The Duals of the 2-Modular Irreducible Modules of the Alternating Groups
We determine the dual modules of all irreducible modules of alternating groups over fields of characteristic 2.
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| citation_txt | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups / J. Murray // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 033, 5 pages
The Duals of the 2-Modular Irreducible Modules
of the Alternating Groups
John MURRAY
Department of Mathematics & Statistics, Maynooth University, Co. Kildare, Ireland
E-mail: John.Murray@mu.ie
Received January 04, 2018, in final form April 04, 2018; Published online April 17, 2018
https://doi.org/10.3842/SIGMA.2018.033
Abstract. We determine the dual modules of all irreducible modules of alternating groups
over fields of characteristic 2.
Key words: symmetric group; alternating group; dual module; irreducible module; charac-
teristic 2
2010 Mathematics Subject Classification: 20C30
1 Introduction and statement of the result
Let Sn be the symmetric group of degree n ≥ 1 and let k be a field of characteristic p > 0. In [7,
Theorem 11.5] G. James constructed all irreducible kSn-modules Dλ where λ ranges over the
p-regular partitions of n. Here a partition is p-regular if each of its parts occurs with multiplicity
less than p.
As the alternating group An has index 2 in Sn, the restriction Dλ↓An is either irreducible
or splits as a direct sum of two non-isomorphic irreducible kAn-modules. Moreover, every
irreducible kAn-module is a direct summand of some Dλ↓An .
Henceforth we will assume, unless stated otherwise, that k is a field of characteristic 2 which
is a splitting field for the alternating group An. For this, it suffices that k contains the finite
field F4. D. Benson [1] has classified all irreducible kAn-modules:
Proposition 1.1. Let λ = (λ1 > λ2 > · · · > λ2s−1 > λ2s ≥ 0) be a strict partition of n. Then
Dλ↓An is reducible if and only if
(i) λ2j−1 − λ2j = 1 or 2, for j = 1, . . . , s, and
(ii) λ2j−1 + λ2j 6≡ 2 (mod 4), for j = 1, . . . , s.
In this note we determine the dual of each irreducible kAn-module. Now Dλ↓An is a self-dual
kAn-module, as Dλ is a self-dual kSn-module. So we only need to determine the dual of an
irreducible kAn-module which is a direct summand of Dλ↓An , when this module is reducible.
Theorem 1.2. Let λ be a strict partition of n such that Dλ↓An is reducible. Then the two
irreducible direct summands of Dλ↓An are self-dual if
s∑
j=1
λ2j is even and are dual to each other
if
s∑
j=1
λ2j is odd.
This paper is a contribution to the Special Issue on the Representation Theory of the Symmetric Groups
and Related Topics. The full collection is available at https://www.emis.de/journals/SIGMA/symmetric-groups-
2018.html
mailto:John.Murray@mu.ie
https://doi.org/10.3842/SIGMA.2018.033
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html
2 J. Murray
For example D(7,5,1)↓A13
∼= S ⊕ S∗, for a non self-dual irreducible kA13-module S, and
D(5,4,3,1)↓A13 decomposes similarly. On the other hand D(7,6)↓A13
∼= S1 ⊕ S2 where S1 and S2
are irreducible and self-dual.
In order to prove Theorem 1.2, we use the following elementary result, which requires the
assumption that k has characteristic 2:
Lemma 1.3. Let G be a finite group and let M be a semisimple kG-module which affords
a non-degenerate G-invariant symmetric bilinear form B. Suppose that B(tm,m) 6= 0, for some
involution t ∈ G and some m ∈M . Then M has a self-dual irreducible direct summand.
Proof. We have M =
⊕n
i=1Mi, for some n ≥ 1 and irreducible kG-modules M1, . . . ,Mn. Write
m =
∑
mi, with mi ∈Mi, for all i. Then
B(tm,m) =
∑
1≤i≤n
B(tmi,mi) +
n∑
1≤i<j≤n
(
B(tmi,mj) +B(tmj ,mi)
)
=
∑
1≤i≤n
B(tmi,mi).
The last equality follows from the fact that char(k) = 2 and
B(tmi,mj) = B
(
mi, t
−1mj
)
= B(mi, tmj) = B(tmj ,mi).
Without loss of generality B(tm1,m1) 6= 0. Then B restricts to a non-zero G-invariant
symmetric bilinear form B1 on M1. As M1 is irreducible, B1 is non-degenerate. So M1 is
isomorphic to its kG-dual M∗1 . �
2 Known results on the symmetric and alternating groups
2.1 The irreducible modules of the symmetric groups
We use the ideas and notation of [7]. In particular for each partition λ of n, James defines the
Young diagram [λ] of λ, and the notions of a λ-tableau and a λ-tabloid.
Fix a λ-tableau x. So x is a filling of [λ] with the symbols {1, . . . , n}. The corresponding
λ-tabloid is {x} := {σ(x) |σ ∈ Rx}, where Rx is the row stabilizer of x. We regard {x} as an
ordered set partition of {1, . . . , n}. The Z-span of the λ-tabloids forms the ZSn-lattice Mλ, and
the set of λ-tabloids is an Sn-invariant Z-basis of Mλ.
Recall from [7, Section 4] that corresponding to each tableau x there is a polytabloid ex :=∑
sgn(σ){σx} in Mλ. Here σ ranges over the permutations in the column stabilizer Cx of
the tableau x. The Specht lattice Sλ is defined to be the Z-span of all λ-polytabloids. In
particular Sλ is a ZSn-sublattice of Mλ; it has as Z-basis the polytabloids corresponding to
the standard λ-tableaux (i.e., the numbers increase from left-to-right along rows, and from
top-to-bottom along columns).
Now James defines 〈 , 〉 to be the symmetric bilinear form on Mλ which makes the tabloids
into an orthonormal basis. As the tabloids are permuted by the action of Sn, it is clear that 〈 , 〉
is Sn-invariant.
Suppose now that λ is a strict partition and consider the unique permutation τ ∈ Rx which
reverses the order of the symbols in each row of the tableau x. In [7, Lemma 10.4] James shows
that 〈τex, ex〉 = 1, as {x} is the only tabloid common to ex and eτx (in fact James proves that
〈τex, ex〉 is coprime to p, if λ is p-regular, for some prime p). Set Jλ := {x ∈ Sλ | 〈x, y〉 ∈ 2Z,
for all y ∈ Sλ}. Then 2Sλ ⊆ Jλ and it follows from [7, Theorem 4.9] that Dλ := (Sλ/Jλ)⊗F2 k
is an absolutely irreducible kSn-module, for any field k of characteristic 2.
The Duals of the 2-Modular Irreducible Modules of the Alternating Groups 3
2.2 The real 2-regular conjugacy classes of the alternating groups
A conjugacy class of a finite group G is said to be 2-regular if its elements have odd order.
R. Brauer proved that the number of irreducible kG-modules equals the number of 2-regular
conjugacy classes of G [4]. Now Brauer’s permutation lemma holds for arbitrary fields [3,
footnote 19]. So it is clear that the number of self-dual irreducible kG-modules equals the
number of real 2-regular conjugacy classes of G.
We review some well known facts about the 2-regular conjugacy classes of the alternating
group. See for example [8, Section 2.5].
Corresponding to each partition µ of n there is a conjugacy class Cµ of Sn; its elements
consist of all permutations of n whose orbits on {1, . . . , n} have sizes {µ1, . . . , µ`} (as multiset).
So Cµ is 2-regular if and only if each µi is odd.
Let µ be a partition of n into odd parts. Then Cµ ⊆ An. If µ has repeated parts then Cµ
is a conjugacy class of An. As Cµ is closed under taking inverses, Cµ is a real conjugacy class
of An.
Now assume that µ has distinct parts. Then Cµ is a union of two conjugacy classes C±µ
of An. Set m := n−`(µ)
2 and let z ∈ Cµ. Then z is inverted by an involution t ∈ Sn of cycle type
(2m, 1n−2m). Since CSn(z) ∼=
∏
Z/µjZ is odd, t generates a Sylow 2-subgroup of the extended
centralizer C∗Sn(z) of z in Sn. It follows that z is conjugate to z−1 in An if and only if t ∈ An.
This shows that C±µ are real classes of An if and only if n−`(µ)
2 is even. This and the discussion
above shows:
Lemma 2.1. The number of self-dual irreducible kAn-modules equals the number of non-strict
odd partitions of n plus twice the number of strict odd partitions µ of n for which n−`(µ)
2 is even.
3 Bressoud’s bijection
We need a special case of a partition identity of I. Schur [9]. This was already used by Benson
in his proof of Proposition 1.1:
Proposition 3.1 (Schur, 1926). The number of strict partitions of n into odd parts equals the
number of strict partitions of n into parts congruent to 0, ±1 (mod 4) where consecutive parts
differ by at least 4 and consecutive even parts differ by at least 8.
D. Bressoud [5] has constructed a bijection between the relevant sets of partitions. We
describe a simplified version of this bijection.
Let µ = (µ1 > µ2 > · · · > µ`) be a strict partition of n whose parts are all odd. We sub-
divide µ into ‘blocks’ of at most two parts, working recursively from largest to smallest parts.
Let j ≥ 1 and suppose that µ1, µ2, . . . , µj−1 have already been assigned to blocks. We form the
block {µj , µj+1} if µj = µj+1 + 2, and the block {µj} otherwise (if µj ≥ µj+1 + 4). Let s be the
number of resulting blocks of µ.
Next we form the sequence of positive integers σ = (σ1, σ2, . . . , σs), where σj is the sum of
the parts in the j-th block of µ. Then the σj are distinct, as the odd parts form a decreasing
sequence, with minimal difference 4, and the even parts form a decreasing sequence, with minimal
difference 8. Moreover, each even σj is the sum of a pair of consecutive odd integers. So σj 6≡ 2
(mod 4), for all j > 0.
We get a composition ζ of n+ 2s(s− 1) by defining
ζ1 = σ1, ζ2 = σ2 + 4, . . . , ζs = σs + 4(s− 1).
The even ζj form a decreasing sequence, with minimal difference 4, and the odd ζj form a weakly
decreasing sequence (ζj = ζj+1 if and only if ζj , ζj+1 represent two singleton blocks {2k − 1}
and {2k − 5} of µ, for some k ≥ 0).
4 J. Murray
Choose a permutation τ such that ζτ1 ≥ ζτ2 ≥ · · · ≥ ζτs. Then we get a strict partition γ
of n by defining
γ1 = ζτ1, γ2 = ζτ2 − 4, . . . , γs = ζτs − 4(s− 1).
By construction, the minimal difference between the parts of γ is 4 and the minimal difference
between the even parts of γ is 8. Moreover, γj ≡ ζτj (mod 4). So γj 6≡ 2 (mod 4). Then µ→ γ
is Bressoud’s bijection.
Finally form a strict partition λ of n which has 2s− 1 or 2s parts, by defining
(λ2j−1, λ2j) =
(γj
2
+ 1,
γj
2
− 1
)
, if γj is even or(
γj + 1
2
,
γj − 1
2
)
, if γj is odd.
Then λ satisfies the constraints (i) and (ii) of Proposition 1.1. Conversely, it is easy to see that
if λ satisfies these constraints, then λ is the image of some strict odd partition µ of n under the
above sequence of operations.
Lemma 3.2. Let µ be a strict-odd partition of n and let λ be the strict partition of n constructed
from µ as above. Then n−`(µ)
2 =
∑
λ2j.
Proof. Each pair of consecutive parts λ2j−1, λ2j of λ corresponds to a block B of µ. Moreover
by our description of Bressoud’s bijection, there are integers q1, . . . , qs, with
∑
qj = 0 such that
(λ2j−1 + 2qj , λ2j + 2qj) =
(
µi + 1
2
,
µi − 1
2
)
, if B = {µi},
(µi, µi+1), if B = {µi, µi+1}.
In case B = {µi, µi+1}, we have µi = µi+1 + 2 and thus µi−1
2 + µi+1−1
2 = λ2j + 2qj . We conclude
that
n− `(µ)
2
=
`(µ)∑
i=1
µi − 1
2
=
s∑
j=1
(λ2j + 2qj) =
s∑
j=1
λ2j . �
4 Proof of Theorem 1.2
Let D(n) be the set of strict partitions of n and let S(n) be the set of strict partitions of n which
satisfy conditions (i) and (ii) in Proposition 1.1. So there are 2|S(n)|+ |D(n)\ S(n)| irreducible
kAn-modules.
Next set S(n)+ := {λ ∈ S(n) |
∑
λ2j is even}. Then it follows from Lemmas 2.1 and 3.2 that
the number of self-dual irreducible kAn-modules equals 2|S(n)+| + |D(n)\ S(n)|. Now Dλ↓An
is an irreducible self-dual kAn-module, for λ ∈ D(n)\S(n). So we can prove Theorem 1.2 by
showing that the irreducible direct summands of Dλ↓An are self-dual for all λ ∈ S(n)+.
Suppose then that λ ∈ S(n)+. Let τ ∈ Sn be the permutation which reverses each row of
a λ-tableau, as discussed in Section 2.1. We claim that τ ∈ An. For τ is a product of
2s∑
i=1
⌊λj
2
⌋
commuting transpositions. Now
⌊λ2j−1
2
⌋
+
⌊λ2j
2
⌋
= λ2j , as λ2j−1 − λ2j = 1, or λ2j−1 − λ2j = 2
and both λ2j−1 and λ2j are odd. So
2s∑
i=1
⌊
λi
2
⌋
=
s∑
j=1
λ2j is even. This proves the claim.
Since Dλ is irreducible and the form 〈 , 〉 is non-zero, 〈 , 〉 is non-degenerate on Dλ. Write
Dλ↓An = S1 ⊕ S2, where S1 and S2 are non-isomorphic irreducible modules. As τ ∈ An, it
follows from Lemma 1.3 that we may assume that S1 is self-dual. Now S∗2 6∼= S∗1
∼= S1 and S∗2 is
isomorphic to a direct summand of Dλ↓An . So S2 is also self-dual. This completes the proof of
the theorem.
The Duals of the 2-Modular Irreducible Modules of the Alternating Groups 5
5 Irreducible modules of alternating groups
over fields of odd characteristic
We now comment briefly on what happens when k is a splitting field for An which has odd
characteristic p. Let sgn be the sign representation of kSn. So sgn is 1-dimensional but non-
trivial. G. Mullineux defined a bijection λ→ λM on the p-regular partitions of n and conjectured
that Dλ ⊗ sgn = DλM for all p-regular partitions λ of n. This was only proved in the 1990’s by
Kleshchev and Ford–Kleshchev. See [6] for details.
Now Dλ↓An
∼= DλM↓An , and Dλ↓An is irreducible if and only if λ 6= λM See [2] for details.
Moreover Dλ and DλM are duals of each other, by [7, Theorem 6.6]. So Dλ↓An is self-dual, if
λ 6= λM . However when λ = λM , we do not know how to determine when the two irreducible
direct summands of Dλ↓An are self-dual.
Acknowledgement
D. Benson told me that it was an open problem to determine the self-dual irreducible kAn-
modules. G.E. Andrews directed me to Bressoud’s paper [5]. We also thank the anonymous
referees for their comments, which helped to improve the clarity of this paper.
References
[1] Benson D., Spin modules for symmetric groups, J. London Math. Soc. 38 (1988), 250–262.
[2] Bessenrodt C., On the representation theory of alternating groups, Algebra Colloq. 10 (2003), 241–250.
[3] Brauer R., On the connection between the ordinary and the modular characters of groups of finite order,
Ann. of Math. 42 (1941), 926–935.
[4] Brauer R., Nesbitt C., On the modular characters of groups, Ann. of Math. 42 (1941), 556–590.
[5] Bressoud D.M., A combinatorial proof of Schur’s 1926 partition theorem, Proc. Amer. Math. Soc. 79 (1980),
338–340.
[6] Ford B., Kleshchev A.S., A proof of the Mullineux conjecture, Math. Z. 226 (1997), 267–308.
[7] James G.D., The representation theory of the symmetric groups, Lecture Notes in Math., Vol. 682, Springer,
Berlin, 1978.
[8] James G.D., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics
and its Applications, Vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.
[9] Schur I., Zur additiven Zahlentheorie, Sitzungsber. Preuß. Akad. Wiss. Phys.-Math. Kl. (1926), 488–495,
Reprinted in Schur I., Gesammelte Abhandlungen, Band III, Springer-Verlag, Berlin – New York, 1973,
43–50.
https://doi.org/10.1112/jlms/s2-38.2.250
https://doi.org/10.2307/1968774
https://doi.org/10.2307/1968918
https://doi.org/10.2307/2043263
https://doi.org/10.1007/PL00004340
https://doi.org/10.1007/BFb0067708
1 Introduction and statement of the result
2 Known results on the symmetric and alternating groups
2.1 The irreducible modules of the symmetric groups
2.2 The real 2-regular conjugacy classes of the alternating groups
3 Bressoud's bijection
4 Proof of Theorem 1.2
5 Irreducible modules of alternating groups over fields of odd characteristic
References
|
| id | nasplib_isofts_kiev_ua-123456789-209539 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-02T23:28:31Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Murray, J. 2025-11-24T10:47:07Z 2018 The Duals of the 2-Modular Irreducible Modules of the Alternating Groups / J. Murray // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C30 arXiv: 1711.06009 https://nasplib.isofts.kiev.ua/handle/123456789/209539 https://doi.org/10.3842/SIGMA.2018.033 We determine the dual modules of all irreducible modules of alternating groups over fields of characteristic 2. D. Benson told me that it was an open problem to determine the self-dual irreducible kAn-modules. G.E. Andrews directed me to Bressoud’s paper [5]. We also thank the anonymous referees for their comments, which helped to improve the clarity of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Duals of the 2-Modular Irreducible Modules of the Alternating Groups Article published earlier |
| spellingShingle | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups Murray, J. |
| title | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups |
| title_full | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups |
| title_fullStr | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups |
| title_full_unstemmed | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups |
| title_short | The Duals of the 2-Modular Irreducible Modules of the Alternating Groups |
| title_sort | duals of the 2-modular irreducible modules of the alternating groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209539 |
| work_keys_str_mv | AT murrayj thedualsofthe2modularirreduciblemodulesofthealternatinggroups AT murrayj dualsofthe2modularirreduciblemodulesofthealternatinggroups |