MAXIMAL IDEALS IN MODULAR GROUP ALGEBRAS OF THE FINITARY SYMMETRIC AND ALTERNATING GROUPS

ALEXANDER BARANOV AND ALEXANDER KLESHCHEV

1. Introduction

A permutation of the set ℕ = {1, 2, 3, … } is called finitary if it fixes all but finitely many elements. The finitary symmetric group Σ_{∞} is the group of all finitary permutations of ℕ. The finitary alternating group A_{∞} is defined as the group of all even finitary permutations of ℕ. Clearly, Σ_{∞} and A_{∞} are locally finite groups. They can be represented as the unions

Σ_{∞} = ∪_{n≥1}Σ_{n}, A_{∞} = ∪_{n≥1}A_{n}

where Σ_{n} and A_{n} are the groups of all permutations and all even permutations of the set {1, 2, …, n}, respectively.

Let F be an arbitrary field of characteristic p > 0.

The main result of this paper is a description of the maximal two-sided ideals of the group algebras F Σ_{∞} and F A_{∞} for the case p > 2. In particular, we show that there are exactly p − 1 of them in F Σ_{∞} and \frac{p−1}{2} in F A_{∞}.

In the last years there has been a noticeable progress in the theory of group algebras of locally finite groups, see the expository paper [23] and references there. A. E. Zalesskii has shown that the ideals (we always mean two-sided ones) of these algebras are closely related with the so-called inductive systems, see Definition 2.1 below. This idea is crucial in our approach here. It allows one to reduce many problems on ideals of F Σ_{∞} and F A_{∞} to the ones on modular representations of finite groups Σ_{n} and A_{n}.

To describe our main results in details we need some facts and terminology from the modular representation theory of the symmetric group. The main reference here is [10].

Let λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{m} > 0) be a partition of n (we write |λ| = n). We define

h(λ) = m (1)

and

χ(λ) = l_{1} − l_{m} + m (2)

1991 Mathematics Subject Classification. 20C05, 16S34.

Supported by the Fundamental Research Foundation of Belarus and the National Science Foundation

The irreducible F Σ_{n}-modules are parametrized by the p-regular partitions of n. If λ is such a partition we denote by D^{λ} the corresponding irreducible. For s = 1, 2, …, p − 1 and n > (p − s)(s − 1) set

Φ(s)_{n} = {D^{μ} | |μ| = n, h(μ) = s, χ(μ) ≤ p} .

Let G be a group and M be an F G-module. We write Ann_{F G}(M) for the annihilator of M in F G. If H is a subgroup of G we denote by M↓H the restriction of M to H.

Theorem 1.1. Let F be a field of characteristic p > 2.

(i) Set

I(s)_{n} = ⋂_{D∈Φ(s)_{n}} Ann_{F Σ_{n}}(D), s = 1, …, p − 1, n > (p − s)(s − 1).

Then I(s)_{n} = I(s)_{n+1} ∩ F Σ_{n} for all s, n. Moreover,

I(s) = ⋃_{n>(p−s)(s−1)} I(s)_{n}, s = 1, …, p − 1,

are exactly all distinct maximal (two-sided) ideals of F Σ_{∞}, and I(s)∩F Σ_{n} = I(s)_{n} for n > (p − s)(s − 1).

(ii) Set

J(t)_{n} = ⋂_{D∈Φ(t)_{n}} Ann_{F A_{n}}(D↓A_{n}), t = 1, …, \frac{p − 1}{2}, n > (p − t)(t − 1).

Then J(t)_{n} = J(t)_{n+1} ∩ F A_{n} for all t, n. Moreover,

J(t) = ⋃_{n>(p−t)(t−1)} J(t)_{n}, t = 1, …, \frac{p − 1}{2},

are exactly all distinct maximal (two-sided) ideals of F A_{∞}, and J(t)∩F A_{n} = J(t)_{n} for n > (p − t)(t − 1).

(iii) I(s) ∩ F A_{∞} = J(t) where t = min(s, p − s), s = 1, …, p − 1.

Remark. (1) Theorem 1.1 was conjectured by A. E. Zalesskii.

(2) It is known that every group algebra F G contains the maximal ideal

Aug(F G) = {∑_{g∈G} α_{g}g | ∑ α_{g} = 0}

called the augmentation ideal. It has codimension 1. One can easily demonstrate another maximal ideal of codimension 1 in F Σ_{∞}:

Aug^{σ}(F Σ_{∞}) = { ∑_{g∈Σ_{∞}} α_{g}g | ∑ sign(g)α_{g} = 0}

It can be shown that

I(1) = Aug(F Σ_{∞}),
J(1) = Aug(F A_{∞}),
I(p − 1) = Aug^{σ}(F Σ_{∞})

and that ideals I(s), 1 < s < p − 1, and J(t), t > 1, are of infinite codimension.

(3) Ideals I(s) of F Σ_{∞} has been essentially described in [15]. Results from [15] are used in this paper.

(4) One can show that all the restrictions D↓A_{n} in the part (ii) of Theorem 1.1 are actually irreducible.

(5) Tensoring F Σ_{n}-modules with the one-dimensional sign representation is one of the crucial methods used in our proofs. This is why the case p = 2 happened to be exceptional in this paper. A.E. Zalesskii observed that if p = 2 there exists at least one maximal ideal in both F Σ_{∞} and F A_{∞} which is different from the augmentation ideal. But it remains unclear if there are other maximal ideals.

(6) We note that for a ground field of characteristic 0 there is a complete description of the ideal lattice in the group algebra of Σ_{∞}, see [6, 18]. In particular, Aug(F Σ_{∞}) and Aug^{σ}(F Σ_{∞}) are the only maximal ideals of F Σ_{∞}. Much less is known about modular group rings. Important information is contained in the papers [7, 19, 21, 22].

(7) The ideals we consider play a role in the theory of identities of algebras, see [1, 18, 19, 20].

(8) In this paper we use recent results from modular representation theory of symmetric groups related with branching rules [13] and tensoring with sign (Mullineux Conjecture) [5], [2].

(9) All results of this paper about the groups Σ_{∞} and A_{∞} can be easily generalized to the finitary symmetric and alternating groups Σ_{Ω} and A_{Ω} of an uncountable set Ω. One just has to use inductive limits instead of unions. We leave details to the reader.

Acknowledgement

We are grateful to A. E. Zalesskii for stimulating and encouraging discussions.

Notation

F a field of characteristic p > 0;

Σ_{n} (resp., A_{n}) the symmetric (resp., alternating) group on {1, …, n};

P_{n} the set of all p-regular partitions of n;

|λ| = n means “λ is a partition of n”;

S^{λ}  the Specht module over F Σ_{n} corresponding to a partition λ of n [10];

D^{λ} the irreducible F Σ_{n}-module corresponding to a p-regular partition λ of n [10]; E^{λ}, E_{±}^{μ} denote the irreducible F A_{n}-modules, see Section 6 below;

sgn the one-dimensional sign representation of F G_{n} (gv = sign(g)v for g ∈ Σ_{n}, v ∈ sgn;

M : P_{n} → P_{n}  the Mullineux bijection. Thus D^{λ} ⊗sgn ≅ D^{M(λ)}, see Section 3 below;

χ(λ), h(λ) are defined in (1) and (2) above;

res A the residue of the node A, see [11] and (4) below;

cont(λ)  the residue content of the Young diagram λ, see [11] and (5) below;
λ_{A}  stands for the Young diagram λ\{A} where λ is a Young dia- gram and A is a removable node of λ, see Section 3 below.

Let G be a group, H a subgroup of G, A an associative algebra, M an F G-module, S a set of F G-modules, N an F H-module. Then we denote:

Irr G (resp., Irr A) the set of the isomorphism classes of the irreducible F G- modules (resp., A-modules);

Irr M the set of isomorphism classes of the composition factors of M;

Irr S stands for ∪_{M∈S} Irr M;

Irr(S↓H) stands for ∪_{M∈S} Irr(M↓H);

M↓H (or M↓_{H}^{G}) the restriction of M from G to H;

N↑^{G} (or N↑_{H}^{G}) the module induced from H to G.

2. Inductive systems (general setting)

In this section we shortly discuss and push a bit further the relation between inductive systems and (two-sided) ideals in the group algebra of a locally finite group, revealed by A. E. Zalesskii. Assume for simplicity that a locally finite group G is countable. Then G can be represented as a union

G = ∪_{i=1}^{∞}G_{i}

where G_{1} ⊂ G_{2} ⊂ … are finite subgroups of G (if G is not countable one has to use an inductive limit).

Definition 2.1. [23, 1.1] Let Φ_{i} be a non-empty subset of Irr G_{i}, i = 1, 2, . . .. We say that the collection Φ = {Φ_{i}}_{i∈ℕ} is an inductive system (for G) if for any i, j ∈ ℕ with i < j we have

Φ_{i} = Irr(Φ_{j}↓G_{i}).

Let X be a (proper) ideal of the group algebra F G. Set

Φ(X)_{i} = Irr((F G/X)↓G_{i}).

One easily checks that Φ(X) = {Φ(X)_{i}}_{i∈ℕ} is an inductive system. Recall that an ideal I of an algebra A is called primitive if it is the annihilator of an irreducible A-module and semiprimitive if it is an intersection of primitive ideals. Equivalently, I is semiprimitive if and only if the Jacobson radical Rad(A/I) is trivial. We shall also use the known fact that the Jacobson radical of a locally finite-dimensional algebra is the largest locally nilpotent ideal. The following result is crucial.

Theorem 2.2. [23, 8.1, 1.25]

(i) Φ(X)_{i} = Irr(F G_{i}/(X ∩ F G_{i})) where F G_{i}/(X ∩ F G_{i}) is considered as an F G_{i}-module.

(ii) The map f : X ↦ Φ(X) from the set of the proper ideals to the set of

inductive systems is surjective. (iii) Let Φ be an inductive system. Then the set f^{−1}(Φ) contains the largest and

the smallest (by inclusion) ideals I(Φ) and K(Φ), respectively. (iv) The ideals I(Φ) are semiprimitive. The quotient algebras I(Φ)/K(Φ) are

locally nilpotent. In particular, the map Φ → I(Φ) is a bijection between the set of the inductive systems and the set of the semiprimitive ideals of F G.

Definition 2.3. Let Φ = {Φ_{i}}_{i∈ℕ} and Ψ = {Ψ_{i}}_{i∈ℕ} be inductive systems. We write Φ ⊆ Ψ if Φ_{i} ⊆ Ψ_{i} for all i.

Lemma 2.4. Let Ψ_{i} ⊆ Irr G_{i}, i ∈ ℕ. Assume

Irr(Ψ_{i}↓G_{i−1}) ⊆ Ψ_{i−1}.

Set

Ψ̃_{i} = ∩_{j≥i} Irr(Ψ_{j}↓G_{i}).

Then Ψ̃ = {Ψ̃_{i}}_{i∈ℕ} is the largest (with respect to ⊆) inductive system contained in Ψ.

Proof. Note that

Ψ_{i} ⊇ Irr(Ψ_{i+1}↓G_{i}) ⊇ Irr(Ψ_{i+2}↓G_{i}) ⊇ …

Since the set Ψ_{i} is finite, there exists n = n(i) such that for any j > n we have

Ψ̃_{i} = Irr(Ψ_{j}↓G_{i}).

Let i ∈ ℕ. Choose j ≥ max(n(i), n(i + 1)). Then

Irr(Ψ̃_{i+1}↓G_{i}) = Irr(Irr(Ψ_{j}↓G_{i+1})↓G_{i}) = Irr(Ψ_{j}↓G_{i}) = Ψ̃_{i},

so Ψ̃ is an inductive system. Clearly, it is the largest one contained in Ψ.

Definition 2.5. Let Φ and Ψ be inductive systems. Set Θ_{i} = Φ_{i} ∩ Ψ_{i}. One can easily see that

Irr(Θ_{i}↓G_{i−1}) ⊆ Θ_{i−1}.

In view of Lemma 2.4, there exists the largest inductive system Θ̃ such that Θ̃ ⊆ Φ and Θ̃ ⊆ Ψ. We shall denote Θ̃ by Φ ∧ Ψ. Moreover, we denote by Φ ∨ Ψ the inductive system with (Φ ∨ Ψ)_{i} = Φ_{i} ∪ Ψ_{i}.

Proposition 2.6.

(i) Φ → I(Φ) is an order-reversing isomorphism of partially ordered sets between the inductive systems and semiprimitive ideals of F G.

(ii) Φ → I(Φ) is a bijection between the set of the minimal (with respect to ⊆)

inductive systems and the set of the maximal ideals of F G. (iii) Semiprimitive ideals of F G satisfy A.C.C. if and only if inductive systems

satisfy D.C.C. (iv) For any ideals I, J of F G we have

Φ(I ∩ J) = Φ(I) ∨ Φ(J)

(v) For any ideals I, J of F G we have

Φ(I + J) = Φ(I) ∧ Φ(J)

Proof. We first prove (iv). Fix k ∈ ℕ and set

I_{k} = I ∩ F G_{k}, J_{k} = J ∩ F G_{k}.

In view of Theorem 2.2 (i), we have to show that

Irr(F G_{k}/(I_{k} ∩ J_{k})) = Irr(F G_{k}/I_{k}) ∪ Irr(F G_{k}/J_{k}).

But this equality follows from the following exact sequences of G_{k}-modules

0 → I_{k}/(I_{k} ∩ J_{k}) → F G_{k}/(I_{k} ∩ J_{k}) → F G_{k}/I_{k} → 0,
0 → J_{k}/(I_{k} ∩ J_{k}) → F G_{k}/(I_{k} ∩ J_{k}) → F G_{k}/J_{k} → 0,

and the embedding I_{k}/(I_{k} ∩ J_{k}) ≅ (I_{k} + J_{k})/J_{k} ⊂ F G_{k}/J_{k}.

Now we prove (i). By 2.2 (iv), Φ → I(Φ) is a bijection between the inductive systems and the semiprimitive ideals. It follows from the definitions that I ⊇ J implies Φ(I) ⊆ Φ(J). This shows that I(Φ) ⊇ I(Ψ) implies Φ ⊆ Ψ. Conversely, let Φ ⊆ Ψ. By (iv),

Φ(I(Φ) ∩ I(Ψ)) = Φ(I(Φ)) ∨ Φ(I(Ψ)) = Φ ∨ Ψ = Ψ,

since Φ ⊆ Ψ. It is known that the intersection of semiprimitive ideals is a semiprimitive ideal. So we have two semiprimitive ideals, I(Φ)∩I(Ψ) and I(Ψ), corresponding to one inductive system Ψ. In view of 2.2 (iv),

I(Φ) ∩ I(Ψ) = I(Ψ).

Hence I(Φ) ⊇ I(Ψ).

(iii) follows immediately from (i).

Let X be a maximal ideal of F G. Then F G/X is simple. Observe that F G/X is not locally nilpotent (for example because it contains the identity). Therefore X is semiprimitive. Now (ii) follows from (i) .

(v) For an ideal X of F G we denote by X̄ the smallest semiprimitive ideal of F G containing X (it coincides with the intersection of the semiprimitive ideals containing X). Since I(Φ(X)) is semiprimitive and contains X, we have X̄ ⊆ I(Φ(X)). The inclusions X ⊆ X̄ ⊆ I(Φ(X)) imply Φ(X̄) = Φ(X). By 2.2 (iv), X̄ = I(Φ(X)) and X̄/X is locally nilpotent. Set

X = I + J, Y = Ī+ J̄.

Observe that

(Ī+ X)/X ≅ Ī/(Ī∩ X)

is a locally nilpotent ideal in Y/X. Analogously (J̄ + X)/X is a locally nilpotent ideal in Y/X. Since

Y/X = (Ī+ X)/X + (J̄ + X)/X.

we conclude that Y/X is locally nilpotent. Hence Rad(F G/X) ⊇ Y/X.

On the other hand, we have Rad(F G/X) ⊆ X̄/X since Rad(F G/X̄) = 0. Thus X̄/X ⊇ Y/X, whence X̄ ⊇ Y. Therefore X̄ ⊇ Ȳ. Since Ȳ ⊇ Y ⊇ X, we conclude that X̄ = Ȳ. So

Φ(Ȳ) = Φ(X̄) = Φ(X).

Since Ȳ is the smallest semiprimitive ideal, containing semiprimitive ideals Ī and J̄, (i) implies

Φ(Ȳ) = Φ(Ī) ∧ Φ(J̄).

Finally, we get

Φ(X) = Φ(Ȳ) = Φ(Ī) ∧ Φ(J̄) = Φ(I) ∧ Φ(J).

Remark. In the proof of Proposition 2.6 (v) we have introduced the operation X → X̄. Note that the set of all (not necessarily proper) semiprimitive ideals of F G is a lattice with respect to the operations I ∩ J and \overline{I + J}. In view of 2.6 (i) this lattice is antiisomorphic to the lattice of all inductive systems with respect to the operations ∨ and ∧, if we admit the empty inductive system (which will correspond to the ideal F G).

Definition 2.7. An inductive system Φ = {Φ_{i}}_{i∈ℕ} is called semisimple if for any i ∈ ℕ and any D ∈ Φ_{i+1} the restriction D↓G_{i} is completely reducible.

Lemma 2.8. Let Φ = {Φ_{i}}_{i∈ℕ} be an inductive system for G. Then Φ is semisimple if and only if

I(Φ) ∩ F G_{i} = ∩_{D∈Φ_{i}} Ann_{F G_{i}} D (3)

for all i ∈ ℕ.

Proof. Set

X_{i} = ∩_{D∈Φ_{i}} Ann_{F G_{i}} D, i = 1, 2, …

Let Φ be semisimple. If i < j and D ∈ Φ_{j} we have D↓G_{i} = D_{1} ⊕ ⋯ ⊕ D_{s} with all D_{k} ∈ Φ_{i}. Moreover, Irr(Φ_{j}↓G_{i}) = Φ_{i}, and the annihilator of a direct sum of modules is the intersection of their annihilators. So we have

X_{j} ∩ F G_{i} = (∩_{D∈Φ_{j}} Ann_{F G_{j}} D) ∩ F G_{i} = ∩_{D∈Φ_{j}}((Ann_{F G_{j}} D) ∩ F G_{i})
= ∩_{D∈Φ_{j}} Ann_{F G_{i}}(D↓G_{i}) = X_{i}.

Therefore X_{i} ⊆ X_{j} (we consider both X_{i} and X_{j} as subsets of F G via embeddings F G_{l} ⊂ F G). So X = ∪_{i=1}^{∞}X_{i} is an ideal of F G such that X ∩ F G_{i} = X_{i}. So, taking into account Theorem 2.2 (i), we get

Φ(X)_{i} = Irr(F G_{i}/(F G_{i} ∩ X)) = Irr(F G_{i}/X_{i}) = Irr(F G_{i}/ Ann_{F G_{i}}(⊕_{D∈Φ_{i}}D)) = Φ_{i},

i.e. Φ(X) = Φ.

Now note that

F G/X = ∪_{i=1}^{∞}F G_{i}/(F G_{i} ∩ X) = ∪_{i=1}^{∞}F G_{i}/X_{i}.

Since every ideal X_{i} is an intersection of primitive ideals (annihilators of irreducible modules) it is semiprimitive, i.e. F G_{i}/X_{i} is semisimple. It follows that Rad(F G/X) = 0, i.e. X is semiprimitive. By Theorem 2.2 (iv), X = I(Φ), which proves the equality (3).

Conversely, let I(Φ) ∩ F G_{i} = X_{i}, i = 1, 2, . . .. For any i ∈ ℕ, we consider an arbitrary module D ∈ Φ_{i+1}. Note that X_{i} ⊆ X_{i+1} ⊆ Ann_{F G_{i+1}}(D) by assumption. Hence the restriction D↓G_{i} is an F G_{i}/X_{i}-module. However, F G_{i}/X_{i} is a semisimple finite-dimensional algebra (because X_{i} ⊇ ∩_{D∈Irr G_{i}} Ann_{F G_{i}} D = Rad F G_{i}). Therefore D↓G_{i} is semisimple.

3. Modular representations of symmetric groups

We start from some notation concerning Young diagrams.

Let us fix an arbitrary partition λ = (l_{1} ≥ ⋯ ≥ l_{m} > 0) of n. We do not distinguish between λ and its Young diagram which is defined as a subset

{(i, j) | 1 ≤ i ≤ m, 1 ≤ j ≤ l_{i}}

of ℕ × ℕ. Elements (i, j) of ℕ × ℕ are called nodes. If A = (i, j) is a node we define

res A = (j − i) (mod p). (4)

If a diagram λ contains exactly c_{i} nodes of residue i, i = 0, 1, …, p − 1, we define

cont(λ) = (c_{0}, c_{1}, …, c_{p−1}). (5)

If l_{i} > l_{i+1} then the node (i, l_{i}) is called a removable node for λ (l_{m+1} is interpreted as 0). If A = (i, l_{i}) is a removable node for λ we denote by λ_{A} the partition (l_{1}, …, l_{i−1}, l_{i} − 1, l_{i+1}, …, l_{m}) of n − 1 whose Young diagram is λ\{A}. We call (i, j) an indent node for λ if l_{i} < l_{i−1}, j = l_{i} + 1 (l_{0} is interpreted as +∞).

We list some known results on representations of Σ_{n} for future reference. All modules over Σ_{n} are F Σ_{n}-modules.

The Specht module S^{λ} labelled by partitions λ of n are defined in [10, Section 4]. The irreducible modules D^{λ} labelled by the p-regular partitions λ of n are defined as the top composition factors of the corresponding S^{λ}, see [10, Section 11]. So

Lemma 3.1. Let λ ∈ P_{n}. Then D^{λ} ∈ Irr(S^{λ}).

The following well known result follows from the “Nakayama Conjecture” [11, 6.1] and Lemma 3.1.

Lemma 3.2. If D^{μ} ∈ Irr(S^{λ}) then

cont(μ) = cont(λ).

We need the following Branching Theorem for the Specht modules.

Theorem 3.3. [10, 9.3] The restriction S^{λ}↓Σ_{n−1} has a filtration whose factors are

{S^{λ_{A}} | A is a removable node for λ}.

We shall use a special case of the branching rule for irreducible modules obtained in [13]. To state the result we need a concept of a normal node.

Definition 3.4. Let A = (i, l_{i}) be a removable node for λ. We call A normal (for λ) if and only if for every j = 1, 2, …, i−1 the following condition holds: the number of the removable nodes B in the rows j, j+1, …, i−1 such that res B = res A is greater than or equal to the number of the indent nodes C in the rows j, j + 1, …, i − 1 such that res C = res A.

Theorem 3.5. [13, 16] Let λ ∈ P_{n} and A be a removable node of λ such that λ_{A} ∈ P_{n−1}. Then D^{λ_{A}} ∈ Irr(D^{λ}↓Σ_{n−1}) if and only if A is normal.

If d ∈ ℕ we define the p-exponent of d as the largest i ∈ ℤ_{+} such that p^{i} divides d.

Theorem 3.6. [9, 12] Let λ ∈ P_{n}. Then S^{λ} is irreducible if and only if the pexponents of the hook length are constant along the columns of λ.

We denote by sgn the one-dimensional sign representation of Σ_{n}. The Mullineux bijection M : P_{n} → P_{n} is defined via

D^{λ} ⊗ sgn ≅ D^{M(λ)}.

The main result of [5] and [2] provides an algorithm for calculating M(λ) which we now describe.

The rim of a Young diagram λ is its south-east border — in other words, a node (i, j) of λ belongs to the rim if and only if (i + 1, j + 1) ∉ λ.

Let us number the nodes of the rim moving from the “top-right” to the “leftbottom” (see the example below). Define the first p-segment of the rim as the set consisting of the nodes with numbers ≤ p. If the last node B of the first p-segment is in the last row of λ then λ has only one p-segment. If not, let r be the row containing B. The first node of the second p-segment is the node which has the smallest number, say d, among the nodes of the rim in row r + 1. The second p-segment is now defined as the set consisting of the nodes whose numbers i satisfy d ≤ i ≤ d + p − 1. Repeating this procedure sufficiently many times we reach the bottom row of the diagram. It is clear that all p-segments except possibly the last one contain p nodes. The p-edge is defined as the union of the p-segments.

Now define diagrams λ^{(1)}, λ^{(2)}, λ^{(3)}, … as follows.

λ^{(1)} = λ, λ^{(i)} = λ^{(i−1)} \ {p − edge of λ^{(i−1)}}

for i > 1. We choose z to be maximal with respect to λ^{(z)} ≠ ∅. The Mullineux symbol of λ is an array

G(λ) = (\begin{matrix}a_{1} & a_{2} & … & a_{z} \\ r_{1} & r_{2} & … & r_{z} \end{matrix})

where a_{i} is the number of nodes of the p-edge of λ^{(i)}, and r_{i} = h(λ^{(i)}) is the number of rows in λ^{(i)}, i = 1, 2, …, z.

The following theorem was first proved in [5]. An easier proof was recently found by Bessenrodt and Olsson [2].

Theorem 3.7. Let λ ∈ P_{n}, and

G(λ) = (\begin{matrix}a_{1} & a_{2} & … & a_{z} \\ r_{1} & r_{2} & … & r_{z} \end{matrix}) .

Put ε_{i} = 0 if p|a_{i}, and ε_{i} = 1 otherwise. Then

G(M(λ)) = (\begin{matrix}a_{1} & a_{2} & … & a_{z} \\ s_{1} & s_{2} & … & s_{z} \end{matrix})

where s_{i} = a_{i} + ε_{i} − r_{i}, i = 1, 2, …, z.

Thus the Mullineux symbol of M(λ) can be easily found from that of λ. On the other hand, a p-regular partition can be reconstructed from its Mullineux symbol— we just add p-edges starting from the last one.

Example. Let λ = (6, 4^{2}, 2, 1). Then the rim of λ contains the nodes represented by numbers in the following picture.

[figure: 3; 2; 1; 4; 7; 6; 5; 9; 8; 10]

Let p = 5. The nodes of the p-edge (which consists of two p-segments) are coloured in black in the following picture.

[figure]

We have λ^{(2)} = (3^{3}), λ^{(3)} = (2^{2}), λ^{(4)} = (1), z = 4, and

G(λ) = (\begin{matrix}8 & 5 & 3 & 1 \\ 5 & 3 & 2 & 1 \end{matrix}) , G(M(λ)) = (\begin{matrix}8 & 5 & 3 & 1 \\ 4 & 2 & 2 & 1 \end{matrix}) .

Finally, M(λ) = (7^{2}, 2, 1).

4. Some lemmas on branching and tensoring with sign

Let m, f ∈ ℕ. Represent m in the form

m = (p − 1)d + r, d ∈ ℤ, 0 < r ≤ p − 1.

We need to consider certain classes of partitions. Set

β(m, f) = ((f + d)^{p−1}, (f + d − 1)^{p−1}, …, (f + 1)^{p−1}, f^{r});
α(m, f) = ((f + d)^{r}, (f + d − 1)^{p−1}, (f + d − 2)^{p−1}, …, f^{p−1});
α(m) = α(m, 1).

Obviously h(β(m, f)) = h(α(m, f)) = m.

If μ = (μ_{1}, …, μ_{t}) is a p-regular partition and f > μ_{1}, put

β(m, f; μ) = ((f + d)^{p−1}, (f + d − 1)^{p−1}, …, (f + 1)^{p−1}, f^{r}, μ_{1}, …, μ_{t});
α(m, f; μ) = ((f + d)^{r}, (f + d − 1)^{p−1}, (f + d − 2)^{p−1}, …, f^{p−1}, μ_{1}, …, μ_{t})

(glue μ to the bottom of β(m, f) and α(m, f), respectively).

Lemma 4.1. The Specht module S^{α(m)} is irreducible.

Proof. This follows from Theorem 3.6.

Let us fix a p-regular partition λ with k distinct parts, i.e.

λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}, …, l_{k}^{a_{k}}), l_{1} > l_{2} > ⋯ > l_{k} > 0, 0 < a_{i} < p.

Then λ has k removable and k+1 indent nodes. Let A_{1}, …, A_{k} (resp., B_{1}, …, B_{k+1}) be all removable (resp., indent) nodes of λ numbered from top to bottom, i.e. A_{i} = (a_{1} + ⋯ + a_{i}, l_{i}), B_{j} = (a_{1} + ⋯ + a_{j−1} + 1, l_{j} + 1), i = 1, …, k, j = 1, …, k + 1 (l_{k+1} is interpreted as 0).

Lemma 4.2. Assume res A_{1} = res A_{2} = ⋯ = res A_{r} for some r ∈ {1, …, k}. Then A_{1}, A_{2}, …, A_{r} are normal nodes.

Proof. Since λ is p-regular, we have res A_{i} ≠ res B_{i} for all i = 1, …, k. Now use 3.4.

Lemma 4.3. Let r ≥ 1 be the largest integer such that

l_{1} = l_{2} + 1 = l_{3} + 2 = ⋯ = l_{r} + (r − 1)

and

a_{2} = a_{3} = ⋯ = a_{r} = (p − 1).

Then r is minimal such that λ_{A_{r}} is p-regular. Moreover, A_{r} is normal for λ.

Proof. The first part of the lemma follows from the definitions. To prove that A_{r} is normal it suffices to note that

res A_{1} = res A_{2} = ⋯ = res A_{r}

and use Lemma 4.2.

Corollary 4.4. Let λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}, …, l_{k}^{a_{k}}) ∈ P_{n}, with l_{k} = 1. Assume that the bottom removable node A_{k} is the only removable node A_{i} such that λ_{A_{i}} is p-regular. Then λ = α(a_{1} + a_{2} + ⋯ + a_{k}).

Proof. It follows from Lemma 4.3 that

l_{i} = l_{k} + (k − i), i = 1, 2, …, k,

and

a_{2} = a_{3} = ⋯ = a_{k} = p − 1,

as desired.

Definition 4.5. Let λ be a p-regular partition. We denote by A(λ) the top removable node of λ such that λ_{A(λ)} is p-regular. Define

φ(λ) = λ_{A(λ)}.

Lemma 4.6. Let λ ∈ P_{n}. Then

D^{φ(λ)} ∈ Irr(D^{λ}↓Σ_{n−1}).

Proof. By Lemma 4.3, A(λ) is normal for λ. So it suffices to apply Theorem 3.5.

Remark. Lemma 4.6 is known [8]. We reprove it here for completeness only. Note however that the result is insufficient to prove Proposition 4.14. There we do have to use Theorem 3.5.

Lemma 4.7. Let μ be a p-regular partition. Assume that for some m we have α(m) ⊆ μ. Set n = |α(m)|. Then

D^{α(m)} ∈ Irr(D^{μ}↓Σ_{n})

Proof. In view of Lemma 4.6, it suffices to show that α(m) = φ^{i}(μ) for some i. Since φ^{|μ|}(μ) = ∅ and (m, 1) ∈ α(m) ⊆ μ, there exists i such that A(φ^{i}(μ)) = (m, 1). So (m, 1) is the top removable node of φ^{i}(μ) leaving a p-regular diagram after its removal. Moreover, (m, 1) belongs to the first column, so it is the bottom removable node of φ^{i}(μ). By Corollary 4.4 we now get φ^{i}(μ) = α(m).

Proposition 4.8. Let n ∈ ℕ and λ be a p-regular partition with h(λ) ≥ n(p − 1). Then

Irr(D^{λ}↓Σ_{n}) = Irr Σ_{n}.

Proof. Since λ is p-regular and h(λ) ≥ n(p − 1) we have α(n(p − 1)) ⊆ λ. Let j = |α(n(p − 1))|. Then Lemma 4.7 implies

D^{α(n(p−1))} ∈ Irr(D^{λ}↓Σ_{j}).

Let μ ∈ P_{n}. Then μ ⊆ α(n(p − 1)). In view of Lemma 4.1,

D^{α(n(p−1))} = S^{α(n(p−1))}.

Now Theorem 3.3 implies that D^{α(n(p−1))}↓Σ_{n} has a filtration such that S^{μ} occurs as its factor. Since μ is p-regular, D^{μ} ∈ Irr(S^{μ}) by Lemma 3.1, and the result follows.

We need to somewhat develop Lemma 4.7. Our goal now is Proposition 4.14.

Lemma 4.9. Let μ = (μ_{1}, …, μ_{s}) be a p-regular partition (μ may be ∅) and f be an integer with f > μ_{1} + 1. Set α = α(m, f; μ), β = β(m, f − 1; μ), n = |β|. Then

D^{β} ∈ Irr(D^{α}↓Σ_{n}).

Proof. Let m = (p − 1)d + r, d ∈ ℤ, 0 < r ≤ p − 1. Define the partitions α^{(0)}, α^{(1)}, …, α^{(r)} and the sets M_{0}, M_{1}, …, M_{r−1} of nodes inductively as follows. Set α^{(0)} = α and α^{(i+1)} = α^{(i)} \ M_{i} where M_{i} is the set of the top (d + 1) removable nodes of α^{(i)}, i = 0, 1, …, r − 1. Note that all nodes of the set M_{i} are of the same residue. So Lemma 4.2 implies that the nodes of M_{i} are normal nodes for the partition α^{(i)} (i = 0, 1, …, r − 1). Therefore

D^{α^{(i+1)}} ∈ Irr (D^{α^{(i)}} ↓ Σ_{|α^{(i+1)}|})

for all i = 0, 1, …, r − 1. It remains to observe that α^{(r)} = β.

Lemma 4.10. Let μ = (μ_{1}, …, μ_{s}) be a p-regular partition (μ may be ∅), f > μ_{1} + 1. Set β = β(m, f; μ), γ = β(m, f − 1; μ), n = |γ|. Then

D^{γ} ∈ Irr(D^{β}↓Σ_{n}).

Proof. The proof is similar to that of Lemma 4.9.

Lemma 4.11. Let μ = (μ_{1}, …, μ_{s}) be a p-regular partition, f > μ_{1} + 1. Set ν = φ(μ) (see 4.5), β = β(m, f; μ), γ = β(m, f − 1; ν), n = |γ|. Then

D^{γ} ∈ Irr(D^{β}↓Σ_{n}).

Proof. Let m = d(p − 1) + r, d ∈ ℤ, 0 < r ≤ p − 1. It follows from the definition of β that in the first m rows there are exactly d + 1 indent nodes for β. Moreover, these indent nodes all have the same residue, say ω. Let A be the removable node for β defined from the equality

β = β(m, f; ν) ⊔ {A}.

Assume res A ≠ ω. Then A is normal for β. Therefore, setting β′ = β(m, f; ν), n′ = |β′|, we have

D^{β′} ∈ Irr(D^{β}↓Σ_{n′}),

in view of Theorem 3.5. Moreover, Lemma 4.10 implies D^{γ} ∈ Irr(D^{β′}↓Σ_{n}), and we are done.

Now let res A = ω. Put β′′ = β(m, f − 1, μ), n′′ = |β′′|. (Actually n′′ = n + 1.) By Lemma 4.10, D^{β′′} ∈ Irr(D^{β}↓Σ_{n′′}). Moreover, the removable nodes for β′′ in the first m rows all have residue ω − 1. So A is normal for β′′. Since γ = β′′_{A}, we get from Theorem 3.5

D^{γ} ∈ Irr(D^{β′′}↓Σ_{n}),

and the result follows.

Lemma 4.12. Let μ be a p-regular partition and f be an integer with f > |μ| + 1. Set β = β(m, f; μ), e = f − |μ|, γ = β(m, e), n = |γ|. Then

D^{γ} ∈ Irr(D^{β}↓Σ_{n}).

Proof. Apply Lemma 4.11 sufficiently many times.

The following result is a generalization of Lemma 4.7.

Lemma 4.13. Let

λ = (l_{1}, l_{2}, …, l_{m}, l_{m+1}, …, l_{s})

be a p-regular partition, with l_{m} > l_{m+1}. Set

μ = (l_{m+1}, …, l_{s})
α = α(m, l_{m}; μ), n = |α|

(μ = ∅ if m = s). Then

D^{α} ∈ Irr(D^{λ}↓Σ_{n}).

Proof. We choose the maximal i ≥ 0 such that φ^{i}(λ) has the form

φ^{i}(λ) = (k_{1}, …, k_{m−1}, l_{m}, …, l_{s}),

see Definition 4.5. This means that the node A(φ^{i}(λ)) lies in the m-th row or below. Since l_{m} ≠ l_{m+1}, it follows from Lemma 4.3 that (k_{1}, …, k_{m−1}, l_{m}) = α(m, l_{m}). Therefore φ^{i}(λ) = α, and the result follows from Lemma 4.6.

Proposition 4.14. Let λ = (l_{1}, l_{2}, …, l_{s}) be a p-regular partition, m ∈ {1, 2, …, s}, and

l_{m} > 2 + ∑_{i=m+1}^{s} l_{i}.

Set f = l_{m} − 1 − ∑_{i=m+1}^{s} l_{i}, β = β(m, f), n = |β|. Then

D^{β} ∈ Irr(D^{λ}↓Σ_{n}).

Proof. Set

μ = (l_{m+1}, …, l_{s}),
α = α(m, l_{m}; μ),
n_{1} = |α|.

By Lemma 4.13,

D^{α} ∈ Irr(D^{λ}↓Σ_{n_{1}}).

Let

β_{1} = β(m, l_{m} − 1; μ), n_{2} = |β_{1}|.

Then

D^{β_{1}} ∈ Irr(D^{α}↓Σ_{n_{2}})

by Lemma 4.9. Now apply Lemma 4.12 with β = β_{1} and γ = β.

Lemma 4.15. Let λ ∈ P_{n} and

cont(λ) = (c_{0}, c_{1}, …, c_{p−1}).

Assume that D^{μ} ∈ Irr(D^{λ}↓Σ_{n−1}) and

cont(μ) = (d_{0}, d_{1}, …, d_{p−1}).

Then there exists j ∈ {0, 1, …, p − 1} such that d_{j} = c_{j} − 1 and d_{i} = c_{i} for i ≠ j.

Proof. By Theorem 3.3, the restriction S^{λ}↓Σ_{n−1} has a filtration with factors S^{λ_{A}} where A is a removable node for λ. Note that

cont(λ_{A}) = (c_{0}, …, c_{j} − 1, …, c_{p−1})

if res A = j. By 3.1, D^{λ} ∈ Irr(S^{λ}) so

Irr(D^{λ}↓Σ_{n−1}) ⊆ Irr(S^{λ}↓Σ_{n−1}) = ∪_{A} Irr(S^{λ_{A}}).

Now it remains to use Lemma 3.2.

Remark. Actually, much more is known about Irr(D^{λ}↓Σ_{n−1}). For example, the contents of the factors Irr(D^{λ}↓Σ_{n−1}) are described in [14]. See also [16] for more information. One can use the argument from the proof of 4.15 to show that D^{μ} ∈ Irr(D^{λ}↓Σ_{n−1}) implies μ dominates some λ_{A}.

The rest of this section is devoted to certain results concerning the Mullineux map described at the end of Section 3.

Lemma 4.16. Let e, m ∈ ℕ, β = β(m, pe). Represent m in the form

m = (p − 1)d + r, d ∈ ℤ, 0 < r ≤ p − 1.

Let M(β) = γ = (γ_{1} ≥ γ_{2} ≥ ⋯ ≥ γ_{s} > 0). Then s = p + d − r and γ_{s} ≥ re.

Proof. One can easily see that the Mullineux symbol of β has the form

G(β) = (\underbrace{{(d + 1)p \atop d(p − 1) + r} {(d + 1)p \atop d(p − 1) + r} … {(d + 1)p \atop d(p − 1) + r}}_{re times} the rest ).

So, by Theorem 3.7, the Mullineux symbol G(γ) has the form

G(γ) = (\underbrace{{(d + 1)p \atop p + d − r} {(d + 1)p \atop p + d − r} … {(d + 1)p \atop p + d − r}}_{re times} the rest ).

It follows from the Mullineux algorithm that h(γ) = p+d−r. Moreover, the p-edge of a partition has nodes in every row. In particular, the p-edge of a partition with s rows has at least one node in the sth row. It follows from the form of G(γ) found above that the partitions γ^{(1)}, γ^{(2)}, …, γ^{(re)} (defined at the end of Section 3) all have s rows. Hence γ_{s} ≥ re.

Corollary 4.17. Let m, e ∈ ℕ, β = β(m, p(p − 1)e), γ = M(β). Represent m in the form

m = (p − 1)d + r, d ∈ ℤ, 0 < r ≤ p − 1.

Define m_{1} = p+d−r. Assume that f ∈ ℤ satisfies 0 < f < re and set δ = β(m_{1}, f), n = |δ|. Then

D^{δ} ∈ Irr(D^{γ}↓Σ_{n}).

Proof. Follows immediately from Lemma 4.16, Proposition 4.14 (with m = s = p + d − r), and Lemma 4.10 (with μ = ∅, m = p + d − r).

Lemma 4.18. Let e ∈ ℕ, β = β(p, p(p − 1)e), γ = M(β). Then

γ = ((p − 1) + pe(p − 1)^{2}, (pe)^{p−1}).

Proof. One can easily see that the Mullineux symbol of β is

G(β) = (\underbrace{{2p \atop p} … {2p \atop p}}_{(p − 1)e times} \underbrace{{p \atop p − 1} … {p \atop p − 1}}_{(p − 1)(p − 2)e times} {p − 1 \atop p − 1}).

So, by Theorem 3.7, the Mullineux symbol of γ is

G(γ) = (\underbrace{{2p \atop p} … {2p \atop p}}_{(p − 1)e times} \underbrace{{p \atop 1} … {p \atop 1}}_{(p − 1)(p − 2)e times} {p − 1 \atop 1}),

and the lemma follows.

Corollary 4.19. Let p > 2, n = (p−2)+pe(p−1)(p−2), e ∈ ℕ, β = β(p, p(p−1)e), γ = M(β). Then

D^{(n)} ∈ Irr(D^{γ}↓Σ_{n}).

Proof. By Lemma 4.18, γ = (γ_{1}, γ_{2}, …, γ_{p}) where γ_{1} = (p − 1) + pe(p − 1)^{2}, γ_{2} = γ_{3} = ⋯ = γ_{p} = pe. Observe that γ_{1} −(γ_{2} +⋯+γ_{p}) = (p−1)+pe(p−1)(p−2) > 2. Moreover, (n) = β(1, γ_{1} −(γ_{2} +⋯+γ_{p})−1). Now, by Proposition 4.14 (with m = 1, λ = γ) we have the desired result.

5. Inductive systems for the finitary symmetric group

Now we apply the general notions of Section 2 to the case G = Σ_{∞} = ∪_{i=1}^{∞}Σ_{i}.

Definition 5.1. Let Φ = {Φ_{i}}_{i∈ℕ} be an inductive system for Σ_{∞}. We define its height h(Φ) as

h(Φ) = sup{h(λ) | D^{λ} ∈ Φ_{i} for some i ∈ ℕ}

(see (1)).

Proposition 5.2. Let Φ = {Φ_{i}}_{i∈ℕ} be an inductive system for Σ_{∞}. Assume that h(Φ) = +∞. Then Φ_{i} = Irr Σ_{i} for all i ∈ ℕ.

Proof. By assumption, for any i ∈ ℕ there exists j > i and D^{λ} ∈ Φ_{j} with h(λ) ≥ i(p − 1). Then

Φ_{i} ⊇ Irr(D^{λ}↓Σ_{i}) = Irr Σ_{i},

by Proposition 4.8.

In [15] the second author has described all semisimple inductive systems for Σ_{∞} (see Definition 2.7). Surprisingly, it will turn out that the minimal inductive systems (i.e. those which correspond to the maximal ideals of F Σ_{∞}) are exactly indecomposable semisimple inductive systems. We formulate some results from [15].

Recall the constant χ(λ) defined in (2). For s = 1, 2, …, p − 1 put

Θ_{i}^{s} = {D^{μ} | |μ| = i, h(μ) = s, χ(μ) ≤ p},
Ω_{i}^{s} = {D^{μ} | μ = (μ_{1}, μ_{2}, …), |μ| = i, h(μ) < s, μ_{1} ≤ p − s}.

Note that Θ_{i}^{s} ∩ Ω_{i}^{s} = ∅ and Ω_{i}^{s} = ∅ for i > (p − s)(s − 1). Set

Φ(s)_{i} = Θ_{i}^{s} ⊔ Ω_{i}^{s}. (6)

Theorem 5.3. [15]

(i) Φ(s) = {Φ(s)_{i}}_{i∈ℕ}, s = 1, 2, …, p − 1, are all distinct semisimple minimal inductive systems for Σ_{∞}.

(ii) Let i ∈ ℕ. If j ≫ i then for any D^{λ} ∈ Φ(s)_{j} we have

Φ(s)_{i} = Irr(D^{λ}↓Σ_{i}).

Proof. (ii) is proved in [15, 2.6, 2.7], (i) is an immediate consequence of (ii) and [15, 2.8].

An important specific feature of the representation theory of symmetric groups is the presence of the sign module (denoted by sgn). Tensoring with sgn is often an effective trick.

Definition 5.4. Let Φ = {Φ_{i}}_{i∈ℕ} be an inductive system for Σ_{∞}. Define the inductive system Φ^{σ} = {Φ_{i}^{σ}}_{i∈ℕ} by

Φ_{i}^{σ} = {D^{λ} ⊗ sgn | D^{λ} ∈ Φ_{i}},

where sgn is the sign representation of Σ_{i}.

Φ^{σ} is well defined in view of the following lemma.

Lemma 5.5. Let Φ = {Φ_{i}}_{i∈ℕ} be an inductive system. Then (i) Φ^{σ} = {Φ_{i}^{σ}}_{i∈ℕ} is an inductive system; (ii) Φ is minimal if and only if Φ^{σ} is so; (iii) Φ is semisimple if and only if Φ^{σ} is so.

Proof. (i) and (iii) follow from the isomorphism

(7) (D ⊗ sgn)↓Σ_{n−1} ≅ (D↓Σ_{n−1}) ⊗ sgn

where D is an F Σ_{n}-module and sgn in the left (resp., right) hand side means the sign representation of Σ_{n} (resp., Σ_{n−1}). The second part follows from the equality (Φ^{σ})^{σ} = Φ.

Lemma 5.6. Let s ∈ {1, 2, …, p − 1}. Then Φ(s)^{σ} = Φ(p − s).

Proof. By Lemma 5.5 (ii), (iii), Φ is minimal and semisimple if and only if Φ^{σ} is so. So Theorem 5.3 (i) implies Φ(s)^{σ} = Φ(t) for some t ∈ {1, 2, …, p − 1}. Note that D^{((pe)^{s})} ∈ Φ(s)_{pes} for any e ∈ ℕ. However h(M((pe)^{s})) = p − s by Lemma 4.16, which implies the demanded result.

Theorem 5.7. Let p > 2, and Φ be an inductive system for Σ_{∞}. Then Φ is minimal if and only if Φ = Φ(s) for some s ∈ {1, 2, …, p − 1}.

Proof. The “if-part” follows from Theorem 5.3 (i).

Let Φ be minimal. If h(Φ) = +∞ then, by Proposition 5.2,

Φ_{i} = Irr Σ_{i}, i ∈ ℕ.

However this system is not minimal (for example because it contains the trivial inductive system Φ(1)). So h(Φ) = h < ∞. It follows that for any N ∈ ℕ there exists i ∈ ℕ and D^{λ} ∈ Φ_{i} with λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{h} ≥ 0) such that l_{1} > N. This shows that the set

V (Φ) = {v ∈ ℕ |  for any N ∈ ℕ there exists i ∈ ℕ and D^{λ} ∈ Φ_{i}
with λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{h} ≥ 0) such that l_{v} > N}

is not empty. Since also V (Φ) ⊆ {1, 2, …, h} we can define

m = m(Φ) = max V (Φ).

It follows from the definition of m that there exists c ∈ ℕ such that

∑_{j=m+1}^{h} l_{j} ≤ c

for any i ∈ ℕ and any D^{(l_{1},...,l_{h})} ∈ Φ_{i}. Let N_{0} ≥ c + 2. By definition of m, there exists i ∈ ℕ and D^{(l_{1},...,l_{h})} ∈ Φ_{i} such that l_{m} > N_{0}. Set

f = l_{m} − ∑_{j=m+1}^{h} l_{j} − 1,
β = β(m, f), n = |β|.

By Proposition 4.14,

D^{β} ∈ Irr(D^{(l_{1},...,l_{h})}↓Σ_{n}) ⊆ Φ_{n}.

Note that f > N_{0} − c − 1. Set

f′ = N_{0} − c − 1,
β′ = β(m, f′), n′ = |β′|.

By Lemma 4.10,

D^{β′} ∈ Irr(D^{β}↓Σ_{n′}) ⊆ Φ_{n′}.

Choosing N_{0} = c + 2, c + 3, … (then f′ = 1, 2, …) we conclude that

D^{β(m,e)} ∈ Φ_{|β(m,e)|} for any e ∈ ℕ. (8)

We call a number t ∈ ℕ admissible for Φ if for any N ∈ ℕ there is f > N such that D^{β(t,f)} belongs to Φ_{|β(t,f)|}. By Lemma 4.10, we have

t ∈ ℕ is admissible for Φ if and only if

D^{β(t,f)} ∈ Φ_{|β(t,f)|} for all f ∈ ℕ. (9)

By (8), m(Φ) is admissible for Φ. This allows us to define a constant s(Φ) as follows

s(Φ) = min{t ∈ ℕ | t is admissible for Φ}.

Let us consider the inductive system Φ^{σ} (see Definition 5.4). By Lemma 5.5 (ii), Φ^{σ} is minimal. So s(Φ^{σ}) is also well defined. We prove the following intermediate fact.

(*) Assume that s(Φ) ≤ s(Φ^{σ}). Then Φ = Φ(t) for some t ∈ {1, 2, …, p − 1}.

Note that (*) immediately implies our theorem. Indeed, if s(Φ) ≤ s(Φ^{σ}) then (*) is just the theorem’s claim. Let s(Φ) > s(Φ^{σ}). Since (Φ^{σ})^{σ} = Φ, (*) implies Φ^{σ} = Φ(t) for some t ∈ {1, 2, …, p − 1}. Then Φ = Φ(p − t) by Lemma 5.6.

Proof of (*). Set s = s(Φ). We consider three cases.

(1) s ≤ p − 1. Then β(s, f) = (f^{s}), and so χ(β(s, f)) = s < p. Hence D^{β(s,f)} ∈ Φ(s)_{|β(s,f)|}, see (6). By (9), the modules D^{β(s,f)}, f ∈ ℕ, belong to the corresponding Φ_{|β(s,f)|}. It now follows from Theorem 5.3 (ii) that Φ(s) ⊆ Φ. Since Φ is minimal, we have Φ = Φ(s).

(2) s = p. Set β(e) = β(p, p(p − 1)e). By (9),

D^{β(e)} ∈ Φ_{|β(e)|}

for any e ∈ ℕ. Set n(e) = (p − 2) + ep(p − 1)(p − 2). Then, by Corollary 4.19,

D^{(n(e))} ∈ Φ_{n(e)}^{σ}

Note that (n) = β(n, 1). So n(e) → +∞ as e → +∞ implies that 1 is admissible for Φ^{σ}. This contradicts the assumption s(Φ^{σ}) ≥ s(Φ).

(3) s > p. Represent s in the form

s = (p − 1)d + r, d ∈ ℤ, 0 < r ≤ p − 1.

By (9),

D^{β(s,p(p−1)e)} ∈ Φ_{|β(s,p(p−1)e)|} for any e ∈ ℕ.

By Corollary 4.17,

D^{β(p+d−r,f)} ∈ Φ_{|β(p+d−r,f)|}^{σ}

for any f < re. This shows that p + d − r is admissible for Φ^{σ}, whence s(Φ^{σ}) ≤ p + d − r. However, this contradicts the assumption s(Φ^{σ}) ≥ s(Φ) because

p + d − r < (p − 1)d + r = s

when s > p. The contradiction obtained completes the proof of (*) and the theorem.

Corollary 5.8. Let p > 2. Set

I(s)_{n} = ⋂_{D∈Φ(s)_{n}} Ann_{F Σ_{n}}(D), s = 1, …, p − 1, n ∈ ℕ.

Then I(s)_{1} ⊂ I(s)_{2} ⊂ . . .. Moreover,

I(s) = ∪_{n=1}^{∞}I(s)_{n}, s = 1, …, p − 1,

are exactly all distinct maximal ideals of F Σ_{∞}, and I(s) ∩ F Σ_{n} = I(s)_{n}.

Proof. In view of Proposition 2.6 (ii) and Theorem 5.7, the maximal ideals of F Σ_{∞} are exactly I(Φ(s)), s = 1, 2, …, p − 1. However, the inductive systems Φ(s) are semisimple. So the result follows from 2.8.

6. Inductive systems for the finitary alternating group

Throughout the section we assume that p > 2.

We first briefly discuss how to label irreducible F A_{n}-modules. It follows easily from the Clifford theory [3, 49.2] (see [4]) that the restriction D^{λ}↓_{A_{n}}^{Σ_{n}} is either irreducible or splits into a direct sum of two irreducibles. In the former case we write

D^{λ}↓_{A_{n}}^{Σ_{n}} = E^{λ},

and in the latter case we write

D^{λ}↓_{A_{n}}^{Σ_{n}} = E_{+}^{λ} ⊕ E_{−}^{λ}.

It is of importance that D^{λ}↓_{A_{n}}^{Σ_{n}} is irreducible provided λ ≠ M(λ). (In fact, if F is a splitting field for A_{n} then D^{λ}↓_{A_{n}}^{Σ_{n}} is irreducible if and only if λ ≠ M(λ). However we want to work over an arbitrary field). Every irreducible F A_{n}-module is of the form E^{λ} or E_{±}^{μ}. Furthermore, E_{+}^{λ} ≅ E̸_{−}^{λ}, E_{±}^{λ} ≅ E̸_{±}^{μ}, E^{λ} ≅ E̸_{±}^{μ}, and E^{λ} ≅ E^{μ} if and only if λ = M(μ) (for all permissible λ ≠ μ).

The following result was first proved in [17].

Lemma 6.1. [17] Let λ ∈ P_{n}, and

cont(λ) = (c_{0}, c_{1}, …, c_{p−1}).

Then

cont(M(λ)) = (d_{0}, d_{1}, …, d_{p−1}).

where d_{0} = c_{0} and d_{i} = c_{p−i} for i = 1, 2, …, p − 1.

Lemma 6.2. Let p > 2. If λ is a Young diagram with cont(λ) = (c_{0}, c_{1}, …, c_{p−1}) then c_{0} ≤ c_{p−1} + 1.

Proof. Let A = (i, j) be an arbitrary node. Set

f(A) = \{ \begin{matrix}(i, j − 1), if j > 1; \\ (i + 1, j), if j = 1. \end{matrix}

If res A = 0 then res f(A) = p − 1. Also, if A ∈ λ but f(A) ∉ λ then A = (m, 1) where m = h(λ). Moreover, A ≠ B and res A = res B imply f(A) ≠ f(B) since p > 2. Thus, to all nodes of λ of residue 0, except possibly one, we associate different nodes of λ of residue p − 1.

Lemma 6.3. Let λ ∈ P_{n}, ν ∈ P_{n+1} be such that

D^{λ}↓_{A_{n}}^{Σ_{n}} = E_{+}^{λ} ⊕ E_{−}^{λ} and D^{ν}↓_{A_{n+1}}^{Σ_{n+1}} = E_{+}^{ν} ⊕ E_{−}^{ν}.

Assume that E_{+}^{λ} ∈ Irr(E_{+}^{ν} ↓_{A_{n}}^{A_{n+1}}) or E_{+}^{λ} ∈ Irr(E_{−}^{ν} ↓_{A_{n}}^{A_{n+1}}). Then D^{λ} ∈ Irr(D^{ν}↓_{Σ_{n}}^{Σ_{n+1}}).

Proof. If D^{λ} ∉ Irr(D^{ν}↓_{Σ_{n}}^{Σ_{n+1}}) then

E_{+}^{λ} ∉ Irr((D^{ν}↓_{Σ_{n}}^{Σ_{n+1}})↓_{A_{n}}^{Σ_{n}}) = Irr((D^{ν}↓_{A_{n+1}}^{Σ_{n+1}})↓_{A_{n}}^{A_{n+1}}) = Irr(E_{+}^{ν}↓_{A_{n}}^{A_{n+1}}) ∪ Irr(E_{−}^{ν}↓_{A_{n}}^{A_{n+1}}),

giving a contradiction.

Lemma 6.4. Let p > 2, and λ ∈ P_{n} be such that D^{λ}↓_{A_{n}}^{Σ_{n}} = E_{+}^{λ} ⊕ E_{−}^{λ}. Assume Φ = {Φ_{i}}_{i∈ℕ} is an inductive system for A_{∞}. Then E_{+}^{λ} ∈ Φ_{n} if and only if E_{−}^{λ} ∈ Φ_{n}.

Proof. Assume E_{+}^{λ} ∈ Φ_{n} and show that E_{−}^{λ} ∈ Φ_{n} (the implication E_{−}^{λ} ∈ Φ_{n} ⇒ E_{+}^{λ} ∈ Φ_{n} is obtained similarly). Note that M(λ) = λ. We shall prove the following intermediate fact.

(**) There is k > n and E^{μ} ∈ Φ_{k} such that E_{+}^{λ} ∈ Irr(E^{μ}↓_{A_{n}}^{A_{k}}).

This will imply the lemma. Indeed, E^{μ} = D^{μ}↓_{A_{k}}^{Σ_{k}}. Therefore

E^{μ}↓_{A_{n}}^{A_{k}} ≅ (D^{μ}↓_{Σ_{n}}^{Σ_{k}})↓_{A_{n}}^{Σ_{n}} (10)

So D^{λ} ∈ Irr(D^{μ}↓_{Σ_{n}}^{Σ_{k}}) because otherwise we would have E_{+}^{λ} ∉ Irr(E^{μ}↓_{A_{n}}^{A_{k}}). It now follows from (10) that E_{−}^{λ} ∈ Irr(E^{μ}↓_{A_{n}}^{A_{k}}).

We now prove (**). Let cont(λ) = (c_{0}, c_{1}, …, c_{p−1}). Since Φ is an inductive system there exists an irreducible module E ∈ Φ_{n+1} such that E_{+}^{λ} ∈ Irr(E↓_{A_{n}}^{A_{n+1}}). If E = E^{ν} we can take μ = ν, k = n + 1. Otherwise E = E_{+}^{ν} or E_{−}^{ν} for some ν with M(ν) = ν. In view of Lemma 6.3, we have

D^{λ} ∈ Irr(D^{ν}↓_{Σ_{n}}^{Σ_{n+1}}).

So by Lemma 4.15,

cont(ν) = (c_{0}, …, c_{j} + 1, …, c_{p−1})

for some j ∈ {0, 1, …, p − 1}. Since M(λ) = λ and M(ν) = ν, Lemma 6.1 forces j = 0. Thus,

cont(ν) = (c_{0} + 1, c_{1}, …, c_{p−1}).

Repeating this argument i times we either arrive to a module E^{μ} ∈ Φ_{n+i} such that E_{+}^{λ} ∈ Irr(E^{μ}↓_{A_{n}}^{A_{n+i}}) or we get a partition ν with

cont(ν) = (c_{0} + i, c_{1}, …, c_{p−1}).

However, Lemma 6.2 shows that i can not be too large.

Lemma 6.5. Let Φ = {Φ_{i}}_{i∈ℕ} be an inductive system for Σ_{∞}. Set

Φ̄_{i} = Irr(Φ_{i}↓_{A_{i}}^{Σ_{i}}).

Then Φ̄ = {Φ̄_{i}}_{i∈ℕ} is an inductive system for A_{∞}.

Proof. Note that

Irr(Φ̄_{i}↓_{A_{i−1}}^{A_{i}}) = Irr(Irr(Φ_{i}↓_{A_{i}}^{Σ_{i}})↓_{A_{i−1}}^{A_{i}}) = Irr(Φ_{i}↓_{A_{i−1}}^{Σ_{i}})
= Irr(Irr(Φ_{i}↓_{Σ_{i−1}}^{Σ_{i}})↓_{A_{i−1}}^{Σ_{i−1}}) = Irr(Φ_{i−1}↓_{A_{i−1}}^{Σ_{i−1}}) = Φ̄_{i−1},

as demanded.

Lemma 6.5 allows us to give the following definition.

Definition 6.6. Let Φ be an inductive system for Σ_{∞}. Define the inductive system Φ↓A_{∞} for A_{∞} as Φ↓A_{∞} = Φ̄ (see Lemma 6.5).

IDEALS IN GROUP ALGEBRAS 23

Definition 6.7. Let S ⊆ Irr(Σ_{n}). Define the set S^{σ} ⊆ Irr(Σ_{n}) as

S^{σ} = {D ⊗ sgn | D ∈ S}.

Lemma 6.8. Let S, T ⊆ Irr(Σ_{n}), S^{σ} = S, T^{σ} = T. If Irr(S↓_{A_{n}}^{Σ_{n}}) = Irr(T ↓_{A_{n}}^{Σ_{n}}) then S = T. Proof. Assume D^{λ} ∈ S but D^{λ} ∉ T. Then D^{M(λ)} ∉ T. Hence E ∈ Irr(S↓_{A_{n}}^{Σ_{n}}) but E ∉ Irr(T ↓_{A_{n}}^{Σ_{n}}) where E = E^{λ} or E = E_{+}^{λ}. The contradiction obtained shows that S ⊆ T. Similarly, T ⊆ S. Lemma 6.9. Let S ⊆ Irr(Σ_{n}). If S^{σ} = S then (Irr(S↓Σ_{n−1}))^{σ} = Irr(S↓Σ_{n−1}). Proof. Follows from (7). Theorem 6.10. Let p > 2. The map Φ → Φ↓A_{∞} defines an isomorphism between the poset of the inductive systems Φ for Σ_{∞} such that Φ^{σ} = Φ and the poset of the inductive systems for A_{∞}. Proof. The map is injective by Lemma 6.8. We construct a right inverse map. Let Ψ = {Ψ_{i}}_{i∈ℕ} be an inductive system for A_{∞}. Set

Ψ̂_{i} = {D^{λ} ∈ Irr Σ_{i} | Irr(D^{λ}↓_{A_{i}}^{Σ_{i}}) ⊆ Ψ_{i}}, i ∈ ℕ.

We first show that Ψ_{i} = Irr(Ψ̂_{i}↓_{A_{i}}^{Σ_{i}}), i ∈ ℕ (11) Indeed, Ψ_{i} ⊇ Irr(Ψ̂_{i}↓_{A_{i}}^{Σ_{i}}) by definition of Ψ̂_{i}. On the other hand, let E ∈ Ψ_{i}. There are two cases:

(1) E = E^{λ};

(2) E = E_{+}^{λ} or E = E_{−}^{λ}.

In the case (1), E = D^{λ}↓_{A_{i}}^{Σ_{i}} hence D^{λ} ∈ Ψ̂_{i} and E ∈ Irr(Ψ̂_{i}↓_{A_{i}}^{Σ_{i}}). In the case (2), by Lemma 6.4, both E_{+}^{λ} and E_{−}^{λ} belong to Ψ. So D^{λ}↓_{A_{i}}^{Σ_{i}} = E_{+}^{λ} ⊕ E_{−}^{λ} implies that D^{λ} ∈ Ψ̂_{i} hence E ∈ Irr(Ψ̂_{i}↓_{A_{i}}^{Σ_{i}}).

It follows from the definition of Ψ̂_{j} that (Ψ̂_{j})^{σ} = Ψ̂_{j}, j ∈ ℕ. (12) Taking j = i − 1 gives (Ψ̂_{i−1})^{σ} = Ψ̂_{i−1}. (13) Taking j = i and using Lemma 6.9, we get

(14) (Irr(Ψ̂_{i}↓_{Σ_{i−1}}^{Σ_{i}}))^{σ} = Irr(Ψ̂_{i}↓_{Σ_{i−1}}^{Σ_{i}}).

Now we show that

(15) Irr(Irr(Ψ̂_{i}↓_{Σ_{i−1}}^{Σ_{i}})↓_{A_{i−1}}^{Σ_{i−1}}) = Irr(Ψ̂_{i−1}↓_{A_{i−1}}^{Σ_{i−1}}).

Indeed, using (11) twice, we obtain

Irr(Irr(Ψ̂_{i}↓_{Σ_{i−1}}^{Σ_{i}})↓_{A_{i−1}}^{Σ_{i−1}}) = Irr(Ψ̂_{i}↓_{A_{i−1}}^{Σ_{i}}) = Irr(Irr(Ψ̂_{i}↓_{A_{i}}^{Σ_{i}})↓_{A_{i−1}}^{A_{i}})
= Irr(Ψ_{i}↓_{A_{i−1}}^{A_{i}}) = Ψ_{i−1} = Irr(Ψ̂_{i−1}↓_{A_{i−1}}^{Σ_{i−1}}).

In view of Lemma 6.8, equations (13), (14) and (15) imply

Irr(Ψ̂_{i}↓_{Σ_{i−1}}^{Σ_{i}}) = Ψ̂_{i−1},

i.e. Ψ̂ is an inductive system. Moreover, (11) shows that Ψ̂↓A_{∞} = Ψ. It remains to note that Ψ ⊆ Φ if and only if Ψ̂ ⊆ Φ̂.

Theorem 6.11. Let p > 2. The inductive systems

Φ(s)↓A_{∞}, s = 1, 2, …, \frac{p − 1}{2},

are exactly all distinct minimal inductive systems for A_{∞}.

Proof. Let S be the set of all inductive systems Φ for Σ_{∞} which satisfy the following two conditions:

(1) Φ^{σ} = Φ.

(2) If Ψ is an inductive system for Σ_{∞}, Ψ ⊆ Φ, and Ψ^{σ} = Ψ then Ψ = Φ.

By Theorem 6.10,

{Φ↓A_{∞} | Φ ∈ S}

is the set of all distinct minimal inductive systems for A_{∞}.

We claim that Φ ∈ S if and only if Φ = Φ(s)∪Φ(s)^{σ} for some s ∈ {1, 2, …, p−1} (see (6)). Indeed, let Φ ∈ S. By Theorem 5.7, {Φ(s) | 1 ≤ s ≤ p − 1} are exactly all minimal inductive systems for Σ_{∞}. So Φ contains some Φ(s). Since Φ^{σ} = Φ, we also have Φ ⊇ Φ(s)^{σ}. Thus Φ(s) ∪ Φ(s)^{σ} ⊆ Φ. Since (Φ(s) ∪ Φ(s)^{σ})^{σ} = Φ(s) ∪ Φ(s)^{σ} and Φ ∈ S, we conclude that Φ = Φ(s) ∪ Φ(s)^{σ}.

On the other hand, if Φ(s) ∪ Φ(s)^{σ} ∉ S then there exists an inductive system Φ with Φ^{σ} = Φ such that Φ ⊂_{≠} Φ(s) ∪ Φ(s)^{σ}. As above, we find t ∈ {1, 2, …, p − 1}

such that Φ(t) ∪ Φ(t)^{σ} ⊆ Φ. Therefore,

Φ(t) ∪ Φ(t)^{σ} ⊂_{≠} Φ(s) ∪ Φ(s)^{σ}

for some s, t ∈ {1, 2, …, p−1}, which is false in view of Lemma 5.6 and the definition of Φ(u). Since (Φ(s) ∪ Φ(s)^{σ})↓A_{∞} = Φ(s)↓A_{∞} it remains to use Lemma 5.6 another time.

Lemma 6.12. The inductive systems Φ(s)↓A_{∞}, s = 1, 2, …, \frac{p−1}{2} are semisimple.

Proof. This follows immediately from the fact that Φ(s) are semisimple.

Corollary 6.13. Let p > 2. Set

J(t)_{n} = ⋂_{D∈Φ(t)_{n}} Ann_{F A_{n}}(D↓A_{n}), t = 1, …, \frac{p − 1}{2}, n ∈ ℕ.

Then (i) J(t)_{1} ⊂ J(t)_{2} ⊂ . . .. Moreover,

J(t) = ∪_{n=1}^{∞}J(t)_{n}, t = 1, …, \frac{p − 1}{2},

are exactly all distinct maximal ideals of F A_{∞}, and J(t) ∩ F A_{n} = J(t)_{n}. (ii) I(s) ∩ F A_{∞} = J(t) where t = min(s, p − s), s = 1, …, p − 1.

Proof. The proof of (i) is similar to that of Corollary 5.8.

(ii) Let t = min(s, p − s). In view of Corollary 5.8, Lemma 5.6 and (i), we have

I(s) ∩ F A_{i} = (I(s) ∩ F Σ_{i}) ∩ F A_{i} = (∩_{D∈Φ(s)_{i}} Ann_{F Σ_{i}}(D)) ∩ F A_{i} = ∩_{D∈Φ(s)_{i}} Ann_{F A_{i}}(D↓A_{i}) = ∩_{D∈Φ(t)_{i}} Ann_{F A_{i}}(D↓A_{i}) = J(t)_{i} = J(t) ∩ F A_{i}

for any i ∈ ℕ. This implies (ii).

Remark. In view of Corollaries 5.8 and 6.13 (i),

I(s) = ∪_{j>(p−s)(s−1)}I(s)_{j}

and

J(t) = ∪_{i>(p−t)(t−1)}J(t)_{i}.

This allows one to describe the ideals I(s) and J(t) in a more compact form, because Ω_{i}^{s} = ∅ for i > (p − s)(s − 1), and so

Ψ(s)_{i} = Θ_{i}^{s},

see (6). This is done in Theorem 1.1.

References

1. S. A. Amitsur, The polynomial identities of associative rings, in “Noetherian Rings and Rings with polynomial Identities”, Proc. Conf. Univ. Durham, 1979, pp. 1–38.

2. C. Bessenrodt and J.B. Olsson. On residue symbols and the Mullineux conjecture, preprint, 1996.

3. C. W. Curtis and I. Reiner, “Representation Theory of Finite Groups and Associative Algebras”. Interscience (Wiley), New York / London, 1962.

4. B. Ford, Irreducible representations of the alternating groups in odd characteristic, Proc. Amer. Math. Soc. (to appear).

5. B. Ford and A. Kleshchev, A proof of the Mullineux Conjecture, Math. Z. (to appear).

6. E. Formanek and D. Lawrence, The group algebra of the infinite symmetric group, Israel J. Math. 23 (1976), 325-331.

7. E. Formanek and C. Procesi, Mumford’s conjecture for the general linear group, Adv. in Math. 19 (1976), 292–305.

8. G. D. James, On the decomposition matrices of the symmetric groups, II, J. Algebra 43 (1976), 45–54.

9. G. D. James, On a conjecture of Carter concerning irreducible Specht modules, Math. Proc. Camb. Phil. Soc. 83 (1978), 11–17.

10. G. D. James, “The Representation Theory of the Symmetric Groups”, Lecture Notes in Mathematics, Vol. 682, Springer-Verlag, Berlin / New York, 1978.

11. G. D. James and A. Kerber, “The Representation Theory of the Symmetric Group”, Encyclopedia of Math. and its Applications, Vol. 16, Addison-Wesley, 1981.

12. G. D. James and G. E. Murphy, The determinant of the Gram matrix for a Specht module, J. Algebra 59 (1979), 222–235.

13. A. S. Kleshchev, Branching rules for modular representations of symmetric groups, II, J. Reine Angew. Math. 459 (1995), 163–212.

14. A. S. Kleshchev, Branching rules for modular representations of symmetric groups, III, J. London Math. Soc. (2) 54(1996), 25-38.

15. A. S. Kleshchev, Completely splittable representations of symmetric groups, J. Algebra 181 (1996), 584–592.

16. A. S. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups , Proc. Lond. Math. Soc. (to appear).

17. G. Mullineux, On the p-cores of p-regular diagrams, J. London Math. Soc. (2), 20 (1979), 222-225.

18. Yu. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR. Izvestiya 8 (1974), 727–760.

19. Yu. P. Razmyslov, “Identities of Algebras and Their Representations”, Nauka, Moscow, 1989 (in Russian).

20. L. H. Rowen, “Polynomial Identities in Ring Theory”, Academic Press, New York, 1980.

21. S. K. Sehgal and A. E. Zalesskii, Induced modules and some arithmetic invariants of the finitary symmetric groups, Nova J. Alg. Geom. 2 (1993), 89–105.

22. A. E. Zalesskii, Modular group rings of the finitary symmetric groups, Isr. J. Math. (1996) (to appear).

23. A. E. Zalesskii, Group rings of simple locally finite groups, in “Finite and locally finite groups”, Kluwer Acad. Publ., 1995, 219–246.

Institute of Mathematics, Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, BELARUS

E-mail address: imanb@imanb.belpak.minsk.by

Department of Mathematics, University of Oregon, Eugene OR 97403, U.S.A. E-mail address: klesh@math.uoregon.edu