Restriction of the irreducible representations of GL_{n} to the symmetric group 𝔖_{n}.

Kyo Nishiyama

Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606-8501, JAPAN

               e-mail: kyo@math.h.kyoto-u.ac.jp

1 Problem

Let λ = (λ_{1}, ⋯, λ_{n}) be a dominant integral weight of GL_{n} and consider the finite dimensional irreducible representation ρ_{λ} with the highest weight λ. The permutation matrices in GL_{n} form a finite subgroup which is isomorphic to the symmetric group 𝔖_{n}. We identify 𝔖_{n} with this subgroup. Then our problem can be stated as follows.

Problem 1.1 Describe the decomposition of ρ_{λ}|_{𝔖_{n}} when restricted to the subgroup 𝔖_{n}.

We will reduce this problem to the decomposition of plethysms in principle.

2 Main result

Let us review GL_{n} ×GL_{m}-duality on the symmetric algebra S(ℂ^{n} ⊗ℂ^{m}). It is well-known that S(ℂ^{n} ⊗ ℂ^{m}) is multiplicity free as a representation of GL_{n} × GL_{m} and decomposes as follows (see, e.g. [Howe]):

S(ℂ^{n} ⊗ ℂ^{m}) ≃ ∑_{length(λ)≤min{m,n}} ρ_{λ}^{(n)} ⊗ ρ_{λ}^{(m)},

where λ = (λ_{1}, λ_{2}, ⋯) is a weight (or a partition) and ρ_{λ}^{(n)} is the irreducible representation of GL_{n} with highest weight λ. Put V = ℂ^{m} and assume m ≥ n. Then we can reformulate the left hand side of the above formula:

S(ℂ^{n} ⊗ V) ≃ ⊗^{n}S(V)
= ⊗^{n} (⨁_{k=0}^{∞} S_{k}(V))
= ⨁_{ν=(ν_{1},··· ,ν_{n})∈ℤ_{≥0}^{n}} S_{ν_{1}}(V) ⊗ ⋯ ⊗ S_{ν_{n}}(V),

where S_{k}(V) denotes a homogeneous component of degree k in S(V). For ν = (ν_{1}, ⋯, ν_{n}) ∈ (ℤ_{≥0})^{n}, put

𝔖_{ν} = 𝔖_{ν_{1}} × ⋯ × 𝔖_{ν_{n}} ⊂ 𝔖_{|ν|} (|ν| = ν_{1} + ⋯ + ν_{n}).

Then we have

S_{ν_{1}}(V) ⊗ ⋯ ⊗ S_{ν_{n}}(V) = (⊗^{|ν|}V )^{𝔖_{ν}} .  (2.1)

By the classical Schur duality (cf. [Weyl]), we know the decomposition

⊗^{|ν|}V ≃ ∑_{|λ|=|ν|,length(λ)≤n} ρ_{λ}^{(m)} ⊗ σ_{λ}

as a GL_{m} × 𝔖_{|ν|}-module. Therefore the above formula (2.1) becomes

(  ∑_{|λ|=|ν|,length(λ)≤n} ρ_{λ}^{(m)} ⊗ σ_{λ})^{𝔖_{ν}} = ∑_{|λ|=|ν|,length(λ)≤n} ρ_{λ}^{(m)} ⊗ (σ_{λ})^{𝔖_{ν}} .

We summarize as

⊗^{m}S(V) ≃ ∑_{ν∈ℤ_{≥0}^{n}} ∑_{|λ|=|ν|,length(λ)≤n} ρ_{λ}^{(m)} ⊗ (σ_{λ})^{𝔖_{ν}}

≃ ∑_{length(λ)≤n} ρ_{λ}^{(m)} ⊗ ( ∑_{ν∈ℤ_{≥0}^{n},|ν|=|λ|}(σ_{λ})^{𝔖_{ν}} ) .

Here, ∑ (σ_{λ})^{𝔖_{ν}} becomes an 𝔖_{n}-module, whose module structure is induced by the original action of GL_{n}. So we obtain

ρ_{λ}^{(n)}|_{𝔖_{n}} ≃ ∑_{ν∈ℤ_{≥0}^{n},|ν|=|λ|}(σ_{λ})^{𝔖_{ν}}

≃ ∑_{μ⊢|λ|,length(μ)≤n} ( ∑_{ν∈𝔖_{n}·μ}(σ_{λ})^{𝔖_{ν}}) .

Let V_{λ} be a representation space on which GL_{n} acts via ρ_{λ}^{(n)} and V_{λ}(μ) its weight space of weight μ. We put V_{λ}(𝔖_{n} · μ) = ∑_{ν∈𝔖_{n}·μ}V_{λ}(ν). Then, clearly V_{λ}(𝔖_{n} · μ) is invariant under

𝔖_{n} and we get the following lemma.

Lemma 2.1 As a representation of 𝔖_{n}, there is an isomorphism:

V_{λ}(𝔖_{n} · μ) ≃ ∑_{ν∈𝔖_{n}·μ}(σ_{λ})^{𝔖_{ν}} .

Take a partition μ ⊢ |λ|. Consider the normalizer 𝔑_{μ} = N_{𝔖_{|μ|}}(𝔖_{μ}) of 𝔖_{μ} in 𝔖_{|μ|};

𝔑_{μ} = {s ∈ 𝔖_{|μ|} | s𝔖_{μ}s^{−1} = 𝔖_{μ}}.

Then there exists a partition α(μ) = α = (α_{1}, ⋯, α_{k}) of length(μ) ≤ n such that

𝔑_{μ}/𝔖_{μ} ≃ 𝔖_{α} = 𝔖_{α_{1}} × ⋯ × 𝔖_{α_{k}}.

Since 𝔑_{μ} normalizes 𝔖_{μ}, it acts on (σ_{λ})^{𝔖_{μ}}. Moreover, by definition, 𝔖_{μ} acts on (σ_{λ})^{𝔖_{μ}} trivially. So we get a representation of 𝔖_{α} ≃ 𝔑_{μ}/𝔖_{μ} on (σ_{λ})^{𝔖_{μ}}.

Proposition 2.2 With the notations above, we have

∑_{μ∈𝔖_{n}·μ}(σ_{λ})^{𝔖_{μ}} ≃ Ind_{𝔖_{α}×𝔖_{n−|α|}}^{𝔖_{n}}(σ_{λ})^{𝔖_{μ}} ⊗ 1,

where 1 means the trivial representation of 𝔖_{n−|α|}.

Proof.

Now we summarize the above results into

Theorem 2.3

ρ_{λ}^{(n)}|_{𝔖_{n}} ≃ ⨁_{μ⊢|λ|} Ind_{𝔖_{α}×𝔖_{n−|α|}}^{𝔖_{n}}(σ_{λ})^{𝔖_{μ}} ⊗ 1,

where α ⊢ length(μ) is determined by μ via

N_{𝔖_{|λ|}}(𝔖_{μ})/𝔖_{μ} ≃ 𝔖_{α}.

Remark. Put k = |λ| = |μ|. By the Frobenius reciprocity, we have

(σ_{λ})^{𝔖_{μ}} ≃ Hom_{𝔖_{μ}}(ℂ, σ_{λ}) ≃ Hom_{𝔖_{k}}(Ind_{𝔖_{μ}}^{𝔖_{k}}ℂ, σ_{λ}),

⟹ dim(σ_{λ})^{𝔖_{μ}} = [σ_{λ} : Ind_{𝔖_{μ}}^{𝔖_{k}}ℂ].

This number is called the Kostka number (cf. [Macdonald]). By Lemma 2.1, it is equal to the weight multiplicity of μ in the representation (ρ_{λ}^{(n)}, V_{λ}) of GL_{n}.

3 Relation to the plethysm

It seems difficult to determine the action of 𝔖_{α} on (σ_{λ})^{𝔖_{μ}}. The reason why it is difficult is as follows. Put k = |λ| (i.e., λ ⊢ k). Note that the normalizer of 𝔖_{μ} = 𝔖_{μ_{1}} × ⋯ × 𝔖_{μ_{n}} in 𝔖_{k} is a wreath product

𝔖_{α} ⋉ 𝔖_{μ} ⊂ 𝔖_{k}, 𝔖_{α} = 𝔖_{α_{1}} × ⋯ × 𝔖_{α_{m}}

for some α ⊢ n. The irreducible representations of a wreath product are well-studied and completely classified (see [JK, Theorem 4.3.34]).

In our case, for π ∈ 𝔖_{α}^{∧}, let (π : 1) be an irreducible representation of 𝔖_{α} ⋉ 𝔖_{μ} which is obtained by the successive application of the natural projection of 𝔖_{α} ⋉ 𝔖_{μ} → 𝔖_{α} and the representation π of 𝔖_{α}. Then we get

Hom_{𝔖_{α}}(π, (σ_{λ})^{𝔖_{μ}}) = Hom_{𝔖_{α}}((π : 1)^{𝔖_{μ}}, (σ_{λ})^{𝔖_{μ}})
= Hom_{𝔖_{α}⋉𝔖_{μ}}((π : 1), σ_{λ})
≃ Hom_{𝔖_{k}}(Ind_{𝔖_{α}⋉𝔖_{μ}}^{𝔖_{k}}(π : 1), σ_{λ}).

If we know (σ_{λ})^{𝔖_{μ}} completely as an 𝔖_{α}-module, then we know Hom_{𝔖_{α}}(π, (σ_{λ})^{𝔖_{μ}}), hence the multiplicity of σ_{λ} in Ind_{𝔖_{α}⋉𝔖_{μ}}^{𝔖_{k}}(π : 1) can be determined. However, the induced representation

Ind_{𝔖_{α}⋉𝔖_{μ}}^{𝔖_{k}}(π : 1)

is a special case of plethysms and its decomposition is not completely known yet (cf. [JK, §5.4]).

Using the above relation, we can rephrase our theorem in terms of the decomposition of plethysm:

(σ_{λ})^{𝔖_{μ}} ≃ ⨁_{π∈𝔖_{α}^{∧}}[σ_{λ} : Ind_{𝔖_{α}⋉𝔖_{μ}}^{𝔖_{k}}(π : 1)] π.

Theorem 3.1

ρ_{λ}^{(n)}|_{𝔖_{n}} ≃ ⨁_{μ⊢|λ|} ⨁_{π∈𝔖_{α}^{∧}}[σ_{λ} : Ind_{𝔖_{α}⋉𝔖_{μ}}^{𝔖_{k}}(π : 1)] Ind_{𝔖_{α}×𝔖_{n−|α|}}^{𝔖_{n}}π ⊗ 1.

Example 3.2 Take λ = (k^{n}) = (k, ⋯, k). In this case, we have ρ_{λ}^{(n)} = (det^{(n)})^{k}. So, we know ρ_{λ}^{(n)}|_{𝔖_{n}} = (sgn)^{k}. Moreover, by Lemma 2.1, we get

(σ_{λ})^{𝔖_{μ}} = \{ \begin{matrix}ℂ & if μ = λ, \\ 0 & otherwise. \end{matrix}

For μ = λ, the normalizer of 𝔖_{λ} = 𝔖_{k} × ⋯ × 𝔖_{k} in 𝔖_{nk} is isomorphic to 𝔖_{n} ⋉ (𝔖_{k})^{n}. So we have

[σ_{(k^{n})}, Ind_{𝔖_{n}⋉(𝔖_{k})^{n}}^{𝔖_{nk}}(π : 1)] = \{ \begin{matrix}1 & if π = (sgn)^{k}, \\ 0 & if π ≠ (sgn)^{k}. \end{matrix}

Example 3.3 |λ| = n and μ = (1, ⋯, 1) = (1^{n}). Then α(μ) = (n) and V(μ) ≃ σ_{λ}. This is a well-known result (cf. [Kostant]).

Example 3.4 |λ| = kn and μ = (k, ⋯, k) = (k^{n}). Then α(μ) = (n) and

V(μ) ≃ ⨁_{π∈𝔖̂_{n}} [σ_{λ} : Ind_{𝔖_{n}⋉(𝔖_{k})^{n}}^{𝔖_{nk}}(π : 1)] π

This is a result of [AMT].

References

[AMT] S. Ariki, J. Matsuzawa and I. Terada, Representations of Weyl groups on zero weight spaces. In “Algebraic and Topological Theories, to the memory of Dr. Takehiko Miyata”, Kinokuniya, 1985, pp. 546 – 568.

[Gutkin] E. Gutkin, Representations of the Weyl group in the space of vectors of zero weight. UspehiMat. Nauka, 28(1973), 237 – 238 (in Russian).

[Kostant] B. Kostant, On Macdonald’s η-function formula, the Laplacian and generalized exponents. Adv. in Math., 20(????), 257 – 285.

[King] R. C. King, Branching rules for GL(N) ⊃ Σ_{m} and the evaluation of inner plethysms, J. Math. Phys., 15(1974).

[Howe] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. In “The Schur lectures (1992)”, Israel Mathematical Conference Proceedings, 8(1995), Bar-Ilan Univ., pp. 1 – 182.

[JK] G. James and A. Kerber, The representation theory of the symmetric group. Encyclopedia of Math. and App. 16, Addison-Wesley, 1981.

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

[Weyl] H. Weyl, The Classical Groups. Princeton University Press, Princeton, N.J., 1946.

First Draft: 1996/10/2 Ver 1.0 : 1996/10/17, Ver 1.1 : 1996/11/5, Ver. 1.2 [00/11/28 10:11]

This version is compiled on November 29, 2000.