BRANCHING RULES FOR SYMMETRIC GROUPS AND APPLICATIONS

A.S. KLESHCHEV

Department of Mathematics University of Oregon Eugene, OR 97403 USA E-mail: klesh@math.uoregon.edu

1. Introduction

Let F be an arbitrary field of characteristic p ≥ 0, and Σ_{n} be the symmetric group on n letters. We shall discuss finite dimensional representations of Σ_{n} over F or, equivalently, finite dimensional F Σ_{n}-modules. Basic facts important for our topic can be found in [18, 21, 20]. We recall here some of them.

An explicit (though complicated) construction associates to every partition λ of n (written λ ⊢ n) a ℤΣ_{n}-module S_{ℤ}^{λ}. This module is free as a ℤ-module. So the dimension of the F Σ_{n}-module S^{λ} := S_{ℤ}^{λ}⊗_{ℤ}F is independent of F. This module is called the Specht module (over F).

The properties of the Specht modules depend on the characteristic p of the ground field F. If p = 0, all modules S^{λ} are irreducible. Moreover, {S^{λ} | λ ⊢ n} is a complete set of the irreducible F Σ_{n}-modules (up to isomorphism). If p > 0, the modules S^{λ} are not irreducible in general. The following result shows that if S^{λ} is not 1-dimensional then it is reducible in some characteristic.

Proposition 1.1 [33] If λ ≠ (n), (1^{n}) then there exists a prime p such that S_{ℤ}^{λ} ⊗_{ℤ} F is reducible for any field F of characteristic p.

However, the modules S^{λ} can be used to get an information on the irreducible F Σ_{n}-modules. Recall that a partition λ is called p-regular if no part in λ is repeated p or more times. It was proved by James that if λ is p-regular then S^{λ} has a simple head, say D^{λ}, and that

{D^{λ} | λ ⊢ n, λ is p-regular}

is a complete set of the irreducible F Σ_{n}-modules up to isomorphism.

The fact that D^{λ} is the simple head of S^{λ}, being very useful, does not provide answers to many important questions about the irreducible modules. For example, the dimension of D^{λ} is not known in general.

Dimension problem. Find dim D^{λ}.

Remark. Carter’s criterion proved by James and Murphy [17, 22] describes when D^{λ} = S^{λ}. In this case we of course know the dimension of D^{λ} from the hook formula. For λ satisfying the Carter’s criterion we also know the dimension of D^{λ} ⊗ sgn. The Young diagram corresponding to D^{λ} ⊗ sgn is given by the Mullineux bijection, see Section 5. Mathieu [35] found another special class of the irreducibles whose dimensions can be described (see also [31] and Section 6 of this paper). Some asymptotic results [19] and lower bounds (see [29, 32] and (9) in Section 6) are also available. For dimensions of the irreducible modules corresponding to the two-part partitions see [8].

Define the decomposition numbers d_{λμ} for Σ_{n} by setting

d_{λμ} = d_{λμ}^{(n)} = [S^{λ} : D^{μ}]_{Σ_{n}} ,

the multiplicity of D^{μ} as a composition factor of S^{λ}. It is well known [18, 12.2] that d_{λλ} = 1 (for any p-regular λ) and d_{λμ} = 0 unless μ dominates λ. Hence

dim D^{λ} = dim S^{λ} − ∑_{μ⊳λ} d_{λμ} dim D^{μ}.

So the knowledge of the decomposition numbers d_{λμ} would provide us with an inductive algorithm for computing the dimensions of the irreducible modules D^{λ}, the inductive base being λ = (n) where dim D^{(n)} = 1.

Thus we have an even more general

Decomposition numbers problem for symmetric groups. Find the decomposition numbers d_{λμ}^{(n)} for all (admissible) λ, μ and n.

This is the central problem in the modular representation theory of Σ_{n}. This problem is also equivalent to a similar problem for simple algebraic groups of type A. To be more precise, assume for the moment that F is infinite and denote by L(λ) the irreducible polynomial representation of the algebraic group GL_{n}(F) with highest weight λ, see e.g. [15]. Then λ can be identified with a partition of a non-negative integer r with at most n parts. The central problem of the representation theory of GL_{n}(F) is to find the formal characters of the irreducible modules L(λ). This problem is equivalent to the problem of determining the “decomposition numbers”

f_{λμ} = f_{λμ}^{(n)} = [V (λ) : L(μ)]_{GL_{n}(F)}

where V (λ) denotes the Weyl module over GL_{n}(F) with highest weight λ. The relation between the decomposition numbers for the general linear and the symmetric groups is as follows.

Denote by λ′ the partition whose Young diagram is the transpose of the Young diagram of λ. We say that λ′ is the partition conjugate to λ.

Theorem 1.2 [15, 6.3,6.4,6.5]. Let λ and μ be partitions of n, μ being pregular. Then for any N ≥ n we have d_{λμ}^{(n)} = f_{λ′μ′}^{(N)}.

This result shows that the knowledge of the decomposition numbers for GL_{n}(F) would give us the knowledge of the decomposition numbers for Σ_{n}. The fact that the converse is in a sense true was recently proved by Erdmann. If ν is a dominant weight for GL_{n}(F) (i.e. a partition of r ∈ ℤ_{≥0} with at most n parts) denote by t(ν) the dominant weight

t(ν) = pν + (p − 1)(n − 1, n − 2, …, 1, 0).

Note that t(ν) is always p-regular.

Theorem 1.3 [9] f_{λμ} = d_{t(λ′),t(μ′)}.

This result shows that the decomposition numbers for GL_{n}(F) are equal to certain decomposition numbers for Σ_{N} when N is allowed to vary. Another interesting corollary of Theorem 1.3 is as follows: to find all decomposition numbers for the symmetric groups it suffices to find only decomposition numbers d_{λμ} with λ and μ in special blocks, namely the blocks corresponding to the cores of the form (p − 1)(k, k − 1, …, 1, 0) (see [21, 2.7]).

Thus the problem of determining the decomposition numbers for the symmetric groups and the problem of determining the formal characters of the irreducible modules over general linear groups are equivalent.

In this paper we are going to concentrate on another problem which is equivalent to these two problems:

Branching problem. Determine the branching multiplicities

b_{λμ} := [D^{λ} ↓_{Σ_{n−1}}: D^{μ}] .

Here λ and μ are arbitrary p-regular partitions of n and n − 1, respectively.

We shall show in Section 3 that, indeed, the numbers d_{λμ} determine the numbers b_{λμ} and vice versa.

Let P^{ν} denote the projective cover of D^{ν}. If μ is a p-regular partition of n − 1 then the induced module P^{μ} ↑^{Σ_{n}} is projective again. We claim that

P^{μ} ↑^{Σ_{n}} ≅ ⨁_{λ} b_{λμ}P^{λ}

where the sum is over all p-regular partitions λ of n. Indeed, this easily follows from the following equalities

b_{λμ} = dim Hom_{Σ_{n−1}}(P^{μ}, D^{λ} ↓_{Σ_{n−1}}) = dim Hom_{Σ_{n}}(P^{μ} ↑^{Σ_{n}}, D^{λ}).

Thus the branching coefficients b_{λμ} can also be interpreted in terms of inducing the projective modules.

Finally note that the problem on the composition factors of the induced module D^{μ} ↑^{Σ_{n}} is connected with but not equivalent to the Branching problem, so it should be considered separately.

2. Some definitions and notation

Fix an arbitrary partition λ = (λ_{1} ≥ λ_{2} ≥ …) ⊢ n. We do not distinguish between λ and its Young diagram

λ = {(i, j) ∈ ℤ_{>0} × ℤ_{>0} | j ≤ λ_{i}}.

Elements (i, j) ∈ ℤ_{>0} × ℤ_{>0} are called nodes. The (p-)residue, resA, of a node A = (i, j) is defined to be the (p-)residue class (j − i) (mod p).

A node (i, λ_{i}) ∈ λ is called a removable node (of λ) if λ_{i} > λ_{i+1}. A node (i, λ_{i} + 1) is called an addable node (for λ) if i = 1 or i > 1 and λ_{i} < λ_{i−1}. The removable (resp., addable) nodes are the nodes which can be removed from (resp., added to) the Young diagram λ to produce a Young diagram with n − 1 (resp., n + 1) nodes.

If A = (i, λ_{i}) is a removable node then

λ_{A} := λ \ {A} = (λ_{1}, …, λ_{i−1}, λ_{i} − 1, λ_{i+1}, …)

is a partition of n − 1 obtained from λ by removing A. Similarly, if B = (i, λ_{i} + 1) is an addable node then

λ^{B} = λ ∪ {B} = (λ_{1}, …, λ_{i−1}, λ_{i} + 1, λ_{i+1}, …)

is a partition of n + 1 obtained from λ by adding B.

A removable node A of λ is called normal if for every addable node B above A with resB = resA there exists a removable node C(B) strictly between A and B with resC(B) = resA, and B ≠ B′ implies C(B) ≠ C(B′). A removable node is called good if it is the lowest among the normal nodes of a fixed residue. An addable node B is called good if it is good as a removable node of λ^{B}.

We denote by sgn the one dimensional sign representation of Σ_{n}.

The residue content of the Young diagram λ is defined to be the p-tuple

cont(λ) = (c_{0}, c_{1}, …, c_{p−1})

where

c_{α} = ♯{nodes in λ of residue α}, α = 0, 1, …, p − 1.

A well known result called the Nakayama Conjecture [21, 2.7.41, 6.1.21] claims that D^{λ} and D^{μ} are in one block if and only if cont(λ) = cont(μ). (One should be a little careful with the reference [21, 6.1.21] here. There p-blocks of the ordinary irreducible representations are considered. So one also have to use the fact that D^{λ} is a composition factor of S^{λ}.)

3. Some history and generalities

If we restrict S^{λ}, rather than D^{λ}, to Σ_{n−1} then the situation is usually treated as well understood, because of the following well known result which we call the Classical branching theorem (see e.g. [18, 9.3]).

Theorem 3.1 (Classical branching theorem). Let

A_{1}, A_{2}, …, A_{k}

be the removable nodes of λ counted from top to bottom. Then there exists a filtration

(0) = S_{0} ⊂ S_{1} ⊂ … ⊂ S_{k} = S^{λ} ↓_{Σ_{n−1}}

of the Σ_{n−1}-module S^{λ} ↓_{Σ_{n−1}} such that S_{i}/S_{i−1} ≅ S^{λ_{A_{i}}}, i = 1, 2, …, k.

In the characteristic 0 case this result immediately yields the classical branching rule. If p > 0 we note that even though the factors of the filtration are described, the submodule structure of S^{λ} ↓_{Σ_{n−1}} is very complicated. For example the socle of this module is not known.

The “dual” result is as follows.

Theorem 3.2 [18, 17.14]. Let B_{1}, B_{2}, …, B_{l} be the addable nodes for λ counted from bottom to top. Then there exists a filtration

(0) = T_{0} ⊂ T_{1} ⊂ … ⊂ T_{l} = S^{λ} ↑^{Σ_{n+1}}

such that T_{i}/T_{i−1} = S^{λ^{B_{i}}} , i = 1, 2, …, l.

The first non-trivial result on the restriction D^{λ} ↓_{Σ_{n−1}} was obtained by James in 1976.

Theorem 3.3 [16]. Let

A_{1}, A_{2}, …, A_{k}

be the removable nodes of a p-regular partition λ counted from top to bottom. Set m ≥ 1 to be the minimal index such that λ_{A_{m}} is p-regular. Then [D^{λ} ↓_{Σ_{n−1}}: D^{μ}] ≠ 0 implies μ ⊳ λ_{A_{m}}, and [D^{λ} ↓_{Σ_{n−1}}: D^{λ_{Am}} ] = m.

This result shows in particular that D^{λ} ↓_{Σ_{n−1}} is not multiplicity free in general. Moreover the multiplicities can take any non-negative integral values.

Now we are in a position to explain why the Branching problem is equivalent to the Decomposition numbers problem.

Assume first that the decomposition numbers d_{λμ}^{(n)} are known for all n. We find the numbers b_{λμ} by induction on λ. If λ = (n) then D^{(n)} is the trivial module, and everything is clear. Let λ ≠ (n), and suppose the numbers b_{νκ} are known for all ν ⊳ λ. In the Grothendieck group we can write

S^{λ} ↓_{Σ_{n−1}}= ∑_{i=1}^{k} S^{λ_{A_{i}}} = ∑_{i=1}^{k} ∑_{μ} d_{λ_{A_{i}},μ}D^{μ},  (1)

i.e. the composition factors of S^{λ} ↓_{Σ_{n−1}} are known. On the other hand, S^{λ} = D^{λ} + ∑_{ν⊳λ} d_{λν}D^{ν}. So

S^{λ} ↓_{Σ_{n−1}}= D^{λ} ↓_{Σ_{n−1}} + ∑_{ν⊳λ}d_{λν}D^{ν} ↓_{Σ_{n−1}},

whence

D^{λ} ↓_{Σ_{n−1}}= S^{λ} ↓_{Σ_{n−1}} − ∑_{ν⊳λ}d_{λν}D^{ν} ↓_{Σ_{n−1}} .

But the second term of the RHS is also known (by the inductive hypothesis).

Conversely, assume that the numbers b_{λμ} are known for all n. We demonstrate how to find the numbers d_{λμ} inductively. Assume d_{λμ} are known for Σ_{n−1}. To find the decomposition numbers for Σ_{n}, note that the composition factors of the restriction S^{λ} ↓_{Σ_{n−1}} are known, see (1), since d_{λ_{A_{i}},μ} are decomposition numbers for Σ_{n−1}. Thus the question on the composition factors of S^{λ} (equivalently the decomposition numbers d_{λμ} with a fixed λ) can be reduced to the following

Questions. Let M be an F Σ_{n}-module. Do the composition factors of the restriction M ↓_{Σ_{n−1}} (together with their multiplicities) determine uniquely the composition factors of M (together with their multiplicities)? If yes, is there an algorithm for determining the composition factors of M, providing the branching coefficients b_{λμ} are known?

In general, already the answer to the first question is negative. For example, we have by the Classical branching theorem,

(S^{(k,k)} + S^{(k+2,k−2)} + …) ↓_{Σ_{n−1}}= (S^{(k+1,k−1)} + S^{(k+3,k−3)} + …) ↓_{Σ_{n−1}},

(in the Grothendieck group of Σ_{n−1}) which shows that the answer to the first question is negative even in characteristic 0.

However, there is a special case, interesting for us, when both answers are positive. This is the case where all the composition factors of M belong to the same block, and this block is known (this assumption is of course satisfied if M = S^{λ}). The idea is that we can always find one composition factor, say D^{μ}, of M. Then we subtract the expression ∑_{ν} b_{μν}D^{ν} = D^{μ} ↓_{Σ_{n−1}} from M ↓_{Σ_{n−1}} and repeat the procedure again.

To make this precise, we need the following lemma.

Lemma 3.4 Let λ be a p-regular partition of n with

cont(λ) = (c_{0}, …, c_{p−1}).

Then for any composition factor D^{ν} of D^{λ} ↓_{Σ_{n−1}} there exists a residue class α ∈ {0, 1, …, p − 1} such that

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

Proof. The composition factors of D^{λ} ↓_{Σ_{n−1}} are among the composition factors of S^{λ} ↓_{Σ_{n−1}}. By the Classical branching theorem, D^{ν} is a composition factor of some S^{λ_{A_{j}}}, whence cont(ν) = cont(λ_{A_{j}}), and the result follows. □

The following proposition provides us with the desired algorithm for determining the composition factors of a Σ_{n}-module M in a given block from the composition factors of M ↓_{Σ_{n−1}}, provided the branching coefficients for the restriction from Σ_{n} to Σ_{n−1} are known.

Proposition 3.5 Assume that we have

M = ∑_{δ⊢n} a_{δ}D^{δ}, and M ↓_{Σ_{n−1}}= ∑_{γ⊢n−1} c_{γ}D^{γ}.

(in the Grothendieck groups of Σ_{n} and Σ_{n−1}, respectively). Suppose also that a_{δ} ≠ 0 implies cont(δ) = (c_{0}, …, c_{p−1}) for some fixed numbers c_{0}, …, c_{p−1}.

Choose ν such that c_{ν} ≠ 0 but γ ⊲ ν implies c_{γ} = 0. By Lemma 3.4, there exists a residue α modulo p such that

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

Then

(i) ν has at least one addable node of residue α; (ii) if we define μ to be ν with the top addable node of residue α added,

then μ is p-regular; (iii) a_{μ} ≠ 0.

Proof. Clearly D^{ν} is a composition factor of some D^{μ} ↓_{Σ_{n−1}} with a_{μ} ≠ 0. Moreover, by the choice of ν, D^{ν} is the bottom composition factor with respect to the dominance order. By Theorem 3.3, ν = μ_{A} where A is the top removable node of μ such that μ_{A} is p-regular. This proves (i). Moreover, it follows from the definitions that A is the top addable node of μ_{A} of residue α, which proves (ii) and (iii). □

We now return to our historic remarks. In 1986, Benson [3] found the decomposition numbers for Σ_{n} for p = 2, n = 14, 15 (for n ≤ 13 these were known [18]). He then used this new information to come up with five conjectures on the branching rules in characteristic 2. Even though these conjectures are now special cases of the theorems presented in Section 4 of this paper, we state the three main conjectures of Benson here. One of the reasons is that the “2-modular combinatorics” is much easier than the general “p-modular combinatorics”, so the results in characteristic 2 have particularly simple form.

Benson’s Conjectures. [3, Conjectures 1,4,5]. Let p = 2 and

λ = (λ_{1} > λ_{2} > … > λ_{s} > 0)

be a 2-regular partition of n.

(i) D^{λ} ↓_{Σ_{n−1}} is irreducible if and only if

λ_{1} ≡ λ_{2} ≡ … ≡ λ_{s} (mod 2),

in which case

D^{λ} ↓_{Σ_{n−1}}= D^{(λ_{1}−1,λ_{2},...,λ_{s})}.

(ii) D^{λ} ↓_{Σ_{n−1}} is completely reducible if and only if

λ_{i} ≡ λ_{i+1} (mod 2),

except possibly several even values of i, in which case

D^{λ} ↓_{Σ_{n−1}}= D^{(λ_{1}−1,λ_{2},...,λ_{s})} ⊕ D^{(λ_{1},...,λ_{j+1}−1,...,λ_{s})}

where j is the minimal index such that λ_{j} ≢ λ_{j+1}. (iii) D^{λ} ↑^{Σ_{n+1}} is completely reducible if and only if

λ_{1} ≡ λ_{2} ≡ … ≡ λ_{s} ≡ 0 (mod 2),

in which case

D^{λ} ↑^{Σ_{n+1}}= D^{(λ_{1},...,λ_{s},1)} ⊕ D^{(λ_{1},λ_{2},...,λ_{s}+1)}.

Remark. Conjecture (ii) above is a corrected form of Conjecture 4 from [3]. The difference is in the set of composition factors which should be as here to make the conjecture true.

In 1992, Jantzen and Seitz [43, 23] have generalized Benson’s conjecture (i) to the case of an arbitrary p. The generalization is not obvious. Gather together equal parts of λ to represent it in the form

λ = (l_{1}^{a_{1}}, l_{2}^{a_{2}}, …, l_{m}^{a_{m}}), l_{1} > l_{2} > … l_{m} > 0, a_{1}, a_{2}, …, a_{m} > 0.

Conjecture of Jantzen-Seitz. [43, 23]. D^{λ} ↓_{Σ_{n−1}} is irreducible if and only if

l_{i} − l_{i+1} + a_{i} + a_{i+1} ≡ 0 (mod p)

for all i = 1, 2, …, m − 1, in which case

D^{λ} ↓_{Σ_{n−1}}= D^{λ_{A}}

where A is the top removable node of λ.

The “if-part” of the conjecture was proved in [23], as well as the “onlyif-part” for m ≤ 3. The “only-if-part” in general was proved in [25, 10, 27]. We note that the papers [43] and [23] were motivated by the problem of describing the maximal subgroups in finite classical groups. The following more general question was also considered there: when is the restriction D^{λ} ↓_{Σ_{k}×Σ_{n−k}} irreducible? It is proved in [23] that dim D^{λ} > 1 implies that such a restriction is always reducible for 1 < k < n − 1 and p > 2. In the case p = 2 this is not true, and as far as we know, the question remains open.

Finally, we mention a recent result of Sheth [44] who obtained a complete description of the branching rule for the two-part partitions, generalizing the dimension formulas of Erdmann [8]. Fix a two-part partition λ = (n − m, m), and let

n − 2m = s_{0} + s_{1}p + … + s_{l}p^{l}

be the p-adic expansion. Set t to be the minimal index such that s_{t} < p−1. In the following theorem we interpret D^{(n−j,j)} as 0 if j < 0.

Theorem 3.6 [44]

(i) If n − 2m = 0 then D^{(n−m,m)} ↓_{Σ_{n−1}}= D^{(n−m,m−1)} (this makes sense only if p > 2);

(ii) If n − 2m ≡ −2 (mod p) and n − 2m > 0 then

D^{(n−m,m)} ↓_{Σ_{n−1}}= D^{(n−m−1,m)};

(iii) If n − 2m ≢ −2, −1 (mod p) and n − 2m > 0 then

D^{(n−m,m)} ↓_{Σ_{n−1}}= D^{(n−m−1,m)} ⊕ D^{(n−m,m−1)};

(iv) If n−2m ≡ −1 (mod p) then D^{(n−m,m)} ↓_{Σ_{n−1}} is indecomposable with composition factors

\{ \begin{matrix}D^{(n−m−1,m)}, \\ 2D^{(n−m−1+p^{i},m−p^{i})}, 0 ≤ i ≤ t − 1, & if s_{t} = p − 2; \\ D^{(n−m−1,m)}, D^{(n−m−1+p^{t},m−p^{t})}, \\ 2D^{(n−m−1+p^{i},m−p^{i})}, 0 ≤ i ≤ t − 1, & if s_{t} < p − 2. \end{matrix}

4. Various versions of branching rules

The general Branching problem formulated in Section 1 remains open. However, there has been a noticeable progress in the area during the last years. In this section we present an exposition of the main results of the papers [25, 27, 28, 29, 30, 32]. We use the notions introduced in Section 2.

Branching Theorems. Let λ be a p-regular partition of n.

(i) For a removable node A of λ such that λ_{A} is p-regular, the irreducible module D^{λ_{A}} is a composition factor of D^{λ} ↓_{Σ_{n−1}} if and only if A is normal. In this case the multiplicity is given by

[D^{λ} ↓_{Σ_{n−1}}: D^{λ_{A}}] = ♯{normal nodes B above A with resB = resA}.

(ii)

soc(D^{λ} ↓_{Σ_{n−1}}) = ⨁_{A good} D^{λ_{A}}.

(iii)

dim_{F} End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) = dim_{F} End_{Σ_{n+1}}(D^{λ} ↑^{Σ_{n+1}}) − 1
= ♯{normal nodes of λ}.

(iv) For a p-regular partition μ of n − 1 the vector space

Hom_{Σ_{n−1}}(S^{μ}, D^{λ} ↓_{Σ_{n−1}})

is non-zero if and only if μ = λ_{A} for some normal node A of λ, in which case this space is 1-dimensional.

(v) Let A_{1}, …, A_{d} be all the good nodes of λ. Then

D^{λ} ↓_{Σ_{n−1}} ≅ I_{1} ⊕ … ⊕ I_{d}

where

soc I_{j} = head I_{j} = D^{λ_{A_{j}}} , j = 1, 2, …, d,

and the modules I_{j} are indecomposable, self-dual and all belong to distinct blocks. A similar result holds for D^{λ} ↑^{Σ_{n+1}}. In particular,

♯{indecomposable summands of D^{λ} ↓_{Σ_{n−1}}}
= ♯{block components of D^{λ} ↓_{Σ_{n−1}}}
= ♯{good nodes of λ},

and

♯{indecomposable summands of D^{λ} ↑^{Σ_{n+1}}}
= ♯{block components of D^{λ} ↑^{Σ_{n+1}}}
= ♯{good addable nodes for λ}.

(vi) D^{λ} ↓_{Σ_{n−1}} is completely reducible if and only if all normal nodes of λ have distinct residues (equivalently, all normal nodes of λ are good), in which case

D^{λ} ↓_{Σ_{n−1}}= ⨁_{A good} D^{λ_{A}}.

(vii) D^{λ} ↑^{Σ_{n+1}} is completely reducible if and only if the number of normal nodes of λ is one less than the number of good addable nodes for λ, in which case

D^{λ} ↑^{Σ_{n+1}}= ⨁_{A good addable} D^{λ^{A}}.

(viii) D^{λ} ↓_{Σ_{n−1}} is irreducible if and only if λ has a unique normal node A, in which case A is the top removable node, λ_{A} is p-regular, and

D^{λ} ↓_{Σ_{n−1}}= D^{λ_{A}}.

Remarks. (1) Note that for a normal node A, the partition λ_{A} may be not p-regular. However when we speak of the number of normal nodes, as in (i), (iii), (vii), we always take into account all normal nodes. On the other hand, one can easily check that λ_{A} is p-regular for any good node A, provided λ itself is p-regular. A similar result holds for good addable nodes. So (ii),(vi) and (vii) make sense.

(2) All branching theorems make sense in characteristic 0. This corresponds to the case “p = ∞”. In particular, (i),(ii),(iv),(vi),(vii) are equivalent to the classical branching rule in this case; (vii) also claims that the number of the removable nodes is always one less than the number of the addable nodes.

(3) Part (iii) can be strongly generalized—it is possible to describe the algebra End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) explicitly, see Theorem 4.1 below.

(4) Let M^{(n−1,1)} denote the natural n-dimensional permutation module over Σ_{n}. Then

End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) ≅ Hom_{Σ_{n}}(D^{λ} ↓_{Σ_{n−1}}↑^{Σ_{n}}, D^{λ})

≅ Hom_{Σ_{n}}(D^{λ} ⊗ M^{(n−1,1)}, D^{λ}) ≅ Hom_{Σ_{n}}(M^{(n−1,1)}, D^{λ} ⊗ D^{λ}).

So Branching theorem (iii) can be interpreted as a fact about tensor squares of irreducible modules.

(5) It is not difficult to verify that a p-regular partition has a unique normal node if and only if it satisfies the condition of Jantzen-Seitz as in Conjecture of Jantzen-Seitz in Section 3.

(6) Branching theorems (ii), (iv), and (vi) have been generalized to the Hecke algebras of type A by Brundan [5].

Now we detalize (iii). Let A_{1}, A_{2}, …, A_{d} be the good nodes of λ. In view of (v), we may decompose

D^{λ} ↓_{Σ_{n−1}}= I_{1} ⊕ I_{2} ⊕ … ⊕ I_{d},

with

soc I_{j} = head I_{j} = D^{λ_{A_{j}}} ,

and the indecomposable modules I_{1}, I_{2}, …, I_{d} belonging to the distinct blocks. Therefore

End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) ≅ End_{Σ_{n−1}}(I_{1}) ⊕ End_{Σ_{n−1}}(I_{2}) ⊕ … ⊕ End_{Σ_{n−1}}(I_{d}).

So we may concentrate on End_{Σ_{n−1}}(I_{j}), 1 ≤ j ≤ d.

Theorem 4.1 [30]. Let 1 ≤ j ≤ d. Assume resA_{j} = α_{j}, and λ has m_{j} normal nodes of residue α_{j}. Then there exists an endomorphism

φ_{j} ∈ End_{Σ_{n−1}}(I_{j})

such that φ_{j}^{m_{j}} = 0, and

{id_{I_{j}}, φ_{j}, φ_{j}^{2}, …, φ_{j}^{m_{j}−1}}

is a basis of End_{Σ_{n−1}}(I_{j}). In particular, the algebra End_{Σ_{n−1}}(I_{j}) is isomorphic to the algebra of truncated polynomials F [x]/(x^{m_{j}}).

Remarks. (1) Note that the irreducible module D^{λ_{A_{j}}} which is isomorphic to the socle and the head of I_{j} appears in I_{j} exactly m_{j} times in view of Branching theorem (i). In this respect, the dimension of the endomorphism algebra of I_{j} behaves exactly like the dimension of the endomorphism algebra of the projective cover P^{λ_{A_{j}}} of D^{λ_{A_{j}}}. If we represent I_{j} as a quotient P^{λ_{A_{j}}} /Q of P^{λ_{A_{j}}} then any endomorphism of P^{λ_{A_{j}}} leaves Q invariant (this observation was made by Dipper and the author, compare with [7]).

(2) It follows from Theorem 4.1 and Branching theorem (i) that I_{j} = D^{λ_{A_{j}}} if and only if m_{j} = 1.

We mention that tensoring with the sign representation and using the Mullineux map often gives an additional and very non-trivial information about the branching rule. The idea is as follows. Let m_{n} denote the Mullineux bijection (see Section 5). Then clearly

[D^{m_{n}(λ)} ↓_{Σ_{n−1}}: D^{m_{n}(λ)_{A}}] = [D^{λ} ↓_{Σ_{n−1}}: D^{m_{n−1}(m_{n}(λ)_{A})}] .  (2)

However, the LHS of (2) is known from Branching theorem (i).

We finish this section with an example.

Example. Let λ = (8, 4, 4, 3, 1), p = 3. Then D^{λ} ↓_{Σ_{19}}= I_{1} ⊕ I_{2} where I_{1} = D^{(8,4,4,3)}, and the following information on I_{2} is available.

− It is an indecomposable and self-dual module, whose endomorphism

algebra is isomorphic to F [x]/(x^{2}). − The socle and the head of I_{2} are isomorphic to D^{(8,4,3,3,1)}. − The partitions corresponding to the composition factors of I_{2} all have

residue content (6, 6, 7) and dominate (7, 4, 4, 3, 1). − [I_{2} : D^{(8,4,3,3,1)}] = 2. − [I_{2} : D^{(7,4,4,3,1)}] = 1. − [I_{2} : D^{(9,5,2,2,1)}] = 1.

The last fact follows from the trick with the Mullineux map, see (2). We know of no other way to get this information.

5. An application to the Mullineux Conjecture

For an arbitrary irreducible F Σ_{n}-module D^{λ}, the tensor product D^{λ} ⊗ sgn is an irreducible module again. So we can write

D^{λ} ⊗ sgn ≅ D^{b_{n}(λ)}.

This defines an involutary bijection b_{n} : P_{n} → P_{n} on the set P_{n} of the p-regular partitions of n.

Mullineux problem. Describe b_{n}.

This problem is important by several reasons. One of them comes from the representation theory of the alternating group A_{n} (see e.g. [11]). If you try to describe the irreducible F A_{n}-modules using representations of Σ_{n} and the Clifford theory, you will face the Mullineux problem!

It is proved in [18, 8.15] that

S^{λ} ⊗ sgn ≅ (S^{λ′})^{∗}  (3)

where λ′ is the partition conjugate to λ. In characteristic zero this provides a complete solution of our problem, since the Specht modules S^{λ} are the irreducible Σ_{n}-modules, and every irreducible module over Σ_{n} is selfdual. However in characteristic p the bijection b_{n} is much more difficult to describe.

The isomorphism (3) implies that D^{λ} ⊗ sgn is a composition factor of S^{λ′}. So, using the Nakayama Conjecture, we obtain:

Let cont(λ) = (c_{0}, …, c_{p−1}). Set d_{α} = c_{−α}, α = 0, 1, …, p − 1.

Then cont(b_{n}(λ)) = (d_{0}, …, d_{p−1})  (4)

Branching rules come into play via the following easy observation.

soc ((D^{λ} ⊗ sgn) ↓_{Σ_{n−1}}) ≅ (soc(D^{λ} ↓_{Σ_{n−1}})) ⊗ sgn.  (5)

It follows from (5) and Branching theorem (ii) that

For any good node A of λ there exists a good

node B of b_{n}(λ) such that b_{n−1}(λ_{A}) = b_{n}(λ)_{B}.  (6)

We can say even more. If resA = α then resB = −α. This follows immediately from (4).

The following combinatorial fact is not difficult to prove.

Lemma 5.1 [29, 1.4]. Let μ be a p-regular partition and α be a fixed presidue. Then μ has at most one good addable node of residue α.

In view of Branching Theorem (ii), Lemma 5.1 has the following curious reformulation:

Corollary 5.2 [29, 3.3]. If soc(D^{λ} ↓_{Σ_{n−1}}) and soc D^{μ} ↓_{Σ_{n−1}} have a common constituent then either λ = μ or cont(λ) ≠ cont(μ).

This surprising result shows that the properties (4) and (6) uniquely determine the bijections b_{n}. More precisely,

Theorem 5.3 [29, 4.2]. Let {x_{n} : P_{n} → P_{n}}, n ≥ 1 be a family of maps satisfying the following two conditions:

(1) for any λ ∈ P_{n}, n ≥ 1, if (c_{0}, …, c_{p−1}) is the content of λ then the content of x_{n}(λ) is (d_{0}, …, d_{p−1}) where d_{α} = c_{−α} for α = 0, 1, …, p − 1;

(2) for any λ ∈ P_{n}, n ≥ 2, there exists a good node A of λ and a good node B of x_{n}(λ) such that x_{n−1}(λ_{A}) = x_{n}(λ)_{B}.

Then x_{n} = b_{n} for all n ≥ 1.

This shows that we can describe b_{n} : P_{n} → P_{n} inductively as follows. Remove any good node A from λ. Assume resA = α. Then b_{n}(λ) is the partition obtained from b_{n−1}(λ_{A}) by adding a good node of residue −α. Note that this is well defined in view of Lemma 5.1 and (6).

In practice we just remove good nodes one after another, until we come to the empty partition. For example, pick any good node A_{1} of λ, then any good node A_{2} of λ_{A_{1}}, and so on. If resA_{j} = α_{j}, we get a sequence (α_{1}, …, α_{n}) of p-residues. Then starting from the empty partition we build a new one by adding the good nodes of residues −α_{n}, −α_{n−1}, …, −α_{1}, starting from −α_{n}. It has been just proved that there exists exactly one way to do it.

In 1979, Mullineux [37] proposed another algorithm for calculating b_{n}. He defined, purely combinatorially, a bijection, say m_{n} : P_{n} → P_{n}, and conjectured that b_{n} = m_{n}. This was called the Mullineux conjecture. (Another version of the Mullineux algorithm can be found in [46].) In calculations by hand the Mullineux algorithm works much faster than the algorithm described above. It has a number of other advantages. In [38], Mullineux has verified that m_{n} satisfies the property (1) of Theorem 5.3. Thus Corollary 5.4 The Mullineux Conjecture is true if and only if the following statement holds:

(*) for any λ ∈ P_{n}, n ≥ 2, there exists a good node A of λ and a good node B of m_{n}(λ) such that m_{n−1}(λ_{A}) = m_{n}(λ)_{B}.

The combinatorial statement (*) turned out to be quite difficult to prove. The first proof was found by Ford and the author [12]. An easier but still quite intricate proof was later found by Bessenrodt and Olsson [4].

We finish this section by considering an important special case:

Example. Which p-regular partition corresponds to the sign representation? In other words, since sgn is irreducible we must have

sgn ≅ D^{κ}

for some p-regular partition κ. Clearly κ = b_{n}((n)), and using either algorithm one finds that

κ = ((d + 1)^{r}, d^{p−1−r}) where n = d(p − 1) + r, 0 ≤ r < p − 1.

Note that this fact also follows from an old paper of James [16]. Another proof, which uses the branching rules was found in [23, 4.3]. The fanciest proof known to the author uses the Schur functors and the fact that a special module of truncated polynomials over GL_{n}(F) has the 1-dimensional 0weight space.

6. An application to the ideals in the group algebras of the finitary symmetric and alternating groups

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. It can be represented as a union Σ_{∞} = ∪_{n≥1}Σ_{n}. Similarly A_{∞} = ∪_{n≥1}A_{n} is the finitary alternating group. Clearly Σ_{∞} and A_{∞} are locally finite groups.

A. Zalesskii has shown (see [47] and the references there) that some ring theoretic questions on the group algebras of locally finite groups are closely related with the asymptotic behavior of the branching rules for finite groups. This fundamental observation accounts for the recent noticeable progress in the ideals theory of the group algebras of locally finite groups.

If a locally finite group G is countable (which is an unnecessary technical assumption) then it can be represented as a union, G = ∪_{n≥1}G_{n}, of its finite subgroups G_{1} ⊂ G_{2} ⊂ . . .. Given a (two-sided) ideal I in the group algebra F G we can form a family of ideals

I_{n} := I ∩ F G_{n} ⊲ F G_{n}, n = 1, 2, ….

This family has the property

I_{n} ∩ F G_{m} = I_{m}, for any 1 ≤ m ≤ n,  (7)

and I can be reconstructed from it as I = ∪_{n≥1}I_{n}. On the other hand, given a family of ideals I_{n} ⊲ F G_{n} satisfying the property (7) we may form a union I = ∪_{n≥1}I_{n}, and then I ∩ F G_{n} = I_{n} for any n. Thus the main problem is how to glue a big ideal from the small ones (equivalently, how to produce and classify the families of ideals satisfying (7)).

The right formalism for doing this comes from the representation theory of finite groups.

Definition (A. Zalesskii). Let Φ_{n} be a set of the isomorphism classes of irreducible F G_{n}-modules, n = 1, 2, . . .. The collection Φ = {Φ_{n}}_{n∈ℕ} is called an inductive system (for G) if for any m, n ∈ ℕ with m < n the following two properties hold:

(1) For any D ∈ Φ_{n}, all composition factors of the restriction D ↓_{G_{m}} belong to Φ_{m};

(2) For any E ∈ Φ_{m}, there exists D ∈ Φ_{n} such that E is a composition factor of D ↓_{G_{m}}.

Theorem 6.1 [47, 8.1,1.25] (see also [2, 2.6]). There is an order reversing bijection of partially ordered sets between the inductive systems and the semi-primitive ideals of F G.

Since the maximal ideals of F G are clearly semi-primitive, we get

Corollary 6.2 There is a 1-1 correspondence between the maximal ideals in F G and the minimal (non-empty) inductive systems for G.

Now we specialize again to the case G = Σ_{∞} or A_{∞}. This special case is of particular importance by several reasons. One of them is the close connection between the ideals in F Σ_{∞} and the theory of PI-rings (see [1, 48, 40, 39, 14, 13]). A. Kemer has informed us that determining the prime ideals in F Σ_{∞} would be a crucial step in a classification of the prime varieties of associative algebras (cf [24]).

The ideal structure of F Σ_{∞} and F A_{∞} is very rich. In case where charF = 0 it was described in [39, 13]. Using Theorem 6.1 and the classical branching rule one can easily rederive such a description. On the other hand, the case of positive characteristic seems to be very difficult since the complete modular branching rule is not known.

Recently Baranov and the author [2] have described the maximal ideals in F Σ_{∞} and F A_{∞}, provided p > 2.

If λ = (λ_{1} ≥ λ_{2} ≥ … ≥ λ_{m} > 0) is a partition of n we define

h(λ) = m, χ(λ) = l_{1} − l_{m} + m.

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} .

Theorem 6.3 [2]. Let F be a field of characteristic p > 2.

(i) Set

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

Then I(s)_{m} = I(s)_{n}∩F Σ_{n} for all m < n, s = 1, 2, …, p−1. 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}), n > (p − t)(t − 1), t = 1, …, \frac{p − 1}{2}.

Then J(t)_{m} = J(t)_{n} ∩F A_{n} for all m < n, t = 1, 2, …, p−1. 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.

The main idea of the proof is to show that Φ(s) = {Φ(s)_{n}}_{n≥(p−s)(s−1)}, s = 1, 2, …, p−1 are exactly all distinct minimal inductive systems for Σ_{∞}, and to use Corollary 6.2. In doing so we had to use the branching rules in maximal available generality together with the Mullineux Conjecture!

Remark. Clearly

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

and

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

are the maximal ideals in F Σ_{∞}. These are I(1) and I(p − 1), respectively. If charF = 0 then Aug(F Σ_{∞}) and Aug^{σ}(F Σ_{∞}) are the only maximal ideals. In the modular case however we have p − 1 maximal ideals, and I(2), I(3), …, I(p − 2) have infinite codimensions.

The modules D^{λ} with χ(λ) ≤ p, which form the inductive systems Φ(s) have appeared before in various contexts, see [41, 35, 31].

Mathieu [35], using theory of tilting modules, found a combinatorial description of their dimensions. Let us consider the version of p-modular Young graph introduced in [29]. The vertices of this directed graph Y are all p-regular partitions, and two p-regular partitions μ and λ are connected by the arrow μ → λ if and only if μ = λ_{A} for some normal node A. Mathieu proved that

dim D^{λ} = ♯{paths from (1) to λ in Y }, provided χ(λ) ≤ p.  (8)

It is not difficult to deduce from the Branching theorems (see [31]) that χ(λ) ≤ p implies that D^{λ} ↓_{Σ_{n−1}} is completely reducible, and every constituent D^{μ} of this restriction also has the property χ(μ) ≤ p. This yields a simple inductive proof of (8), and also shows that when considering the dimensions of the modules D^{λ} with χ(λ) ≤ p we can restrict ourselves to the paths in the complete subgraph of Y whose vertices are the partitions ν with χ(ν) ≤ p. This is how the result is formulated in [35].

A second version of a p-modular Young graph has the same vertices but fewer arrows: we connect μ → λ if and only if μ = λ_{A} for some good node A. This version plays a fundamental role in the recently discovered LLT-algorithm [34].

We define a third version of a p-modular Young graph. The vertices of this directed graph Y′ are, as usual, all p-regular partitions, but Y′ has much more arrows. We draw i arrows from μ to λ if μ = λ_{A} for some normal node A such that there are (i − 1) normal nodes B above A with resB = resA or if λ = m_{n−1}(m_{n}(λ)_{C}) for some normal node C of m_{n}(λ) such that there are (i − 1) normal nodes D above C with resD = resC. (Recall that we denote by m_{n} the Mullineux map, see Section 5). Now it follows immediately from Branching theorem (i) and the Mullineux conjecture that

dim D^{λ} ≥ ♯{paths from (1) to λ in Y′}, for any p-regular λ. (9)

7. Some related problems on algebraic groups

In this section we assume that the basic field F is algebraically closed.

The proofs of the Branching theorems presented in Section 4 are obtained in two steps. First, the Schur functors [15, 23] are used to translate the problem into a problem on GL_{n}(F) (or SL_{n}(F)), and then this latter problem is solved using the algebraic groups methods. Both steps have rather subtle points. For example, to ensure that Schur functors “work properly”, we need the following result, interesting in its own right.

Theorem 7.1 [25]. Let L(λ) be the irreducible (rational) SL_{n}(F)-module with p-restricted highest weight λ. Then every irreducible constituent of

soc L(λ) ↓_{SL_{n−1}(F)}≅ head L(λ) ↓_{SL_{n−1}(F)}

has a p-restricted highest weight.

This fact is rather surprising since in general the restricted module L(λ) ↓_{SL_{n−1}(F)} has a lot of non-p-restricted composition factors. Theorem 7.1 can be developed as follows.

Theorem 7.2 [26]. Let λ be an arbitrary dominant weight, and

L(λ) = L(λ^{(0)}) ⊗ L(λ^{(1)})^{[1]} ⊗ … ⊗ L(λ^{(d)})^{[d]}

be the presentation of L(λ) as the Steinberg tensor product, with every λ^{(d)} p-restricted. Then

soc L(λ) ↓_{SL_{n−1}(F)}≅ soc L(λ^{(0)}) ↓_{SL_{n−1}(F)}
⊗ (soc L(λ^{(1)}) ↓_{SL_{n−1}(F)})^{[1]} ⊗ … ⊗ (soc L(λ^{(d)}) ↓_{SL_{n−1}(F)})^{[d]} . (10)

Remarks. (1) Note that the tensor product in (10) is completely reducible in view of Theorem 7.1. We can reformulate the statement of Theorem 7.2 as follows: the irreducible constituents of soc L(λ) ↓_{SL_{n−1}(F)} are all the modules of the form

L(μ^{(0)}) ⊗ L(μ^{(1)})^{[1]} ⊗ … ⊗ L(μ^{(d)})^{[d]}

where μ^{(j)} runs over the highest weights of all irreducible components of the module soc L(λ^{(j)}) ↓_{SL_{n−1}(F)}, j = 0, 1, …, d.

(2) Theorem 7.1 can be generalized to the embeddings of algebraic groups of types C_{l−1} < C_{l}, D_{l} < B_{l}, B_{l} < D_{l+1}. On the other hand, if we restrict ourselves to the Levi-type subgroups then Theorem 7.1 is only true for the embeddings A_{l−1} < A_{l} and C_{l−1} < C_{l}. On the other hand Theorem 7.2 is true for the embeddings A_{l−1} < A_{l}, D_{l} < B_{l}, B_{l} < D_{l+1}. We refer the reader to [26] for more details.

(3) Theorem 7.1 is true not only for L(λ) but also for any submodule of the co-Weyl module ∇(λ), in particular for ∇(λ) itself. (This was observed by Brundan and the author).

Now we consider a problem which, modulo some intricate combinatorics [32], is essentially equivalent to Branching theorem (i). The problem is to describe the dimension of the weight space L(λ)_{λ−α} over SL_{n}(F) for any positive root α.

We introduce some notation. It is convenient to work with the group SL_{l+1}(F) considered as an algebraic group of type A_{l}. Let

{α_{1}, α_{2}, …, α_{l}}

be a base in the root system A_{l}, and

{ω_{1}, ω_{2}, …, ω_{l}}

be the corresponding fundamental weights. Any positive root in A_{l} has a form

α_{i} + α_{i+1} + … + α_{j} with 1 ≤ i ≤ j ≤ l.

One can use Smith’s theorem [45] to reduce our problem to the case where

α = α_{1} + α_{2} + … + α_{l}

is the highest root (otherwise we just “live” in a smaller SL_{n}). Let

λ = λ_{1}ω_{1} + λ_{2}ω_{2} + … + λ_{l}ω_{l}.

Those λ_{i} which are non-zero modulo p play a particular role in our problem. So put

{i_{1} < i_{2} < … < i_{k}} = {i | 1 ≤ i ≤ l, λ_{i} ≢ 0}.

The difficulty of our problem “grows quickly” with the growth of k. Let us denote

d(λ) = dim L(λ)_{λ−α_{1}−λ_{2}−...−α_{l}};
x_{j} = i_{j} − i_{j−1}, j = 2, 3, …, k;

B_{j,m} = ∑_{q=j}^{m} λ_{i_{q}} + i_{m} − i_{j}, 1 ≤ j ≤ m ≤ k.

Then (writing “≡” for “≡ (mod p)”) we have:

(i) If k = 0 then d(λ) = 0 (clear). (ii) If k = 1 then d(λ) = 1 (clear). (iii) Let k = 2. Then

d(λ) = \{ \begin{matrix}x_{2}, & if B_{1,2} ≡ 0, \\ x_{2} + 1, & otherwise. \end{matrix}

(this is an easy calculation, see [42, 8.6])

1022 A.S. KLESHCHEV

(iv) Let k = 3. Then

d(λ) = \{ \begin{matrix}x_{2}x_{3} + 1, & if B_{1,2} ≡ 0, B_{2,3} ≡ 0; \\ x_{2}(x_{3} + 1) + 1, & if B_{1,2} ≡ 0, B_{2,3} ≢ 0; \\ (x_{2} + 1)x_{3} + 1, & if B_{1,2} ≢ 0, B_{2,3} ≡ 0; \\ (x_{2} + 1)(x_{3} + 1) − 1, & if B_{1,2} ≢ 0, B_{2,3} ≢ 0, B_{1,3} ≡ 0; \\ (x_{2} + 1)(x_{3} + 1), & if B_{1,2} ≢ 0, B_{2,3} ≢ 0, B_{1,3} ≢ 0. \end{matrix}

(this is a difficult calculation)

(v) Let k = 4. Then the value of d(λ) can be found from Table 1.

TABLE 1.

d(λ)  B_{1,2} B_{1,3} B_{1,4} B_{2,3} B_{3,4} B_{2,4}
x_{2}x_{3}x_{4} + x_{2} + x_{4}  ≡ 0 any any ≡ 0 ≡ 0 any
(x_{2} + 1)x_{3}x_{4} + x_{2} + x_{4} + 1  ≢ 0 any any ≡ 0 ≡ 0 any
x_{2}x_{3}(x_{4} + 1) + x_{2} + x_{4} + 1  ≡ 0 any any ≡ 0 ≢ 0 any
(x_{2} + 1)x_{3}(x_{4} + 1) + x_{2} + x_{4} + 1  ≢ 0 any ≢ 0 ≡ 0 ≢ 0 any
as above  ≢ 0 ≢ 0 ≡ 0 ≡ 0 ≢ 0 ≡ 0
(x_{2} + 1)x_{3}(x_{4} + 1) + x_{2} + x_{4}  ≢ 0 ≡ 0 ≡ 0 ≡ 0 ≢ 0 any
as above  ≢ 0 ≢ 0 ≡ 0 ≡ 0 ≢ 0 ≢ 0
x_{2}(x_{3} + 1)x_{4} + x_{2} + x_{4}  ≡ 0 any any ≢ 0 ≡ 0 any
x_{2}(x_{3} + 1)x_{4} + x_{2} + x_{3}x_{4} + 1  ≢ 0 ≡ 0 any ≢ 0 ≡ 0 any
(x_{2} + 1)(x_{3} + 1)x_{4} + x_{2} + 1  ≢ 0 ≢ 0 any ≢ 0 ≡ 0 any
x_{2}x_{3}(x_{4} + 1) + (x_{2} + 1)x_{4} + 1  ≡ 0 any any ≢ 0 ≢ 0 ≡ 0
x_{2}(x_{3} + 1)x_{4} + x_{2}x_{3} + x_{3}(x_{4} + 1) + 1 ≢ 0 ≡ 0 any ≢ 0 ≢ 0 ≡ 0
(x_{2} + 1)x_{3}(x_{4} + 1) + (x_{2} + 1)x_{4} + 1  ≢ 0 ≢ 0 any ≢ 0 ≢ 0 ≡ 0
x_{2}(x_{3} + 1)(x_{4} + 1) + (x_{4} + 1)  ≡ 0 any any ≢ 0 ≢ 0 ≢ 0
x_{2}(x_{3} + 1)(x_{4} + 1) + x_{3}(x_{4} + 1) + 1  ≢ 0 ≡ 0 any ≢ 0 ≢ 0 ≢ 0
x_{2}(x_{3} + 1)(x_{4} + 1) + x_{3}(x_{4} + 1) + x_{4} ≢ 0 ≢ 0 ≡ 0 ≢ 0 ≢ 0 ≢ 0
(x_{2} + 1)(x_{3} + 1)(x_{4} + 1)  ≢ 0 ≢ 0 ≢ 0 ≢ 0 ≢ 0 ≢ 0

Part (v) is very difficult to get by elementary calculations. We have deduced it from the general result which we now present.

The following definition resembles the definition of normal nodes in Young diagrams. This is not a coincidence. Definition 7.3 Let 1 ≤ j ≤ m ≤ k. We want to define a number σ_{j,m}. This number is 0 or 1 depending on whether a certain combinatorial condition holds:

if for every q, j < q ≤ m, with B_{j,q} ≡ 0 (mod p) there exists t(q), j < t(q) < q such that B_{t(q),q} ≡ 0 (mod p) and such that q ≠ q′ implies t(q) ≠ t(q′), we put σ_{j,m} = 1.

otherwise we put σ_{j,m} = 0.

Now we define a k × k matrix M(λ) = ∥a_{tu}∥_{1≤t,u≤k} by setting

a_{t,u} = \{ \begin{matrix}σ_{t,u−1} + (i_{u+1} − i_{u} − 1)σ_{t,u}, & if 1 ≤ t < u < k; \\ σ_{tk}, & if u = k; \\ −1, & if u = t − 1; \\ 0, & otherwise. \end{matrix}

For example if k = 4 we get

M(λ) = [ \begin{matrix}x_{2} & σ_{1,1} + (x_{3} − 1)σ_{1,2} & σ_{1,2} + (x_{4} − 1)σ_{1,3} & σ_{1,4} \\ −1 & x_{3} & σ_{2,2} + (x_{4} − 1)σ_{2,3} & σ_{2,4} \\ 0 & −1 & x_{4} & σ_{3,4} \\ 0 & 0 & −1 & σ_{4,4} \end{matrix} ] .

Now we can formulate the following important result Theorem 7.4 [32, 1.17] dim L(λ)_{λ−α_{1}−α_{2}−...−α_{l}} = det M(λ). Remarks. (1) Cofactor expansion of det M(λ) along the first row yields a recurrent formula for d(λ).

(2) Theorem 7.4 is just one of the different descriptions of d(λ) found in [32]. There we also present various bases for the weight space

L(λ)_{λ−α_{1}−α_{2}−...−α_{l}}.

Now consider the following, more special question. Assume the positions i_{1} < i_{2} < … < i_{k} for which λ_{i_{j}} ≢ 0 (mod p) are fixed. What is the minimal possible value for d(λ) and for which λ this minimal value takes place? Definition 7.5 We define the integers f(i_{j}, i_{j+1}, …, i_{k}), j = k, k−1, …, 1, by setting

f(i_{k}) = 1;
f(i_{k−1}, i_{k}) = i_{k} − i_{k−1};
f(i_{j}, i_{j+1}, …, i_{k}) = (i_{j+1} − i_{j})f(i_{j+1}, i_{j+2}, …, i_{k})
+f(i_{j+2}, i_{j+3}, …, i_{k}),
j = k − 2, k − 3, …, 1.

Example. Let i_{1} = 1, i_{2} = 2, …, i_{l} = l. Then f(1, 2, …, l) = f_{l} where f_{j} denotes the j-th Fibonacci number defined by f_{1} = 1, f_{2} = 1 and f_{j} = f_{j−1} + f_{j−2} for j ≥ 3.

Theorem 7.6 [32, 1.21] Assume that

{i_{1} < i_{2} < … < i_{k}} = {i | 1 ≤ i ≤ l, λ_{i} ≢ 0 (mod p)} ≠ ∅.

Then

dim L(λ)_{λ−α_{1}−α_{2}−...−α_{l}} ≥ f(i_{1}, i_{2}, …, i_{k}).

Moreover,

dim L(λ)_{λ−α_{1}−α_{2}−...−α_{l}} = f(i_{1}, i_{2}, …, i_{k}).

if and only if

B_{j,j+1} ≡ 0 (mod p) for all j = 1, 2, …, k − 1.

Dominant weights λ = λ_{i_{1}}ω_{1}+λ_{i_{2}}ω_{2}+…+λ_{i_{k}}ω_{k} with λ_{i_{j}} ≢ 0 (mod p) for j = 1, 2, …, k and B_{j,j+1} ≡ 0 (mod p) for j = 1, 2, …, k −1, have first appeared in the paper [23]. Since then they showed up in various contexts. They also play a role in a special branching rule for general linear groups which we now describe.

It is natural to expect that by analogy with symmetric groups, one might be able to obtain certain branching theorems for general linear groups. However this problem turns out to be much more difficult. We believe that the first problem which should be considered is a description of the socle of the restriction, L ↓_{GL_{n−1}(F)}, of an irreducible GL_{n}(F)-module L to the subgroup GL_{n−1}(F). This problem is unsolved at the moment. The best result available now is due to Brundan, the author and I. Suprunenko [6]. In this paper we get a criterion for the complete reducibility of L ↓_{GL_{n−1}(F)}, and in the case where the restriction L ↓_{GL_{n−1}(F)} is completely reducible we describe its composition factors. To explain the results obtained it is convenient to return to the partition notation.

If λ is a partition of a non-negative integer r with at most n parts we can identify it with a dominant weight for GL_{n}(F). The (polynomial) irreducible GL_{n}(F)-module with highest weight λ will be denoted by L(λ) or L_{n}(λ).

Using Theorems 7.1 and 7.2 we can reduce all the questions under consideration to the case where λ is p-restricted (i.e. λ_{i} − λ_{i+1} < p for i = 1, 2, …, n − 1). More precisely, these theorems imply that a module

L(λ) = L(λ^{(0)}) ⊗ L(λ^{(1)})^{[1]} ⊗ … ⊗ L(λ^{(d)})^{[d]}

has a completely reducible restriction to GL_{n−1}(F) if and only if every module L(λ^{(j)}), j = 1, 2, …, d, does. Moreover, in this case Theorem 7.2 tells us how to find the composition factors of the restriction L(λ) ↓_{GL_{n−1}(F)} if the composition factors of the restrictions L(λ^{(j)}) ↓_{GL_{n−1}(F)}, j = 1, 2, …, d, are known.

So from now on we assume that λ is p-restricted.

Let us give the necessary combinatorial definitions. We fix a p-restricted dominant weight λ (i.e. λ = (λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n} ≥ 0), and λ_{i} − λ_{i−1} < p for i = 1, 2, …, n − 1). Let

{i_{1} < i_{2} < … < i_{k}} = {i | 1 ≤ i < n, λ_{i} > λ_{i+1}}.  (11)

Put

B^{jm} := i_{m} − i_{j} + λ_{i_{j}} − λ_{i_{m}+1}, 1 ≤ j ≤ m ≤ k,  (12)
R_{j} := {res(i_{j}, l) | λ_{i_{j}+1} < l ≤ λ_{i_{j}}}, j = 1, 2, …, k.  (13)

Then B^{jm} is the length of a special hook, and R_{j} is the set of the residues of such nodes in the row i_{j} which are the bottoms of their columns.

We say that λ is GJS (generalized Jantzen-Seitz) if at least one of the following two conditions holds:

(1) B^{j,j+1} ≡ 0 (mod p) for j = 1, 2, …, k − 1;

(2) R_{j} ∩ R_{m} = ∅ as long as 1 ≤ j ≠ m ≤ k.

Let μ = (μ_{1} ≥ … ≥ μ_{n−1} ≥ 0) be a GL_{n−1}(F)-weight. We say μ is allowable (for λ) if the following two conditions hold:

(a)λ_{i+1} ≤ μ_{i} ≤ λ_{i} for all 1 ≤ i < n;

(b) if μ_{i_{j}} < λ_{i_{j}} and B^{jm} ≡ 0 (mod p) for some m > j (k ≥ m > j ≥ 1) then μ_{i_{m}} = λ_{i_{m}+1}.

Theorem 7.7 [6] Let L_{n}(λ) be an irreducible (polynomial) GL_{n}(F)-module with p-restricted highest weight λ. Then the restriction of L_{n}(λ) to the natural subgroup GL_{n−1}(F) is completely reducible if and only if λ is GJS, in which case

L_{n}(λ) ↓_{GL_{n−1}(F)}≅ ⨁_{μ allowable} L_{n−1}(μ).

Now we make a much stronger assumption: we want L_{n}(λ) to be completely reducible under the restrictions to any Levi subgroups of GL_{n}(F). The representation L_{n}(λ) satisfying this assumption will be called completely splittable. The analogous notion for the symmetric groups was introduced in [31]. It was proved there that an irreducible F Σ_{n}-module D^{λ} is completely splittable (i.e. is completely reducible under the restrictions to all Young subgroups) if and only if χ(λ) ≤ p, see Section 6. Surprisingly, we get essentially the same answer for the general linear groups (up to the conjugation and modulo Steinberg’s tensor products).

First of all note that using Theorems 7.1 and 7.2 as above one can show that a module

L(λ) = L(λ^{(0)}) ⊗ L(λ^{(1)})^{[1]} ⊗ … ⊗ L(λ^{(d)})^{[d]}

is completely splittable if and only if every module L(λ^{(j)}), j = 1, 2, …, d, is. So we may assume that λ is p-restricted.

Set

ψ(λ) = B^{1k},

see (11) and (12).

Theorem 7.8 [6] An irreducible GL_{n}(F)-module L_{n}(λ) with p-restricted highest weight λ is completely splittable if and only if ψ(λ) ≤ p.

Example. One can use Theorem 7.8 and Steinberg’s tensor product to show that L(aω_{j}) is a completely splittable SL_{n}(F)-module for any a ∈ ℤ_{≥0} and 1 ≤ j < n.

Completely splittable representations are of particular importance because they admit an explicit description of their formal characters. The reason is that not only the restriction of such a representation to GL_{n−1} is completely reducible but every composition factor of this restriction is completely splittable again. So we may apply induction and the known branching rule. This leads to the following result.

Let λ be p-restricted, and ψ(λ) ≤ p. We define a p-hook to be a pair of nodes (i, j) and (i′, j′) in λ with i − j + p − 1 = i′ − j′. We refer to (i, j) as the top-right node in the p-hook, and (i′, j′) as the bottom-left node. Given a standard λ-tableau t, we say a p-hook is bad (in t) if

(1) the entry in the top-right node of the hook is strictly larger than the entry in the bottom-left node;

(2) no entry in the column containing the bottom-left node is equal to the entry in the top-right node.

Remark. We note that the top-right node of a p-hook may be to the left of its bottom-left node! However this never happens for bad p-hooks, which justifies the terminology.

Theorem 7.9 [6] Let λ be p-restricted, and ψ(λ) ≤ p. Then

dim L(λ)_{μ} = ♯{standard λ-tableaux of type μ without bad p-hooks}.

BRANCHING RULES FOR SYMMETRIC GROUPS 1027

We mention that an equivalent character formula was obtained independently by Mathieu and Papadopoulo [36] using a tilting modules approach.

Acknowledgement

I am grateful to M. Broue, R. Carter and J. Saxl who organized a remarkable program on representation theory in Cambridge. This article is an extended version of my talks in the Isaac Newton Institute.

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. A. Baranov and A. Kleshchev, Maximal ideals in modular group algebras of the finitary symmetric and alternating groups, Trans. Amer. Math. Soc., to appear.

3. D. Benson, Some remarks on the decomposition numbers for the symmetric groups, in Proceedings Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47, Part 1, Amer. Math. Soc., Providence, 1987, p. 381–394.

4. C. Bessenrodt and J.B. Olsson, On residue symbols and the Mullineux conjecture, preprint, University of Copenhagen, 1996 (to appear in J. Alg. Comb.).

5. J. W. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. London Math. Soc., to appear.

6. J. Brundan, A. Kleshchev and I. Suprunenko, Semisimple restrictions from GL(n) to GL(n − 1), preprint, 1997.

7. R. Dipper, On quotients of Hom-functors and representations of finite general linear groups. I, J. Algebra 130 (1990), 235–259.

8. K. Erdmann, Tensor products and dimensions of simple modules for symmetric groups, Manuscripta Math. 88 (1995), 357–386.

9. K. Erdmann, Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316–320.

10. B. Ford, Irreducible restrictions of representations of the symmetric groups, Bull. London Math. Soc., 27 (1995), 453–459.

11. B. Ford, Irreducible representations of the alternating groups in odd characteristic, Proc. Amer. Math. Soc. 125 (1997), 375–380.

12. B. Ford and A. Kleshchev, A proof of the Mullineux conjecture, preprint, 1995 (to appear in Math. Z.).

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

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

15. J. A. Green, Polynomial representations of GL_{n}, Lecture Notes in Math., vol. 830, Springer-Verlag, 1980.

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

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

18. G. D. James, The representation theory of the symmetric groups, Springer Lecture Notes 682, Berlin, Heidelberg, New York, 1978.

19. G. D. James, On the minimal dimensions of irreducible representations of symmetric groups. Math. Proc. Cambridge Philos. Soc. 94 (1983), 417–424.

20. G. D. James., The representation theory of the symmetric groups, in The Arcata

1028 A.S. KLESHCHEV

Conference on Representations of Finite Groups, Proc. Sympos. Pure Math. 47, Amer. Math. Soc., 1987, pp. 111–126.

21. G. D. James and A. Kerber, The Representation Theory of the Symmetric Group, Addsion-Wesley, London, 1981.

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

23. J. C. Jantzen and G. M. Seitz, On the representation theory of the symmetric groups, Proc. London Math. Soc. 65 (1992), 475–504.

24. A. Kemer, Remarks on the prime varieties, Israel J. Math. 96 (1996), 341–356.

25. A. S. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, I, Proc. London Math. Soc. 69 (1994), 515–540.

26. A. S. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II, Comm. Alg. 22 (1994), 6175–6208.

27. A. S. Kleshchev, Branching rules for modular representations of symmetric groups, I. J. Algebra 178 (1995), 493-511.

28. A. S. Kleshchev, Branching rules for modular representations of symmetric groups, II, J. reine angew. Math. 459 (1995), 163–212.

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

30. A. S. Kleshchev, Branching rules for modular representations of symmetric groups, IV. J. Algebra (submitted).

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

32. A. S. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. Lond. Math. Soc. (3) 75 (1997), to appear.

33. A. Kleshchev and A.Premet, The globally irreducible representations of symmetric groups, Math. Proc. Camb. Phil. Soc., submitted.

34. A. Lascoux, B. Leclerc, and J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys., 181 (1996), 205– 263.

35. O. Mathieu, On the dimension of some modular irreducible representations of the symmetric group, Lett Math. Phys. 38 (1996), 23-32.

36. O. Mathieu and G. Papadopoulo, A character formula for a family of simple modular representations of GL_{n}, preprint, 1997.

37. G. Mullineux, Bijections of p-regular partitions and p-modular irreducibles of the symmetric groups, J. London Math. Soc. (2) 20 (1979), 60–66.

38. G. Mullineux, On the p-cores of p-regular diagrams, J. London Math. Soc. (2) 20) (1979), 222–226.

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

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

41. A. Ryba, Fibonacci representations of the symmetric groups, J. Algebra bf 170 (1994), 678–686.

42. G.M Seitz, The Maximal Subgroups of Classical Algebraic Groups. Mem. Amer. Math. Soc. 67 (1987), 1–286.

43. G.M. Seitz, Subgroups of finite and algebraic groups, in Groups, Combinatorics, and Geometry, Lond. Math. Soc. Lecture Notes Series, 165, Part 1, Camb. Univ. Press., Cambridge, 1992, pp. 316–326.

44. J. Sheth, Branching rules for two-part partitions and inductive systems, preprint, University of Oregon, 1997.

45. S.D. Smith, Irreducible modules and parabolic subgroups. J. Algebra 74 (1982), 286–289.

BRANCHING RULES FOR SYMMETRIC GROUPS 1029

46. M. Xu, On Mullineux’ conjecture in the representation theory of symmetric groups, Comm Alg. 25(6)(1997), 1797–1803.

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

48. A. E. Zalesskii, Modular group rings of the finitary symmetric groups, Isr. J. Math. 96 (1996), 609-621.