Quantized Symmetric Powers

R. M. Green & R. J. Marsh

September 9, 1996

1 Introduction

Let g = sl_{n}(ℂ) be the Lie algebra of n×n matrices over ℂ of trace zero, and let U be the q-analogue of its universal enveloping algebra defined by Drinfel’d [2] and Jimbo [4]. According to [7, 3.5.6, 6.2.3 & 6.3.4], for each dominant weight μ in the weight lattice of g there is an irreducible, finite-dimensional highest weight U-module V (μ) with highest weight μ. Kashiwara [5] and Lusztig [7, 14.4.12] have independently shown the existence of a certain canonical basis B(μ) for V (μ). In [5], Kashiwara proves the existence of a certain crystal basis associated with V (μ) using certain operators, and defines a graph (the crystal graph) which encodes how these operators act on the crystal basis. Using this, he defines B(μ). In §2 we define a certain subspace of the r-th tensor power of the basic module for U, for each integer r ≥ 1, and show that it is a module for U, with highest weight (r, 0, 0, …, 0). (We use the same numbering as [1, Planche I]). In §3 we study the irreducible finite-dimensional highest weight module for U with highest weight (r, 0, 0, …, 0), and find its canonical basis in terms of monomials in U acting on its highest weight vector. We then bring all of this together by showing that the module in §2 is isomorphic to the module studied in §3, and as a result we have two nice descriptions of the canonical basis for this module. Using these we determine how the generators of U act on this basis.

We use the treatment in [7, §§1-3]. Let I = {1, 2, …, n − 1}. Let g be a semisimple Lie algebra of type A_{n−1}, with root system Φ, simple roots α_{1}, α_{2}, …, α_{n−1}, and Killing form ( , ). Let h_{1}, h_{2}, …, h_{n−1} be a basis for a Cartan subalgebra h of g, satisfying (h_{i}, h) = α_{i}^{∗}(h) for all h in h and all i ∈ I. Let Y be the ℤ-lattice spanned by h_{1}, h_{2}, …, h_{n−1}. Let ω_{1}, ω_{2}, …, ω_{n−1} be the fundamental weights of g, defined by ω_{i}(h_{j}) = δ_{ij}, and let X be the ℤ-lattice spanned by them (the weight lattice). Define i · j to be −1 if i ≠ j in I and 2 if i = j ∈ I. (So i · j is a scalar multiple of (α_{i}, α_{j}).) Then (I, ·) is a Cartan datum as in [7, 1.1.1]. Note that we have (i · i)/2 = 1 for all i ∈ I. For h ∈ Y and μ ∈ X, define ⟨h, μ⟩ to be μ(h). Define an imbedding of I into Y by i ↦ h_{i} and into X by i ↦ α_{i} for all i ∈ I. We then have a root datum of type (I, ·) as in [7, 2.2.1], with ⟨h_{i}, α_{j}⟩ = α_{j}(h_{i}) = A_{ij} the corresponding Cartan matrix.

Let ℚ(v) be the field of rational functions in an indeterminate v, and A ⊆ ℚ(v) the ring ℤ[v, v^{−1}]. For N, M ∈ ℕ we define the following (which all lie in A):

[N] = \frac{v^{N} − v^{−N}}{v − v^{−1}} , [N]! = [N][N − 1] ⋯ [1], [ \begin{matrix}M \\ N \end{matrix} ] = \frac{[M]!}{[N]![M − N]!}.

These are referred to as quantized integers, quantized factorials and quantized binomial coefficients, respectively. If v is specialized to 1 they specialize to the usual integers, factorials and binomial coefficients.

We define the quantized enveloping algebra U corresponding to the above data (as in [7, 3.1.1 & 33.1.5]) to be the ℚ(v)-algebra U with generators 1, E_{1}, E_{2}, …, E_{n−1}, F_{1}, F_{2}, …, F_{n−1}, and K_{h} for h ∈ Y, subject to the relations: (for each i, j ∈ I and h, h′ ∈ Y)

K_{0} = 1,
K_{h}K_{h′} = K_{h+h′},

K_{h}E_{i} = v^{α_{i}(h)}E_{i}K_{h},

K_{h}F_{i} = v^{−α_{i}(h)}F_{i}K_{h},

E_{i}F_{i} − F_{i}E_{i} = \frac{K_{i} − K_{i}^{−1}}{v − v^{−1}} ,

E_{i}F_{j} − F_{j}E_{i} = 0, i ≠ j,

∑_{{p+p′=1−A_{ij} \atop p,p′≥0}} (−1)^{p′} [ \begin{matrix}1 − A_{ij} \\ p′ \end{matrix} ] E_{i}^{p}E_{j}E_{i}^{p′} = 0, i ≠ j,

∑_{{p+p′=1−A_{ij} \atop p,p′≥0}} (−1)^{p′} [ \begin{matrix}1 − A_{ij} \\ p′ \end{matrix} ] F_{i}^{p}F_{j}F_{i}^{p′} = 0, i ≠ j,

(where, for i ∈ I, we put K_{i} = K_{h_{i}} and K_{i}^{−1} = K_{−h_{i}}).

Note that U is in fact generated by the elements E_{i}, F_{i}, K_{i} and K_{i}^{−1} for i ∈ I. We make the following definitions (see [7, 3.1.1 & 3.1.13]). For M ∈ ℕ, and i ∈ I, we put E_{i}^{(M)} = E_{i}^{M}/[M]!, and F_{i}^{(M)} = F_{i}^{M}/[M]!, which are called divided powers. Let U_{A} be the A-subalgebra of U generated by the elements E_{i}^{(N)}, F_{i}^{(N)}, K_{h} for i ∈ I, N ∈ ℕ and h ∈ Y. It is called the integral form of U. Let U^{+} be the ℚ(v)-subalgebra of U generated by the E_{i}, i ∈ I, and U_{A}^{+} the A-subalgebra of U generated by E_{i}^{(N)}, i ∈ I, N ∈ ℕ. Let U^{−} be the ℚ(v)-subalgebra of U generated by the F_{i}, i ∈ I, and U_{A}^{−} the A-subalgebra of U generated by F_{i}^{(N)}, i ∈ I, N ∈ ℕ. Let U^{0} be the ℚ(v)-subalgebra generated by the K_{h}, h ∈ Y.

Let X^{+} ⊆ X be the set of dominant weights, i. e. those of the form μ_{1}ω_{1}+μ_{2}ω_{2}+⋯+μ_{n−1}ω_{n−1} ∈ X where ω_{1}, ω_{2}, …, ω_{n−1} are the fundamental weights of g and μ_{1}, μ_{2}, …, μ_{n−1} ∈ ℕ. If L is a Umodule, x ∈ L and μ = μ_{1}ω_{1} + μ_{2}ω_{2} + ⋯ + μ_{n−1}ω_{n−1} ∈ X, we say that x has weight μ (or weight (μ_{1}, μ_{2}, …, μ_{n−1})), if K_{h}x = v^{μ(h)}x for all h ∈ Y. We call the subspace of L consisting of all of the elements of weight μ the μ-weight space of L. As in [7, 3.4.1], we restrict our attention to U-modules which are direct sums (as ℚ(v)-vector spaces) of their weight spaces. We say that x ∈ L, x ≠ 0, is a highest (respectively, lowest) weight vector if x has weight μ, for some μ ∈ X, E_{i}x = 0 (respectively, F_{i}x = 0) for each i ∈ I and U^{−}x = L (respectively, U^{+}x = L). Such a vector is uniquely determined up to a non-zero scalar multiple. We say that L is a highest weight module with highest weight μ if it contains a highest weight vector of weight μ. Let μ = μ_{1}ω_{1} + μ_{2}ω_{2} + ⋯ + μ_{n−1}ω_{n−1} be a dominant weight. We follow the construction in [7, 3.4.5 & 3.5.6]. Let J be the left ideal of U generated by the elements E_{i} for i ∈ I and the elements K_{μ} − v^{(μ(h))} for h ∈ Y. Then the map from U^{−} to U/J taking x ∈ U^{−} to x + J is a ℚ(v)-vector space isomorphism, which can be used to transfer the left U-module structure of U/J to U^{−}. The resulting U-module we denote by M(μ); it is the Verma module. Let T (μ) be the left ideal of M(μ) (as a ℚ(v)-algebra) generated by the elements F_{i}^{μ_{i}+1}, for i ∈ I, and let V (μ) be the quotient module M(μ)/T (μ). Then, by [7, 6.2.3 & 6.3.4], V (μ) is an irreducible, finite-dimensional highest weight U-module with highest weight μ, unique up to isomorphism. We fix x_{1} as the image of 1 ∈ M(μ) under the natural map from M(μ) to V (μ). Then x_{1} is a highest weight vector for V (μ). It is known that V (μ) is the direct sum of its weight spaces (see [7, 3.4.1 & 3.5.6]). We also write V (μ)_{A} = U_{A}^{−}x_{1}, the integral form of V (λ) (see [7, 19.3.1]). By [7, 19.3.2], V (λ)_{A} is a U_{A}-module.

The algebra U is also equipped with a coassociative comultiplication, Δ : U → U ⊗ U, which has the following effect on the generators:

Δ(E_{i}) = 1 ⊗ E_{i} + E_{i} ⊗ K_{i},
Δ(F_{i}) = K_{i}^{−1} ⊗ F_{i} + F_{i} ⊗ 1,
Δ(K_{h}) = K_{h} ⊗ K_{h}.

The map Δ is also an algebra homomorphism. We put Δ^{(2)} = Δ(Δ⊗1), etc., so that for r ∈ ℕ, r ≥ 1, Δ^{(r−1)} is an algebra homomorphism from U to U^{⊗r}. Thus if M is a U-module, we can make M^{⊗r} into a U-module via

u.(m_{1} ⊗ ⋯ ⊗ m_{r}) = Δ^{(r−1)}(u)(m_{1} ⊗ ⋯ ⊗ m_{r}).

It can be checked by a simple inductive argument that the effect of Δ^{(r−1)} on the generators E_{i}, F_{i}, K_{h}, is as follows:

Δ^{(r−1)}(E_{i}) = (1 ⊗ ⋯ ⊗ 1 ⊗ E_{i}) + (1 ⊗ ⋯ ⊗ 1 ⊗ E_{i} ⊗ K_{i}) + ⋯
⋯ + (E_{i} ⊗ K_{i} ⊗ ⋯ ⊗ K_{i}),
Δ^{(r−1)}(F_{i}) = (K_{i}^{−1} ⊗ ⋯ ⊗ K_{i}^{−1} ⊗ F_{i}) + (K_{i}^{−1} ⊗ ⋯ ⊗ K_{i}^{−1} ⊗ F_{i} ⊗ 1) + ⋯
⋯ + (F_{i} ⊗ 1 ⊗ ⋯ ⊗ 1),
Δ^{(r−1)}(K_{h}) = K_{h} ⊗ ⋯ ⊗ K_{h}.

We shall need the following definition of the Kashiwara operators (see [5, 2.2]):

Definition 1.1 Suppose that M is an integrable U-module. Fix i ∈ [1, n−1]. Any element x ∈ M can be written uniquely x = ∑_{0≤k≤k′} F_{i}^{(k)}x_{k,k′}, where the x_{k,k′} satisfy E_{i}x_{k,k′} = 0 and K_{i}x_{k,k′} = v^{k′}x_{k,k′}. We then set F̃_{i}(x) = ∑_{0≤k≤k′} F_{i}^{(k+1)}x_{k,k′} and Ẽ_{i}(x) = ∑_{1≤k≤k′} F_{i}^{(k−1)}x_{k,k′}.

Following Kashiwara [5, 2.3.1] we make the following definition:

Definition 1.2 Suppose that M is an integrable U-module. Let R be the ring of rational functions in v regular at v^{−1} = 0. A pair (L, B) is called a crystal basis of M if the following conditions hold:

(1) L is a free R-submodule of M, and ℚ(v) ⊗_{R} L ≅ M.

(2) B is a basis of the ℚ-vector space L/v^{−1}L.

(3) L = ⨁_{ν} L_{ν} and B = ⊔_{ν}B_{ν}, where M_{ν} is the ν-weight space of M, L_{ν} = L ∩ M_{ν}, and B_{ν} = B ∩ (L_{ν}/v^{−1}L_{ν}).

(4) F̃_{i}L ⊆ L, Ẽ_{i}L ⊆ L, where Ẽ_{i} and F̃_{i} are the Kashiwara operators (see Definition 1.1).

(5) F̃_{i}B ⊆ B ∪ {0} and Ẽ_{i}B ⊆ B ∪ {0}, for all i ∈ [1, n − 1].

(6) For b_{1}, b_{2} ∈ B and i ∈ [1, n − 1], b_{1} = Ẽ_{i}b_{2} if and only if b_{2} = F̃_{i}b_{1}.

We use (6) to draw the corresponding crystal graph, which is an indication of how the Kashiwara operators act on the crystal basis. There is one vertex corresponding to each element of B. If b_{1} and b_{2} are as in (6), we draw an edge from the vertex corresponding to b_{1} to the vertex corresponding to b_{2}, with the arrow from b_{1} to b_{2}, and the label i on it.

Theorem 1.3 Suppose that μ is a dominant weight and V (μ) is the irreducible finite-dimensional highest weight U-module with highest weight μ, and highest weight vector x_{1}, as defined above. Let L(μ) be the R-submodule generated by the vectors of the form F̃_{i_{1}} F̃_{i_{2}} ⋯ F̃_{i_{k}}x_{1} and let B(μ) be the subset of L(μ)/v^{−1}L(μ) consisting of the non-zero images of these vectors under the natural projection L(μ) → L(μ)/v^{−1}L(μ). Then (L(μ), B(μ)) is a crystal basis of V (μ), unique up to an automorphism of V (μ).

Proof: See [5, 2.6]. □

We also have:

Theorem 1.4 Suppose (L_{j}, B_{j}) is a crystal basis for an integrable U-module M_{j}, for j = 1, 2. Set L = L_{1} ⊗ L_{2} ⊆ M_{1} ⊗ M_{2} and B = {b_{1} ⊗ b_{2} : b_{j} ∈ B_{j}, j = 1, 2} ⊆ L/v^{−1}L. Then (L, B) is a crystal basis of M_{1} ⊗ M_{2}.

Proof: See [5, 2.4]. □

Following Kashiwara and Nakashima (see [6, 2.1.2]), we make the following definition:

Definition 1.5 Fix i ∈ [1, n−1]. Suppose that M is an integrable U-module with crystal basis (L, B), and b_{1}, b_{2} ∈ B. Suppose further that Ẽ_{i}b_{1} = F̃_{i}b_{2} = 0 and F̃_{i}b_{1} = b_{2}. We say that b_{1} is of type u_{+} for i and that b_{2} is of type u_{−} for i. If b ∈ B and Ẽ_{i}b = F̃_{i}b = 0, we say that b is of type u_{0} for i.

Proposition 1.6 Fix i ∈ [1, n − 1]. Suppose that M_{j}, j = 1, 2, …, r, are integrable U-modules, and that (L_{j}, B_{j}) is a crystal basis for each M_{j}, with b_{j} ∈ B_{j}, j = 1, 2, …, r. Suppose also that each b_{j} is of type u_{+}, u_{−} or u_{0} for i. Then the following procedure describes how Ẽ_{i} and F̃_{i} act on b_{1} ⊗b_{2} ⊗⋯⊗b_{r}, an element of a crystal basis for M_{1} ⊗ M_{2} ⊗ ⋯ ⊗ M_{r}:

(a) Rewrite the tensor formally by replacing each b_{j} with u_{+} if it is of type u_{+} for i, or with u_{−} or u_{0} similarly.

(b) Neglect u_{0} and also any pair u_{+} ⊗ u_{−}. Repeatedly remove pairs of the form u_{+} ⊗ u_{−} until there are no more left. Rewrite the tensor without these elements, and call this new tensor t. We call this the i-reduced formof b.

(c) To apply Ẽ_{i}: Change the rightmost u_{−} in t to u_{+}. (If there is no u_{−} in the i-reduced form, then Ẽ_{i} acts as zero on the original crystal basis element.)

To apply F̃_{i}: Change the leftmost u_{+} in t to u_{−}. (If there is no u_{+} in the i-reduced form, then F̃_{i} acts as zero on the original crystal basis element.)

(d) Return to t all of the elements deleted in (b). Then replace each u_{+}, u_{−} or u_{0} with the b_{j} it was originally. This is well-defined except when a u_{+} or u_{−} has been changed as in step (b). In the case when a u_{−} has been replaced with a u_{+}, suppose that originally the u_{−} was b_{j}. Replace the u_{+} in t with Ẽ_{i}b_{j}. Similarly if a u_{+} has been replaced with a u_{−} and the u_{+} originally was b_{j}, replace the u_{−} in t with F̃_{i}b_{j}.

Proof: See Remarks 2.1.2 and 2.1.3 in [6]. □

Let ¯ be the ℚ-algebra automorphism from U to U taking E_{i} to E_{i}, F_{i} to F_{i}, and K_{h} to K_{−h}, for each i ∈ [1, n − 1] and h ∈ Y, and v to v^{−1} (see [7, 3.1.12]). There is an induced automorphism (also denoted ¯ ) of any module V (μ) for U defined by \overline{ux_{1}} = \overline{u}x_{1} for any u ∈ U^{−} (see [7, 19.3.4]).

Theorem 1.7 Suppose V (μ), B(μ) and L(μ) are as in Theorem 1.3. Then for each b ∈ B(μ) there is a unique b̃ ∈ L(μ) such that b̃ ↦ b under the canonical projection L(μ) → L(μ)/v^{−1}L(μ) and b̃ ∈ L(μ) ∪ \overline{L(μ)}. Furthermore, the set \{b̃ : b ∈ B(μ)\} forms a ℚ(v)-basis B(μ) for V (μ) (the canonical basis).

Proof: See [5, §0] □

From any Coxeter system with a finite Coxeter group W, we can construct the so called Hecke algebra, denoted by H(W). This is a ℚ(v)-algebra with basis elements T_{w} parametrized by w ∈ W, and the following multiplication rules, for any elements s of length 1 in W:

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & if ℓ(sw) > ℓ(w); \\ qT_{sw} + (q − 1)T_{w} & if ℓ(sw) < ℓ(w). \end{matrix}

In this paper, we take W to be the symmetric group on r letters, and s to be a simple transposition (p, p + 1). We identify the indeterminate q with v^{2}.

The following properties of Hecke algebras will be used in this paper.

Lemma 1.8 Let d ∈ W, and let s(i_{1}) ⋯ s(i_{t}) be a reduced expression for d (i.e. one of minimal length). Then T_{d} = T_{s(i_{1})} ⋯ T_{s(i_{t})} and

T_{d}. ( ∑_{w∈W} T_{w}) = q^{ℓ(d)}. ( ∑_{w∈W} T_{w}) .

Proof: The first assertion follows by a very easy induction. The proof of the second assertion is by induction on d, and the base case follows by splitting the sum into

∑_{w:ℓ(sw)<ℓ(w)} T_{w} + ∑_{w:ℓ(sw)>ℓ(w)} T_{w}. □

Let V be the ℚ(v)-vector space with basis (e_{1}, …, e_{n}). Then T^{r}(V) = V^{⊗r} can be made into a U–H bimodule as follows. We first make V into a left U-module via

E_{i}e_{i+1} = e_{i},
E_{i}e_{j} = 0 (for j ≠ i + 1),
F_{i}e_{i} = e_{i+1},
F_{i}e_{j} = 0 (for j ≠ i),
K_{i}e_{i} = ve_{i},
K_{i}e_{i+1} = v^{−1}e_{i+1},
K_{i}e_{j} = e_{j} (j ≠ i, i + 1).

We now make T^{r}(V) into a left U-module via the homomorphism Δ^{(r−1)}. Following [3], we also make T^{r}(V) into a right H-module via

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}T_{p,p+1} := \{ \begin{matrix}ve_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}} & if i_{p} < i_{p+1}; \\ v^{2}e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} & if i_{p} = i_{p+1}; \\ ve_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}} + (v^{2} − 1)e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} & if i_{p} > i_{p+1}. \end{matrix}

2 The Quantized Symmetric Powers

Definitions 2.1 Fix r ∈ ℕ, r ≥ 1. Let i be an r-tuple (i_{1}, …, i_{r}) satisfying 1 ≤ i_{1} ≤ ⋯ ≤ i_{r} ≤ n.

For each i ∈ [1, n], let λ_{i} be the multiplicity of i in (i_{1}, …, i_{r}). Thus λ is a composition of r. Let W_{λ} be the corresponding parabolic subgroup of S_{r} generated by all simple transpositions (p, p + 1) satisfying i_{p} = i_{p+1}, and let D_{λ} be the associated set of distinguished right coset representatives.

We define the element [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}] ∈ V^{⊗r} to be

v^{−m_{i}} ∑_{σ∈D_{λ}} v^{ℓ(σ)}e_{i_{σ1}} ⊗ ⋯ ⊗ e_{i_{σr}},

where m_{i} := |{1 ≤ a < b ≤ r : i_{a} < i_{b}}|, i.e. the length of the longest element in D_{λ}.

We call the span of all such elements [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}] the r-th quantized symmetric power of the vector space V, and denote it by S_{q}^{r}(V). Note. This quantized symmetric power can also be obtained as a quotient of V^{⊗r} (see [8, p166]). Consider the subspace I_{r} of V^{⊗r} spanned by the elements of the form

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{s}} ⊗ e_{i_{s+1}} ⊗ ⋯ ⊗ e_{i_{r}} − v^{−1}e_{i_{1}} ⊗ ⋯ e_{i_{s+1}} ⊗ e_{i_{s}} ⊗ ⋯ ⊗ e_{i_{r}},

where i_{s} < i_{s+1}. Then it is easy to see that I_{r} is a U-submodule of V^{⊗r}. The quotient V^{⊗r}/I_{r} is isomorphic as a U-module to S_{q}^{r}(V). (It is an irreducible finite-dimensional highest weight module for U with highest weight (r, 0, 0, …, 0). We shall see in Lemma 3.2 that S_{q}^{r}(V) is also such a module, so they must be isomorphic.)

Lemma 2.2 The vector space S_{q}^{r}(V) coincides with the vector space S spanned by all vectors

e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.x_{W} ,

where x_{W} = ∑_{w∈S_{r}} T_{w}, and the elements j_{a} are integers between 1 and n inclusive. (Note that there is no monotonicity assumption on the sequence (j_{1}, …, j_{r}).)

Proof: We first claim that S is spanned by all vectors of the form

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.x_{W} ,

where the i_{a} are integers between 1 and n inclusive, and i_{1} ≤ ⋯ ≤ i_{r}. To see this, consider an arbitrary vector

e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.x_{W} ,

as given in the statement of the Lemma. Let i = (i_{1}, …, i_{r}) be a rearrangement of (j_{1}, …, j_{r}) satisfying i_{1} ≤ ⋯ ≤ i_{r}. Let W_{λ} be the parabolic subgroup corresponding to the sequence i, and let D_{λ} be the associated set of distinguished right coset representatives. Then there exists w ∈ D_{λ}, satisfying i_{s.w} = j_{s} for all 1 ≤ s ≤ r. It follows from Lemma 1.8 that

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.T_{w} = v^{ℓ(w)}e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.

Since T_{w}.x_{W} = q^{ℓ(w)}.x_{W}, we find that e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.x_{W} is a multiple of e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.x_{W}. It now follows that S = S_{q}^{r}(V) because if i_{1} ≤ ⋯ ≤ i_{r}, then

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.x_{W} = (e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.x_{W_{λ}}). ∑_{σ∈D_{λ}} T_{σ}

and

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.x_{W_{λ}} = P_{λ}(q).e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}},

where P_{λ} is the Poincaré polynomial associated with the parabolic subgroup W_{λ}. □

We now show that S_{q}^{r}(V) is a U-module.

Lemma 2.3 (i) The vector space S_{q}^{r}(V) is a U-module.

(ii) Assume i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r}. Then there is only one natural basis vector e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}} of V^{⊗r} occurring with nonzero coefficient in the expression for [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}] which satisfies j_{1} ≥ ⋯ ≥ j_{r}. We call such a basis vector the leading term of [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}]. It occurs with coefficient 1.

(iii) The elements [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}], where i_{1} ≤ ⋯ ≤ i_{r}, form a basis for S_{q}^{r}(V).

Proof: We start by proving (i). Using Lemma 2.2 it is enough to show that S is a U-module, via the usual comultiplication action. We know that V^{⊗r} is a U-module. The result now follows from the fact that the actions of U and H on V^{⊗r} commute (see [3]). We have:

u. (e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.x_{W}) = (u.e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}) .x_{W} .

The right-hand side is now clearly an element of S, as required.

For the proof of (ii), consider i = (i_{1}, …, i_{r}) where i_{1} ≤ ⋯ ≤ i_{r} as usual. Denote the associated parabolic subgroup and set of distinguished coset representatives by W_{λ} and D_{λ} respectively. It is immediate that

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}. ∑_{σ∈D_{λ}} T_{w} = ∑_{σ∈D_{λ}} v^{ℓ(σ)}e_{i_{σ1}} ⊗ ⋯ ⊗ e_{i_{σr}}.

Denote by m the length of the longest element of D_{λ}, and multiply both sides of the above equation by v^{−m}. The result (ii) follows because all the terms on the right are multiples of distinct basis elements.

To prove (iii), we only need prove independence, because the spanning property comes from the definition of quantized symmetric power. We observe that distinct elements of the form [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}] have distinct leading terms, and each term e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}} where j_{1} ≥ ⋯ ≥ j_{r} corresponds to a unique element of this form (i.e. [e_{j_{1}} ∨⋯∨e_{j_{r}}]). The independence of the spanning set is a direct consequence of this. □

Using Lemma 2.3, we can explicitly calculate the coefficients of the matrices corresponding to the left actions of E_{i}, F_{i} and K_{i} on S_{q}^{r}(V).

Let 1 ≤ i_{1} ≤ ⋯ ≤ i_{r} ≤ n, and write i = (i_{1}, …, i_{r}). Consider the basis element e_{i} = [e_{i_{r}} ∨⋯∨e_{i_{1}}] of S_{q}^{r}(V). Let λ_{j} be the multiplicity with which j occurs in the sequence (i_{1}, …, i_{r}).

If λ_{j} > 0, we define i_{j}^{−} to be the (monotone increasing) r-tuple which is the same as i except that the rightmost occurrence of j has been replaced by a j + 1.

If λ_{j+1} > 0, we define i_{j}^{+} to be the (monotone increasing) r-tuple which is the same as i except that the leftmost occurrence of j + 1 has been replaced by a j.

Examples

Consider n = r = 4 and i = (1, 2, 2, 4). Then i_{2}^{−} = (1, 2, 3, 4), i_{3}^{+} = (1, 2, 2, 3) and i_{2}^{+} is not defined.

Lemma 2.4 Maintain the above notation. Then the following identities hold:

(i)

F_{j}.e_{i} = \{ \begin{matrix}[λ_{j+1} + 1]e_{i_{j}^{−}} & if e_{i_{j}^{−}} exists, \\ 0 & otherwise, \end{matrix}

(ii)

K_{j}.e_{i} = v^{λ_{j}−λ_{j+1}}e_{i},

E_{j}.e_{i} = \{ \begin{matrix}[λ_{j} + 1]e_{i_{j}^{+}} & if e_{i_{j}^{+}} exists, \\ 0 & otherwise. \end{matrix}

Proof: The proof of (ii) is easy and follows from the definition and basic properties of e_{i}, and the action of K_{i} under the comultiplication Δ.

The proof of (iii) is analogous to the proof of (i), so we only prove (i).

By consideration of the action of F_{j} on tensor space, we find that each basis element of V^{⊗r} occurring with nonzero coefficient in F_{j}.e_{i} is of form e_{k_{1}} ⊗ ⋯ ⊗ e_{k_{r}}, where the sequence (k_{1}, …, k_{r}) is a permutation of i_{j}^{−}. (If i_{j}^{−} is not defined, then F_{i} must act as zero as required. We may therefore assume λ_{j} > 0.) Lemma 2.3 now implies that F_{j}.e_{i} is some scalar multiple of e_{i_{j}^{−}}, so it remains to prove that the scalar is as claimed.

Lemma 2.3 (ii) shows that it is enough to show that the coefficient of the leading term l_{k} := e_{k_{r}} ⊗ ⋯ ⊗ e_{k_{1}} in F_{j}.e_{i} is equal to [λ_{j+1} + 1].

Denote the leading term of e_{i} by l_{i} := e_{i_{r}} ⊗ ⋯ ⊗ e_{i_{1}}. Consider the set of basis elements occurring in e_{i} with nonzero coefficients which yield a nonzero coefficient of l_{i} after the action of F_{i}. This set consists of l_{i} together with certain elements l_{i,p}, as p ranges from 1 to λ_{j+1} inclusive. (If λ_{j+1} = 0 then this set consists only of l_{i}.) The elements l_{i,p} are permutations of l_{i}, identical except that the leftmost occurrence of e_{j} (note that there must be one since we are assuming λ_{j} > 0) is shifted p places to the left. Note also that l_{i} = l_{i,0}. We illustrate this with an example.

Suppose n = 2, r = 5, l_{i} = e_{2} ⊗ e_{2} ⊗ e_{2} ⊗ e_{1} ⊗ e_{1} and j = 1. Then l_{k} = e_{2} ⊗ e_{2} ⊗ e_{2} ⊗ e_{2} ⊗ e_{1}, λ_{1} = 2, λ_{2} = 3, and we have

l_{i,0} = e_{2} ⊗ e_{2} ⊗ e_{2} ⊗ e_{1} ⊗ e_{1},
l_{i,1} = e_{2} ⊗ e_{2} ⊗ e_{1} ⊗ e_{2} ⊗ e_{1},
l_{i,2} = e_{2} ⊗ e_{1} ⊗ e_{2} ⊗ e_{2} ⊗ e_{1},
l_{i,3} = e_{1} ⊗ e_{2} ⊗ e_{2} ⊗ e_{2} ⊗ e_{1}.

One can see from the proof of Lemma 2.3 that the coefficient of l_{i,p} in e_{i} is v^{−p}. By consideration of the action of F_{j} under the comultiplication, we see that l_{k} appears in F_{j}.l_{i,p} with coefficient v^{λ_{j+1}}.v^{−p}.

This means that the overall coefficient of l_{k} in F_{j}.e_{i} is

∑_{p=0}^{λ_{j+1}} v^{λ_{j+1}}v^{−2p},

i.e. [λ_{j+1} + 1]. The result now follows. □

We now show how S_{q}^{r}(V) may be generated as a U-module by a single vector. Denote by y_{1} the vector [e_{1} ∨ ⋯ ∨ e_{1}] = e_{1} ⊗ ⋯ ⊗ e_{1}.

Lemma 2.5 Let x = [e_{i_{r}} ∨ ⋯ ∨ e_{i_{1}}] satisfy i_{1} ≤ ⋯ ≤ i_{r}. For each integer j satisfying 1 ≤ j ≤ n, let λ_{j} be the multiplicity with which e_{j} appears in x. For each k satisfying 1 ≤ k < n, let c_{k} = ∑_{j=k+1}^{n} λ_{j}. Then

F_{n−1}^{(c_{n−1})} ⋯ F_{1}^{(c_{1})}.y_{1} = x.

Proof: We proceed by induction on t = ∑_{j=1}^{n−1} c_{j}. The case t = 0 is trivial. For the general case, let l be the largest index such that c_{l} is nonzero. (We may assume l ≥ 1, otherwise we are in the case t = 0 which has been dealt with.) This implies that e_{k} does not occur in x unless k ≤ l + 1. Let x′ be the basis element of S_{q}^{r}(V) obtained from x by replacing all the occurrences of e_{l+1} by e_{l}. Then by the inductive hypothesis we have

F_{l−1}^{(c_{l−1})} ⋯ F_{1}^{(c_{1})}.y_{1} = x′.

We need to show that F_{l}^{(c_{l})}.x′ = x. It is enough to prove that

F_{l}^{c_{l}}.x′ = [c_{l}]! x,

and this follows from repeated application of Lemma 2.4 (i). □

We now find that the quantized symmetric power is a highest weight U-module.

Corollary 2.6 The space S_{q}^{r}(V) is a highest weight U-module of highest weight (r, 0, …, 0) and with highest weight vector y_{1}.

Proof: Lemma 2.5 shows that S_{q}^{r}(V) = U^{−}.y_{1}. It is easy to check that K_{i}.y_{1} = v^{δ_{i,1}r}.y_{1}. So y_{1} has weight (r, 0, …, 0). We see from Lemma 2.4 (iii) that E_{i}.y_{1} = 0 for all i. This completes the proof. □

3 The Canonical Basis

Let L be the R-submodule of V generated by e_{1}, e_{2}, …, e_{n}, and let B = {b_{1}, b_{2}, …, b_{n}} be the set of images of these elements under the natural projection from L to L/v^{−1}L. Then (L, B) is a crystal basis for V. The crystal graph is:

∘  ^{∘}  ^{∘}  ^{∘}  ^{∘} ........... ...........  ........... ...........  ........... ...........^{1}  ^{2}  ^{3}  ^{n − 1}  ^{n}^{1}  ^{2}  ^{n − 1}

The edge labels are written above the edges, and vertex j corresponds to the crystal basis element b_{j} (see [6, §3.2]). We know from §1 that for r ∈ ℕ, r ≥ 1, there is an irreducible finite-dimensional highest weight module V^{r} for U with highest weight (r, 0, 0, …, 0), and highest weight vector x_{1}. According to Kashiwara and Nakashima (see [6, 3.4.2]), a crystal basis for V^{r} is (L^{r}, B^{r}), where L^{r} = L^{⊗r} and

B^{r} = {b_{i_{r}} ⊗ b_{i_{r−1}} ⊗ ⋯ ⊗ b_{i_{1}} : 1 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r} ≤ n}.

Note that, if k ∈ [1, n − 1] then in B, b_{k} is of type u_{+} for k, b_{k+1} is of type u_{−} for k, and all other b_{p}’s are of type u_{0} for k. For 1 ≤ j ≤ n, suppose that j appears with multiplicity λ_{j} in the sequence i_{1}, i_{2}, …, i_{r}. Define c_{k} = ∑_{j=k+1}^{n} λ_{j}, for k = 1, 2, … n − 1. Put x̂_{1} = b_{1} ⊗ b_{1} ⊗ ⋯ ⊗ b_{1} in B^{r}. Then x̂_{1} is the vector of highest weight in B^{r}. It is clear from Lemma 1.6 that:

F̃_{1}^{c_{1}} x̂_{1} = b_{2} ⊗ b_{2} ⊗ ⋯ ⊗ b_{2} ⊗ b_{1} ⊗ b_{1} ⊗ ⋯ b_{1}  (1)

(where there are c_{1} b_{2}’s in the expression). Since λ_{1} is equal to the number of 1’s in i_{1}, i_{2}, … i_{r}, and c_{1} = r − λ_{1}, it can be seen that in the expression on the right hand side in equation (1) there are the same number of b_{1}’s as in the expression b_{i_{r}} ⊗ b_{i_{r−1}} ⊗ ⋯ ⊗ b_{i_{1}}. If we now apply F̃_{2}^{c_{2}} to both sides of (1) we get in a similar way an expression with the same number of b_{1}’s and b_{2}’s as b_{i_{r}} ⊗ b_{i_{r−1}} ⊗ ⋯ ⊗ b_{i_{1}}. Repeating this argument, we see that:

b_{i_{r}} ⊗ b_{i_{r−1}} ⊗ ⋯ ⊗ b_{i_{1}} = F̃_{n−1}^{c_{n−1}} F̃_{n−2}^{c_{n−2}} ⋯ F̃_{1}^{c_{1}} x̂_{1}.

Note that here we must have 0 ≤ c_{n−1} ≤ c_{n−2} ≤ ⋯ ≤ c_{1} ≤ r, and also that if c′_{1}, c′_{2}, … c′_{n−1} are any integers satisfying 0 ≤ c′_{n−1} ≤ c′_{n−2} ≤ ⋯ ≤ c′_{1} ≤ r, then there exist integers i_{1}, i_{2}, …, i_{r} satisfying 1 ≤ i_{1} ≤ i_{2} ⋯ ≤ i_{r} ≤ n such that if we define the c’s the same way as above, then c_{j} = c′_{j} for j = 1, 2, …, n − 1. So, we have

B^{r} = {F̃_{n−1}^{c_{n−1}} F̃_{n−2}^{c_{n−2}} ⋯ F̃_{1}^{c_{1}} x̂_{1} : 0 ≤ c_{n−1} ≤ c_{n−2} ≤ ⋯ ≤ c_{1} ≤ r}  (2)

Suppose that 1 ≤ i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{r} ≤ n, and c_{k} = ∑_{j=k+1}^{n} λ_{j}, for k = 1, 2, …, n, where λ_{j} is the number of occurrences of j in i_{1}, i_{2}, …, i_{r}. Fix i ∈ [1, n − 1]. Then since E_{i} commutes with F_{j} if i ≠ j, we have that in V^{r}, E_{i}(F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1}) = 0. Also, if i = 1 then K_{i}F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1} = K_{1}x_{1} = v^{r}x_{1}, where r ≥ c_{1}, and if i ≠ 1 then we have that K_{i}F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1} = v^{c_{i−1}}F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1}, and c_{i−1} ≥ c_{i}. (We use the relations of U and the definition of the module V^{r}). So in either case, the Kashiwara decomposition with respect to i (see Definition 1.1) of F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1} is just F_{i}^{(0)}x_{0,c}, where c ≥ c_{i}. It follows that F̃_{i}^{c_{i}}F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1} = F_{i}^{(c_{i})}F_{i−1}^{(c_{i−1})}F_{i−2}^{(c_{i−2})} ⋯ F_{1}^{(c_{1})}x_{1}. Therefore,

F̃_{n−1}^{c_{n−1}} F̃_{n−2}^{c_{n−2}} ⋯ F̃_{1}^{c_{1}}x_{1} = F_{n−1}^{(c_{n−1})}F_{n−2}^{(c_{n−2})} ⋯ F_{1}^{(c_{1})}x_{1}.

Thus these elements are all fixed under ¯ , so by their definition, (2) and Theorem 1.7 form the canonical basis for V^{r}. We have proved the following theorem:

Theorem 3.1 Suppose r ∈ ℕ, r ≥ 1. Then the canonical basis for V^{r}, the irreducible finitedimensional highest weight module of highest weight (r, 0, 0, …, 0) is given by the following set:

{F_{n−1}^{(c_{n−1})}F_{n−2}^{(c_{n−2})} ⋯ F_{1}^{(c_{1})}x_{1} : 0 ≤ c_{n−1} ≤ c_{n−2} ≤ ⋯ ≤ c_{1} ≤ r}.□

We are now ready to formalize the link between the quantized symmetric power and the canonical basis discussed in §3.

Lemma 3.2 The quantized symmetric power S_{q}^{r}(V) is isomorphic as a U-module to the irreducible finite-dimensional highest weight U-module V^{r} of highest weight (r, 0, …, 0). The isomorphism can be chosen to send [e_{1} ∨ ⋯ ∨ e_{1}] to x_{1}.

Proof: We know from Corollary 2.6 that S_{q}^{r}(V) is a highest weight U-module with highest weight (r, 0, …, 0). We deduce from the representation theory of quantized enveloping algebras in characteristic zero that it has as a quotient the irreducible U-module of highest weight (r, 0, …, 0). The basis described in Lemma 2.5 shows that the dimensions of the two modules are the same, so the modules are isomorphic. Since they are both highest weight modules, the isomorphism must send highest weight vectors to highest weight vectors, and thus we can choose the isomorphism as stated. □

We can now state the actions of the generators of U on the canonical basis.

Theorem 3.3 Let x = F_{n−1}^{(c_{n−1})} ⋯ F_{1}^{(c_{1})}x_{1} be an element of the canonical basis described in §3. Define c_{0} = 0 and c_{n} = r. Then the actions of E_{i}, F_{i} and K_{i} on x are given as follows:

(i)

F_{i}.x = \{ \begin{matrix}[c_{i} − c_{i+1} + 1] F_{n−1}^{(c_{n−1})} ⋯ F_{i+1}^{(c_{i+1})}F_{i}^{(c_{i}+1)}F_{i−1}^{(c_{i−1})} ⋯ F_{1}^{(c_{1})}.x_{1} & if c_{i} < c_{i−1}, \\ 0 & otherwise, \end{matrix}

(ii)

K_{i}.x = v^{c_{i−1}−2c_{i}+c_{i+1}}x,

(iii)

E_{i}.x = \{ \begin{matrix}[c_{i−1} − c_{i} + 1] F_{n−1}^{(c_{n−1})} ⋯ F_{i+1}^{(c_{i+1})}F_{i}^{(c_{i}−1)}F_{i−1}^{(c_{i−1})} ⋯ F_{1}^{(c_{1})}.x_{1} & if c_{i} > c_{i+1}, \\ 0 & otherwise. \end{matrix}

Proof: Since x is an element of the canonical basis, we must have r ≥ c_{1} ≥ ⋯ ≥ c_{n−1} ≥ 0. Let λ_{k} := c_{k−1} − c_{k}, for each 1 ≤ k ≤ n.

Using Lemma 2.5, we can identify x with the element of S_{q}^{r}(V) given by x′ := [e_{i_{r}} ∨⋯∨e_{i_{1}}], where i_{1} ≤ ⋯ ≤ i_{r} and e_{j} occurs in x′ with multiplicity λ_{j}. Also note that replacing an occurrence of e_{i} in x′ by e_{i+1} corresponds to increasing c_{i} by one; this is possible if and only if c_{i} < c_{i−1}. Similarly, replacing an occurrence of e_{i+1} by e_{i} corresponds to decreasing c_{i} by one; this is possible if and only if c_{i} > c_{i+1}. The assertions now follow from Lemma 2.4. □

Remark 3.4 It is worth noting that E_{i} and F_{i} take canonical basis elements to A-multiples of other canonical basis elements, or to zero, so the matrices representing them with respect to the canonical basis have elements in A. The fact that the matrices can be written over A is also a consequence of [7, Theorems 14.4.11(a) & 14.4.12].

The authors would like to thank Dr. K. Erdmann for bringing this problem to their attention. This paper was written while the authors were supported by grants from the Engineering and Physical Science Research Council, and supervised by Professor R. Carter at the University of Warwick.

References

[1] N. Bourbaki. Groupes et algèbres de Lie, Chapitres 4,5 et 6. Masson, Paris, 1981.

[2] V. G. Drinfeld. Hopf algebras and the Yang–Baxter equation. Soviet Math. Dokl., 32:254–258, 1985.

[3] J. Du. A note on Quantized Weyl Reciprocity at roots of unity. Algebra Colloq., 4:363–372, 1995.

[4] M. Jimbo. A q-difference analogue of U(g) and the Yang–Baxter equation. Lett. Math. Phys., 10:63–69, 1985.

[5] M. Kashiwara. On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J., 63(2):465–516, 1991.

[6] M. Kashiwara and T. Nakashima. Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Alg., 165(2):295–345, 1994.

[7] G. Lusztig. Introduction to Quantum Groups. Birkhäuser, Boston, 1993.

[8] S. Martin. Schur Algebras and Representation Theory. C. U. P., 1993.

Richard Green,  Robert Marsh,
Mathematical Institute, Sonderforschungsbereich 343,
24-29 St. Giles’,  Universität Bielefeld,
Oxford,  Postfach 100131,
OX1 3LB,  33501 Bielefeld,
United Kingdom.  Germany.
greenr@maths.ox.ac.uk sfb20@mathematik.uni-bielefeld.de