Decomposition of the Canonical Representation of W (1) × End[m] on Λ(m)

Dedicated to Professor Takeshi Hirai on his 60th birthday

Haiquan Wang

Introduction. There are several ways to obtain finite dimensional irreducible representations of the general linear group GL(n, ℂ), or equivalently of its Lie algebra 𝔤𝔩(n, ℂ). Among them the method introduced by Schur and Weyl is classical. They considered the m-fold tensor product V^{⊗m} of the defining representation of GL(V) (V = ℂ^{n}), and the action of the permutation group 𝔖_{m} on it. These actions are commutative each other, and V^{⊗m} decomposes multiplicity freely as a GL(V) × 𝔖_{m}-module:

V^{⊗m} = ∑_{D} ρ_{D} ⊗ σ_{D},

where D runs over Young diagrams of size m with depth at most n, and ρ_{D} (resp.σ_{D}) is an irreducible representation of GL(n) (resp.𝔖_{m}). Furthermore, through the decomposition, ρ_{D} determines σ_{D} and vice versa (e.g., see [1]). If m varies in the set of non-negative integers, each irreducible representation of GL(n, ℂ) appears in this decomposition up to multiplication by a suitable power of the determinant character.

The Cartan-type Lie algebras are ℤ-graded, simple, infinite-dimensional Lie algebras, whose properties and representations have been discussed extensively. I.A.Kostrikin ([6]) proved that all the finite type graded representations are either representations of height 1 (in the sense of A.N.Rudakov [11]) or their conjugate except the algebra W (1). In the case of W (1), Kostrikin gave models of all the irreducible graded modules of finite type with onedimensional homogeneous components. On the other hand, K.Nishiyama ([8]) considered Schur-Weyl duality for the natural representation of Cartan-type Lie algebra W (n). In particular, he suggested to use End[m] instead of 𝔖_{m}, which is a semigroup consisted of all the mappings from a finite set [m] = {1, 2, ⋯, m} into itself.

For Cartan-type Lie superalgebras, which is a “superanalogue” of Cartan-type Lie algebras, we also want to have an analogue of the Schur-Weyl duality. As a first step, we consider one of Cartan-type Lie superalgebras: Lie superalgebra of all the superderivations on the Grassmann algebra Λ(n), which is denoted by W (n) (see, e.g.,[4]). In [9] and [10], using the semigroup End[m], the author and Nishiyama have determined the commutant algebra of W (n) in the m-fold tensor product of the natural representation under the condition m ≤ n.

Let us explain it more precisely. Let ψ be the natural representation of W (n) on Λ(n) and φ be a representation of End[m] on ⊗^{m}Λ(n) (the definition is given in Section 1). If m ≤ n, then the commutant algebra of ψ^{⊗m}(W (n)) in End(⊗^{m}Λ(n)) is the algebra generated by φ(End[m]) in End(⊗^{m}Λ(n))([10]). Moreover, if n = 1 or (n, m) = (2, 3), the same conclusion also holds (see [9]).

Along the idea of Schur and Weyl, we want to decompose the space ⊗^{m}Λ(n) as W (n) × End[m]-module. However, in our case, the representation of W (n) on ⊗^{m}Λ(n) is not semisimple, and it seems difficult to decompose it. Therefore we are forced to consider the quotient representations. Nishiyama suggests the following conjectures:

Conjecture 1 Let ρ ⊗ σ be a finite dimensional irreducible representation of W (n) × End[m]. Then we have

dim Hom_{W (n)×End[m]}(⊗^{m}Λ(n), ρ ⊗ σ) ≤ 1.

Conjecture 2 Put ℜ_{W (n)}(U) := {ρ ∈ W (n)^{∧} | Hom_{W (n)}(U, ρ) ≠ (0)} and ℜ_{End[m]}(U) := {σ ∈ End[m]^{∧} | Hom_{End[m]}(U, σ) ≠ (0)}. Then for any ρ ∈ ℜ_{W (n)}(⊗^{m}Λ(n)), there is one and only one σ ∈ ℜ_{End[m]}(⊗^{m}Λ(n)) such that

dim Hom_{W (n)×End[m]}(⊗^{m}Λ(n), ρ ⊗ σ) ≠ 0.

Therefore, the following mapping is a bijection:

W (n)^{∧} ⊇ ℜ_{W (n)}(⊗^{m}Λ(n)) ∋ ρ ⟷ σ ∈ ℜ_{End[m]}(⊗^{m}Λ(n)) ⊆ End[m]^{∧},

where W (n)^{∧} (resp. End[m]^{∧}) is the set of all the irreducible modules of W (n) (resp. End[m]).

For general integer n, we could not prove the above conjectures, but in the simplest case n = 1, we have affirmative results (Theorems 3.1, 3.2 and 3.3). These are main results of this article.

Let us describe the contents of each section briefly. In the first section, we give the basic notations and preliminary results. In the second section, we give the decomposition of ⊗^{m}Λ(1) as W (1) × 𝔖_{m}-module (Proposition 2.3). In the third section, we give the decomposition of ⊗^{m}Λ(1) as W (1) × 𝔖_{m}-module (Theorem 3.1); and the decomposition of ⊗^{m}Λ(1) as 𝔤𝔩(1) × End[m]-module (Theorem 3.2). In Theorem 3.3 we give affirmative answers to Conjectures 1 and 2 in the case of n = 1. In the last section, we give models of some indecomposable modules of W (1).

Acknowledgment The author expresses sincere thanks to Professors T.Hirai and K.Nishiyama. They introduced him into this field, and in the past three years, gave many instructions and encouragements. For this article, they have given many suggestions.

1. Notations and Preliminaries

1.1. Symmetric group 𝔖_{m}.

In this article, we denote by D a Young diagram, by (λ_{1}, λ_{2}, ⋯ λ_{k}) its associated partition and by B(D) or by B one of Young tableaux of shape D. We denote by 𝔖_{m} the symmetric group of degree m. For a Young tableau B of size m, we put

P_{B} := {g ∈ 𝔖_{m} : g preserves each row of B},
Q_{B} := {g ∈ 𝔖_{m} : g preserves each column of B},

a_{B} := ∑_{g∈P_{B}} g ; b_{B} := ∑_{g∈Q_{B}} sgn(g)g.

The Young symmetrizer c_{B} of B is defined by c_{B} := a_{B}b_{B}. Denote the group ring of 𝔖_{m} by R := ℂ𝔖_{m}. It is well-known that R_{B} := R · c_{B} is an irreducible 𝔖_{m}-module. Further, R_{B} ≅ R_{B′} as 𝔖_{m}-modules if and only if Young diagrams of B and B′ coincide. We denote this 𝔖_{m}-module by σ_{D}.

A Young tableau B is called standard if the numbers in each row of B are increasing from left to right and the ones in each column are increasing from top to bottom. For example, for Young diagram D = (3, 2), we list all the standard tableaux of shape D below.

1 2 3 4 5 1 2 4 3 5 1 2 5 3 4 1 3 4 2 5 1 3 5 2 4

Denote the set of all Young tableaux of shape D by B(D) and the set of all standard Young tableaux by B^{S}(D).

Lemma 1.1. (1) The number of standard tableaux of shape D is equal to the dimension of the irreducible representation σ_{D} of 𝔖_{m} corresponding to D.

(2) If B and B′ are two different standard tableaux of shape D, then a_{B}b_{B′} = b_{B′}a_{B} = c_{B}c_{B′} = 0.

(3) Let B^{S}(D) = {B_{1}, B_{2}, ⋯, B_{r}} be the set of all the standard tableaux of shape D, and put 𝔞(D) := ∑_{B∈B(D)} Rc_{B}; then 𝔞(D) = Rc_{B_{1}} ⨁ ⋯ ⨁ Rc_{B_{r}}.

For proof of the above lemma, see [2],[3],⋯, for example.

1.2. Cartan-type Lie superalgebra W (n) and its natural representation.

Let Λ(n) be a Grassmann algebra over ℂ in n variables {ξ_{1}, ξ_{2}, ⋯, ξ_{n}}, and Λ^{k} be the subspace of k-homogeneous elements of Λ(n). Let W (n) be the set of all the superderivations of Λ(n). It becomes naturally a Lie superalgebra. Every superderivation D ∈ W (n) can be written in the form D = ∑_{i=1}^{n} P_{i}∂/∂ξ_{i} with P_{i} ∈ Λ(n) (1 ≤ i ≤ n), where ∂/∂ξ_{i} is a superderivation of degree 1 defined by (∂/∂ξ_{i})ξ_{j} = δ_{ij}. By definition, the Lie superalgebra W (n) acts on Grassmann algebra Λ(n) as follows: for any homogeneous D ∈ W (n) and ∀ξ_{i_{1}} ∧ ⋯ ∧ ξ_{i_{r}},

D(ξ_{i_{1}} ∧ ⋯ ∧ ξ_{i_{r}}) = ∑_{s=1}^{r}(−1)^{(s−1)degD}ξ_{i_{1}} ∧ ⋯ ∧ D(ξ_{i_{s}}) ∧ ⋯ ∧ ξ_{i_{r}}.

We call it the natural representation of W (n), and denote it by (ψ, Λ(n)) or simply by ψ.

Let us consider the m-fold tensor product ⊗^{m}Λ(n) of (ψ, Λ(n)). We have a natural isomorphism as W (n)-modules:

⊗^{m}Λ(n) ≃ Λ[ξ_{ij}|1 ≤ i ≤ n, 1 ≤ j ≤ m] =: Λ(n, m),

where Λ[ξ_{ij}|1 ≤ i ≤ n, 1 ≤ j ≤ m] is a Grassmann algebra generated by {ξ_{ij} (1 ≤ i ≤ n, 1 ≤ j ≤ m)}. In the following, we identify ⊗^{m}Λ(n) with Λ(n, m). By means of tensor product, W (n) is imbedded into End(⊗^{m}Λ(n)) ≅ End(Λ(n, m)). More precisely, an element

D = ∑_{i=1}^{n} P_{i}(ξ_{1}, ⋯, ξ_{n})\frac{∂}{∂ξ_{i}} ∈ W (n)

corresponds to a superderivation

ψ^{⊗m}(D) = ∑_{i=1}^{n} ∑_{α=1}^{m} P_{i}(ξ_{1α}, ⋯, ξ_{nα})\frac{∂}{∂ξ_{iα}} ∈ DerΛ(n, m)

via the m-fold tensor product ψ^{⊗m} of ψ.

In particular, in the case of n = 1, we have

W (1) = ⟨\frac{∂}{∂ξ} , ξ \frac{∂}{∂ξ} ⟩, deg(\frac{∂}{∂ξ} ) = 1, deg(ξ \frac{∂}{∂ξ} ) = 0.

For convenience, we use the isomorphism Λ(1, m) ≅ Λ[ξ_{1}, ξ_{2}, ⋯, ξ_{m}]. Then we have

D_{−1} := ψ^{⊗m}(\frac{∂}{∂ξ} ) = ∑_{i=1}^{m} \frac{∂}{∂ξ_{i}} , D_{0} := ψ^{⊗m}(ξ \frac{∂}{∂ξ} ) = ∑_{i=1}^{m} ξ_{i} \frac{∂}{∂ξ_{i}} .

Obviously, D_{−1}(Λ^{k}) ⊆ Λ^{k−1}, D_{0}(Λ^{k}) ⊆ Λ^{k} for any k.

Denote by [m] the set {1, 2, ⋯, m} of integers, and put End[m] = {φ : [m] → [m]} the set of all the maps from [m] into itself. By composition of maps, End[m] becomes a semigroup with unit, whose group elements form the permutation group 𝔖_{m} of degree m. We call End[m] the permutation semigroup. Denote its semigroup ring by 𝔈_{m}. An element φ ∈ End[m] acts on Λ(n, m) as (φP)(ξ_{i,j}) = P (ξ_{i,φ(j)})(P ∈ Λ(n, m)) and we extend it to 𝔈_{m} by linearity (see [10]). Thus, we have a representation of 𝔈_{m} on Λ(n, m). Denote the image algebra of this representation by E_{m} ⊂ EndΛ(n, m).

Let C_{m} denote the commutant algebra of ψ^{⊗m}(W (n)) in End Λ(n, m). We have the following fundamental results.

Theorem 1.2. ([9],[10]) Assume m ≤ n, or n = 1, or (n, m) = (2, 3). The commutant algebra C_{m} of ψ^{⊗m}(W (1)) coincides with the representation image E_{m} of the semigroup ring 𝔈_{m} of the permutation semigroup End[m]:

E_{m} = C_{m}.

Theorem 1.3. ([10]) The bicommutant algebra of the m-fold tensor product ψ^{⊗m} of the natural representation ψ of W (1) is equal to the image ψ^{⊗m}(U(W (1))) of the universal enveloping algebra U(W (1)).

1.3. Composition series.

Let R be an arbitrary ring with unit and V an R-module. A descending chain

V = V_{1} ⊃ V_{2} ⊃ ⋯ ⊃ V_{k} ⊃ V_{k+1} = (0)

of submodules of V is called a composition series of V if all the quotient modules V_{i}/V_{i+1} are irreducible. These quotient modules V_{i}/V_{i+1} are called the composition factors. Two composition series are said to be equivalent if they have the same length and if the factors can be paired off in such a way that the corresponding factors are R-isomorphic.

Lemma 1.4. (Jordan-Hölder) If an R-module V possesses a composition series, any two composition series of V are equivalent.

An R-module V ≠ {0} is said to be indecomposable if it is impossible to express V as a direct sum of two non-trivial submodules.

2. The Decomposition of Λ(m) as W (1) × 𝔖_{m}-module

In this section, we give the decomposition of Λ(m) as W (1) × 𝔖_{m}-module explicitly. The decomposition of Λ(m) according to the action of 𝔖_{m} is well-known (e.g., see [5, p.55]). However, we give the explicit structure of the decomposition here because it is used in the subsequent sections. Let us begin with the determination of the subspace c_{B}Λ(m).

Lemma 2.1. Let λ = (λ_{1}, λ_{2}, ⋯, λ_{k}) be a partition of m corresponding to a Young diagram D.

(1) If D is not of hook type, then we have c_{B}Λ(m) = 0 for B ∈ B(D).

(2) Let D be a hook type diagram and λ = (λ_{1}, 1, ⋯, 1). We have c_{B}Λ^{k}(m) ≠ 0 (B ∈ B(D)) if and only if k = μ_{1} or k = μ_{1} − 1, where μ_{1} + λ_{1} = m + 1.

(3) Under the same assumption as in (2), we have

c_{B}Λ(m) = c_{B}Λ^{μ_{1}}(m) ⊕ c_{B}Λ^{μ_{1}−1}(m) (B ∈ B(D)).

The both subspaces are of one-dimension. Moreover, we have an explicit description of each subspace as follows:

c_{B}Λ^{μ_{1}}(m) = ℂa_{B}(ξ_{b(1,1)} ∧ ⋯ ∧ ξ_{b(μ_{1},1)}), c_{B}Λ^{μ_{1}−1} = ℂa_{B}(D_{−1}(ξ_{b(1,1)} ∧ ⋯ ∧ ξ_{b(μ_{1},1)})),

where b(i, j) denotes the number in the (i, j)-box in the Young tableau B ∈ B(D).

Proof. Take B ∈ B(D) and put

a_{i} := ∑_{g∈P_{B,i}} g, b_{j} := ∑_{g∈Q_{B,j}} sgn(g)g,

where

P_{B,i} := {g | g ∈ 𝔖_{m}, g preserves the i-th row and fixes the other letters},
Q_{B,j} := {g | g ∈ 𝔖_{m}, g preserves the j-th column and fixes the other letters}.

Then we can write down a_{B} and b_{B} as follows:

a_{B} = a_{k}a_{k−1} ⋯ a_{2}a_{1}, b_{B} = b_{s}b_{s−1} ⋯ b_{2}b_{1},

where s is the number of the column of the diagram D.

Note that, for any φ ∈ Q_{B} and Ξ ∈ Λ(m), we have c_{B}φ(Ξ) = sgn(φ)c_{B}(Ξ). Therefore, in order to calculate c_{B}Λ(m), we only need to consider c_{B}(Ξ) for the following type of Ξ with appropriate integers h_{1}, h_{2}, ⋯, h_{s} after translation by a certain φ ∈ Q_{B}:

Ξ = (ξ_{b(1,1)} ∧ ⋯ ∧ ξ_{b(h_{1},1)}) ∧ (ξ_{b(1,2)} ∧ ⋯ ∧ ξ_{b(h_{2},2)}) ∧ · · · .

Let us consider (1). Since D is not of hook type, λ_{2} ≥ 2. Let μ = (μ_{1}, μ_{2}, ⋯, μ_{s}) be the transposition of λ (see e.g.,[7], §1). Note that μ_{2} ≥ 2. If h_{i} ≤ μ_{i} −2 for some i, then it is easy to see that b_{i}(Ξ) = 0, hence c_{B}(Ξ) = 0. So we assume that μ_{i} ≥ h_{i} > μ_{i} − 2. In particular, we assume that h_{2} ≥ μ_{2} − 2 ≥ 1. We consider the expansion of b_{B}(Ξ). For any term of this expansion (e.g., ξ_{i_{u}} ∧ ⋯ ∧ ξ_{i_{v}}), if there are two subscripts (e.g., i_{u}, i_{v}) from the same row, then the result of the operator a_{B} acting on this term is zero. Hence we only consider the terms which do not have two subscripts from the same row. Therefore we can assume that h_{3} = h_{4} = ⋯ = 0, hence μ_{3} ≤ 1.

If h_{2} ≥ 2, then h_{1} + h_{2} ≥ μ_{1} − 1 + 2 = μ_{1} + 1. This means that, in each term, there are at least two subscripts from the same row, and so c_{B}(Ξ) = 0. Therefore we can assume μ_{2} − 1 ≤ h_{2} < 2, which means μ_{2} < 3. Combining with the assumption λ_{2} ≥ 2, we need only to consider the situation μ_{2} = 2 and h_{2} = 1.

(a) If h_{1} = μ_{1}, then in each term in the expansion of b_{B}(ξ_{b(1,1)} ∧⋯∧ξ_{b(μ_{1},1)} ∧ξ_{b(1,2)}), there are at least two subscripts from the same row, hence c_{B}(Ξ) = 0.

(b) If h_{1} = μ_{1}−1, then b_{1}(ξ_{b(1,1)}∧⋯∧ξ_{b(μ_{1}−1,1)}∧ξ_{b(1,2)}) = D_{−1}(ξ_{b(1,1)}∧⋯∧ξ_{b(μ_{1},1)}∧ξ_{b(1,2)}), hence c_{B}(Ξ) = a_{B}b_{B}(Ξ) = D_{−1}a_{B}b_{2}(ξ_{b(1,1)} ∧ ⋯ ∧ ξ_{b(μ_{1},1)} ∧ ξ_{b(1,2)}). By the same reason as in (a), we have c_{B}(Ξ) = 0.

For (2) and (3), we can do the similar calculation and easily get the desired results. Q.E.D.

Next we consider the 𝔖_{m}-module structure.

Lemma 2.2. Let λ = (λ_{1}, 1, ⋯, 1) be a partition of m and D the corresponding Young diagram of hook-type. The subspace

∑_{r=1}^{s} c_{B_{r}}(Λ^{μ_{1}}(m)) (λ_{1} + μ_{1} = m + 1)

is irreducible under 𝔖_{m}, where B^{S}(D) = {B_{1}, B_{2}, ⋯, B_{s}} are the set of all the standard Young tableaux of shape D.

Proof. We use the notations in Lemma 2.1. By Lemma 2.1, we have

c_{B}Λ^{μ_{1}}(m) = ℂa_{B}(ξ_{b(1,1)} ∧ ⋯ ∧ ξ_{b(h_{1},1)}).

Since c_{B_{i}}c_{B_{j}} = const · δ_{ij}c_{B_{i}} (const ≠ 0), {c_{B_{i}}(ξ_{b_{i}(1,1)} ∧ ⋯ ∧ ξ_{b_{i}(μ_{1},1)}) | i = 1, 2, ⋯, s} are linearly independent. Put

H_{D} := ⟨c_{B_{i}}(ξ_{b_{i}(1,1)} ∧ ⋯ ∧ ξ_{b_{i}(μ_{1},1)}) | i = 1, 2, ⋯, s⟩.

Then {c_{B_{i}}(ξ_{b_{i}(1,1)} ∧ ⋯ ∧ ξ_{b_{i}(μ_{1},1)}) | i = 1, 2, ⋯, s} is a basis of H_{D}.

From ∑_{i=1}^{s} Rc_{B_{i}} = ∑_{i=1}^{s} c_{B_{i}}R, we know that H_{D} is an invariant space of 𝔖_{m}. We prove the space H_{D} is irreducible under 𝔖_{m}. Take an element x = ∑_{i} k_{i}c_{B_{i}}(ξ_{b_{i}(1,1)} ∧⋯∧ξ_{b_{i}(μ_{1},1)}) ∈ H_{D}. If k_{h} ≠ 0, then

c_{B_{h}}(x) = c_{B_{h}}(∑_{i} k_{i}c_{B_{i}}(ξ_{b_{i}(1,1)} ∧ ⋯ ∧ ξ_{b_{i}(μ_{1},1)})) = k_{h}c_{B_{h}}(ξ_{b_{h}(1,1)} ∧ ⋯ ∧ ξ_{b_{h}(μ_{1},1)}) ≠ 0.

Clearly c_{B_{h}}(ξ_{b_{h}(1,1)} ∧ ⋯ ∧ ξ_{b_{h}(μ_{1},1)}) generates H_{D} as an 𝔖_{m}-module, and we have done. Q.E.D.

Using the above lemma, we can decompose Λ(m) as a W (1) × 𝔖_{m}-module.

Proposition 2.3. Let W (1) and 𝔖_{m} act on Λ(m) naturally. We have the following decomposition of Λ(m):

Λ(m) = ⨁_{D} ⨁_{B∈B^{S}(D)} c_{B}Λ(m),

where D runs over the Young diagrams of hook type corresponding to the partition (k, 1, ⋯, 1) = (k, 1^{m−k}) (k = 1, 2, ⋯ m) and B^{S}(D) is the set of all the standard tableaux of shape D.

For any Young tableau B, c_{B}Λ(m) is an indecomposable module of W (1) with composition series

c_{B}Λ(m) ⊇ c_{B}Λ^{m−k}(m) ⊇ {0}; dim c_{B}Λ(m) = 2, dim c_{B}Λ^{m−k}(m) = 1.

For any D, the sum ∑_{B∈B^{S}(D)}^{⊕} c_{B}Λ^{m−k+1}(m) is an irreducible module of 𝔖_{m} of type D. Furthermore, D_{−1}(∑_{B∈B^{S}(D)}^{⊕} c_{B}Λ^{m−k+1}(m)) = ∑_{B∈B^{S}(D)}^{⊕} c_{B}Λ^{m−k}(m) is also an irreducible module of the same type D.

Proof. Let D be a Young diagram of hook type with partition (k, 1, ⋯, 1). By the above lemmas, for any B ∈ B(D), we have

c_{B}Λ(m) = c_{B}Λ^{m−k}(m) ⊕ c_{B}Λ^{m−k+1}(m).

Let V_{1}(B) := c_{B}Λ(m) and V_{2}(B) := c_{B}Λ^{m−k}(m). By a simple calculation we have D_{−1}(V_{1}) = V_{2}, and D_{−1}(V_{2}) = 0 because D_{−1}^{2} = 0. Clearly, D_{0}(V_{i}) ⊆ V_{i} (i = 1, 2), so V_{1} and V_{2} are invariant spaces of W (1). From the proof of Lemma 2.2, we can deduce that dim (V_{2}(B)) = 1, therefore V_{2} is an irreducible module of W (1). On the other hand, dim kerD_{−1} |_{V_{1}(B)}=dim V_{2}(B) = 1, we have V_{1}(B) is an indecomposable module of W (1). Again from the proof of Lemma 2.2, dim (V_{1}(B)/V_{2}(B)) = 1, hence {0} ⊆ V_{2}(B) ⊆ V_{1}(B) is a composition series of W (1).

The rest of the proof is clear from Lemmas 2.1 and 2.2. Q.E.D.

3. The Duality between W (1) and End[m]

Along the idea of Schur and Weyl, we want to obtain the decomposition of ⊗^{m}Λ(n) as a W (n) × End[m]-module. Because the representation of W (n) on ⊗^{m}Λ(n) is not semisimple, it seems difficult to do it. However, for the case n = 1, thanks to Proposition 2.3, we can obtain the decompositions of ⊗^{m}Λ(1) as a W (1) × 𝔖_{m}-module, and as a W_{0}(1) × End[m] = 𝔤𝔩(1) × End[m]-module. Using these two decompositions we can get the further result which is similar to the classical duality between GL(n, ℂ) and 𝔖_{m}. To be more precise, for any ρ ⊗ σ ∈ W (1)^{∧} ⊗ End[m]^{∧}, where W (1)^{∧} (resp. End[m]^{∧}) is the set of equivalence classes of irreducible finite dimensional representations of W (1) (resp. End[m]), we have

dim Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ) ≤ 1.

If the equality holds,

dim Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ) = 1,

then ρ ∈ W (1)^{∧} uniquely determines σ ∈ End[m]^{∧} satisfying the above equality. Based on these explicit calculations and results, Conjectures 1 and 2 are suggested by Nishiyama for general n and m.

Let us begin to state the main results of this paper.

Let D be a Young diagram corresponding to the partition (λ_{1}, λ_{2}, ⋯, λ_{m}) and B be one of the Young tableaux of shape D. Put V_{1}(B) := c_{B}Λ(m) and V_{2}(B) = c_{B}Λ^{k−1}(m) (λ_{1} + k = m + 1). By the results in Section 2, V_{1}(B) = V_{2}(B) = 0 if D is not of hook type. If D is one of the hook type diagrams corresponding to the partition (λ_{1}, 1, ⋯, 1), then V_{1}(B) = c_{B}Λ^{k}(m) ⊕ c_{B}Λ^{k−1}(m) (B ∈ B^{S}(D)) and dim c_{B}Λ^{k}(m) = dim c_{B}Λ^{k−1}(m) = 1. Using these facts, we have

Theorem 3.1. (1) As W (1)-modules, we have isomorphisms:

V_{1}(B) ≅ Hom_{𝔖_{m}}(σ_{D}, Λ(m)), V_{2}(B) ≅ Hom_{𝔖_{m}}(σ_{D}, Λ^{k−1}(m)),

where B is a Young tableaux of shape D with partition (λ_{1}, λ_{2}, ⋯, λ_{m}) and σ_{D} is the irreducible representation of 𝔖_{m} with diagram D.

(2) Let ρ̃_{D_{m,k}} = Hom_{𝔖_{m}}(σ_{D_{m,k}}, Λ(m)). As a W (1)-module, ρ̃_{D_{m,k}} is an indecomposable module and has a composition series:

ρ̃_{D_{m,k}} ≅ V_{1}(B) ⊇ V_{2}(B) ⊃ {0},

where D_{m,k} denotes the hook type Young diagram with partition (m − k + 1, 1, ⋯, 1) and B ∈ B(D_{m,k}). Furthermore, the composition factors are 1-dimensional modules.

(3) The W (1) × 𝔖_{m}-module Λ(m) has the following decomposition:

Λ(m) ≅ ⨁_{k=1}^{m} ρ̃_{D_{m,k}} ⊗ σ_{D_{m,k}}.

Proof. We want to prove (1). As a realization of σ_{D}, we take Rc_{B} (B ∈ B(D)). If D is not of hook type, then the both hands side are zero. Therefore we can assume that D = D_{m,k}. We define W (1)-equivariant map G : Hom_{𝔖_{m}}(Rc_{B}, Λ(m)) ⟶ V_{1}(B) by G(f) = f(c_{B}) (∀f ∈ Hom_{𝔖_{m}}(Rc_{B}, Λ(m)). It is clear that G is an injection. Take a non-zero element x_{1} ∈ c_{B}Λ(m) and define f_{1} ∈ Hom_{𝔖_{m}}(Rc_{B}, Λ(m)) by f_{1}(φc_{B}) = φ · x_{1} (∀φ ∈ R). By definition, we have G(f_{1}) = x_{1}, hence G is surjective. This proves the first isomorphism in (1). For the second isomorphism in (1), it is enough to note the following facts:

D_{−1}(Hom_{𝔖_{m}}(σ_{D}, Λ^{k−1}(m))) = 0, D_{−1}(V_{2}(B)) = 0.

Combining Proposition 2.3 and (1), we can easily complete the proof of (2) and (3). Q.E.D.

The commutant algebra of W (1) in the representation Λ(m) = ⊗^{m}Λ(1) is E_{m}, which is the representation image of the semigroup ring of End[m]. Following the idea of the classical situation, we want to get the decomposition of Λ(m) as a module of W (1) × E_{m}. As an intermediate step, we describe the decomposition of Λ(m) as W_{0}(1) × End[m] ≅ 𝔤𝔩(1)×End[m]-module.

Put U_{1}(k) := Λ^{k}(m), and U_{2}(k) := ker(D_{−1} |_{Λ^{k}(m)}) (0 ≤ k ≤ m). Then U_{1}(k) is an indecomposable End[m]-module which we denote by σ̃_{m,k}. We denote the irreducible representation of 𝔤𝔩(1) with weight k by ρ_{k}.

Theorem 3.2. We use the same notations as in Theorem 3.1. As a 𝔤𝔩(1) × End[m]- module, we have the decomposition:

Λ(m) = ⨁_{k=0}^{m} ρ_{k} ⊗ σ̃_{m,k}.

If 1 ≤ k ≤ m − 1, σ̃_{m,k} has a composition series U_{1}(k) ⊇ U_{2}(k) ⊃ {0}. For k = 0, m, σ̃_{m,k} is irreducible.

Proof. Assume that 1 ≤ k ≤ m. Because D_{−1} and End[m] commutes with each other, U_{2}(k) = ker(D_{−1} |_{Λ^{k}(m)}) is an invariant subspace of End[m]. By the relation ker(D_{−1}|_{Λ^{k}}) = Im(D_{−1}|_{Λ^{k+1}}) and Proposition 2.3, U_{2}(k) is irreducible under 𝔖_{m}, hence it is irreducible under End[m]. The only thing to be proved is that U_{1}(k)/U_{2}(k) is irreducible under End[m]. By the definition of U_{1}(k) and U_{2}(k), U_{1}(k)/U_{2}(k) is linearly isomorphic to U_{2}(k − 1) by operator D_{−1} (see [10], Lemma 3.1), furthermore, it is an intertwining operator between End[m]- module U_{1}(k)/U_{2}(k) and End[m]-module U_{2}(k−1). Using the irreducibility of End[m]-module U_{2}(k − 1), we complete the proof of the theorem. Q.E.D.

We denote the irreducible representation of End[m] on Λ^{k}(m)/U_{2}(k) by σ_{D_{m,k}} (k ≥ 1) and the irreducible representation of W (1) on V_{1}(B(D_{m,k}))/V_{2}(B(D_{m,k})) by ρ_{D_{m,k}} (k ≥ 1). Then we have the following duality.

Theorem 3.3. (1) For any ρ ⊗ σ ∈ W (1)^{∧} × End[m]^{∧}, we have

dim Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ) ≤ 1.

The equality holds if and only if ρ ⊗ σ = ρ_{D_{m,k}} ⊗ σ_{D_{m,k}} for some 1 ≤ k ≤ m.

(2) Put

ℜ_{W (1)}(Λ(m)) := {ρ ∈ W (1)^{∧} | Hom_{W (1)}(Λ(m), ρ) ≠ (0)},
ℜ_{End[m]}(Λ(m)) := {σ ∈ End[m]^{∧} | Hom_{End[m]}(Λ(m), σ) ≠ (0)}.

For any ρ ∈ ℜ_{W (1)}(Λ(m)), there is one and only one σ ∈ ℜ_{End[m]}(Λ(m)) such that

dim Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ) = 1.

Therefore, the following mapping is a bijection:

W (1)^{∧} ⊇ ℜ_{W (1)}(Λ(m)) ∋ ρ ⟷ σ ∈ ℜ_{End[m]}(Λ(m)) ⊆ End[m]^{∧}.

Furthermore, this correspondence is explicitly realized by ρ_{D_{m,k}} ⟷ σ_{D_{m,k}}.

Proof. We keep to the notation in §2 and §3. From Proposition 2.3, we have Λ(m) = ∑_{k=0}^{m}(H_{D_{m,k}} ⊕ U_{2}(k)) (for the definition of H_{D_{m,k}} (k ≥ 1), see the proof of Lemma 2.2 and we set H_{D_{m,0}} = {0}). Note that D_{−1}(H_{D_{m,k+1}}) = U_{2}(k).

Assume that dim Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ) ≠ 0. We take a non-zero element f ∈ Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ). As W_{0}(1) ≅ 𝔤𝔩(1)-module, Λ(m) decomposed into the direct sum of one-dimensional irreducible representations of weight k (0 ≤ k ≤ m). Note that ρ ∈ W (1)^{∧} is one-dimensional and completely determined by the weight of W_{0}(1). Since Λ(m)/kerf ≅ ρ ⊗ σ ≅ (dimσ)ρ as a W (1)-module, ρ must be an irreducible module of W (1) with highest weight k (0 ≤ k ≤ m). As a consequence, there exists a unique k such that f(Λ^{k}(m)) ≠ (0). We put k = k(ρ).

Assume that 1 ≤ k(ρ) ≤ m − 1. Since f is W (1) × End[m]-equivariant, kerf is invariant under the action of W (1) × End[m]. Hence we have

U_{2}(k(ρ)) = D_{−1}(H_{D_{m,k(ρ)+1}}) ⊂ D_{−1}(Λ^{k(ρ)+1}) ⊂ kerf.

On the other hand, by Lemma 2.2, H_{D_{m,k(ρ)}} is an irreducible module of 𝔖_{m}, therefore, kerf ∩ H_{D_{m,k(ρ)}} = {0}. Summarizing above, we know that for any f ∈ Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ),

kerf = ∑_{k≠k(ρ)} Λ^{k}(m) ⊕ D_{−1}(H_{D_{m,k(ρ)+1}}),

hence,

dim Hom_{W (1)×End[m]}(Λ(m), ρ ⊗ σ) = 1.

If k(ρ) = 0, then by the same argument as above, we can conclude Λ^{0} = U_{2}(0) ⊂ kerf. This means f = 0 and we get a contradiction.

Finally, assume k(ρ) = m. Since ∑_{i=0}^{m−1} Λ^{i}(m) is invariant under W (1) × End[m], Λ^{m} ≅ Λ(m)/ ∑_{i=0}^{m−1} Λ^{i}(m) is irreducible under W (1) × End[m]. Therefore f is uniquely determined up to constant multiplication.

Note that Λ(m)/kerf ≅ U_{2}(k(ρ) − 1) as End[m]-module. This proves that ρ ⊗ σ ≅ ρ_{D_{m,k}} ⊗ σ_{D_{m,k}}. (2) is obvious from the above arguments. Q.E.D.

We call the correspondence in (2) Howe’s correspondence for W (1) × End[m].

Appendix: Indecomposable Representations of W (1)

In this appendix, for the sake of completeness, we prove that the representations of W (1) on Λ(m) give a model of one principal class of representations of W (1). Let us begin with a general representation theory of W (1).

Take a finite-dimensional indecomposable graded representation ρ of W (1) on a representation space V = V¯0 ⊕ V¯1. Put T_{λ} := λ − ρ(D_{0}) and V_{λ} := {v | T_{λ}^{k}v = 0, ∃k ∈ ℤ_{≥0}}, the generalized eigenvector space with eigenvalue λ. Then we have the generalized eigenspace decomposition of V:

V = ⨁_{λ} V_{λ}.

Obviously, ρ(D_{0})(V_{λ}) ⊆ V_{λ}. Let us consider the action of ρ(D_{−1}) on V_{λ}.

Since we have the relation [D_{0}, D_{−1}] = −D_{−1} and 2D_{−1}^{2} = [D_{−1}, D_{−1}] = 0, so T_{λ}ρ(D_{−1}) = ρ(D_{−1})T_{λ+1}. Using induction, we get ρ(D_{−1})T_{λ}^{k} = T_{λ−1}^{k}ρ(D_{−1}).

Let λ be any generalized eigenvalue of ρ(D_{0}). Clearly H_{λ} := ⊕_{k∈ℤ}V_{λ+k} is invariant under W (1). By the finite-dimensionality of V, we can assume that H_{λ} = V_{λ} ⊕ V_{λ−1} ⊕ ⋯ ⊕ V_{λ−μ} with some μ ∈ ℤ_{≥0}. Again by the finite-dimensionality of V and the generalized eigenspace decomposition of V, we have V = H_{λ_{1}} ⊕ H_{λ_{2}} ⊕ ⋯ H_{λ_{r}}, By the indecomposability of the representation, we see that r = 1 and conclude V = H_{λ}.

Now, we assume ρ(D_{0}) is semisimple. Then V_{λ−s} (0 ≤ s ≤ μ) is the eigenspace with eigenvalue λ − s. In this case, we get a W (1)-invariant decomposition V = (V_{λ} ⊕ D_{−1}(V_{λ})) ⊕ (Vλ−1^{′} ⊕ V_{λ−2} ⊕ ⋯ V_{λ−μ}) , where Vλ−1^{′} is any complement space of D_{−1}(V_{λ}) in V_{λ−1}. By the indecomposability of W (1), we conclude that V = V_{λ}⊕D_{−1}(V_{λ}). Using the similar argument, we can deduce that

V = ℂv ⊕ ℂD_{−1}v.

Summarizing the above results, we have the following proposition.

Proposition A. Let (ρ, V) be an indecomposable graded representation of W (1), then

V = V_{λ} ⊕ V_{λ−1} ⊕ ⋯ ⊕ V_{λ−μ},

where μ is an integer. If moreover ρ(D_{0}) acts semisimply on V, then V has the following decomposition:

V = ℂv ⊕ ℂρ(D_{−1})v,

where v is an eigenvector of ρ(D_{0}) with eigenvalue λ. If ρ(D_{−1})v = 0, then V = ℂv, and so the representation is irreducible. If ρ(D_{−1})v ≠ 0, then the representation is an indecomposable representation with composition series V ⊇ V_{1} ⊇ {0}, where V_{1} = ℂρ(D_{−1})v.

Combining the above proposition with results in §3, we conclude that any indecomposable representation (ρ, V) of W (1) for which ρ(D_{0}) is semisimple with positive integral eigenvalue is realized in the tensor product Λ(m) of the natural representation of W (1) for a certain m.

For an arbitrary integer μ in the proposition, there really exists an indecomposable W (1)- module (ρ, V) such that ρ(D_{0}) is not semisimple and V = V_{λ} ⊕ V_{λ−1} ⊕ ⋯ ⊕ V_{λ−μ}.

Example. Let V = ℂ^{2(μ+1)} and n(λ) = [ \begin{matrix}λ & 0 \\ 1 & λ \end{matrix} ]. Note that n(λ) is not semisimple. An

action of W (1) on V is given as follows:

ρ(D_{0}) = [ \begin{matrix}n(λ) & 0 & 0 & ⋯ & 0 \\ 0 & n(λ − 1) & 0 & ⋯ & 0 \\ 0 & 0 & n(λ − 2) & ⋯ & 0 \\ ⋯ & ⋯ & ⋯ & ⋯ & 0 \\ 0 & 0 & 0 & ⋯ & n(λ − μ) \end{matrix} ] ,

ρ(D_{−1}) = [ \begin{matrix}0 & 0 & 0 & ⋯ & 0 & 0 \\ n(0) & 0 & 0 & ⋯ & 0 & 0 \\ 0 & n(0) & 0 & ⋯ & 0 & 0 \\ ⋯ & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & 0 & 0 & ⋯ & n(0) & 0 \end{matrix} ] ,

where ρ(D_{0}) and ρ(D_{−1}) are (2μ + 2) × (2μ + 2) matrices. It is easy to check by calculation that (ρ, V) is an indecomposable module of W (1) and has the properties just stated as above.

Department of Mathematics Faculty of Science Kyoto University

References

[1] R.Howe, The classical groups and invariants of binary forms, Proc. Symp. Pure Math. 48(1988), 133-166.

[2] N.Iwahori, Representation theory of Symmetric Groups and General Linear Groups, (in Japanese), Iwanami Shoten, 1978.

[3] G.James and A.Kerbe The representation theory of the symmetric groups, Encyclopedia of Mathematics and Its Applications, vol. 16, Addison-Wesley, Reading, MA, 1981

[4] V.G.Kac, Lie superalgebras, Advances in Mathematics, 26(1977), 8 – 96.

[5] A.A.Kirillov(ed), Representation theory and noncommutative harmonic analysis.I, Encyclopaedia of Mathematical Sciences, Vol.22, Springer-Verlag, 1991.

[6] I.A.Kostrikin, Irreducible graded representations of Lie algebras of Cartan type, Soviet Math. Dokl., 19(1978), No.6, 1369 – 1371.

[7] I.G.Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.

[8] K.Nishiyama, Commutant algebra and harmonic polynomials of a Lie algebra of vector fields, J. Alg., 183 (1996), 545-559.

[9] K.Nishiyama and H.Wang, Commutant algebra of superderivations on a Grassmann algebra, Proc. Japan Acad. 72. Ser.A.(1996), 8 – 11.

[10] K.Nishiyama and H.Wang, Commutant algebra of Cartan-type Lie superalgebra W (n), J. Math. Kyoto Univ., 36-1 (1996), 129 – 142.

[11] A.N.Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type, Math. USSR, Izv. 8 (1974), No.4, 836 – 866.

[12] H.Weyl, The classical groups, Princeton Univ. Press, Princeton, NJ, 1946.