ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Pages 35–46 (September 19, 2002) S 1079-6762(02)00103-8

QUANTUM AFFINE ALGEBRAS, COMBINATORICS OF YOUNG WALLS, AND GLOBAL BASES

SEOK-JIN KANG AND JAE-HOON KWON

(Communicated by Efim Zelmanov)

Abstract. We construct the Fock space representation of quantum affine algebras using combinatorics of Young walls. We also show that the crystal basis of the Fock space representation can be realized as the abstract crystal consisting of proper Young walls. We then generalize Lascoux-Leclerc-Thibon algorithm to obtain an effective algorithm for constructing the global bases of basic representations.

1. Introduction

Let U_{q}(𝔤) be a quantum affine algebra of type A_{n}^{(1)}, A_{2n−1}^{(2)}, D_{n}^{(1)}, A_{2n}^{(2)}, D_{n+1}^{(2)} and B_{n}^{(1)}, and let Λ be a dominant integral weight of level 1. In [4], Kang introduced a new family of combinatorial objects called the Young walls. The Young walls consist of colored blocks with various shapes built on a given ground state wall Y_{Λ} and can be viewed as generalizations of (colored) Young diagrams.

The rules and patterns for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. Then the set Y(Λ) of proper Young walls becomes an abstract crystal for the quantum affine algebra U_{q}(𝔤), and the subcrystal Y_{∘}(Λ) consisting of reduced proper Young walls is isomorphic to the crystal B(Λ) of the basic representation V (Λ).

In this paper, we construct the Fock space representation F(Λ) of U_{q}(𝔤) in a purely combinatorial way. More precisely, we take F(Λ) to be the ℚ(q)-vector space spanned by the proper Young walls, and define the U_{q}(𝔤)-module action on F(Λ) using combinatorics of Young walls. Then F(Λ) becomes an integrable U_{q}(𝔤)-module in the category O^{int}. The Fock space F(Λ) can be regarded as the q-deformed wedge space arising from a level 1 perfect representation [9].

We also show that the crystal of F(Λ) coincides with the abstract crystal Y(Λ) consisting of proper Young walls. Thus, we get an explicit decomposition of the Fock space F(Λ) into a direct sum of irreducible highest weight modules over U_{q}(𝔤) by locating the maximal vectors in the U_{q}(𝔤)-crystal Y(Λ).

Finally, we generalize Lascoux-Leclerc-Thibon algorithm [10] to obtain an effective algorithm for constructing the global basis G(Λ) of the basic representation

\begin{matrix}c \\ ◯ \end{matrix}2002 American Mathematical Society

Received by the editors December 14, 2001.

2000 Mathematics Subject Classification. Primary 17B37; Secondary 17B10.

Key words and phrases. Quantized universal enveloping algebra, crystal basis, global basis.

This research was supported by KOSEF Grant # 98-0701-01-05-L and the Young Scientist Award, Korean Academy of Science and Technology.

V (Λ). To each reduced proper Young wall Y ∈ Y_{∘}(Λ), the corresponding global basis element G(Y) can be expressed as a linear combination of proper Young walls whose coefficient polynomials form an upper triangular matrix with the diagonal entries 1. We expect that there exist some interesting algebraic structures such that the irreducible modules at some specializations are parametrized by the reduced proper Young walls and that the decomposition matrices are determined by the polynomials giving the global basis elements.

2. Quantum affine algebras

Let I = { 0, 1, …, n } be an index set and let (A, P^{∨}, P, Π^{∨}, Π) be an affine Cartan datum:

(i) A = (a_{ij})_{i,j∈I} is a generalized Cartan matrix of affine type, (ii) P^{∨} = ℤh_{0} ⊕ ⋯ ⊕ ℤh_{n} ⊕ ℤd is the dual weight lattice, (iii) P = { λ ∈ 𝔥^{∗} | λ(P^{∨}) ⊂ ℤ } is the weight lattice, where 𝔥 = ℚ ⊗_{ℤ} P^{∨} is the

Cartan subalgebra, (iv) Π^{∨} = { h_{i} | i ∈ I } is the set of simple coroots,

(v) Π = { α_{i} | i ∈ I } is the set of simple roots defined by α_{j}(h_{i}) = a_{ij}, α_{j}(d) = δ_{0,j} (i, j ∈ I).

Set P^{+} = { λ ∈ P | λ(h_{i}) ∈ ℤ_{≥0} for all i ∈ I }. The elements of P^{+} are called the dominant integral weights.

To each affine Cartan datum, we can associate an infinite-dimensional Lie algebra 𝔤 called the affine Kac-Moody algebra. The canonical central element and the null root of 𝔤 will be expressed as c = ∑_{i=0}^{n} c_{i}h_{i} and δ = ∑_{i=0}^{n} d_{i}α_{i}, respectively, where c_{i} and d_{i} (i ∈ I) are positive integers given in [3]. The fundamental weights Λ_{i} (i ∈ I) are the linear functionals defined by Λ_{i}(h_{j}) = δ_{ij}, Λ_{i}(d) = 0 (i, j ∈ I).

We denote by q^{h} (h ∈ P^{∨}) the basis elements of the group algebra ℚ(q)[P^{∨}] with the multiplication q^{h}q^{h′} = q^{h+h′} (h, h′ ∈ P^{∨}). Let ( |) be a nondegenerate symmetric bilinear form on 𝔥^{∗} satisfying \frac{2(α_{i}|α_{j})}{(α_{i}|α_{i})} = a_{ij} for i, j ∈ I. Set q_{i} = q^{\frac{(α_{i}|α_{i})}{2}} ,

K_{i} = q^{\frac{(α_{i}|α_{i})}{2} h_{i}}, [k]_{i} = \frac{q_{i}^{k}−q_{i}^{−k}}{q_{i}−q_{i}^{−1}} and [n]_{i}! = ∏_{k=1}^{n}[k]_{i}. We also use the notation

e_{i}^{(n)} = e_{i}^{n}/[n]_{i}! and f_{i}^{(n)} = f_{i}^{n}/[n]_{i}!.

The quantum affine algebra U_{q}(𝔤) is the associative algebra with 1 over ℚ(q) generated by e_{i}, f_{i} (i ∈ I) and q^{h} (h ∈ P^{∨}) subject to the defining relations given, for example, in [6]. In this paper, we will focus on the quantum affine algebras of type A_{n}^{(1)}, A_{2n−1}^{(2)}, D_{n}^{(1)}, A_{2n}^{(2)}, D_{n+1}^{(2)} and B_{n}^{(1)}.

3. Crystal bases

Recall that the category O^{int} consists of U_{q}(𝔤)-modules M satisfying the properties:

(i) M = ⨁_{λ∈P} M_{λ}, where M_{λ} = { v ∈ M | q^{h}v = q^{λ(h)}v for all h ∈ P^{∨} }, (ii) for each i ∈ I, M is a direct sum of finite-dimensional irreducible U_{(i)}-

modules, where U_{(i)} = ⟨e_{i}, f_{i}, K_{i}^{±}⟩ ≅ U_{q}(𝔰𝔩_{2}), (iii) for any v ∈ M, there exists l ≥ 1 such that e_{i_{1}} ⋯ e_{i_{l}}v = 0 for any i_{1}, …, i_{l} ∈

I. For each i ∈ I, every element u ∈ M_{λ} can be written uniquely as u = ∑_{k≥0} f_{i}^{(k)}u_{k}, where k ≥ −λ(h_{i}) and u_{k} ∈ ker e_{i} ∩ M_{λ+kα_{i}}. The Kashiwara operators are the endomorphisms ẽ_{i} and f̃_{i} on M defined by

(1)  ẽ_{i}u = ∑_{k≥1} f_{i}^{(k−1)}u_{k}, f̃_{i}u = ∑_{k≥0} f_{i}^{(k+1)}u_{k}.

Let 𝔸_{0} = { f/g ∈ ℚ(q) | f, g ∈ ℚ[q], g(0) ≠ 0 } be the localization of ℚ[q] at q = 0.

Definition 3.1. A crystal basis of M is a pair (L, B), where

(i) L is a free 𝔸_{0}-module of M such that M ≅ ℚ(q) ⊗_{𝔸_{0}} L, (ii) B is a ℚ-basis of L/qL, (iii) L = ⨁_{λ∈P} L_{λ}, where L_{λ} = L ∩ M_{λ}, (iv) B = ⨆_{λ∈P} B_{λ}, where B_{λ} = B ∩ (L_{λ}/qL_{λ}),

(v) ẽ_{i}L ⊂ L, f̃_{i}L ⊂ L for all i ∈ I, (vi) ẽ_{i}B ⊂ B ∪ {0}, f̃_{i}B ⊂ B ∪ {0} for all i ∈ I, (vii) for i ∈ I and b, b′ ∈ B, f̃_{i}b = b′ if and only if b = ẽ_{i}b′.

The set B becomes a colored oriented graph, called the crystal graph, with the arrows defined by b →^{i} b′ if and only if f̃_{i}b = b′.

By extracting the properties of crystal graphs, we can define the notion of abstract crystals [7, 8]. An affine crystal is a set B together with the maps wt : B → P, ε_{i}, φ_{i} : B → ℤ∪{−∞}, ẽ_{i}, f̃_{i} : B → B ∪{0} satisfying the conditions given in [7, 8].

Theorem 3.2 ([6]).

(a) Let V (λ) be the irreducible highest weight U_{q}(𝔤)-module with highest weight λ ∈ P^{+} and highest weight vector u_{λ}. Let L(λ) be the free 𝔸_{0}-submodule of V (λ) spanned by the vectors of the form f̃_{i_{1}} ⋯ f̃_{i_{r}}u_{λ} (i_{k} ∈ I, r ∈ ℤ_{≥0}) and set

B(λ) = { f̃_{i_{1}} ⋯ f̃_{i_{r}}u_{λ} + qL(λ) ∈ L(λ)/qL(λ) }\{0}.

Then (L(λ), B(λ)) is a crystal basis of V (λ), and every crystal basis of V (λ) is isomorphic to (L(λ), B(λ)).

(b) Define a ℚ-algebra automorphism of U_{q}(𝔤) by \overline{e}_{i} = e_{i}, \overline{f}_{i} = f_{i}, \overline{q^{h}} = q^{−h} and \overline{q} = q^{−1} for i ∈ I and h ∈ P^{∨}. Set 𝔸 = ℚ[q, q^{−1}] and let V (λ)^{𝔸} = U_{𝔸}^{−}(𝔤)u_{λ}, where U_{𝔸}^{−}(𝔤) is the 𝔸-subalgebra of U_{q}(𝔤) generated by f_{i}^{(n)} (i ∈ I, n ∈ ℤ_{≥0}). Then there exists a unique 𝔸-basis G(λ) = { G(b) | b ∈ B(λ) } of V (λ)^{𝔸} such that

G(b) ≡ b mod qL(λ) and \overline{G(b)} = G(b)

for all b ∈ B(λ).

The basis G(λ) is called the global basis or canonical basis of V (λ) associated with the crystal graph B(λ).

4. Combinatorics of Young walls

We briefly review the notion of Young walls and their combinatorics introduced by Kang [4]. The Young walls are built of colored blocks of three different shapes.

Type  Shape  Width Thickness Height Volume
I  =  1  1  1  1
II  =  1  1  \frac{1}{2}  \frac{1}{2}
III  =  ,  =  1  \frac{1}{2}  1  \frac{1}{2}

Given a dominant integral weight Λ of level 1, i.e. Λ(c) = 1, we fix a frame Y_{Λ} called the ground state wall of weight Λ and on this frame, we build a wall of thickness less than or equal to one unit. The rules for building the walls are as follows:

(1) The colored blocks should be stacked in columns. No block can be placed on top of a column of half-unit thickness.

(2) Except for the right-most column, there should be no free space to the right of any block.

(3) The colored blocks should be stacked in a specified pattern which is determined by the type of the quantum affine algebra and Λ.

The coloring of blocks, description of ground state walls and the patterns for building the walls are given in [4]. For example, if 𝔤 = B_{3}^{(1)}, and Λ = Λ_{0}, we will use the colored blocks

0  1  2  _{3}

to build the walls on the ground state wall Y_{Λ_{0}} = 1 0 1 0 = _{0 1 0 1} following the pattern:

1  ^{1} ^{1}  ^{1}^{0} ^{0}  ^{0} ^{0}^{1}  ^{1} ^{1}  ^{1}^{0} ^{0}  ^{0} ^{0} 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3

A wall built on Y_{Λ} following the above rules is called a Young wall on Y_{Λ}, for the heights of its columns are weakly decreasing as we proceed from right to left. We often write Y = (y_{k})_{k=0}^{∞} = (…, y_{2}, y_{1}, y_{0}) as an infinite sequence of its columns.

Definition 4.1.

(1) A column of a Young wall is called a full column if its height is a multiple of the unit length and its top is of unit thickness.

(2) For quantum affine algebras of type A_{2n−1}^{(2)}, D_{n}^{(1)}, A_{2n}^{(2)}, B_{n}^{(1)} and D_{n+1}^{(2)}, a Young wall is said to be proper if none of the full columns have the same heights.

(3) For quantum affine algebras of type A_{n}^{(1)}, every Young wall is defined to be proper.

We will denote by Y(Λ) the set of all proper Young walls on Y_{Λ}.

Definition 4.2. Let Y be a proper Young wall on Y_{Λ}.

(1) A block of color i (in short, i-block) in Y is called a removable i-block if Y remains a proper 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 proper Young wall.

(3) A column in Y is said to be i-removable (resp. i-admissible) if there is a removable i-block (resp. an admissible i-slot) in that column.

We now define the abstract Kashiwara operators Ẽ_{i}, F̃_{i} on Y(Λ) as follows. Fix i ∈ I and let Y = (y_{k})_{k=0}^{∞} be a proper Young wall on Y_{Λ}.

(1) To each column y_{k} of Y, we assign

−− if y_{k} is twice i-removable,
− if y_{k} is once i-removable but not i-admissible,
−+ if y_{k} is once i-removable and once i-admissible,
+ if y_{k} is once i-admissible but not i-removable,
++ if y_{k} is twice i-admissible,
· otherwise.

(2) From this sequence of +’s and −’s, we cancel out every (+, −)-pair to obtain a finite sequence of −’s followed by +’s, reading from left to right. This finite sequence (− ⋯ −, + ⋯ +) is called the i-signature of Y.

(3) We define Ẽ_{i}Y to be the proper Young wall obtained from Y by removing the i-block corresponding to the right-most − in the i-signature of Y. We define Ẽ_{i}Y = 0 if there is no − in the i-signature of Y.

(4) We define F̃_{i}Y to be the proper Young wall obtained from Y by adding an i-block to the column corresponding to the left-most + in the i-signature of Y. We define F̃_{i}Y = 0 if there is no + in the i-signature of Y.

Next, we define

wt(Y) = Λ − ∑_{i∈I} k_{i}α_{i} ∈ P,
ε_{i}(Y) = the number of −’s in the i-signature of Y ,
φ_{i}(Y) = the number of +’s in the i-signature of Y ,

where k_{i} denotes the number of i-blocks in Y that have been added to Y_{Λ}.

Proposition 4.3 ([4]). The set Y(Λ) together with the maps wt, ε_{i}, φ_{i}, Ẽ_{i} and F̃_{i} (i ∈ I) becomes an affine crystal.

Let δ = d_{0}α_{0} + ⋯ + d_{n}α_{n} be the null root of U_{q}(𝔤), and set a_{i} = d_{i} if 𝔤 ≠ D_{n+1}^{(2)}, a_{i} = 2d_{i} if 𝔤 = D_{n+1}^{(2)}. The part of a column with a_{i}-many i-blocks for each i ∈ I in some cyclic order is called a δ-column. A δ-column in a proper Young wall is called removable if it can be removed to yield another proper Young wall.

Definition 4.4. A proper Young wall Y is said to be reduced if none of its columns contain a removable δ-column.

Let Y_{∘}(Λ) ⊂ Y(Λ) be the set of all reduced proper Young walls on Y_{Λ}. Then we have:

Theorem 4.5 ([4]). The set Y_{∘}(Λ) is an affine crystal. Moreover, there exists an affine crystal isomorphism Y_{∘}(Λ) ⟶^{∼} B(Λ), where B(Λ) is the crystal of the basic representation V (Λ).

Remark 4.6. If H_{N}(q) is a Hecke algebra of type A_{N−1} with q a primitive nth root of unity, then the irreducible representations of H_{N}(q) are parametrized by the set of reduced proper Young walls of type A_{n}^{(1)} with N blocks [2]. We expect that there exist some interesting algebraic structures whose irreducible representations (at some specialization) are parametrized by reduced proper Young walls. In [1], Brundan and Kleshchev verified our speculation by showing that the irreducible representations of the Heck-Clifford superalgebra H_{N}(q) with q a primitive (2n + 1)th root of unity are parametrized by the set of reduced proper Young walls of type A_{2n}^{(2)} with N blocks.

5. Fock space representation

Let F(Λ) = ⨁_{Y ∈Y(Λ)} ℚ(q)Y be the ℚ(q)-vector space with a basis Y(Λ). The goal of this section is to define a U_{q}(𝔤)-module structure on F(Λ), the Fock space representation of U_{q}(𝔤). Moreover, we also show that the affine crystal Y(Λ) is exactly the crystal of F(Λ).

For this purpose, we need to define the action of e_{i}, f_{i} (i ∈ I) and q^{h} (h ∈ P^{∨}) on proper Young walls in Y(Λ). Let Y = (y_{k})_{k=0}^{∞} be a proper Young wall on Y_{Λ}. We denote by |y_{k}| the number of blocks in y_{k} added to Y_{Λ}. Then the associated partition is defined to be |Y | = (…, |y_{k}|, …, |y_{1}|, |y_{0}|). For Y = (y_{k})_{k=0}^{∞}, Z = (z_{k})_{k=0}^{∞} in Y(Λ), we define |Y | ⊵ |Z| if and only if ∑_{k=l}^{∞} |y_{k}| ≥ ∑_{k=l}^{∞} |z_{k}| for all l ≥ 0.

The action of q^{h} (h ∈ P^{∨}) on Y ∈ Y(Λ) is easily defined by q^{h}Y = q^{wt(Y)(h)}. We now focus on the action of e_{i} and f_{i} on Y (i ∈ I).

Case 1. Suppose that the i-blocks are of type I.

If b is a removable i-block in y_{k} of Y, then let Y_{R}(b) = (y_{k−1}, …, y_{1}, y_{0}) be the wall consisting of the columns lying on the right of b, and set R_{i}(b; Y) = φ_{i}(Y_{R}(b)) − ε_{i}(Y_{R}(b)). (The wall Y_{R}(b) should be regarded as a U_{(i)}-crystal, where no block can be added on y_{l} for l ≥ k.) We denote by Y ↗ b the Young wall obtained by removing b from Y. Then we define

(2)  e_{i} Y = ∑_{b} q_{i}^{−R_{i}(b;Y)}(Y ↗ b),

where b runs over all removable i-blocks in Y.

On the other hand, if b is an admissible i-slot in y_{k} of Y, then let Y_{L}(b) = (…, y_{k+2}, y_{k+1}) be the Young wall consisting of the columns in Y lying on the left of b, and set L_{i}(b; Y) = φ_{i}(Y_{L}(b)) − ε_{i}(Y_{L}(b)). (The wall Y_{L}(b) may be a proper Young wall on another ground state wall Y_{Λ′}.) We denote by Y ↙ b the Young wall obtained by adding an i-block at b. Then we define

(3)  f_{i} Y = ∑_{b} q_{i}^{L_{i}(b;Y)}(Y ↙ b),

where b runs over all admissible i-slots in Y.

Case 2. Suppose that the i-blocks are of type II (in this case, q = q_{i}).

Let b be a removable i-block in y_{k} of Y. If the i-signature of y_{k} is −−, or if the i-signature of y_{k} is − and there is another i-block below b, define Y ↗ b to be the Young wall obtained by removing b from Y. If the i-signature of y_{k} is −+, or if the i-signature of y_{k} is − and there is no i-block below b, define Y ↗ b = q^{−1}(1 − (−q^{2})^{l(b)+1})Z, where Z is the Young wall obtained by removing b from Y and l(b) is the number of y_{l}’s with l < k such that |y_{l}| = |y_{k}|.

Y = Y_{R}(b) ⋯_{b} ^{l(b)}

Y ↗ b = \frac{(1 − (−q^{2})^{l(b)+1})}{q} × ⋯

In either case, set Y_{R}(b) = (y_{l−1}, …, y_{0}), where l is the integer such that |y_{k}| = |y_{k−1}| = ⋯ = |y_{l+1}| < |y_{l}|, and let R_{i}(b; Y) = φ_{i}(Y_{R}(b)) − ε_{i}(Y_{R}(b)). Then we define

(4)  e_{i} Y = ∑_{b} q_{i}^{−R_{i}(b;Y)}(Y ↗ b),

where b runs over all removable i-blocks in Y.

On the other hand, suppose that b is an admissible i-slot in y_{k} of Y. If the i-signature of y_{k} is ++, or if the i-signature of y_{k} is + and there is no i-block below b, then we define Y ↙ b to be the Young wall obtained by adding an i-block at b. If the i-signature of y_{k} is −+, or if the i-signature of y_{k} is + and there is another i-block below b, then we define Y ↙ b = q^{−1}(1 − (−q^{2})^{l(b)+1})Z, where Z is the Young wall obtained by adding an i-block at b and l(b) is the number of y_{l}’s with l > k such that |y_{l}| = |y_{k}|. That is

Y = Y_{L}(b) ⋯ _{b}^{l(b)}

Y ↙ b = \frac{(1 − (−q^{2})^{l(b)+1})}{q} × ⋯

In either case, set Y_{L}(b) = (…, y_{l+2}, y_{l+1}), where l is the integer such that |y_{l+1}| < |y_{l}| = |y_{l−1}| = ⋯ = |y_{k}|, and let L_{i}(b; Y) = φ_{i}(Y_{L}(b)) − ε_{i}(Y_{L}(b)). Then we define

(5)  f_{i} Y = ∑_{b} q_{i}^{L_{i}(b;Y)}(Y ↙ b),

where b runs over all admissible i-slots in Y.

Case 3. Suppose that the i-blocks are of type III.

If b is a removable i-block in y_{k} of Y, then we define Y ↗ b to be the Young wall obtained by removing b from Y. We also consider the following i-block b in y_{k} of Y, which we call a virtually removable i-block.

Y_{R}(b)^{···}  _{b}_{l(b)}  or  Y_{R}(b)^{···}  _{b}_{l(b)}

In this case, we define Y ↗ b to be

(−q_{i})^{l(b)}×  ^{···}  and ^{(−q_{i})^{l(b)}×  ^{···}}

respectively, where l(b) is given in the above figure. In either case, set Y_{R}(b) = (y_{k−1}, …, y_{0}) and R_{i}(b; Y) = φ_{i}(Y_{R}(b)) − ε_{i}(Y_{R}(b)). Then we define

(6)  e_{i} Y = ∑_{b} q_{i}^{−R_{i}(b;Y)}(Y ↗ b),

where b runs over all removable and virtually removable i-blocks in Y.

On the other hand, if b is an admissible i-slot in y_{k} of Y, then we define Y ↙ b to be the Young wall obtained by adding an i-block at b. We also consider the following i-slot b in y_{k} of Y, which we call a virtually admissible i-slot

Y_{L}(b)  ^{···}^{b}  ^{l(b)}  or  Y_{L}(b)  ^{···}^{b}  ^{l(b)}

In this case, we define Y ↙ b to be

(−q_{i})^{l(b)}×  ^{···}  or ^{(−q_{i})^{l(b)}×  ^{···}}

respectively, where l(b) is given in the above figure. In either case, set Y_{L}(b) = (…, y_{k+2}, y_{k+1}) and L_{i}(b; Y) = φ_{i}(Y_{L}(b)) − ε_{i}(Y_{L}(b)). Then we define

(7)  f_{i} Y = ∑_{b} q_{i}^{L_{i}(b;Y)}(Y ↙ b),

where b runs over all admissible and virtually admissible i-slots in Y.

With these actions, we can verify that all the defining relations for the quantum affine algebra U_{q}(𝔤) are satisfied. The verification is quite lengthy and based on the case-by-case check argument (see [5] for the details). Hence we obtain:

Theorem 5.1. The Fock space F(Λ) is an integrable U_{q}(𝔤)-module in the category O^{int}.

It still remains to show that the affine crystal Y(Λ) is the crystal of F(Λ). Let L(Λ) = ⨁_{Y ∈Y(Λ)} 𝔸_{0}Y. It is easy to see that the pair (L(Λ), Y(Λ)) satisfies the first four conditions in Definition 3.1. For the remaining three conditions, the main step is to show that the Kashiwara operators on Y(Λ) induced by the U_{q}(𝔤)-module action on F(Λ) coincide with the abstract Kashiwara operators on Y(Λ) given in Section 4. The proof of this step is based on the crystal basis theory for U_{q}(𝔰𝔩_{2})- modules and the tensor product rule.

Theorem 5.2. The pair (L(Λ), Y(Λ)) is a crystal basis of the Fock space F(Λ), where Y(Λ) is the affine crystal given in Section 4.

Using Theorem 5.2, one can decompose the Fock space F(Λ) into a direct sum of irreducible highest weight modules over U_{q}(𝔤) by locating the maximal vectors in the affine crystal Y(Λ).

Corollary 5.3.

(8)  F(Λ) = \begin{cases}⨁_{m=0}^{∞} V (Λ − mδ)^{⊕p(m)} & if 𝔤 ≠ D_{n+1}^{(2)}, \\ ⨁_{m=0}^{∞} V (Λ − 2mδ)^{⊕p(m)} & if 𝔤 = D_{n+1}^{(2)}. \end{cases}

6. Generalized LLT algorithm

In this section, we generalize Lascoux-Leclerc-Thibon algorithm to obtain an effective algorithm for constructing the global basis G(Λ) of the basic representation V (Λ). Note that V (Λ) is realized as the U_{q}(𝔤)-submodule of F(Λ) generated by Y_{Λ} and that the crystal B(Λ) of V (Λ) is isomorphic to the affine crystal Y_{∘}(Λ) consisting of reduced proper Young walls. Thus our goal is the following: for each reduced proper Young wall Y ∈ Y_{∘}(Λ), we would like to find an algorithm for the corresponding global basis element G(Y) as a linear combination of proper Young walls in Y(Λ).

First, we describe the action of f_{i}^{(r)} (i ∈ I) on Y(Λ). For a proper Young wall Y ∈ Y(Λ), write

(9)  f_{i}^{(r)}Y = ∑_{\begin{matrix}Z∈Y(Λ) \\ wt(Z)=wt(Y)−rα_{i} \end{matrix}} Q_{Y,Z}(q)Z,

where Q_{Y,Z}(q) ∈ ℚ(q). For each Z = (z_{k})_{k=0}^{∞} ∈ Y(Λ) with Q_{Y,Z}(q) ≠ 0, there exists a unique sequence of Young walls Y = Y^{(0)}, Y^{(1)}, …, Y^{(r)} = Z such that

(i) c_{k+1}Y_{k+1} = Y_{k} ↙ b_{k+1} for a (virtually) admissible i-slot b_{k+1} of Y_{k} and

c_{k+1} ∈ ℤ[q, q^{−1}],

(ii) b_{k+1} is placed on b_{k} or to the right of b_{k}.

We suppose that b_{k} is located in the i_{k}th column of Y_{k−1} (1 ≤ k ≤ r). Let Q_{Y^{(k)},Y^{(k+1)}}(q) be the coefficient of Y^{(k+1)} in the expression of f_{i}Y^{(k)}, and define

(10)  Q_{Y,Z}^{∘}(q) = ∏_{k=0}^{r−1} Q_{Y^{(k)},Y^{(k+1)}}(q) ∈ ℤ[q, q^{−1}].

If the i-blocks are of type II, consider

J_{1} = { k | b_{k−1} is beneath b_{k} },
J_{2} = { k | there exists an i-block (≠ b_{k−1}) beneath b_{k} },
J_{3} = { k | there exists no i-block on and beneath b_{k} },
S = { k ∈ J_{2} | k − 1 ∈ J_{3} and |z_{i_{k−1}}| = |z_{i_{k}}| − 1 }.

Put l = |J_{1}|, m = |J_{2}| and n = |J_{3}|. Note that 2l + m + n = r. For each k ∈ S, set d_{k} = qc_{k}.

Lemma 6.1. For Y ∈ Y(Λ), suppose that Q_{Y,Z}(q) ≠ 0 and wt(Z) = wt(Y) − rα_{i} with r ≥ 1 and i ∈ I. Let □ be the type of the i-colored blocks. Then we have

(11)  Q_{Y,Z}(q) = \begin{cases}Q_{Y,Z}^{∘}(q)q\binom{r}{2}_{i} & if □ = I, III, \\ Q_{Y,Z}^{∘}(q)\frac{q^{σ(l,m,n)}}{[2]^{l} ∏_{k∈S} d_{k}} & if □ = II, \end{cases}

where σ(l, m, n) = 4\binom{l}{2} + \binom{m}{2} + \binom{n}{2} + 2l(m + n) + mn. In particular, Q_{Y,Z}(q) ∈ ℤ[q, q^{−1}].

Next, for a reduced proper Young wall Y ∈ Y_{∘}(Λ), choose the left-most column y_{l} such that |y_{l}| ≠ 0 and denote by b the top of y_{l}. Then, since Y is reduced, b must be a removable block of color, say, i. Remove b from Y and denote by Y_{1} the resulting Young wall (if y_{l} is twice i-removable, remove two i-blocks from y_{l}). If Y_{1} is reduced, we stop there and set \overline{Y} = Y_{1}. If Y_{1} is not reduced, then take the next i-removable block in y_{l−1} and remove it from Y_{1} to get Y_{2} (if y_{l−1} is twice i-removable, remove two i-blocks from y_{l−1}). We continue this process from left to right until we get a reduced proper Young wall and denote it by \overline{Y}. Suppose \overline{Y} is obtained by removing r-many i-blocks from Y. If we write f_{i}^{(r)}\overline{Y} = ∑_{Z} Q_{\overline{Y} ,Z}(q)Z, then by Lemma 6.1, it is easy to verify that Q_{\overline{Y} ,Y} (q) = 1.

By repeating the above process, we obtain a sequence of reduced proper Young walls \{Y^{(k)}\}_{k=0}^{N}, where Y^{(0)} = Y, Y^{(1)} = \overline{Y^{(0)}} = \overline{Y}, …, Y^{(k+1)} = \overline{Y^{(k)}}, …, Y^{(N)} = \overline{Y^{(N−1)}} = Y_{Λ}. Suppose that Y^{(k+1)} = \overline{Y^{(k)}} is obtained by removing r_{k+1}- many i_{k+1}-blocks from Y^{(k)} (0 ≤ k ≤ N − 1). Then we define

(12)  A(Y) = f_{i_{1}}^{(r_{1})} ⋯ f_{i_{N}}^{(r_{N})}Y_{Λ} ∈ V (Λ)^{𝔸}.

By definition, we have \overline{A(Y)} = A(Y). Write A(Y) = ∑_{Z∈Y(Λ)} A_{Y,Z}(q)Z, where A_{Y,Z}(q) ∈ ℚ(q). Then the coefficients A_{Y,Z}(q) satisfy the following properties:

Proposition 6.2.

(a) A_{Y,Z}(q) ∈ ℤ[q, q^{−1}].

(b) A_{Y,Z}(q) = 0 unless |Y | ⊵ |Z|.

(c) A_{Y,Z}(q) ≠ 0 and |Y | = |Z| imply Y = Z and A_{Y,Y} (q) = 1.

Recall that there exists a lexicographic ordering > on the set of partitions. For the proper Young walls with the same associated partition, we fix an arbitrary total ordering ≻. We now introduce a total ordering > on the set Y(Λ) as follows: we define Y > Z if either |Y | > |Z| or |Y | = |Z| and Y ≻ Z. Note that if |Y | ⊵ |Z| and |Y | ≠ |Z|, then Y > Z.

By Proposition 6.2, A(Λ) = { A(Y) | Y ∈ Y_{∘}(Λ) } are linearly independent over ℚ(q). Hence it is a ℚ(q)-basis of V (Λ). We claim that A(Λ) is actually an 𝔸-basis of V (Λ)^{𝔸}.

For a reduced proper Young wall Y ∈ Y_{∘}(Λ), consider the corresponding global basis element

(13)  G(Y) = ∑_{Z∈Y(Λ)} G_{Y,Z}(q)Z.

Since G(Λ) is an 𝔸-basis of V (Λ)^{𝔸}, G(Y) is an 𝔸-linear combination of the vectors f_{i_{1}}^{(r_{1})} ⋯ f_{i_{N}}^{(r_{N})}Y_{Λ}. Hence Lemma 6.1 implies G_{Y,Z}(q) ∈ ℚ[q, q^{−1}]. Since G(Y) ≡ Y mod qL(Λ), we may assume that G_{Y,Z}(q) ∈ ℚ[q] and G_{Y,Z}(q) ∈ qℚ[q] unless Y = Z. Write G(Y) = ∑_{W ∈Y_{∘}(Λ)} H_{Y,W} (q)A(W), where H = (H_{Y,W} (q))_{Y,W ∈Y_{∘}(Λ)} is the basis change matrix from A(Λ) to G(Λ) over ℚ(q).

Since \overline{G(Y)} = G(Y) and \overline{A(W)} = A(W) for all Y, W ∈ Y_{∘}(Λ), we get H_{Y,W} (q) = H_{Y,W} (q^{−1}). Consider G = (G_{Y,Z}(q)), H = (H_{Y,W} (q)) and A = (A_{W,Z}(q)), which are square matrices indexed by Y_{∘}(Λ) in decreasing total ordering. Since G = HA and A is upper unit-triangular, we have H = GA^{−1}, which implies H_{Y,W} (q) ∈ 𝔸. Hence we obtain:

Proposition 6.3. A(Λ) = { A(Y) | Y ∈ Y_{∘}(Λ) } is an 𝔸-basis of V (Λ)^{𝔸}.

Recall that G(Y) = ∑_{W ∈Y_{∘}(Λ)} H_{Y,W} (q)A(W), and let W be the Young wall which is maximal among the ones with H_{Y,W} (q) ≠ 0 with respect to the total ordering >. By the maximality of W and Proposition 6.2 (b), we have G_{Y,W} (q) = H_{Y,W} (q)A_{W,W} (q) = H_{Y,W} (q). Since G_{Y,W} (q) ∈ ℚ[q] and G_{Y,W} (q) = H_{Y,W} (q) = H_{Y,W} (q^{−1}) = G_{Y,W} (q^{−1}), G_{Y,W} (q) must be a constant. Moreover, since G(Y) ≡ Y mod qL(Λ), we conclude that Y = W and G_{Y,Y} (q) = H_{Y,Y} (q) = 1. Therefore, by Proposition 6.2 (b), we can write

(14)  G(Y) = ∑_{Y ≥W} H_{Y,W} (q)A(W) = Y + ∑_{Y >Z} G_{Y,Z}(q)Z.

Now, we can apply the algorithm for computing G(Y) as was done in [10]. Fix a weight λ ≤ Λ, and let Y_{1} > ⋯ > Y_{l} be all the reduced proper Young walls in Y_{∘}(Λ)_{λ}. Note that (14) implies G(Y_{l}) = A(Y_{l}). Suppose that we have computed G(Y_{k+1}), …, G(Y_{l}). Also, from (14), A(Y_{k}) is an 𝔸-linear combination of G(Y_{k}), G(Y_{k+1}), …, G(Y_{l}). For k + 1 ≤ s ≤ l, if the coefficient of Y_{s} in A(Y_{k}) − ∑_{p=k+1}^{s−1} γ_{p}(q)G(Y_{p}) (or A(Y_{k}) if s = k + 1) is ∑_{i=0}^{r} a_{i}q^{−i} + ∑_{j=1}^{r′} b_{j}q^{j}, then define γ_{s}(q) = ∑_{i=1}^{r} a_{i}(q^{i} + q^{−i}) + a_{0}. Then we obtain

G(Y_{k}) = A(Y_{k}) − γ_{k+1}(q)G(Y_{k+1}) − ⋯ − γ_{l}(q)G(Y_{l}).

To summarize, we have Theorem 6.4. For a reduced proper Young wall Y ∈ Y_{∘}(Λ), the above algorithm yields the corresponding global basis element

(15)  G(Y) = ∑_{Z∈Y(Λ)} G_{Y,Z}(q)Z.

Moreover, the coefficients G_{Y,Z}(q) satisfy the following conditions :

(i) G_{Y,Z}(q) ∈ ℤ[q], (ii) G_{Y,Z}(q) = 0 unless |Y | ⊵ |Z|, (iii) G_{Y,Y} (q) = 1 and G_{Y,Z}(0) = 0 if Y ≠ Z.

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School of Mathematics, Korea Institute for Advanced Study, Seoul 130-012, Korea

E-mail address: sjkang@kias.re.kr

School of Mathematics, Korea Institute for Advanced Study, Seoul 130-012, Korea

E-mail address: jhkwon@kias.re.kr