Séminaire Lotharingien de Combinatoire 50 (2004), Article B50e

A Littlewood-Richardson rule for evaluation representations of U_{q}(𝔰̂𝔩_{n})

Bernard Leclerc

Abstract

We give a combinatorial description of the composition factors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra of type GL_{m}. Using quantum affine Schur-Weyl duality, this yields a combinatorial description of the composition factors of the tensor product of two evaluation modules of the quantum affine algebra U_{q}(𝔰̂𝔩_{n}).

1 Introduction

1.1 Let H_{m} denote the Iwahori-Hecke algebra of type A_{m−1} over ℂ(t). This is a semisimple associative algebra isomorphic to the group algebra ℂ(t)[𝔖_{m}] of the symmetric group. Hence its simple modules S(λ) are parametrized by the partitions λ of m. Consider a decomposition m = m_{1} + m_{2}, and two partitions λ^{(1)} and λ^{(2)} of m_{1} and m_{2}, respectively. Then we have a H_{m_{1}}-module S(λ^{(1)}) and a H_{m_{2}}-module S(λ^{(2)}), and we can form the induced module

S(λ^{(1)}) ⊙ S(λ^{(2)}) := Ind_{H_{m_{1}}⊗H_{m_{2}}}^{H_{m}} (S(λ^{(1)}) ⊗ S(λ^{(2)})) .

Here, H_{m_{1}} ⊗H_{m_{2}} is identified to a subalgebra of H_{m} in the standard way. Using again the isomorphism H_{m} ≅ ℂ(t)[𝔖_{m}], we see that the multiplicity of a simple H_{m}-module S(μ) in S(λ^{(1)}) ⊙ S(λ^{(2)}) is equal to the classical Littlewood-Richardson coefficient c_{λ^{(1)}λ^{(2)}}^{μ} (see e.g. [Mcd]).

1.2 Let now Ĥ_{m} be the affine Iwahori-Hecke algebra over ℂ(t) (see 2.1 below). For each invertible z ∈ ℂ(t) we have a surjective evaluation homomorphism τ_{z} : Ĥ_{m} → H_{m}. Pulling back the simple H_{m}-module S(λ) via τ_{z} we obtain a simple Ĥ_{m}-module S(λ; z) called an evaluation module. In analogy with 1.1, given two invertible elements z_{1} and z_{2} of ℂ(t), we can then form the induced Ĥ_{m}-module

S(λ^{(1)}; z_{1}) ⊙ S(λ^{(2)}; z_{2}) := IndĤ_{m} ; Ĥ_{m_{1}}⊗Ĥ_{m_{2}} (S(λ^{(1)}; z_{1}) ⊗ S(λ^{(2)}; z_{2})) .

It turns out that if we fix λ^{(1)}, λ^{(2)} and vary the spectral parameters z_{1}, z_{2}, this module is generically irreducible, that is, it is simple except for a finite number of values of the ratio z_{1}/z_{2}. In [LNT, Theorem 36] a combinatorial description of these special values was given.

In this note we shall make this result more precise by describing all the composition factors of S(λ^{(1)}; z_{1}) ⊙ S(λ^{(2)}; z_{2}) at these critical values z_{1}/z_{2}. We shall also prove that, in contrast with the classical Littlewood-Richardson rule, all the composition factors appear with multiplicity one. The composition factors occuring in a product will be described using the combinatorics of Lusztig’s symbols, that is, of certain two-row arrays introduced by Lusztig for parametrizing the irreducible complex representations of the classical reductive groups over finite fields [Lu1, Lu2].

1.3 We will derive our combinatorial formula from some explicit calculations of canonical bases in level 2 representations of the quantum algebra U_{v}(𝔰𝔩_{n+1}) performed in [LM]. More precisely, by dualizing [LM, Theorem 3], we get a formula for the expansion of the product of two quantum flag minors on the dual canonical basis of U_{v}(𝔰𝔩_{n+1}) (Theorem 5). Using then Ariki’s theorem as in [LNT], we obtain immediately the above-mentioned Littlewood-Richardson rule for induction products of two evaluation modules over affine Hecke algebras (Theorem 2).

Finally, by means of the quantum affine analogue of the Schur-Weyl duality developed by Cherednik, Chari-Pressley and Ginzburg-Reshetikhin-Vasserot, we can deduce from this rule a similar one for the tensor product of two evaluation modules over the quantum affine algebra U_{q}(𝔰̂𝔩_{N}).

2 Composition factors of induced Ĥ_{m}-modules

2.1 Let Ĥ_{m} be the affine Hecke algebra of type GL_{m} over ℂ(t). It has invertible generators T_{1}, …, T_{m−1}, y_{1}, …, y_{m} subject to the relations

T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}, (1 ⩽ i ⩽ m − 2),

T_{i}T_{j} = T_{j}T_{i}, (|i − j| > 1),
(T_{i} − t)(T_{i} + 1) = 0, (1 ⩽ i ⩽ m − 1),
y_{i}y_{j} = y_{j}y_{i}, (1 ⩽ i, j ⩽ m),
y_{j}T_{i} = T_{i}y_{j}, (j ≠ i, i + 1),
T_{i}y_{i}T_{i} = t y_{i+1}, (1 ⩽ i ⩽ m − 1).

The subalgebra H_{m} generated by the T_{i}’s is the Iwahori-Hecke algebra of type A_{m−1}.

For any invertible z ∈ ℂ(t) we have a unique algebra homomorphism τ_{z} : Ĥ_{m} → H_{m} such that

τ_{z}(T_{i}) = T_{i}, τ_{z}(y_{1}) = z, (i = 1, …, m − 1).

This is called the evaluation at z.

We also have an algebra automorphism σ_{z} : Ĥ_{m} → Ĥ_{m} such that

σ_{z}(T_{i}) = T_{i}, σ_{z}(y_{i}) = zy_{i}, (i = 1, …, m − 1).

This is called the shift by z.

2.2 As mentioned in the introduction, given two partitions λ^{(1)} and λ^{(2)}, the structure of the induced Ĥ_{m}-module

S(λ^{(1)}; z_{1}) ⊙ S(λ^{(2)}; z_{2})

depends essentially on the ratio z_{1}/z_{2}. Indeed, by twisting this module with the shift automorphism σ_{z} we obtain the induced module

S(λ^{(1)}; zz_{1}) ⊙ S(λ^{(2)}; zz_{2}).

For example, it is known that if z_{1}/z_{2} ∉ t^{ℤ} then S(λ^{(1)}; z_{1}) ⊙ S(λ^{(2)}; z_{2}) is irreducible. Therefore, we can assume without loss of generality that

z_{i} = t^{a_{i}}, a_{i} ∈ ℤ, a_{i} ⩾ ℓ(λ^{(i)}),  (i = 1, 2),  (1)

where as usual ℓ(λ) denotes the length of the partition λ. Since S(λ^{(1)}; z_{1}) ⊙ S(λ^{(2)}; z_{2}) and S(λ^{(2)}; z_{2}) ⊙ S(λ^{(1)}; z_{1}) have the same composition factors with the same multiplicities, we can also assume that a_{1} ⩽ a_{2}.

2.3 It will be convenient to write partitions in weakly increasing order. Given a partition λ and an integer a ⩾ ℓ(λ) we can make λ into a non-decreasing sequence (λ_{1}, …, λ_{a}) of length a by setting λ_{j} = 0 for j = 1, …, a − ℓ(λ). We can then associate to (λ, a) the increasing sequence

β = (β_{1}, …, β_{a}), β_{j} = j + λ_{j}.  (2)

In this way, given (λ^{(i)}, a_{i}) (i = 1, 2) as in 2.2, we obtain a symbol

S = ( \begin{matrix}β^{(2)} \\ β^{(1)} \end{matrix} ) = ( \begin{matrix}β_{1}^{(2)}, …, β_{a_{2}}^{(2)} \\ β_{1}^{(1)}, …, β_{a_{1}}^{(1)} \end{matrix} ) .  (3)

For example, the symbol attached to the pairs ((1, 1, 2), 3) and ((2, 3), 5) is

S = \binom{1 2 3 6 8}{2 3 5} .

Conversely, given a symbol S, i.e. a two-row array as in Eq. (3) with

1 ⩽ β_{1}^{(i)} < ⋯ < β_{a_{i}}^{(i)} (i = 1, 2),

there is a unique pair (λ^{(i)}, a_{i}) (i = 1, 2) whose symbol is S.

2.4 The symbol S of Eq. (3) is said to be standard if β_{k}^{(2)} ⩽ β_{k}^{(1)} for k ⩽ a_{1}. In [LM, §2.5] we have defined the pairs of a standard symbol S, and the set C(S) of all symbols Σ obtained from S by permuting some of its pairs. As shown in [LM, Lemma 9], these notions are equivalent to the notion of admissible involution of Lusztig [Lu4].

For the convenience of the reader we shall recall these definitions. Let S = \binom{β}{γ} be a standard symbol. We define an injection ψ : γ ⟶ β such that ψ(j) ⩽ j for all j ∈ γ. To do so it is enough to describe the subsets

γ^{l} = {j ∈ γ | ψ(j) = j − l}, (0 ⩽ l ⩽ n).

We set γ^{0} = γ ∩ β and for l ⩾ 1 we put

γ^{l} = {j ∈ γ − (γ^{0} ∪ ⋯ ∪ γ^{l−1}) | j − l ∈ β − ψ(γ^{0} ∪ ⋯ ∪ γ^{l−1})}.

Observe that the standardness of S implies that ψ is well-defined.

Example 1 Take

S = \binom{1 3 5 8 9}{3 6 7 10} .

Then

γ^{0} = {3}, γ^{1} = {6, 10}, γ^{2} = ⋯ = γ^{5} = ∅, γ^{6} = {7}.

Hence

ψ(3) = 3, ψ(6) = 5, ψ(7) = 1, ψ(10) = 9.

The pairs (j, ψ(j)) with ψ(j) ≠ j (that is, j ∉ β ∩ γ) will be called the pairs of S. Given a standard symbol S with p pairs, we denote by C(S) the set of all symbols obtained from S by permuting some pairs in S and reordering the rows. We consider S itself as an element of C(S), hence C(S) has cardinality 2^{p}.

2.5 Given a partition λ and an integer a we call Young diagram of (λ, a) the Young diagram of λ in which each cell (i, j) is filled with the integer i − j + a. For instance, if λ = (2, 3) and a = 5 then the Young diagram of (λ, a) is

4 5 5 6 7

The rows of the Young diagram of (λ, a) yield a multisegment

m(λ, a) := ∑_{1⩽k⩽a}[k, k + λ_{k} − 1].

This is a formal sum (or multiset) of intervals in ℤ, in which we discard the empty intervals corresponding to the k’s with λ_{k} = 0. Thus, continuing with the same example, we have

m((2, 3), 5) = [4, 5] + [5, 7].

Similarly, we attach to a pair (λ^{(i)}, a_{i}) (i = 1, 2) or to its symbol S the multisegment

m(S) = m(λ^{(1)}, a_{1}) + m(λ^{(2)}, a_{2}).

2.6 To each multisegment

m := ∑_{k}[α_{k}, β_{k}]

is attached an irreducible Ĥ_{m}-module L_{m}, where m = ∑_{k}(β_{k} + 1 − α_{k}) (see e.g. [LNT, §2.1]).

2.7 Let us assume that the pair (λ^{(i)}, a_{i}) (i = 1, 2) satisfies the conditions of 2.2. Let Σ denote the symbol attached to this pair. We can now state:

Theorem 2 The composition factors of S(λ^{(1)}; t^{a_{1}}) ⊙ S(λ^{(2)}; t^{a_{2}}) are the modules L_{m(S)} where S runs through the set of standard symbols such that Σ ∈ C(S). Each of them occurs with multiplicity one.

Theorem 2 will be deduced from Theorem 5 below.

Example 3 Let (λ^{(1)}, a_{1}) = ((1, 4), 2) and (λ^{(2)}, a_{2}) = ((1, 2, 3), 4). The corresponding symbol is

Σ = \binom{1 3 5 7}{2 6} .

The standard symbols S such that Σ ∈ C(S) are

\binom{1 2 5 6}{3 7} , \binom{1 2 5 7}{3 6} , \binom{1 3 5 6}{2 7} , \binom{1 3 5 7}{2 6} .

It follows that the composition factors of S((1, 4); t^{2}) ⊙ S((1, 2, 3); t^{4}) are the L_{m} where m is one the following multisegments:

n_{1} = [1, 2] + [2, 6] + [3, 4] + [4, 5], n_{2} = [1, 2] + [2, 5] + [3, 4] + [4, 6],

n_{3} = [1, 1] + [2, 2] + [2, 6] + [3, 4] + [4, 5], n_{4} = [1, 1] + [2, 2] + [2, 5] + [3, 4] + [4, 6].

2.8 By restriction to the finite Hecke algebra H_{m} the irreducible Ĥ_{m}-modules L_{m(S)} decompose into direct sums of Specht modules. The sum of all these Specht modules is given by the (classical) Littlewood-Richardson rule for the product S(λ^{(1)}) ⊙ S(λ^{(2)}). It would be interesting to find a combinatorial description of the splitting of S(λ^{(1)})⊙S(λ^{(2)}) thus obtained.

Example 4 Let us continue Example 3. The restrictions to H_{11} of the 4 irreducible Ĥ_{11}- modules are as follows:

L_{n_{1}} ↓ = S(1, 3, 7) ⊕ S(2, 2, 7) ⊕ S(2, 3, 6) ⊕ S(1, 1, 3, 6)
⊕ S(1, 2, 2, 6) ⊕ S(1, 2, 3, 5) ⊕ S(2, 2, 2, 5),
L_{n_{2}} ↓ = S(1, 4, 6) ⊕ S(2, 3, 6) ⊕ S(2, 4, 5) ⊕ S(1, 1, 4, 5) ⊕ S(1, 2, 3, 5)
⊕ S(1, 2, 4, 4) ⊕ S(3, 3, 5) ⊕ S(1, 3, 3, 4) ⊕ S(2, 2, 3, 4),
L_{n_{3}} ↓ = S(1, 1, 2, 7) ⊕ S(1, 2, 2, 6) ⊕ S(1, 1, 1, 2, 6) ⊕ S(1, 1, 2, 2, 5),
L_{n_{4}} ↓ = S(1, 1, 3, 6) ⊕ S(1, 2, 3, 5) ⊕ S(1, 1, 1, 3, 5) ⊕ S(1, 1, 2, 3, 4).

This gives a splitting of S(1, 4) ⊙ S(1, 2, 3).

3 Canonical bases

3.1 Fix n ⩾ 2 and let 𝔤 = 𝔰𝔩_{n+1}. We consider the quantum enveloping algebra U_{v}(𝔤) over ℚ(v) with Chevalley generators e_{j}, f_{j}, t_{j} (1 ⩽ j ⩽ n). The simple roots and the fundamental weights are denoted by α_{k} and Λ_{k} (1 ⩽ k ⩽ n) respectively. The irreducible representation of U_{v}(𝔤) with highest weight Λ is denoted by V (Λ). We denote by U_{v}(𝔫) the subalgebra of U_{v}(𝔤) generated by e_{j} (1 ⩽ j ⩽ n).

3.2 Let B (resp. B^{∗}) denote the canonical basis (resp. the dual canonical basis) of U_{v}(𝔫) ([Lu3], [BZ]; see also [LNT, §3]). The elements of B and B^{∗} are naturally labelled by the multisegments m supported on [1, n]. We shall denote them by b_{m} and b_{m}^{∗} respectively.

The vectors b_{m}^{∗} for which m is of the form

m = m(λ, a)

for some partition λ and some integer a are called quantum flag minors. Indeed, by [BZ], they can be expressed as quantum minors of a triangular matrix whose entries are iterated brackets of the e_{i}’s (see [LNT, §5.2]).

3.3 Let (λ^{(i)}, a_{i}) (i = 1, 2) be as in 2.2. We also assume that the multisegments

m_{i} = m(λ^{(i)}, a_{i}) (i = 1, 2)

are supported on [1, n]. Let Σ be the symbol attached to the pair (λ^{(i)}, a_{i}) (i = 1, 2). For a standard symbol S such that Σ ∈ C(S) we denote by n(S, Σ) the number of pairs of S which are permuted to get Σ. Finally, we denote by N_{j}(λ, a) the number of cells of the Young diagram of (λ, a) containing the integer j.

Theorem 5 We have

b_{m_{1}}^{∗}b_{m_{2}}^{∗} = v^{−N_{a_{1}}(λ^{(2)}, a_{2})} ∑_{S} v^{n(S, Σ)} b_{m(S)}^{∗}

where the sum runs through all standard symbols S such that Σ ∈ C(S).

Example 6 We take (λ^{(1)}, a_{1}) and (λ^{(2)}, a_{2}) as in Example 3. Hence

m_{1} = [1, 1] + [2, 5], m_{2} = [2, 2] + [3, 4] + [4, 6].

Then N_{a_{1}}(λ^{(2)}, a_{2}) = N_{2}((1, 2, 3), 4) = 1, and we obtain, using the notation of Example 3,

b_{m_{1}}^{∗} b_{m_{2}}^{∗} = v^{−1}(v^{2} b_{n_{1}}^{∗} + v b_{n_{2}}^{∗} + v b_{n_{3}}^{∗} + b_{n_{4}}^{∗}).

3.4 Proof of Theorem 5. Following [LNT, §7.2], we will replace calculations of products of elements of B^{∗} by calculations of dual canonical bases of finite-dimensional representations of U_{v}(𝔤).

3.4.1 Let U_{v}(𝔫^{−}) denote the subalgebra of U_{v}(𝔤) generated by the f_{i}’s, and let x ↦ x^{♯} denote the algebra isomorphism from U_{v}(𝔫) to U_{v}(𝔫^{−}) defined by e_{i}^{♯} = f_{i} (i = 1, …, n). Let Λ be a dominant integral weight and let u_{Λ} be a highest weight vector of the irreducible module V (Λ). Then the map π_{Λ} : x ↦ x^{♯}u_{Λ} projects the canonical basis B of U_{v}(𝔫) to the union of the canonical basis B(Λ) of V (Λ) with the set {0}. The dual map π_{Λ}^{∗} gives an embedding of the dual canonical basis B^{∗}(Λ) of V (Λ) ≃ V (Λ)^{∗} into the dual canonical basis B^{∗} of U_{v}(𝔫) ≃ U_{v}(𝔫)^{∗}.

3.4.2 In particular the subset of B^{∗} obtained by embedding the bases B^{∗}(Λ_{a}) (1 ⩽ a ⩽ n) of the fundamental representations is precisely the subset of quantum flag minors. It is well known that V (Λ_{a}) is a minuscule representation whose bases B^{∗}(Λ_{a}) and B(Λ_{a}) coincide. Moreover the elements of these bases are naturally labelled by the pairs (λ, a) whose Young diagram (as defined in 2.5) contains only cells numbered by integers between 1 and n. Denoting them by b_{(λ,a)}^{∗} we have

π_{Λ_{a}}^{∗}(b_{(λ,a)}^{∗}) = b_{m(λ,a)}^{∗}

Equivalently, we can also label the elements of B^{∗}(Λ_{a}) by one-row symbols β as in Eq. (2) with β_{i} ⩽ n + 1.

3.4.3 Similarly, the basis B^{∗}(Λ_{a_{1}}) ⊗ B^{∗}(Λ_{a_{2}}) is naturally labelled by the set of symbols S as in Eq. (3) with β_{a_{i}}^{(i)} ⩽ n + 1 (i = 1, 2). Using the theory of crystal bases [K1, K2] one can see that the basis B^{∗}(Λ_{a_{1}} + Λ_{a_{2}}) has a natural labelling by the subset of standard symbols [LM, §2.3]. Moreover, denoting by b_{S}^{∗} the element of B^{∗}(Λ_{a_{1}} + Λ_{a_{2}}) labelled by the standard symbol S we have, using also the notation of 2.5,

π_{Λ_{a_{1}}+Λ_{a_{2}}}^{∗}(b_{S}^{∗}) = b_{m(S)}^{∗}.

3.4.4 Let ι : V (Λ_{a_{1}} +Λ_{a_{2}}) → V (Λ_{a_{1}})⊗V (Λ_{a_{2}}) be the U_{v}(𝔤)-module embedding which maps u_{Λ_{a_{1}}+Λ_{a_{2}}} to u_{Λ_{a_{1}}} ⊗ u_{Λ_{a_{2}}}, and let ι^{∗} : V (Λ_{a_{1}}) ⊗ V (Λ_{a_{2}}) → V (Λ_{a_{1}} + Λ_{a_{2}}) be its dual. Let b_{i}^{∗} ∈ B^{∗}(Λ_{a_{i}}) (i = 1, 2) and denote by b_{m_{i}}^{∗} = π_{Λ_{a_{i}}}^{∗}(b_{i}^{∗}) (i = 1, 2) the corresponding quantum flag minors. It is shown in [LNT, §7.2.7] that the image of b_{1}^{∗} ⊗ b_{2}^{∗} under the composition of maps π_{Λ_{a_{1}}+Λ_{a_{2}}}^{∗} ∘ ι^{∗} coincides up to a power of v with the product b_{m_{1}}^{∗}b_{m_{2}}^{∗}. Hence to calculate the B^{∗}-expansion of b_{m_{1}}^{∗}b_{m_{2}}^{∗} it is enough to calculate the matrix of the map ι^{∗} with respect to the bases B^{∗}(Λ_{a_{1}}) ⊗ B^{∗}(Λ_{a_{1}}) and B^{∗}(Λ_{a_{1}} + Λ_{a_{2}}).

3.4.5 The matrix of ι with respect to the bases B(Λ_{a_{1}} + Λ_{a_{2}}) and B(Λ_{a_{1}}) ⊗ B(Λ_{a_{1}}) was calculated in [LM, Theorem 3] in terms of Lusztig’s symbols. Transposing this matrix we obtain the desired matrix of ι^{∗}. Using 3.4.2 and 3.4.3, we then get the formula of Theorem 5. □

3.5 Proof of Theorem 2. By [LNT, §3.7] the multisegments m indexing the composition factors L_{m} of S(λ^{(1)}; t^{a_{1}}) ⊙ S(λ^{(2)}; t^{a_{2}}) are those occuring in the right-hand side of the formula of Theorem 5. Moreover the composition multiplicities are obtained by specializing v to 1 in the coefficients of this formula. Hence they are all equal to 1. □

4 Tensor products of U_{q}(𝔰̂𝔩_{N})-modules

4.1 Let U_{q}(𝔰̂𝔩_{N}) be the quantized affine algebra of type A_{N−1}^{(1)} with parameter q a square root of t (see for example [CP] for the defining relations of U_{q}(𝔰̂𝔩_{N})). The quantum affine Schur-Weyl duality between Ĥ_{m} and U_{q}(𝔰̂𝔩_{N}) [CP, Ch, GRV] gives a functor F_{m,N} from the category of finite-dimensional Ĥ_{m}-modules to the category of level 0 finitedimensional representations of U_{q}(𝔰̂𝔩_{N}). If N ⩾ m, F_{m,N} maps the simple modules of Ĥ_{m} to simple modules of U_{q}(𝔰̂𝔩_{N}). However, the image of a non-zero simple Ĥ_{m}-module may be the zero U_{q}(𝔰̂𝔩_{N})-module. More precisely, the simple Ĥ_{m}-module L_{m} is mapped to a non-zero simple U_{q}(𝔰̂𝔩_{N})-module if and only if all the segments occuring in m have length ⩽ N − 1. In this case the Drinfeld polynomials of F_{m,N}(L_{m}) are easily calculated from m (see [CP]).

The functor F_{m,N} transforms induction product into tensor product, that is, for M_{1} in C_{m_{1}} and M_{2} in C_{m_{2}} one has

F_{m_{1}+m_{2},N}(M_{1} ⊙ M_{2}) = F_{m_{1},N}(M_{1}) ⊗ F_{m_{2},N}(M_{2}) .

4.2 The image under F_{m,N} of an evaluation module for Ĥ_{m} is an evaluation module for U_{q}(𝔰̂𝔩_{N}), and all evaluation modules of U_{q}(𝔰̂𝔩_{N}) can be obtained in this way, by varying m ∈ ℕ^{∗}.

4.3 By application of the Schur functor F_{m,N} to Theorem 2 we thus obtain a combinatorial description of all composition factors of the tensor product of two evaluation modules of U_{q}(𝔰̂𝔩_{N}).

Example 7 We continue Example 3 and Example 6. The image of the Ĥ_{5}-module L_{m_{1}} under F_{5,N} is the evaluation module V (m_{1}) of U_{q}(𝔰̂𝔩_{N}) with Drinfeld polynomials

P_{1}(u) = u − q^{−2},
P_{2}(u) = P_{3}(u) = 1,
P_{4}(u) = u − q^{−7},
P_{k}(u) = 1, (5 ⩽ k ⩽ N − 1).

This is a non-zero module if and only if N ⩾ 5. Similarly, the image of the Ĥ_{6}-module L_{m_{2}} under F_{6,N} is the evaluation module V (m_{2}) of U_{q}(𝔰̂𝔩_{N}) with Drinfeld polynomials

P_{1}(u) = u − q^{−4},
P_{2}(u) = u − q^{−7},
P_{3}(u) = u − q^{−10},
P_{k}(u) = 1, (4 ⩽ k ⩽ N − 1).

This is a non-zero module if and only if N ⩾ 4. The images of the Ĥ_{11}-modules L_{m_{1}}, L_{m_{2}}, L_{m_{3}}, L_{m_{4}} under F_{11,N} are the modules V (n_{1}), V (n_{2}), V (n_{3}), V (n_{4}) with respective Drinfeld polynomials

P_{1}(u) = 1,
P_{2}(u) = (u − q^{−3})(u − q^{−7})(u − q^{−9}),
P_{3}(u) = P_{4}(u) = 1,
P_{5}(u) = u − q^{−8},
P_{k}(u) = 1, (6 ⩽ k ⩽ N − 1);
P_{1}(u) = 1,
P_{2}(u) = (u − q^{−3})(u − q^{−7}),
P_{3}(u) = u − q^{−10},
P_{4}(u) = u − q^{−7},
P_{k}(u) = 1, (5 ⩽ k ⩽ N − 1);
P_{1}(u) = (u − q^{−2})(u − q^{−4}),
P_{2}(u) = (u − q^{−7})(u − q^{−9}),
P_{3}(u) = P_{4}(u) = 1,
P_{5}(u) = u − q^{−8},
P_{k}(u) = 1, (6 ⩽ k ⩽ N − 1);

P_{1}(u) = (u − q^{−2})(u − q^{−4}),
P_{2}(u) = u − q^{−7},
P_{3}(u) = u − q^{−10},
P_{4}(u) = u − q^{−7},
P_{k}(u) = 1, (5 ⩽ k ⩽ N − 1).

The modules V (n_{1}) and V (n_{3}) are non-zero only if N ⩾ 6. Hence V (m_{1}) ⊗ V (m_{2}) has only two composition factors V (n_{2}) and V (n_{4}) for N = 5, and four composition factors V (n_{1}), V (n_{2}), V (n_{3}), V (n_{4}) for N ⩾ 6.

4.4 We note that our result implies the following Theorem 8 All composition factors of the tensor product of two evaluation modules of U_{q}(𝔰̂𝔩_{N}) occur with multiplicity one.

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B. Leclerc :  Laboratoire de Mathématiques Nicolas Oresme,

Université de Caen, Campus II, Bld Maréchal Juin, BP 5186, 14032 Caen cedex, France

email : leclerc@math.unicaen.fr