COMPLETELY SPLITTABLE REPRESENTATIONS OF SYMMETRIC GROUPS

ALEXANDER KLESHCHEV

Introduction

This note is motivated by papers [6] and [7].

Let K be a field of characteristic p > 0, P_{n} the set of all p-regular partitions of n, Σ_{n} the symmetric group on n letters, and D^{λ} the irreducible KΣ_{n}-module corresponding to a partition λ ∈ P_{n} ([2]). For a composition μ = (μ_{1}, μ_{2}, …, μ_{r}) of n, the subgroup

Σ^{μ} = Σ_{μ_{1}} × Σ_{μ_{2}} × ⋯ × Σ_{μ_{r}} < Σ_{n}

is called a (standard) Young subgroup.

If λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{m} > 0) is a partition, let

h(λ) = m, χ(λ) = l_{1} − l_{m} + m.

In fact χ(λ) is the length of a special hook.

Example. Let λ = (8, 7, 7, 5, 3, 3). Then χ(λ) = 11. The nodes of the corresponding hook are coloured in black in the following picture.

∘ ∘ • • • • • •
∘ ∘ • ∘ ∘ ∘ ∘
∘ ∘ • ∘ ∘ ∘ ∘
∘ ∘ • ∘ ∘
∘ ∘ •
∘ ∘ •

In [7], a nice family of KΣ_{n}-modules for the case p = 5 is constructed. It consists of pairs, {D_{n}^{+}, D_{n}^{−}}, n ≥ 1 (denoted by {R_{n}, R′_{n}} in [7]). If n = 1, D_{n}^{+} ≅ D_{n}^{−} is the trivial module over Σ_{1}, and D_{n}^{+}, D_{n}^{−} are non-isomorphic irreducible KΣ_{n}-modules for n > 1. The following remarkable property of the family is stated in [7]. The restriction of D_{n}^{±} to any Young subgroup Σ_{μ_{1}} × Σ_{μ_{2}} × ⋯ × Σ_{μ_{r}} is semi-simple, and, moreover, every simple component of this restriction has form D_{μ_{1}}^{±} ⊗D_{μ_{2}}^{±} ⊗⋯⊗D_{μ_{r}}^{±}.

The paper [6] is devoted to the description of the dimensions of the D^{λ} for λ satisfying χ(λ) ≤ p. The answer is obtained in terms of the paths in a modular version of the Young graph (see below).

1991 Mathematics Subject Classification. 20C05.

This modular Young graph has been also considered in [5]. It was proved there that

(1) dim D^{λ} ≥ ♯{paths from (1) to λ in the modular Young graph}

for an arbitrary λ ∈ P_{n}. This inequality was obtained using results on the restrictions of modular irreducibles D^{λ} to Σ_{n−1}. It follows from the proof of Proposition 2.2 in [5] (cf. also Theorem 2.3 below) that we have equality in (1) if and only if the restriction D^{λ} ↓_{Σ_{i}} is semi-simple for any i < n. This is equivalent to the condition that D^{λ} ↓_{Σ^{μ}} is semi-simple for any Young subgroup Σ^{μ} < Σ_{n}, cf. Lemma 1.7 below. Thus we are led to the following definition.

Definition 0.1. An irreducible KΣ_{n}-module D^{λ} is called completely splittable if and only if the restriction D^{λ} ↓_{Σ^{μ}} to any Young subgroup Σ^{μ} < Σ_{n} is semi-simple.

In this paper we first describe the completely splittable representations. It turns out that D^{λ} is completely splittable if and only if χ(λ) ≤ p. As a consequence, we reprove Mathieu’s result on dim D^{λ} for λ with χ(λ) ≤ p.

We then develop the result of Ryba mentioned above describing all families of representations defined below.

Definition 0.2. Let Φ^{n} = {D_{1}^{n}, D_{2}^{n}, …, D_{z(n)}^{n}} be a set of irreducible KΣ_{n}-modules, n = 1, 2, . . .. The set Φ = ⋃_{n≥1} Φ^{n} is called a semi-simple inductive system if and only if the following two conditions hold.

(1) The restriction D_{j}^{n} ↓_{Σ_{n−1}} is a direct sum of modules from Φ^{n−1} (for all n = 2, 3, … and j = 1, 2, …, z(n)).

(2) For any element D_{i}^{n−1} of Φ^{n−1} there exists an element D_{j}^{n} of Φ^{n} such that D_{i}^{n−1} is a component of the restriction D_{j}^{n} ↓_{Σ_{n−1}} (n = 2, 3, …).

Remarks (1) If Φ = ⋃_{n≥1} Φ^{n} is a semi-simple inductive system as in 0.2 then it follows that any D_{j}^{n} ∈ Φ^{n} is a completely splittable module (cf. 0.1 and 1.7). Moreover, it follows that any simple component of the restriction D_{j}^{n} ↓_{Σ^{(μ_{1},μ_{2},...,μr)}} to a Young subgroup has the form D_{i_{1}}^{μ_{1}} ⊗D_{i_{2}}^{μ_{2}} ⊗⋯⊗D_{i_{r}}^{μ_{r}} where D_{i_{v}}^{μ_{v}} ∈ Φ^{μ_{v}}, v = 1, …, r.

(2) The system D_{n}^{±} from [7] is an example of a semi-simple inductive system.

(3) I am grateful to A.E. Zalesskii for the following remarks. If, in Definition 0.2, we do not demand restrictions to be semi-simple we obtain the usual definition of an inductive system. This was introduced by A.E. Zalesskii as a tool for the investigation of the lattice of the two-sided ideals in group rings of the locally finite groups, cf. [10]. It follows from our classification of the semi-simple inductive systems given below that for p > 3 there exist those which do not contain trivial and sign modules for sufficiently large n. This fact can be interpreted in the following terms. Let A_{∞} be the finitary alternating group (i.e. A_{∞} = ∪_{n≥1}A_{n} where A_{n−1} < A_{n} is the standard embedding). For p > 3 there exists a two-sided ideal of the group algebra KA_{∞} which is not contained in the augmentation ideal. It was known before that a similar result holds for p = 2. By contrast, if the characteristic of K is zero then any two-sided ideal of KA_{∞} is contained in the augmentation ideal, cf. [9].

1. Preliminaries

Throughout λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{m} > 0) is an arbitrary p-regular partition of n. We write it in the form

λ = (λ_{1}^{a_{1}}, λ_{2}^{a_{2}}, …, λ_{k}^{a_{k}}), λ_{1} > λ_{2} > ⋯ > λ_{k} > 0, a_{1}, a_{2}, …, a_{k} > 0. (2)

Then χ(λ) = λ_{1} − λ_{k} + ∑_{v=1}^{k} a_{v}, h(λ) = ∑_{v=1}^{k} a_{v}. For 1 ≤ i ≤ j ≤ k we put

β(i, j) = λ_{i} − λ_{j} + ∑_{v=i}^{j} a_{v}, γ(i, j) = λ_{i} − λ_{j} + ∑_{v=i+1}^{j} a_{v}.

We consider the Young diagram for λ as the set (denoted by the same symbol λ) of nodes

λ = {(i, j) ∈ ℤ × ℤ | i ≥ 1, 1 ≤ j ≤ l_{i}}.

Let A_{i} = (∑_{v=1}^{i} a_{v}, λ_{i}), i = 1, 2, …, k. The nodes A_{i} are by definition the removable nodes of λ. If A = A_{i} is a removable node then we denote by λ_{A} the partition of n − 1 with the Young diagram λ \ A. For any integer i with 1 ≤ i ≤ k we put

M_{i} = {j ∈ ℤ | 1 ≤ j < i, β(j, i) ≡ 0 (mod p)}.

Definition 1.1. A removable node A_{i} of λ is called normal if and only if for any j ∈ M_{i} there exists an integer d(j) such that

j < d(j) < i,  β(j, d(j)) ≡ 0 (mod p),  |{d(j)|j ∈ M_{i}}| = |M_{i}|.

We state a result from [4] for future reference.

Theorem 1.2. [4] The restriction D^{λ} ↓_{Σ_{n−1}} is semi-simple if and only if γ(i, j) ≢ 0 (mod p) for any i < j with A_{i} and A_{j} normal. In this case λ_{A_{v}} is p-regular for any normal A_{v}, and

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁{D^{λ_{Av}} |A_{v} is normal}.

Corollary 1.3. Let χ(λ) ≤ p. Then D^{λ} ↓_{Σ_{n−1}} is semi-simple. Moreover

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁_{v=1}^{k} D^{λ_{Av}} if χ(λ) < p,

and

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁_{v=1}^{k−1} D^{λ_{Av}} if χ(λ) = p.

Proof. Clearly χ(λ) = β(1, k). Let 1 ≤ i < j ≤ k. Then γ(i, j) > 0, and γ(i, j) < β(1, k) implies γ(i, j) < p. So γ(i, j) ≢ 0 (mod p). By Theorem 1.2, D^{λ} ↓_{Σ_{n−1}} is semi-simple, and

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁{D^{λ_{Av}} |A_{v} is normal}.

Since β(i, j) < β(1, k) = p unless (i, j) = (1, k), we get β(i, j) ≢ 0 (mod p) for all 1 ≤ i < j ≤ k, (i, j) ≠ (1, k). If β(1, k) < p then by Definition 1.1 all nodes A_{v}, v = 1, 2, …, k, are normal. If β(1, k) = p then similarly all of them but A_{k} are normal.

Lemma 1.4. Let λ ≠ (n).

(i) χ(λ_{A_{v}}) > χ(λ) if and only if v = k and λ_{k} > 1. In this case χ(λ_{A_{v}}) = χ(λ) + 1.

(ii) χ(λ_{A_{v}}) < χ(λ) if and only if v = k, λ_{k} = 1 or v = 1, a_{1} = 1.

Proof. This follows immediately from the definitions.

Corollary 1.5. If χ(λ) ≤ p and D^{μ} is a simple component of D^{λ} ↓_{Σ_{n−1}} then χ(μ) ≤ p.

Proof. By 1.3,

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁_{v=1}^{x} D^{λ_{Av}}

where x = k if χ(λ) < p, and x = k − 1 if χ(λ) = p. (Note that k > 1 if χ(λ) = p since λ is p-regular throughout the paper.)

We claim that χ(λ_{A_{v}}) ≤ p for any v such that D^{λ_{Av}} is a component of D^{λ} ↓_{Σ_{n−1}}. Indeed, by Lemma 1.4(i), χ(λ_{A_{v}}) > p ≥ χ(λ) implies v = k, λ_{k} > 1, and χ(λ) = p. But if χ(λ) = p then D^{λ_{A_{k}}} is not a component of D^{λ} ↓_{Σ_{n−1}}.

Lemma 1.6. Let G_{1} and G_{2} be finite groups, and M be a K[G_{1} × G_{2}]-module such that M ↓_{G_{v}} is semi-simple for v = 1, 2. Assume K is algebraically closed. Then M is semi-simple.

Proof. The proof goes along the lines of the proof of Theorem 5 in [8]. Since G_{2} centralises G_{1} then G_{2} preserves the G_{1}-homogeneous components of M. So we may assume that

M ↓_{G_{1}} ≅ \underbrace{L ⊕ ⋯ ⊕ L}_{t times}

where L is an irreducible KG_{1}-module. Then the centraliser of G_{1} in M is GL(W) where W is the t-dimensional vector space, and, moreover, M ≅ L⊗W as a module over G_{1} ×GL(W). It remains to notice that G_{2} < GL(W) acts completely reducibly on W.

Lemma 1.7. The following conditions are equivalent.

(i) D^{λ} ↓_{Σ^{μ}} is semi-simple for any Young subgroup Σ^{μ} < Σ_{n}.

(ii) D^{λ} ↓_{Σ_{i}} is semi-simple for any i < n (here Σ_{i} is considered as the subgroup of Σ_{n} consisting of the permutations of the first i letters).

Proof. (i) implies (ii) because Σ_{i} is the Young subgroup Σ^{(i,1,...,1)}. Assume that D^{λ} ↓_{Σ_{i}} is semi-simple. Then the restriction of D^{λ} to any subgroup consisting of the permutations of a fixed set of i letters is semi-simple since such a subgroup is conjugate to Σ_{i}. So (ii) implies (i) in view of Lemma 1.6 (and the fact that GF (p) is a splitting field for Σ_{n} so we may assume that K is algebraically closed).

2. Main Results

Theorem 2.1. D^{λ} is completely splittable if and only if χ(λ) ≤ p.

Proof. Proceed by induction on n. If n < p the assertion of the theorem is clear. Let n ≥ p and suppose the result holds for n − 1.

Assume χ(λ) ≤ p. Then D^{λ} ↓_{Σ_{n−1}} is semi-simple by 1.3. Moreover, in view of 1.5, χ(μ) ≤ p as soon as D^{μ} is a component of D^{λ} ↓_{Σ_{n−1}}. By the inductive hypothesis, D^{μ} is completely splittable. Now it suffices to apply Lemma 1.7 to prove that D^{λ} is completely splittable.

In the other direction, assume χ(λ) > p. If D^{λ} ↓_{Σ_{n−1}} is not semi-simple we are done. So we may suppose that

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁{D^{λ_{Av}} |A_{v} is normal},

cf. Theorem 1.2. If there exists a normal node A with χ(λ_{A}) > p then by the inductive hypothesis D^{λ_{A}} is not completely splittable, hence D^{λ} is not completely splittable.

So we may assume that χ(λ_{A}) ≤ p for any normal node A of λ. Note that A_{1} is normal, by definition, so by Lemma 1.4 either k = 1, λ_{1} = 1 or a_{1} = 1. In the former case λ = (1^{n}) is not p-regular because n > p, giving a contradiction. So a_{1} = 1. If λ = (n) then χ(λ) = 1 which contradicts the assumption χ(λ) > p. So k ≥ 2. Moreover, χ(λ_{A_{1}}) = χ(λ) − 1 so χ(λ) = p + 1.

Assume first that k = 2. Then β(1, 2) = χ(λ) = p + 1, hence A_{2} is normal. Moreover γ(1, 2) = β(1, 2) − a_{1} = p. Therefore D^{λ} ↓_{Σ_{n−1}} is not semi-simple by 1.2.

So we may assume k ≥ 3. Note that

β(1, 2) = β(1, 3) − (λ_{2} − λ_{3}) − a_{3} ≤ β(1, 3) − 2 ≤ β(1, k) − 2 = χ(λ) − 2 = p − 1.

So β(1, 2) ≢ 0 (mod p) and hence A_{2} is normal. However χ(λ_{A_{2}}) = χ(λ) = p + 1, by virtue of Lemma 1.4, giving a contradiction.

Definition 2.2. We define a graph Y as follows. Vertices of Y are labelled by the p-regular partitions λ of all n ≥ 1 with χ(λ) ≤ p. There exists an arrow μ → λ if and only if μ = λ_{A} for some removable node A of λ.

Remark. The graph Y just defined is a complete subgraph of the p-Young graph defined in [5].

The following result is proved in [6] by different methods.

Proposition 2.3. Let n ≥ 2, χ(λ) ≤ p. Then

dim D^{λ} = ♯{paths from (1) to λ in Y }.

Proof. It follows from 1.3, 1.4 and 2.2 that

D^{λ} ↓_{Σ_{n−1}} ≅ ⨁{D^{μ}|there exists an arrow μ → λ in Y }.

Now the proposition is obtained by induction.

We reformulate Proposition 2.3 in terms of λ-tableaux. If t is a λ-tableau we denote by t(i, j) the entry of t in the i-th row and j-th column. A λ-tableau is called standard if its entries increase in rows from left to right and in columns from top to bottom. We shall call a standard λ-tableau p-standard if for any two of its entries t(i, j) and t(i′, j′) with i > i′, j < j′, and i + j′ + 1 − i′ − j = p one has t(i′, j′) > t(i, j).

Corollary 2.4. Let n ≥ 2, χ(λ) ≤ p. Then

dim D^{λ} = ♯{p-standard λ-tableaux}.

Proof. The proof can be obtained by induction similarly to the proof of 2.3. Alternatively, one may notice that there exists a 1-1-correspondence between paths from (1) to λ in Y and p-standard λ-tableaux and apply 2.3.

Notation. For s = 1, 2, …, p − 1 put

ξ_{n}^{s} = {D^{μ}|μ ∈ P_{n}, h(μ) = s, χ(μ) ≤ p},
ω_{n}^{s} = {D^{μ}|μ = (μ_{1}, μ_{2}, …, μ_{r}) ∈ P_{n}, h(μ) < s, μ_{1} ≤ p − s}.

Of course, ξ_{n}^{s} = ∅ if and only if n < s, and ω_{n}^{s} = ∅ if and only if n > (p − s)(s − 1). Set

Φ_{n}^{s} = ξ_{n}^{s} ∪ ω_{n}^{s},
Φ(s) = ⋃_{n≥1} Φ_{n}^{s}.

For any M ⊆ {1, 2, …, p − 1} put

Φ(M) = ⋃_{s∈M} Φ(s) = ⋃_{n≥1}( ⋃_{s∈M} Φ_{n}^{s}).

Lemma 2.5. Let D^{μ} ∈ ξ_{n}^{s}, D^{θ} ∈ ξ_{N}^{s}. If μ = (m_{1}, …, m_{s}), θ = (t_{1}, …, t_{s}), and m_{v} ≤ t_{v} for v = 1, 2, …, s then D^{μ} is a component of D^{θ} ↓_{Σ_{n}}.

Proof. This follows from 1.3 (by induction on N − n).

Lemma 2.6. Let D^{μ} ∈ ξ_{n}^{s}, D^{θ} ∈ ξ_{N}^{s}. If N ≫ 0, then D^{μ} is a component of D^{θ} ↓_{Σ_{n}}.

Proof. Let μ = (m_{1}, …, m_{s}), θ = (t_{1}, …, t_{s}). If N is big enough then it follows from the assumptions that m_{v} ≤ t_{v}, v = 1, 2, …, s. Now apply Lemma 2.5.

Lemma 2.7. Let n ≤ (p − s)(s − 1), D^{μ} ∈ ω_{n}^{s}, D^{θ} ∈ Φ_{N}^{s}. If N ≫ 0, then D^{μ} is a component of D^{θ} ↓_{Σ_{n}}.

Proof. Let η = ((p − s)^{(s−1)}, 1), d = (p − s)(s − 1) + 1. Then D^{η} ∈ ξ_{d}^{s} so by Lemma 2.6, D^{η} is a component of D^{θ} ↓_{Σ_{d}}. Now it remains to notice that by 1.3, D^{μ} is a component of D^{η} ↓_{Σ_{n}}.

Theorem 2.8. (i) Any Φ(s), s = 1, 2, …, p − 1 is a semi-simple inductive

system. (ii) For any M ⊆ {1, 2, …, p − 1}, Φ(M) is a semi-simple inductive system.

(iii) Any semi-simple inductive system coincides with Φ(M) for some subset M ⊆ {1, 2, …, p − 1}.

Proof. (i) Let D^{μ} ∈ Φ_{n}^{s}. Then χ(μ) ≤ p. By 2.1, the restriction D^{μ} ↓_{Σ_{n−1}} is semisimple. Let D^{ν} be any of its components. We prove that D^{ν} ∈ Φ_{n−1}^{s}.

If D^{μ} ∈ ω_{n}^{s} then it follows from 1.3 that D^{ν} ∈ ω_{n−1}^{s} ⊆ Φ_{n−1}^{s}.

Let D^{μ} ∈ ξ_{n}^{s} and μ = (μ_{1}, μ_{2}, …, μ_{s}). If μ_{s} > 1 or μ_{s} = 1, ν ≠ (μ_{1}, μ_{2}, …, μ_{s−1}) then by 1.3 and 1.5 we have D^{ν} ∈ ξ_{n−1}^{s} ⊆ Φ_{n−1}^{s}. Assume μ_{s} = 1 and ν = (μ_{1}, μ_{2}, …, μ_{s−1}). Then it follows from 1.3 that μ_{1}+s−1 = χ(μ) < p, i.e. μ_{1} ≤ p−s. Thus D^{ν} ∈ ω_{n−1}^{s} ⊆ Φ_{n−1}^{s}.

Now we prove that there exists D^{η} ∈ Φ_{n+1}^{s} such that D^{μ} is a component of D^{η} ↓_{Σ_{n}}. In view of what has already been proved, it suffices to show that for some N > n there is D^{θ} ∈ Φ_{N}^{s} such that D^{μ} is a component of D^{θ} ↓_{Σ_{n}}. But this follows from 2.6 and 2.7.

(ii) follows from (i).

(iii) Let Φ = ∪_{n≥1}Φ_{n} be an arbitrary semi-simple inductive system. By 2.1, D^{μ} ∈ Φ_{n} implies χ(μ) ≤ p. In particular, h(μ) ≤ p − 1 since μ is p-regular. Define

M = {s|h(μ) = s for infinitely many μ with D^{μ} ∈ Φ}.

Then M ⊆ {1, 2, …, p − 1}. We shall prove that Φ = Φ(M).

First we prove that s ∈ M implies Φ(s) ⊆ Φ. Let D^{μ} ∈ Φ_{n}^{s} for some s ∈ M. Since every Φ_{d}^{s} is finite then by definition of M, for any d there is N > d and θ ∈ P_{N} with h(θ) = s, χ(θ) ≤ p, D^{θ} ∈ Φ_{N}. By definition of inductive system, any component of D^{θ} ↓_{Σ_{n}} belongs to Φ_{n}. But D^{μ} is such a component provided N is big enough, in view of Lemmas 2.6 and 2.7.

Now we prove that Φ ⊆ ∪_{s∈M}Φ(s). Let D^{μ} ∈ Φ_{n}. Then for any N > n we have that D^{μ} is a component of D^{θ} ↓_{Σ_{n}} for some D^{θ} ∈ Φ_{N}. However it follows from the definition of the set M that for N ≫ 0, D^{θ} ∈ Φ_{N} implies h(θ) ∈ M. Since we also have χ(θ) ≤ p, this means that D^{θ} ∈ ∪_{s∈M}Φ(s). Hence D^{μ} ∈ ∪_{s∈M}Φ(s).

Remarks (1) The system Φ^{1} is clearly the trivial inductive system, i.e. Φ_{n}^{1} = {D^{(n)}} for any n. The system Φ^{p−1} is the sign system, i.e. Φ_{n}^{p−1} = {sgn_{n}} where sgn_{n} is the sign representation of Σ_{n}. Indeed, Φ_{n}^{p−1} = {D^{ε_{n}}} where ε_{n} = ((d + 1)^{r}, d^{p−1−r}) if n = (p − 1)d + r, d, r ∈ ℤ, 0 ≤ r < p − 1. The fact that D^{ε_{n}} ≅ sgn_{n} can be found in [3] or easily deduced from the main result of [1].

(2) In view of the previous remark and Theorem 2.8 there are only three semisimple inductive systems in case p = 3, namely, the trivial one, the sign one, and the union of the trivial and the sign systems. They do not provide non-trivial inductive systems for the alternating groups. That is why for p = 3 the question about two-sided ideals mentioned in Remark (3) after Definition 0.2 remains open.

(3) The system D_{n}^{±} from [7] coincides with Φ(2) for p = 5.

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Department of Mathematics, University of Oregon, Eugene OR97403, U.S.A. E-mail address: klesh@math.uoregon.edu