PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 2, February 1997, Pages 375–380 S 0002-9939(97)03621-6

IRREDUCIBLE REPRESENTATIONS

OF THE ALTERNATING GROUP IN ODD CHARACTERISTIC

BEN FORD

(Communicated by Ronald M. Solomon)

Abstract. We use the recently-proved conjecture of Mullineux to determine which modular irreducible representations of the symmetric group Σ_{n} split on restriction to A_{n}, and which remain irreducible (everything taking place over a splitting field for A_{n} of characteristic p > 2). An indexing of the absolutely irreducible representations of A_{n} is thus obtained. A modular analogue of the Frobenius symbol for a partition is introduced, which makes the Mullineux map somewhat more intuitive.

1. Introduction and notation

Parameterizations of the irreducible representations of a class of groups by some combinatorial objects provide some of the most appealing results and useful tools in group theory — labeled Dynkin diagrams (groups of Lie type), partitions and tableaux (symmetric groups), etc. It is Alfred Young’s parameterization of the ordinary representations of the symmetric group Σ_{n} by partitions of n (see [9] for an overview) that began the line of research we continue here. In particular, Young’s constructions were extended by James ([6]) to representations of Σ_{n} over an arbitrary field K of characteristic p, and we investigate these modular representations which James constructed.

As soon as one has a result about the symmetric group, it is natural to ask what the result implies for the alternating group A_{n}. Young’s constructions can be used to give a nice parameterization of the ordinary irreducible representations of the alternating group A_{n} by describing those irreducible representations of Σ_{n} which split upon restriction to A_{n} (see [7, §2.5]). But James’s parameterization of the modular irreducible representations of Σ_{n} has proved much more difficult to use to obtain information about A_{n}. Benson in [3] used a reduction modulo two of the spin representations to determine which 2-modular irreducible representations of Σ_{n} are irreducible upon restriction to A_{n}, and which split as the sum of two (non-isomorphic) irreducible modules (that each Σ_{n}-module must do one of these things is clear as A_{n} is normal of index 2 in Σ_{n}).

With the Mullineux conjecture ([8]) in the representation theory of the symmetric group now proved ([5]), a reasonably nice parameterization of the absolutely

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors August 28, 1995.

1991 Mathematics Subject Classification. Primary 20C20, 20C30.

Supported in part by the NSA.

Thanks to Jens C. Jantzen for suggesting this question.

irreducible representations of the alternating group A_{n} over a field of odd characteristic is now available. We will assume that K is a splitting field for A_{n} (n > 1), of characteristic p > 2.

The standard approach in descending from Σ_{n} to A_{n} is to use Clifford’s theory (see Curtis and Reiner [4, §49]), of which we need only an easy application since Σ_{n}/A_{n} has order 2.

First some notation: Let λ be a partition of n (written λ ⊢ n); that is,

λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{m} > 0), ∑ l_{i} = n.

We usually collect like parts together and write

λ = (λ_{1}^{a_{1}}, …, λ_{k}^{a_{k}}),

with λ_{i} > λ_{i+1} > 0 for i = 1, …, k − 1; a_{i} ≠ 0; and ∑ a_{i}λ_{i} = n. That the number of partitions of n is equal to the number of possible cycle structures for an element of Σ_{n}, and thus to the number of conjugacy classes in Σ_{n}, is clear. But the number of conjugacy classes in Σ_{n} is the same as the number of ordinary irreducible representations. Young showed how to construct (over Q) an irreducible module S^{λ} for each partition λ, such that if μ and λ are distinct partitions, then S^{λ} and S^{μ} are not isomorphic.

We use the symbol [λ] to represent the partition’s Young diagram, which consists of n nodes • placed in decreasing rows, as illustrated by the following example. Let λ = (5^{2}, 3, 1) (so n = 14); then the Young diagram is:

[λ] = \begin{matrix}• & • & • & • & • \\ • & • & • & • & • \\ • & • & • \\ • \end{matrix}

By the rim of [λ] we shall mean the collection of nodes which are either at the bottom of a column, at the right end of a row, or both.

Given a partition λ, one obtains another partition λ′, called the conjugate partition to λ, by transposing the Young diagram [λ] about its main diagonal. So if λ = (5^{2}, 3, 1) as above, then the Young diagram for λ′ is

[λ′] = \begin{matrix}• & • & • & • \\ • & • & • \\ • & • & • \\ • & • \\ • & • \end{matrix}

and λ′ = (4, 3^{2}, 2^{2}).

Finally, If H is a subgroup of a group G, and M is an F G-module for some field F, by M ↓_{H} we shall mean M restricted to H, that is, M considered as a module over F H ⊆ F G. If N is an F H-module, then N ↑^{G} will be the induced module F G ⊗_{F H} N (see [4, §38] for more).

2. Mullineux: A modular λ′

REPRESENTATIONS OF A_{n} 377

Of course, if λ is a partition of n, then so is λ′, and so λ and λ′ correspond to (perhaps different, if λ ≠ λ′) irreducible Q-representations of Σ_{n}. The crucial fact in descending to A_{n} is that S^{λ} ⊗ sgn_{n} = S^{λ′}, where for σ ∈ Σ_{n}, sgn_{n}(σ) = 1 if σ is an even permutation, and sgn_{n}(σ) = −1 if σ is odd. Thus S^{λ} ↓_{A_{n}}= S^{λ′} ↓_{A_{n}} (as the two representations differ only on odd elements of Σ_{n}). Clifford’s theory then allows us to tell fairly easily when S^{λ} ↓_{A_{n}} is irreducible (this happens if λ ≠ λ′), and when S^{λ} ↓_{A_{n}} is the sum of two inequivalent irreducible A_{n}-modules (if λ = λ′).

If we now consider representations over K instead of over Q, things don’t work as nicely. Assume λ is a p-regular partition, and let D^{λ} be the irreducible KΣ_{n}- module corresponding to λ. It is still true that (D^{λ} ⊗ sgn_{n}) ↓_{A_{n}}= D^{λ} ↓_{A_{n}}, but it is no longer true that D^{λ} ⊗ sgn_{n} = D^{λ′}. Mullineux’s algorithm helps out here and gives a modular analogue of λ′. That is, Mullineux in [8] defined a bijection m on the set of p-regular partitions of n, and his conjecture that D^{λ} ⊗ sgn_{n} = D^{m(λ)} has now been verified ([5]). With this in hand, we can describe the irreducible K-representations of A_{n}.

Clearly D^{λ} ↓_{A_{n}}= D^{m(λ)} ↓_{A_{n}}, as in the characteristic zero case. Also, D^{λ} ↓_{A_{n}} is completely reducible (since A_{n} is a normal subgroup of Σ_{n}). Consider an irreducible summand D of D^{λ} ↓_{A_{n}} (= D^{m(λ)} ↓_{A_{n}}). If we consider the action of an odd element of Σ_{n}, say (12), on D, the possibilities are that D(12) = D (the inertia group of D is Σ_{n}), or D(12) ≠ D (the inertia group of D is A_{n}).

In the first case, we have that D = D^{λ} ↓_{A_{n}} is irreducible, since D^{λ} is an irreducible Σ_{n}-module and D is an invariant submodule. Then it is easy to see that D^{λ} and D^{λ} ⊗ sgn_{n} = D^{m(λ)} are not isomorphic as Σ_{n}-modules: Assume that σ is an Σ_{n}-isomorphism. Then σ is also an A_{n}-isomorphism; but D is an irreducible A_{n}-module and K is a splitting field for A_{n}. Thus σ must be a scalar multiple of the identity map; but this contradicts the assumption that it is an isomorphism between D^{λ} and D^{λ} ⊗ sgn_{n}.

If, on the other hand, D(12) ≠ D, then Clifford’s theorem says

D^{λ} ↓_{A_{n}}= D ⊕ D(12) = D^{m(λ)} ↓_{A_{n}},

and this implies (by Mackey’s subgroup theorem; see for example [4, §44]) that D^{λ} ≅ D ↑^{Σ_{n}} ≅ D^{m(λ)}.

Summarizing, we have:

Theorem 2.1. 1. If λ ≠ m(λ), then D^{λ} ↓_{A_{n}} is irreducible.

2. If λ = m(λ), then D^{λ} ↓_{A_{n}} is the sum of two irreducible, non-equivalent, representations of A_{n}, say D_{+}^{λ} and D_{−}^{λ}.

Finally,

{D^{λ} ↓_{A_{n}}| λ ≠ m(λ)} ∪ {D_{+}^{λ}, D_{−}^{λ} | λ = m(λ)}

is a complete system of inequivalent irreducible KA_{n}-modules.

3. Fixed points and a modular Frobenius symbol

So the issue is identifying the fixed points of the Mullineux map. Andrews and Olsson in [2] counted the fixed points of m and described them to some extent. Unfortunately, the fixed points are not easy to “see” from the diagram or the usual presentation of the partitions, as they are in the ordinary case. We shall describe (briefly) the Mullineux map and describe its fixed points. We will use an alternative method of recording the data involved, which makes the map a bit more appealing.

First let us recall the Frobenius symbol of a partition (see for example [1]), a method of representing a partition to make the conjugation action obvious. Given a partition λ, we draw its Young diagram [λ], and then construct a 2 × k array

Fr(λ) = (\begin{matrix}a_{1} & a_{2} & ⋯ & a_{k} \\ b_{1} & b_{2} & ⋯ & b_{k} \end{matrix}) ,

where k is the number of nodes on the main diagonal of the diagram, a_{i} is the number of nodes to the right of the ith node on the diagonal, and b_{i} is the number of nodes below the ith node on the diagonal. For example, the Frobenius symbol

for λ = (5^{2}, 3, 1) is Fr(λ) = (\begin{matrix}4 & 3 & 0 \\ 3 & 1 & 0 \end{matrix}), as we see from the diagram below.

[figure]

Notice that n is the sum of all the entries in the array, plus the number of columns. Also notice that another way to obtain the symbol is to set

a_{1} = (the length of the rim of [λ]) − (the number of rows in [λ]),
b_{1} = (the number of rows in [λ]) − 1;

then remove the rim and repeat to obtain a_{2}, b_{2}, etc.

The advantage of the notation is that Fr(λ′) is obtained from Fr(λ) by simply interchanging the two rows. In particular, the fixed points of the conjugation map become obvious. We shall see that with an appropriately defined modular analogue of Fr(λ), the same is true of the Mullineux map.

Given a partition λ, we define the p-rim of the associated Young diagram [λ] as follows (see [5] for a more rigorous definition): Beginning at the top right-hand corner of [λ], the first p nodes of the rim are in the p-rim. Then skip to the next row, and take the next p nodes of the rim. Continue until we reach the end of the rim; the last of these “p-segments” may contain fewer than p nodes. For example, the 3-rim and the 5-rim of the diagram for λ = (5^{2}, 3, 1) are shown as open circles:

p = 3 : \begin{matrix}• & • & • & • & ∘ \\ • & • & • & ∘ & ∘ \\ ∘ & ∘ & ∘ \\ ∘ \end{matrix} p = 5 : \begin{matrix}• & • & • & • & ∘ \\ • & • & ∘ & ∘ & ∘ \\ • & • & ∘ \\ ∘ \end{matrix}

Now let h_{1} be the number of nodes in the p-rim of λ, and let r_{1} be the number of rows in λ. Delete the p-rim and repeat to obtain sequences h_{1}, h_{2}, … and r_{1}, r_{2}, . . .. Let k be such that h_{k+1} = r_{k+1} = 0, but h_{k} ≠ 0 ≠ r_{k}. The standard way of keeping track of this data is via the Mullineux symbol

M(λ) = (\begin{matrix}h_{1} & h_{2} & ⋯ & h_{k} \\ r_{1} & r_{2} & ⋯ & r_{k} \end{matrix});

but we want to give a different formulation which encodes the same information but makes the desired map more intuitive.

The p-modular Frobenius symbol for λ, Fr_{p}(λ), will be a 3 × k array

Fr_{p}(λ) = (\begin{matrix}a_{1} & a_{2} & ⋯ & a_{k} \\ b_{1} & b_{2} & ⋯ & b_{k} \\ ε_{1} & ε_{2} & ⋯ & ε_{k} \end{matrix}) ,

constructed as follows:

ε_{i} = \begin{cases}0 if p|h_{i}, \\ 1 if p |̸h_{i}, \end{cases}
a_{i} = h_{i} − r_{i},
b_{i} = r_{i} − ε_{i}.

Notice that n is the sum of all the entries in Fr_{p}(λ).

The fact that the Mullineux symbol uniquely determines λ (proved by Mullineux in [8]) means that the modular Frobenius symbol uniquely determines λ (as it is possible to reconstruct M(λ) from Fr_{p}(λ)).

Finally, if λ has modular Frobenius symbol

Fr_{p}(λ) = (\begin{matrix}a_{1} & a_{2} & ⋯ & a_{k} \\ b_{1} & b_{2} & ⋯ & b_{k} \\ ε_{1} & ε_{2} & ⋯ & ε_{k} \end{matrix}) ,

then the Mullineux map m is defined by

Fr_{p}(m(λ)) = (\begin{matrix}b_{1} & b_{2} & ⋯ & b_{k} \\ a_{1} & a_{2} & ⋯ & a_{k} \\ ε_{1} & ε_{2} & ⋯ & ε_{k} \end{matrix}) .

So to perform the p-analogue of conjugation, we just interchange the top two rows of the modular Frobenius symbol for the partition — just as in the ordinary case, we switch the two rows of Fr(λ).

Now it is easy to see the fixed points of m: λ = m(λ) if and only if the top two rows of Fr_{p}(λ) are the same. For example, if λ = (5^{2}, 3, 1), then

Fr_{3}(λ) = (\begin{matrix}3 & 1 & 1 & 0 \\ 3 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix}) Fr_{5}(λ) = (\begin{matrix}2 & 2 & 0 \\ 3 & 3 & 2 \\ 1 & 0 & 1 \end{matrix}) .

Thus if char K = 3, D^{λ} ≅ D^{m(λ)} and thus D^{λ} ↓_{A_{n}} splits as a sum of two irreducible modules; while if char K = 5, D^{λ} ≅ D̸^{m(λ)}, and so D^{λ} ↓_{A_{n}} is irreducible.

References

[1] George E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 49 (1984), no. 301, 1–44. MR 85m:11063

[2] George E. Andrews and Jørn B. Olsson, Partition identities with an application to group representation theory, J. Reine Angew. Math. 413 (1991), 198–212. MR 91k:20019

[3] Dave Benson, Spin modules for symmetric groups, J. London Math. Soc. (2) 38 (1988), 250– 262. MR 89k:20020

[4] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley, New York, 1962. MR 26:2519

[5] Ben Ford and Alexander Kleshchev, A proof of the Mullineux conjecture, Math. Z. (to appear).

[6] Gordon D. James, The irreducible representations of the symmetric groups, Bull. London Math. Soc. 8 (1976), 229–232. MR 54:5329

[7] Gordon D. James and Adalbert Kerber, The representation theory of the symmetric group, Addison-Wesley, London, 1981. MR 83k:20003 [8] Glen Mullineux, Bijections of p-regular partitions and p-modular irreducibles of the symmetric

groups, J. London Math. Soc. (2) 20 (1979), 60–66. MR 80j:20016

[9] Daniel Edwin Rutherford, Substitutional analysis, Edinburgh University Press, 1948. MR 10:280i

Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350

   E-mail address: ford@math.washington.edu

Current address: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106

   E-mail address: bjf6@po.cwru.edu