CRYSTAL GRAPHS FOR BASIC REPRESENTATIONS OF QUANTUM AFFINE ALGEBRAS

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1 Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number, June, Pages 8 CRYSTAL GRAPHS FOR BASIC REPRESENTATIONS OF QUANTUM AFFINE ALGEBRAS SEOK-JIN KANG Abstract. We give a realization of crystal graphs for basic representations of quantum affine algebras in terms of new combinatorial objects called the Young walls.. Introduction In [6, 7], Kashiwara developed the theory of crystal bases for integrable modules over the quantum groups associated with symmetrizable Kac-Moody algebras. The crystal bases can be viewed as bases at q = and they are given a structure of colored oriented graph, called the crystal graphs, with arrows defined by Kashiwara operators. The crystal graphs have many nice combinatorial features reflecting the internal structure of integrable representations of quantum groups. For instance, the characters of integrable representations can be computed by counting the elements in the crystal graphs with a given weight. Moreover, the tensor product decomposition of integrable modules into a direct sum of irreducible submodules is equivalent to decomposing the tensor product of crystal graphs into a disjoint union of connected components. Therefore, to understand the combinatorial nature of integrable representations, it is essential to find realizations of crystal graphs in terms of nice combinatorial objects. In [9], Misra and Miwa constructed the crystal graphs for basic representations of quantum affine algebras U q (A () n ) using Young diagrams with colored boxes. Their idea was extended to construct crystal graphs for irreducible highest weight U q (A () n )-modules of arbitrary higher level []. The crystal graphs constructed in [9] can be parametrized by certain paths which arise naturally in the theory of solvable lattice models. Motivated by this observation, Kang, Kashiwara, Misra, Miwa, Nakashima and Nakayashiki developed the theory of perfect crystals for general quantum affine algebras and gave a realization of crystal graphs for irreducible highest weight modules of arbitrary higher level in terms of paths. In this way, the theory of vertex models can be explained in the language of representation theory of quantum affine algebras and the -point function of the vertex model was expressed as the quotient of the string function and the character of the corresponding irreducible highest weight representation. In [3], Kang gives a realization of crystal graphs for basic representations of quantum affine algebras of classical type using some new combinatorial objects Mathematics Subject Classification. 7B65, 7B7. Key words and phrases. quantum groups, crystal bases, Young walls mathematics. Received August 3,. c Information Center for Mathematical Sciences

2 S.-J. KANG which are called the Young walls. The Young walls consist of colored blocks that are built on the given ground-state and can be viewed as generalizations of Young diagrams. The rules for building Young walls are quite similar to playing with LEGO blocks and the Tetris game. The crystal graphs for basic representations are characterized as the set of all reduced proper Young walls. The weight of a Young wall can be computed easily by counting the number of colored blocks that have been added to the ground-state. Hence the weight multiplicity is just the number of all reduced proper Young walls of given weight. In this article, we illustrate this construction of crystal graphs for the quantum affine algebra U q (C () ). The readers can find more details in [] and [3].. The quantum affine algebra U q (C () ) and crystal bases Let I = {,, } be the index set. Consider the generalized Cartan matrix A = (a ij ) i,j I of affine type C () and its Dynkin diagram: A = and. Let P = Zh Zh Zh Zd be a free abelian group, called the dual weight lattice and set h = C Z P. We define the linear functionals α i and Λ i (i I) on h by α i (h j ) = a ji, α i (d) = δ,i, Λ i (h j ) = δ ij, Λ i (d) = (i, j I). The α i (resp. h i ) are called the simple roots (resp. simple coroots) and the Λ i are called the fundamental weights. We denote by Π = {α i i I} (resp. Π = {h i i I}) the set of simple roots (resp. simple coroots). Let c = h + h + h and δ = α + α + α. Then we have α i (c) =, δ(h i ) = for all i I and δ(d) =. We call c (resp. δ) the canonical central element (resp. null root). The free abelian group P = ZΛ ZΛ ZΛ Zδ is called the weight lattice and the elements of P are called the affine weights. We denote by q h (h P ) the basis elements of the group algebra C(q)[P ] with the multiplication q h q h = q h+h (h, h P ). Set q = q = q, q = q and K = q h, K = q h, K = q h. We will also use the following notations. [k] i = qk i q k i q i q, [n] i! = i n [k] i, and e (n) k= i = e n i /[n] i!, f (n) i = f n i /[n] i!. Definition.. The quantum affine algebra U q (C () ) of type C() is the associative algebra with over C(q) generated by the symbols e i, f i (i I) and q h (h P ) subject to the following defining relations:

3 CRYSTAL GRAPHS FOR QUANTUM AFFINE ALGEBRAS 3 q =, q h q h = q h+h (h, h P ), q h e i q h = q α i(h) e i, q h f i q h = q α i(h) f i (h P, i I), K i K i e i f j f j e i = δ i,j q i q (i, j I), i e e (q + q )e e e + e e =, f f (q + q )f f f + f f =, e 3 e (q + + q )e e e + (q + + q )e e e e e 3 =, f 3 f (q + + q )f f f + (q + + q )f f f f f 3 =, e 3 e (q + + q )e e e + (q + + q )e e e e e 3 =, f 3 f (q + + q )f f f + (q + + q )f f f f f 3 =, e e (q + q )e e e + e e =, f f (q + q )f f f + f f =, e e = e e, f f = f f. We call (A, Π, Π, P, P ) the Cartan datum associated with the quantum affine algebra U q (C () ). We now briefly review the basics of crystal basis theory. A U q (C () )-module M is called integrable if (i) M = λ P M λ (resp. M = λ P M λ), where M λ = {v M q h v = q λ(h) v for all h P (resp. h P )}, (ii) for each i I, M is a direct sum of finite dimensional irreducible U i -modules, where U i denotes the subalgebra generated by e i, f i, K ± i which is isomorphic to U q (sl ). Fix i I. By the representation theory of U q (sl ), any element v M λ may be written uniquely as v = k f (k) i v k, where v k ker e i M λ+kαi. We define the endomorphisms ẽ i and f i on M, called the Kashiwara operators, by ẽ i v = k f (k ) i v k, fi v = f (k+) i v k. k Let A be the subring of C(q) consisting of the rational functions in q that are regular at q =. Definition.. (a) A free A-submodule L of an integrable U q -module M, stable under ẽ i and f i, is called a crystal lattice if M = C(q) A L and L = λ P L λ, where L λ = L M λ. (b) A crystal basis of an integrable module M is a pair (L, B) such that (i) L is a crystal lattice of M, (ii) B is a C-basis of L/qL, (iii) B = λ P B λ, where B λ = B (L λ /ql λ ),

4 4 S.-J. KANG (iv) ẽ i B B {}, f i B B {}, (v) for b, b B, b = f i b if and only if b = ẽ i b. The set B is given a colored oriented graph structure by defining b i b if and only if b = f i b. The graph B is called the crystal graph of M and it reflects the combinatorial structure of M. For instance, we have dim C(q) M λ = #B λ for all λ P (or λ P ). By extracting properties of the crystal graphs, we define the notion of abstract crystals as follows. Definition.3 ([8]). An affine crystal is a set B together with the maps wt : B P, ε i : B Z { }, ϕ i : B Z { }, ẽ i : B B {}, and f i : B B {}, satisfying the following conditions: (i) wt(b), h i = ϕ i (b) ε i (b) for all b B, (ii) for b B with ẽ i b B, wt(ẽ i b) = wt(b) + α i, ε i (ẽ i b) = ε i (b), ϕ i (ẽ i b) = ϕ i (b) +, (iii) for b B with f i b B, wt( f i b) = wt(b) α i, ε i ( f i b) = ε i (b) +, ϕ i ( f i b) = ϕ i (b), (iv) b = f i b if and only if b = ẽ i b for b, b B, (v) ẽ i b = f i b = if ε i (b) =. The crystal graph of an integrable U q (C () )-module is an affine crystal. Let B and B be affine crystals. A morphism ψ : B B of crystals is a map ψ : B {} B {} such that (i) ψ() =, (ii) if b B and ψ(b) B, then wt(ψ(b)) = wt(b), ε i (ψ(b)) = ε i (b), ϕ i (ψ(b)) = ϕ i (b), (iii) if b, b B, ψ(b), ψ(b ) B and f i b = b, then f i ψ(b) = ψ(b ). The tensor product B B of B and B is the set B B whose crystal structure is defined by wt(b b ) = wt(b ) + wt(b ), ε i (b b ) = max(ε i (b ), ε i (b ) wt(b ), h i ), ϕ i (b b ) = max(ϕ i (b ), ε i (b ) + wt(b ), h i ), { ẽ i b b if ϕ i (b ) ε i (b ), ẽ i (b b ) = b ẽ i b if ϕ i (b ) < ε i (b ), { fi b b if ϕ i (b ) > ε i (b ), f i (b b ) = b f i b if ϕ i (b ) ε i (b ). 3. The Young walls In this section, we will explain the notion of Young walls. The Young walls will be built of three kinds of blocks; the -block ( ), the -block ( ), and the -block ( ). They are supposed to be colored by elements from the index set I and we do not allow rotations of the blocks. The -block and -block are of unit height, unit width, and half-unit thickness. The -block is of half-unit height, unit width, and unit thickness. With these blocks, we will build a wall of unit thickness, extending infinitely to the left, like playing with LEGO blocks. The base of the

5 CRYSTAL GRAPHS FOR QUANTUM AFFINE ALGEBRAS 5 wall may not be arbitrary, but must be chosen from one of the following three. Y Λ = Y Λ = Y Λ = The drawings are meant to extend infinitely to the left. The dotted lines in the front parts of Y Λ and Y Λ signify where other blocks that will build up the wall of unit thickness may be placed. These will be called the ground-state walls. As it is quite awkward drawing these, and since we can t see the blocks lying to the back, we will simplify the drawing as follows. The drawing on the right shows an example. We will now list the rules for building the wall with colored blocks. Rules 3.. (i) The wall must be built on top of one of the ground-state walls. (ii) The blocks should be stacked in columns. No block may be placed on top of a column of half-unit thickness. (iii) The top and the bottom of the -block and -block may only touch a -block. The sides of the -block may only touch a -block and vice versa. (iv) Placement of -block and -block to either the front or the back of the wall must be consistent for each column. The -block and -block should always be placed at heights which are multiples of the unit length. (v) Stacking more than two -blocks consecutively on top of another is not allowed. (vi) Except for the right-most column, there should be no free space to the right of any block. (vii) In the right-most column of a wall built on Y Λ, the -block should be placed to the front and the -block should be placed to the back. We will give some examples illustrating the rules for building the walls. From now on, the ground-state wall extending infinitely to the left will be omitted and what remains will be shaded in the drawings. Example 3.. Good walls.

6 6 S.-J. KANG Example 3.3. Bad walls. A column in the wall is a full column if the height of the column is a multiple of the unit length and the top of the column is of unit thickness. The first wall in Example 3. has two full columns. The right-most column in the second wall is not full but the other two are. Neither the third nor the forth wall contain full columns. Definition 3.4. (a) A wall satisfying Rules 3. is called a Young wall of ground state Λ i, if it is built on the ground state wall Y Λi and the height of the columns are weakly decreasing as we go to the left. (b) A Young wall is said to be proper if no two full columns of the wall are of the same height. 4. The crystal structure In this section, we will define a crystal structure on the set of all proper Young walls. The action of Kashiwara operators will be described in a way similar to playing the Tetris game. Definition 4.. (a) A block in a proper Young wall is removable if the wall remains a proper Young wall after removing the block. (b) A place in a proper Young wall, where one may add a block to obtain another proper Young wall is called an admissible slot. (c) A column in a proper Young wall is said to contain a removable δ, if we may remove a -block, a -block, and two -blocks from the column in some order and still obtain a proper Young wall. (d) A proper Young wall is reduced if none of its columns contain a removable δ. The second and the last wall of Example 3. are reduced. The first wall is not proper and the third wall contains one removable δ. The set of all reduced proper Young walls of ground-state Λ i is denoted by Y(Λ i ). We now defined the action of the Kashiwara operators ẽ i and f i. A column in a proper Young wall is called i-removable if the top block of that column is a removable i-block. A column is i-admissible if the top of that column is an i-admissible slot. The action of Kashiwara operators is defined as follows.. Go through each column and write a under each i-admissible column and a under each i-removable column.

7 CRYSTAL GRAPHS FOR QUANTUM AFFINE ALGEBRAS 7. If we are dealing with the i = case, there could be columns from which two -blocks may be removed. Place under them. Under the columns that are -removable and at the same time -admissible, place. Write under the columns that are twice -admissible. 3. From the (half-)infinite list of s and s, cancel out each pair to obtain a finite sequence of s followed by some s (reading from left to right). 4. For ẽ i, remove the i-block corresponding to the right-most remaining. Set it to zero if no remains. 5. For f i, add an i-block to the column corresponding to the left-most remaining. Set it to zero if no remains. With this definition of Kashiwara operators, the crystal graphs for basic representations of U q (C () ) are realized as the affine crystal consisting of all reduced proper Young walls. Theorem 4.. [] (a) The set of all reduced proper Young walls is stable under the Kashiwara operators defined above. That is, if Y is a reduced proper Young wall, then ẽ i Y and f i Y are either reduced proper Young walls or for all i I. (b) For each i =,,, we have the isomorphism of crystals Y(Λ i ) = B(Λ i ). In the next examples, we draw the crystal graphs B(Λ ) and B(Λ ) in terms of reduced proper Young walls. Example 4.3. Crystal graph of B(Λ ). Y Λ

8 8 S.-J. KANG Example 4.4. Crystal graph of B(Λ ). Y Λ References [] J. Hong, S.-J. Kang, Crystal graphs for basic representations of the quantum affine algebra U q(c () ), RIM-GARC preprint (999). [] M. Jimbo, K. C. Misra, T. Miwa, and M. Okado, Combinatorics of representations of U q ( bsl(n)) at q =, Comm. Math. Phys. 36 (99), no. 3, [3] S.-J. Kang, Combinatorial representation theory of quantum affine algebras, in preparation. [4] S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A. Suppl. A (99), [5], Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (99), no. 3, [6] M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 33 (99), no., [7], On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (99), no., [8], Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (994), no., [9] K. Misra and T. Miwa, Crystal base for the basic representation of U q ( bsl(n)), Comm. Math. Phys. 34 (99), no., Department of Mathematics, Seoul National University, Seoul 5-74, Korea address: sjkang@math.snu.ac.kr