +) is called the i-signature of Y . (c) We defineẽ i Y to be the Young wall obtained from Y by removing the i-block corresponding to the rightmost − in the i-signature of Y . If there is no − in the i-signature, we setẽ i Y = 0. (d) We definef i Y to be the Young wall obtained from Y by adding an i-block to the column corresponding to the leftmost + in the i-signature of Y . If there is no + in the i-signature, We also define where k ij is the number of i-blocks in the j-th column y j of Y . Note that the height of y j is ht(wt(y j )).
which maps the highest weight element u Λ k to the empty Young wall ∅.
is the smallest j-th column in P 1 satisfying the following conditions: where a j ≡ ht(wt(y j )) − j + k (mod n + 1) for all j ≥ 0.
In a similar manner, given a Λ k -path , otherwise y j (p n ) is the smallest j-th column in P n satisfying the following conditions: One can prove that the Young wall Y n k (p n ) is contained in Y n (Λ k ), and the map where u Λ0 is the highest weight element of B(Λ 0 ). Then the Λ 0 -paths p 1 ∈ P 1 (Λ 0 ), p n ∈ P n (Λ 0 ) and p ad ∈ P ad (Λ 0 ) corresponding to b are given as which yield the Young walls Y 1 := Y 1 0 (p 1 ) and Y n := Y n 0 (p n ) as follows:

Geometric Constructions of Crystal Graphs
In this section, we review geometric constructions of crystal bases via quiver varieties. See [3,12,14,17,18,19] for more details.
Let I = Z/(n + 1)Z and H the set of the arrows such that i → j with i, j ∈ I, i − j = ±1 . For h ∈ H, we denote by in(h) (resp. out(h)) the incoming (resp. outgoing) vertex of h. Define an involution − : H → H to be the map interchanging i → j and j → i. Let so that H = Ω ⊔ Ω; i.e., Note that our graph is an affine Dynkin graph of type A (1) n . We take the map ǫ : H → {−1, 1} given by In a similar manner, for λ = n i=0 w i Λ i ∈ P + , we define the I-graded vector space and let Let us denote by π Ω (resp. π Ω ) the natural projection from E(α) to E Ω (α) (resp. E Ω (α)). For π Ω (χ)). The matrix representation of x ∈ E Ω (α) in the ordered basis v(α) is given as We also may consider the matrix representation of x ∈ E Ω (α) in the same manner: , the composition map χ hN · · · χ h1 is zero. We define Lusztig's quiver variety to be We denote by IrrΛ(α) the set of all irreducible components of Λ(α) k)) of the quiver (I, Ω) is indecomposable and nilpotent. Note that the isomorphism class of (V (k ′ , k), x(k ′ , k)) does not change when k ′ and k are simultaneously translated by a multiple n + 1. Moreover, any indecomposable nilpotent finite-dimensional representation of the quiver (I, Ω) is isomorphic to (V (k ′ , k), x(k ′ , k)) for some pair (k ′ ≤ k). Let Z be the set of all pairs (k ′ ≤ k) of integers defined up to simultaneous translation by a multiple of n + 1 and letZ be the set of all functions from Z to Z ≥0 with finite support. Note thatZ naturally corresponds to isomorphism classes of nilpotent finite-dimensional representations of the quiver (I, Ω). The set of G(α)-orbits on the set of nilpotent elements in E Ω (α) is naturally indexed by the subsetZ(α) ofZ such that, for f ∈Z(α), Here the sum is taken over all k ′ ≤ k up to simultaneous translation by a multiple of n+1. An element f ∈Z(α) is aperiodic if, for any k ′ ≤ k, not all integers f (k ′ , k), f (k ′ + 1, k + 1), . . . , f (k ′ + n, k + n) are greater than zero. For any f ∈Z(α), let C f be the conormal bundle of the G(α)-orbit corresponding to f , and let C f be the closure of C f . Then we have Theorem 4.1 ( [14]). The map f → C f is a 1-1 correspondence between the set of aperiodic elements inZ(α) and IrrΛ(α).
In a similar manner, for a pair (k ≥ k ′ ) of integers, let x(k, k ′ ) : V (k ′ , k) → V (k ′ , k) be the Clinear map sending e i to e i−1 , where e k ′ −1 = 0. Then the representation (V (k ′ , k), x(k, k ′ )) of the quiver (I, Ω) is indecomposable and nilpotent, and the isomorphism class of (V (k ′ , k), x(k, k ′ )) does not change when k and k ′ are simultaneously translated by a multiple n + 1. Any indecomposable nilpotent finite-dimensional representation of the quiver (I, Ω) is isomorphic to (V (k ′ , k), x(k, k ′ )) for some pair (k ≥ k ′ ). Let Z be the set of all pairs (k ≥ k ′ ) of integers defined up to simultaneous translation by a multiple of n + 1 and letZ be the set of all functions from Z to Z ≥0 with finite support. Then the set of G(α)-orbits on the set of nilpotent elements in E Ω (α) is naturally indexed by the subsetZ(α) ofZ such that, for f ∈Z(α), Here the sum is taken over all k ≥ k ′ up to simultaneous translation by a multiple of n+1. An element f ∈Z(α) is aperiodic if, for any k ≥ k ′ , not all integers f (k, k ′ ), f (k + 1, k ′ + 1), . . . , f (k + n, k ′ + n) are greater than zero. Then one can show that there is a 1-1 correspondence between the set of aperiodic elements inZ(α) and IrrΛ(α).
In [18], Saito gave a crystal structure on and proved the following theorem.  For each irreducible component X 0 ∈ ι Λ k (B(Λ k )), Frenkel and Savage constructed a special point in X 0 × i∈I Hom(V i , W i ) which is not killed by the stability condition, and showed that there is a 1-1 correspondence between the set of such special points and the set of (n + 1)-reduced colored Young diagrams. Savage later established a crystal isomorphism between B(λ) and Young walls for quantum affine algebras of type A (1) n and D (1) n in [19]. We briefly recall the result of [3] for type A (1) n in terms of Young walls. Note that the orientation appeared in [3] is Ω. Take a Young wall Y n ∈ Y n (Λ k ) such that wt(Y n ) = α. Let l i be the length of the i-th row of the Young wall Y n (i ≥ 1) and let N be the height of Y n . Set and consider the function f ∈Z(α) given by Note that f is aperiodic. Let C f be the closure of the conormal bundle of the G(α)-orbit O f in E Ω (α) corresponding to f , and define the irreducible component By [3, Theorem 5.5], the map Y n → X Y n is a 1-1 correspondence between Moreover, it is proved in [19,Theorem 8.4.] that the map Y n → X Y n from Y n (Λ k ) to B(Λ k ) is a crystal isomorphism. We would like to point out that C f = ι Λ k (X Y n ). Now we construct an element in the G(α)-orbit O f in E Ω (α) from the Young wall Y n . Let b ij be the i-th block from bottom in the j-th column of Y n . Let Color(b ij ) be the color of b ij , which is an element in I. Define where ≺ is the lexicographical order; i.e., (r, s) ≺ (i, j) if and only if r < i or (r = i and s < j). We define By construction, one can show that J i is invariant under x(Y n ) and the representation (J i , x(Y n )| Ji ) of the quiver (I, Ω) is isomorphic to the representation (V (1 − i + k, l i − i + k), x(l i − i + k, 1 − i + k)) of the quiver (I, Ω) for 1 ≤ i < N . Here, x(Y n )| Ji is the restriction of x(Y n ) on the invariant subspace J i . Hence x(Y n ) is contained in the G(α)-orbit O f corresponding f , which yields ι Λ k (X Y n ) = the closure of the conormal bundle of the G(α)-orbit of x(Y n ). (4.4) By a direct computation, for t ∈ Z ≥0 , we have In the same manner, we take a Young wall Y 1 ∈ Y 1 (Λ k ) such that wt(Y 1 ) = α. Denote by X Y 1 the image of Y 1 under the crystal isomorphism Let b ij be the i-th block from bottom in the j-th column of Y 1 , and Color(b ij ) the color of b ij . Set where ≺ is the lexicographical order, and define Moreover, we obtain x(Y n ) = (E and ι Λ0 (X) is the closure of the conormal bundle of the G(α)-orbit of x(Y 1 ) (resp. x(Y n )). However, we note that since [x(Y 1 ), x(Y n )] = 0.

Quiver Varieties and the Perfect Crystals B 1 , B n
In this section, we give an explanation of the 1-1 correspondence between the geometric realization B(Λ k ) and the path realization of the crystal B(Λ k ) associated with the perfect crystals B 1 and B n , and give a geometric interpretation of the fundamental theorem of perfect crystals in the case of the perfect crystals B 1 and B n . Let α ∈ Q + and let λ be a dominant integral weight of level 1. Choose an irreducible component X in IrrΛ(λ, α). For a generic point χ = x + x ∈ ι λ (X), we will give an explicit description of the λ-path in B 1 (resp. B n ) corresponding to X using the dimensions of the spaces ker x i+1 / ker x i (resp. ker x i+1 / ker x i ) for i ≥ 0. For this purpose, we need a couple of lemmas.
Proof. Let χ = x + x ∈ X 0 and k ∈ Z ≥0 . By (4.1) we have [x, x] = 0, which yields Our assertion follows from the fact that ker(xx) k , ker x k and ker x k are I-graded vector spaces.
Proof. By Theorem 4.1, there is an open subset U 1 ⊂ X 0 such that π Ω (U 1 ) is contained in the G(α)orbit of some element in E Ω (α). In the same manner, there is an open subset U 2 ⊂ X 0 such that π Ω (U 2 ) is contained in the G(α)-orbit of some element in E Ω (α). Set U = U 1 ∩ U 2 ⊂ X 0 . Then, by construction, for any χ = x + x, χ ′ = x ′ + x ′ ∈ U , there exist g, g ∈ G(α) such that which yield, for any k ∈ Z ≥0 , ker x k = ker(gx ′ g −1 ) k = g(ker x ′ k ) and ker x k = ker(gx ′ g −1 ) k = g(ker x ′ k ).
An element χ ∈ X 0 in the open subset U ⊂ X 0 in Lemma 5.2 will be called a generic point. Thanks to Lemma 5.2, we may consider dim(ker x k ) and dim(ker x k+1 / ker x k ) (resp. dim(ker x k ) and dim(ker x k+1 / ker x k )) for a generic point χ = x + x in an irreducible component X 0 ∈ IrrΛ(α). Recall the injective map given in (4.2) for 0 ≤ k ≤ n. Applying Lemma 5.1 and Lemma 5.2 to (4.6) and (4.3), we obtain the following theorem.
be the unique crystal isomorphism given by Theorem 2.2 and Theorem 4.3, and take an irreducible component X ∈ B(Λ k ). Then, for a generic point Proof. Let U be an open subset of ι Λ k (X) as in Lemma 5.2. By Lemma 5.2, it suffices to show that (a) and (b) hold for some Let Y 1 be the Young wall in Y 1 (Λ k ) corresponding to X under the crystal isomorphism Y 1 (Λ k ) ∼ = B(Λ k ), and b ij the i-th block from bottom in the j-th column of Y 1 . By (4.7), there exists χ = x+x ∈ U such that x = gx(Y 1 )g −1 for some g ∈ G(α). Then, by the equation (4.8), for t ∈ Z ≥0 , we have Let y t be the t-th column of Y 1 for t ∈ Z ≥0 . Note that y t = {b it ∈ Y 1 | i ≥ 1}. Then, we have wt(y t ) = bit∈yt α Color(bit) (5.2) which implies that the height of y t is ht(wt(y t )) = dim(ker x t+1 / ker x t ).
The remaining assertion (b) can be proved in the same manner.
Proposition 5.4. Let be the unique crystal isomorphism given by Theorem 2.2 and Theorem 4.3, and take a Λ k -path p 1 ∈ P 1 (Λ k ) (resp. p n ∈ P n (Λ k )). Let α = Λ k − wt(p 1 ) and X 1 = q 1 k (p 1 ) (resp. β = Λ k − wt(p n ) and X n = q n k (p n )). Then Recall the fundamental isomorphism theorem of perfect crystals (2.1). From Theorem 4.3, we have the following crystal isomorphisms: for 0 ≤ k ≤ n. We would like to give a geometric interpretation to the crystal isomorphisms ψ 1 i , ψ n i in terms of quiver varieties. To do that, we need a couple of lemmas. Let V be an I-graded vector space and χ an element of Hom(V, V ). If W is a χ-invariant I-graded subspace of V , then χ can be viewed as an element in Hom(V /W, V /W ) (resp. Hom(W, W )), which is denoted by χ| V /W (resp. χ| W ).
Lemma 5.5. Let x ∈ i∈I Hom(V i−1 , V i ) for an I-graded vector space V := i∈I V i , and set W := ker x and y := x| V /W .

Take an element
where W i is the i-subspace of W for i ∈ I. Then there exists an element such that [x, x] = 0 and x| V /W = y.
Proof. Let r = dim V − dim W and s = dim W . Take an ordered basis for W and extend it to be an ordered basis for V so that the matrix representations of x, y and y are given as follows: has full rank, the following equation By construction, we have Lemma 5.6. Let U be an open subset of X 0 ∈ IrrΛ(α) as in Lemma 5.2. Set β = dim(ker x) (resp. γ = dim(ker x)) (a) There exists an irreducible component where φ : V (α)/ ker x → V (α − γ) is an I-graded vector space isomorphism.
Proof. Note that β and γ are well-defined by Lemma 5.2. We first deal with the case (a). For an element χ = x + x ∈ U , let is an I-graded vector space isomorphism. Since χ ∈ Λ(α), we have Take two elements χ = x + x, χ ′ = x ′ + x ′ ∈ U , and choose two I-graded vector space isomorphisms φ : . From the properties of U described in the proof of Lemma 5.2, we have for some g ∈ G(α), which yields that π Ω (χ φ ) and π Ω (χ ′ φ ′ ) are in the same G(α − β)-orbit. Therefore, there exists an irreducible component Since χ, χ ′ are arbitrary, our assertion follows. The remaining case (b) can be proved in the same manner.
(a) There exists a unique irreducible component X ′ ∈ IrrΛ(Λ k−1 , α − β) satisfying the following conditions: such that any element χ ′ ∈ U ′ can be written as for some χ = x + x ∈ X 0 and some I-graded vector space isomorphism φ : where a ≡ d + k (mod n + 1). (b) There exists a unique irreducible component X ′′ ∈ IrrΛ(Λ k+1 , α − γ) satisfying the following conditions: there is an open subset U ′′ ⊂ ι Λ k+1 (X ′′ ) such that any element χ ′′ ∈ U ′′ can be written as for some χ = x + x ∈ X 0 and some I-graded vector space isomorphism φ : Proof. We first deal with the case (a) of the crystal isomorphism ψ 1 k : Let Y be the Young wall in Y 1 (Λ k ) corresponding to X and Y ′ the Young wall obtained by removing the 0-th column from Y . Then Y ′ can be viewed as an element in Y 1 (Λ k−1 ). Take the irreducible component X ′ in B(Λ k−1 ) corresponding to Y ′ . By Theorem 5.3 and (5.2), we have where a ≡ d + k (mod n + 1) and wt(b a ) = Λ k − Λ k−1 − cl(β). Let U be an open subset of X 0 as in Lemma 5.6, and take an element χ = x + x ∈ U . Since x(Y )| V (α)/ ker(x(Y )) is naturally identified with x(Y ′ ) and x is contained in the G(α)-orbit of x(Y ), by Lemma 5.6, we have for an I-graded vector space isomorphism φ : V (α)/ ker x → V (α − β).
Take an element χ ′ = x ′ + x ′ in an open subset U ′ of ι Λ k−1 (X ′ ) given as in Lemma 5.2. Then x ′ can be written as . The assertion (ii) follows from Lemma 5.5.
The remaining case (b), ψ n k : B(Λ k ) → B(Λ k+1 ) ⊗ B n , can be proved in the same manner.
Here,  15 otherwise, and dim(ker Hence we obtain

Quiver Varieties and Adjoint Crystals
In this section, we will prove the main theorem of this paper, Theorem 6.3, which shows that there exists an explicit crystal isomorphism between the geometric realization B(Λ 0 ) and the path realization P ad (Λ 0 ) of B(Λ 0 ) arising from the adjoint crystal B ad . By Theorem 2.2 and Theorem 4.3, we have the crystal isomorphism Let α ∈ Q + and let X be an irreducible component of IrrΛ(Λ 0 , α). For a generic point χ = x + x ∈ ι Λ0 (X), we will give an explicit description of the Λ 0 -path p ad (X) in terms of dimension vectors of ker(xx) k+1 / ker(xx) k for k ≥ 0.
Let α, β ∈ Q + with β ≤ α. Consider the diagram given in [14]: where F ′′ is the variety of all pairs (χ, W ) such that and F ′ is the variety of all quadruples (χ, W, f, g) such that Then we have is the I-graded vector space isomorphism induced by g, p 2 (χ, W, f, g) = (χ, W ), p 3 (χ, W ) = χ, and π is the natural first projection. Note that p 2 is a G(α − β) × G(β)-principal bundle and an open map.
Let U be an open subset of X 0 ∈ IrrΛ(α) as in Lemma 5.2, and β = dim(ker x) for χ = x + x ∈ U . Define the map ı : U → F ′′ by ı(χ) = (χ, ker x) for χ = x + x ∈ U . Note that p 3 • ı = id| U . By Lemma 5.6, there exists an irreducible component Given an open subset U ′ ⊂ Λ(α − β) with U ′ ∩ X ′ 0 = ∅, by Lemma 5.5, is a nonempty open subset of X 0 . Therefore, given an open subset U ′ ⊂ X ′ 0 , there exists an open subsetŨ ⊂ X 0 such that, for any element In the same manner, let γ = dim(ker x), and consider the diagram Define the map ı : U → F ′′ by ı(χ) = (χ, ker x) for χ = x + x ∈ U , and let X ′′ 0 be an irreducible component as in Lemma 5.6. Then one can deduce that, given an open subset U ′ ⊂ X ′′ 0 , there exists an open subsetŨ ⊂ X 0 such that, for any element . Consequently, we have the following lemma.
Finally, we are ready to state the main theorem in this paper.
The following corollary, which is an immediate consequence of Theorem 5.7 and Theorem 6.3, can be regarded as a geometric interpretation of the fundamental isomorphism theorem for perfect crystals Corollary 6.4. Let X 0 = ι Λ0 (X) for some X ∈ IrrΛ(Λ 0 , α). For a generic point χ = x + x ∈ X 0 , set θ = dim(ker(xx)) and c = dim(ker x).