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Modular functions and resolvent problems

With an appendix by Nate Harman

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Abstract

The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of “resolvent problems” formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension at \(p=2\) for the symmetric groups \(S_n\) is equal to the essential dimension at 2 of certain \(S_n\)-coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic p, specifically Serre-Tate theory, as well as a family of remarkable mod 2 symplectic \(S_n\)-representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the \(p=2\) case. In the second half of this paper we introduce the notion of \(\mathcal {E}\)-versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are \(\mathcal {E}\)-versal. We use these \(\mathcal {E}\)-versality result to deduce the equivalence of Hilbert’s 13th Problem (and related conjectures) with problems about congruence covers.

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Notes

  1. See e.g. [11,12,13, 25, 26, 42, 43, 45, 51], as well as [48, 49].

  2. When the functions are understood, we denote an algebraic function simply by the cover \(\tilde{X}\rightarrow X\).

  3. The reader may think of any version of the theory of p-adic analytic spaces they prefer (Tate, Raynaud, Berkovich, or Hüber’s adic spaces), as this will have no bearing on our arguments.

  4. The results of Dickson [18] and Wagner [71, 72] show that the permutation irrep is a minimal-dimensional faithful irrep for \(n>8\) and \(p=2\), or for \(n>6\) and p odd.

  5. Mutatis mutandis, this follows by the same reasoning as in [21].

  6. Recall that there are exceptional isomorphisms \(\text {Sp}_4(\mathbb {F}_2)\cong O_4^+(\mathbb {F}_3)\cong S_6\).

  7. Recall that there is an exceptional isomorphism of \(O_5^+(\mathbb {F}_3)\) with the Weyl group of \(E_6\).

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Acknowledgements

The authors would like to thank Robert Guralnick for pointing out a mistake in an earlier version of this paper. We also thank Igor Dolgachev, Bert van Geemen, Bruce Hunt, Aaron Landesman, Curt McMullen, Zinovy Reichstein and Ron Solomon for helpful correspondence. We thank the anonymous referee for helpful comments and suggestions.

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Correspondence to Jesse Wolfson.

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Communicated by Vasudevan Srinivas.

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The authors are partially supported by NSF grants DMS-1811772 and Jump Trading Mathlab Research Fund (BF), DMS-1601054 (MK) and DMS-1811846 (JW).

Appendix by Nate Harman Department of Mathematics, University of Michigan, Ann Arbor MI; Email nharman@umich.edu.

Appendices

Appendix by Nate Harman

On quadratic representations of \(S_n\)

1.1 Statement of results

Recall that any linear representation of a p-group G over a field k of characteristic p contains a non-zero invariant vector, in particular this implies that the only irreducible representation of G over k is the trivial representation. This does not mean that all representations are trivial though, there are non-split extensions of trivial representations and understanding their structure is a central part of modular representation theory.

In a non-semisimple setting, one basic invariant of a representation is its Lowey length. For representations of p-groups in characteristic p it can be defined as follows: Start with a representation V and then quotient it by its space of invariants to obtain a new representation \(V' = V / V^G\), then repeat this process until the quotient is zero. The Lowey length is the number of steps this takes.

In the above work Farb, Kisin, and Wolfson analyze certain special representations of symmetric groups in characteristic 2, the so-called Dickson embeddings. Typically denoted \(D^{(n-1,1)}\) in the representation theory literature, these representations have the following key property: Let \(n = 2m\) or \(2m+1\), these representations have Lowey length 2 when restricted to the rank m (which is the maximum possible) elementary abelian 2-subgroup \(H_n\) generated by \((1,2), (3,4), \dots , \text { and } (2m-1,2m)\).

This motivates the following definition: We say that an irreducible representation of a \(S_n\) in characteristic p is quadratic with respect to a maximal rank elementary abelian p-subgroup H if it has Lowey length 2 upon restriction to H. The purpose of this note is to prove first that this is only a characteristic 2 phenomenon, and second that these representations \(D^{(n-1,1)}\) are the only representations which are quadratic with respect to some maximal rank elementary abelian p-subgroup for n sufficiently large (\(n\ge 9\)).

In characteristic \(p > 2\), the maximal rank elementary abelian p-subgroups in \(S_n\) are just those generated by a maximal collection of disjoint p-cycles. Our first main theorem tells us that there are no quadratic representations in characteristic \(p >2\), and in fact we can detect the failure to be quadratic here by restricting to a single p-cycle.

Theorem A.1

Any irreducible representation of \(S_n\) with \(n \ge p\) in characteristic \(p>2\) which is not a character has Lowey length at least 3 upon restriction to the copy of \(C_p\) generated by \((1,2,\dots , p)\), and therefore is not quadratic with respect to any maximal rank elementary abelian p-subgroup.

Note that in any characteristic \(p >2\) the characters of \(S_n\) are just the trivial and sign representations.

In characteristic 2 things are a bit more complicated. While the subgroup \(H_n\) of \(S_{n}\) is a maximal rank elementary 2-subgroup, it is no longer the unique such subgroup up to conjugation. Recall that in \(S_4\) there is the Klein four subgroup \(K = \{e, (12)(34), (13)(24), (14)(23) \}\), which is a copy of \(C_2^2\) not conjugate to \(H_4\).

We can construct other maximal rank elementary 2-subgroups of \(S_{n}\) by taking products

$$\begin{aligned} \underbrace{K \times K \times \dots \times K}_{m \text { times}} \times H_{n-4m} \ \subset \ \underbrace{S_4 \times S_4 \times \dots \times S_4}_{m \text { times}} \times S_{n-4m} \ \subset \ S_{n} \end{aligned}$$

and up to conjugacy though these are all the maximal rank elementary abelian 2-subgroups inside \(S_{n}\).

\(S_8\) has a special irreducible representation \(D^{(5,3)}\) of dimension 8 which upon restriction \(A_8\) decomposes as a direct sum \(D^{(5,3)+} \oplus D^{(5,3)-}\) of two representations of dimension 4. These representations realize the “exceptional" isomorphism \(A_8 \cong GL_4(\mathbb {F}_2)\), or rather they realize two different isomorphisms differing by either by conjugating \(A_8\) by a transposition in \(S_8\) or by the inverse-transpose automorphism of \(GL_4(\mathbb {F}_2)\). Under this isomorphism the subgroup \(K\times K \subset A_8\) gets identified with the subgroup of matrices of the form

$$\begin{aligned} \begin{bmatrix} 1 &{}\quad 0 &{}\quad a &{}\quad b \\ 0 &{}\quad 1 &{}\quad c &{}\quad d \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0&{}\quad 1 \end{bmatrix} \end{aligned}$$

which is manifestly quadratic. Our second main theorem will be to show that there are no other quadratic representations other than the Dickson embedding once n is at least 9.

Theorem A.2

Suppose V is a non-trivial irreducible representation of \(S_n\) with \(n \ge 9\) over a field of characteristic 2 which is quadratic with respect to a maximal rank elementary abelian 2-subgroup H. Then \(V \cong D^{(n-1,1)}\), and H is conjugate to \(H_n\).

1.2 Proofs of main theorems

We will be assuming a familiarity with the modular representation theory of symmetric groups. A standard reference for this material the book [38] of James, which we will be adopting the notation from and referring to for all the basic results we need. The irreducible representations of \(S_n\) in characteristic p are denoted by \(D^\lambda \), for p-regular partitions \(\lambda \) of n. These arise as quotients of the corresponding Specht modules \(S^\lambda \), which are well behaved reductions of the ordinary irreducible representations in characteristic zero.

1.2.1 Proof of Theorem A.1

First we will reduce the problem to just looking at representations of \(S_p\). For that we have the following lemma:

Lemma A.1

  1. (1)

    Every irreducible representation V of \(S_n\) with \(n \ge p\) in characteristic \(p > 3\) which is not a character has a composition factor when restricted to \(S_p\) which is not a character.

  2. (2)

    Every irreducible representation V of \(S_n\) with \(n \ge 4 \) in characteristic 3 which is not a character has a composition factor when restricted to \(S_4\) which is not a character.

Proof

For part (a) suppose V only has composition factors which are characters when restricted to \(S_p\). If we restrict this to the alternating group \(A_p\) all the composition factors must be trivial, as \(A_p\) only has the trivial character. If we further restrict to \(A_{p-1}\) the whole action must be trivial because representations of \(A_{p-1}\) are semisimple in characteristic p. However if the action of \(A_{p-1}\) is trivial on V then so is the action of the entire normal subgroup generated by \(A_{p-1}\) inside \(S_n\), which we know is all of \(A_n\) if \(n>3\). So V must be the trivial as a representation of \(A_n\), and is therefore a character of \(S_n\).

For part (b), let’s again suppose V only has composition factors that are characters when restricted to \(S_4\), which implies it only has trivial composition factors when restricted to \(A_4\). If we further restrict to the Klein four subgroup K the whole action must be trivial because representations of K are semisimple in characteristic \(p \ne 2\). As before we see V must be trivial for the normal subgroup of \(S_n\) generated by K, which we know is all of \(A_n\) for \(n > 4\). Therefore V is a character. \(\square \)

Remark: The modification for characteristic 3 is necessary because in characteristic 3 the only irreducible representations of \(S_3\) are the trivial and sign representations. Theorem A.1 holds vacuously in this case.

It is now enough to prove Theorem A.1 for \(S_p\) in characteristic \(p >3\), and for \(S_4\) in characteristic 3. Let’s first focus on the case where \(p>3\). If \(\lambda \) is a p-core, then Nakayama’s conjecture (which is actually a theorem, see [38] Theorem 21.11) tells us \(D^\lambda = S^\lambda \) is projective, and hence remains projective when restricted to \(C_p\) and therefore has Lowey length p. This leaves those irreducible representations corresponding to hook partitions \( \lambda = (p-k, 1^k)\).

In the simplest case where \(\lambda = (p-1,1)\) then \(D^\lambda \) is the \((p-2)\)-dimensional quotient of the standard \((p-1)\)-dimensional representation \(S^{(p-1,1)}\) by its one dimensional space of invariants, and one can easily verify this forms a single \((p-2)\)-dimensional indecomposable representation of \(C_p\). Peel explicitly computed the decomposition matrices for \(S_p\) in characteristic p (see [38] Theorem 24.1), and it follows from his calculation that the remaining irreducible representations \(D^\lambda \) with \(\lambda = (p-k, 1^k)\) for \(1 < k \le p-2\) are just exterior powers \(\Lambda ^k D^{(p-1,1)}\) of this \((p-2)\)-dimensional representation.

Since \(k < p\) we know that \(\Lambda ^k D^{(p-1,1)}\) is a direct summand of \((D^{(p-1,1)})^{\otimes k}\), which as a representation of \(C_p\) is just the unique \((p-2)\)-dimensional indecomposable representation tensored with itself k times. Tensor product decompositions for representations of cyclic groups are known explicitly ([29] Theorem 3), and in particular it is known that a tensor product of two odd dimensional indecomposable representations of \(C_p\) always decomposes as a direct sum of odd dimensional indecomposable representations. So we see \((D^{(p-1,1)})^{\otimes k}\) and \(\Lambda ^k D^{(p-1,1)} = D^{(p-k, 1^k)}\) only have odd length indecomposable factors when restricted to \(C_p\). If it had Lowey length 1 when restricted to \(C_p\) that means the action is trivial, which implies the action of \(A_p\) must also be trivial as \(A_p\) is simple, but that would imply the original representation of \(S_n\) was a character.

In the characteristic 3 case there are only two irreducible representations of \(S_4\), they are the standard 3-dimensional representation \(S^{(3,1)} = D^{(3,1)}\) and its sign twisted version \(S^{(2,1,1)} = D^{(2,1,1)}\). These are 3-core partitions so again by Nakayama’s conjecture they are both projective and therefore remain projective when restricted to \(C_3\) and have Lowey length 3. \(\square \)

1.2.2 Proof of Theorem A.2

The overall structure of the proof will be to successively rule different classes of representations and maximal rank elementary abelian 2-subgroups through a sequence of lemmas. The first such lemma will let us rule out those irreducible representations \(D^\lambda \) where \(\lambda \) is a 2-regular partition with at least 3 parts.

Lemma A.2

If \(\lambda \) is a 2-regular partition with at least 3 parts, then the irreducible representation \(D^\lambda \) of \(S_n\) contains a projective summand when restricted to \(S_6\).

Proof: Note that any 2-regular partition \(\lambda \) with at least 3 parts can be written as \((3,2,1) + \mu = (\mu _1 +3, \mu _2 + 2 , \mu _3 + 1, \mu _4, \dots , \mu _\ell )\) for some partition \(\mu = (\mu _1, \mu _2, \dots , \mu _\ell )\). James and Peel [39] constructed explicit Specht filtrations for \(Ind_{S_6 \times S_{n-6}}^{S_n}(S^{(3,2,1)} \otimes S^\mu )\), which have \(S^\lambda \) as the top filtered quotient. In particular this implies \(Ind_{S_6 \times S_{n-6}}^{S_n}(S^{(3,2,1)} \otimes S^\mu )\) has \(D^\lambda \) as a quotient. However by Frobenius reciprocity we know that

$$\begin{aligned} \text {Hom}_{S_n}(Ind_{S_6 \times S_{n-6}}^{S_n}(S^{(3,2,1)} \otimes S^\mu ), D^\lambda ) \cong \text {Hom}_{S_6 \times S_{n-6}}(S^{(3,2,1)} \otimes S^\mu , Res_{S_6 \times S_{n-6}}^{S_n}(D^\lambda )). \end{aligned}$$

So since the left hand side is nonzero, the right hand side is as well.

Now if we look at \(S^{(3,2,1)} \otimes S^\mu \) as a representation of \(S_6\) it is just a direct sum of \(\text {dim} (S^\mu )\) copies of \(S^{(3,2,1)}\), which we know is irreducible and projective by Nakayama’s conjecture. In particular the image under any nonzero homomorphism is also just a direct sum of copies of \(S^{(3,2,1)}\), so \(D^\lambda \) must contain at least one copy of \(S^{(3,2,1)}\) as a direct summand. \(\square \)

Corollary A.1

If \(\lambda \) is a 2-regular partition with at least 3 parts, then \(D^\lambda \) is not quadratic with respect to any maximal rank elementary abelian 2-subgroup of \(S_n\).

Proof. After conjugating we may assume that our maximal rank elementary abelian subgroup intersects \(S_6\) in an elementary abelian 2-group of rank at least 2. The previous lemma says any such irreducible representation must contain projective summand when restricted to \(S_6\), and then this summand remains projective upon restriction to the intersection of \(S_6\) with our maximal rank elementary 2-subgroup. Projective representations of \(C_2^2\) have Lowey length 3, so the Lowey length for the entire maximal rank elementary abelian 2-subgroup of \(S_n\) must be at least that big. \(\square \)

This reduces the problem to understanding what happens for two-part partitions \(\lambda = (a,b)\). These representations are much better understood then the general case. For one thing, the branching rules for restriction are completely known in this case, although we’ll just need the following simplified version:

Lemma A.3

(See [52] Theorem 3.6, following [66]). If (ab) is a two-part partition of n with \(a-b > 1\) then \(D^{(a-1,b)}\) appears as a subquotient with multiplicity one inside the restriction of \(D^{(a,b)}\) to \(S_{n-1}\), and the other composition factors are all of the form \(D^{(a-1+r,b-r)}\) with \(r>0\).

Recall that we defined \(H_{2k} \subset S_{2k}\) to be the elementary abelian 2-subgroup of \(S_{2k}\) generated by the odd position adjacent transpositions \((2i-1, 2i)\) for \(1 \le i \le k\), we will also consider \(H_{2k}\) as a subgroup of \(S_n\) for \(n>2k\) via the standard inclusions of \(S_{2k}\) into \(S_n\). The next lemma will be to settle for us exactly which representations are have Lowey length 2 when restricted to the standard maximal rank elementary abelian subgroup \(H_n\).

Lemma A.4

\(D^{(n-k,k)}\) contains a projective summand when restricted to \(H_{2k}\).

Proof: We know from the branching rules (Lemma A.3) that \(D^{(n-k,k)}\) contains a copy of \(D^{(k+1,k)}\) as a subquotient when restricted to \(S_{2k+1}\), so it is enough to verify it for \(D^{(k+1,k)}\). Moreover \(H_{2k} \subset S_{2k}\) so really this calculation is taking place inside \(M(2k) := \text {Res}^{S_{2k+1}}_{S_{2k}}D^{(k+1,k)}\).

These representations \(D^{(k+1,k)}\) and M(2k) are well studied. Benson proved \(D^{(k+1,k)}\) is a reduction modulo 2 of the so-called basic spin representation of \(S_{2k+1}\) in characteristic zero ([5] Theorem 5.1). Nagai and Uno ([69] Theorem 2, or see [60] Proposition 3.1 for an account in English), gave explicit matrix presentations for M(2k) and showed that they have the following recursive structure:

$$\begin{aligned} \text {Res}^{S_{2m}}_{S_{2i} \times S_{2m-2i}} M(2m) \cong M(2i)\otimes M(2m-2i) \end{aligned}$$

In particular since M(2) can easily be seen to be the regular representation of \(S_2 = H_2\), it follows by induction that M(2k) is projective (and just a single copy of the regular representation) for \(H_{2k}\). \(\square \)

Corollary A.2

The only nontrivial irreducible representation of \(S_n\) which is quadratic with respect to \(H_n\) is \(D^{(n-1,1)}\).

Proof: Corollary A.1 tells us that if \(\lambda \) has at least 3 parts, \(D^\lambda \) has Lowey length at least 3 when restricted to \(H_n\). Then Lemma A.4 tells us that \(D^{(n-k,k)}\) has Lowey length at least \(k+1\) as a \(H_n\) representation and is therefore not quadratic for \(k>1\). \(\square \)

To finish the proof of Theorem A.2 we need to show that for n at least 9 that there are no representations which are quadratic with respect to to any of these other maximal rank elementary abelian 2-subgroups \(K^m \times H_{n-4k}\) with \(m \ge 1\). Lemma A.1 rules out \(D^\lambda \) for \(\lambda \) of length at least 3, so again we will just need to address the case when \(\lambda \) is a length 2 partition.

We do this through a series of lemmas ruling out different cases, but first will state the following well-known fact from the modular representation theory of symmetric groups:

Lemma A.5

([38] Theorem 9.3). If \(\lambda \) is a partition of n, then \(S^\lambda \) restricted to \(S_{n-1}\) admits a filtration

$$\begin{aligned} 0=M_0 \subset M_1 \subset \dots \subset M_N \cong S^\lambda \end{aligned}$$

such that the successive quotients \(M_i / M_{i-1}\) are isomorphic to Specht modules \(S_\mu \), and \(S^\mu \) appears if and only if \(\mu \) is obtained from \(\lambda \) by removing a single box, in which case it appears with multiplicity one.

Lemma A.6

\(D^{(n-1,1)}\), for \(n \ge 5\), contains a projective summand when restricted to K, and is therefore not quadratic with respect to any group containing K.

Proof: It suffices to prove it for \(D^{(4,1)}\) as every \(D^{(n-1,1)}\) for \(n >5\) contains it as a composition factor upon restriction to \(S_5\) by Lemma A.3. This representation \(D^{(4,1)}\) is just the 4 dimensional subspace of \(\mathbb {F}_2^5\) where the sum of the coordinates is zero. If we restrict this representation to \(S_4\) this can be identified with the standard 4-dimensional permutation representation via the map \((a,b,c,d) \rightarrow (a,b,c,d,-a-b-c-d)\). The restriction of the standard action of \(S_4\) on a 4-element set to K is simply transitive, so this representation is just a copy of the regular representation. \(\square \)

Lemma A.7

\(D^{(n-2,2)}\) for \(n \ge 7\) and \(D^{(n-3,3)}\) for \(n \ge 9\) each contain a projective summand when restricted to K, and are therefore not quadratic with respect to any group containing K.

Proof: It suffices to prove it for \(D^{(5,2)}\) and \(D^{(6,3)}\) as every \(D^{(n-2,2)}\) for \(n >7\) contains \(D^{(5,2)}\) as a composition factor upon restriction to \(S_7\), and similarly every \(D^{(n-2,2)}\) for \(n >7\) contains \(D^{(6,3)}\) as a composition factor upon restriction to \(S_9\) by Lemma A.3.

For \(S_7\) and \(S_9\) the decomposition matrices are known explicitly and we have that \(D^{(5,2)} = S^{(5,2)}\) and \(D^{(6,3)} = S^{(6,3)}\) (see the appendix of [38]). For Specht modules the branching rules are given by Lemma A.5 and \(S^{(5,2)}\) and \(S^{(6,3)}\) both contain \(S^{(4,1)}\) as a subquotient upon restriction to \(S_5\). The result then follows from the previous lemma. \(\square \)

Lemma A.8

\(D^{(n-k,k)}\) for \(k \ge 4\) and \(n \ge 2k+1\) is not quadratic when restricted to \(K^m \times H_{n-4m}\) for any \(m \ge 1\).

Proof: We know by Lemma A.4 these are projective upon restriction to \(H_{2k}\), and are therefore projective when restricted to the intersection of \(H_{2k}\) with \(K^m \times H_{n-4m}\). This intersection has rank at least 2 since \(k\ge 4\), and therefore projective objects have Lowey length at least 3. This completes the proof of Theorem A.2. \(\square \)

1.3 Modifications for \(A_n\)

We will now briefly describe what changes if we work with alternating groups instead of symmetric groups, but we will omit some of the details of the calculations. First a quick summary of the modular representation theory of alternating groups in terms of the theory for symmetric groups:

Upon restriction from \(S_n\) to \(A_n\), the irreducible representations \(D^\lambda \) either remain irreducible, or split as a direct sums \(D^\lambda \cong D^{\lambda +} \oplus D^{\lambda -}\) of two irreducible non-isomorphic representations of the same dimension; all irreducible representations of \(A_n\) are uniquely obtained this way. We’ll note that in characteristic \(p > 2\) this is a standard application of Clifford theory, but in characteristic 2 it is a difficult theorem of Benson ([5] Theorem 1.1). Moreover it is known exactly which \(D^\lambda \) split this way, but we won’t go into the combinatorics here.

When \(p >2\) the maximum rank abelian p-groups in \(S_n\) all lie in \(A_n\), and the proof of Theorem A.1 goes through without modification to give the following theorem.

Theorem A.1’. Any non-trivial irreducible representation of \(A_n\) with \(n \ge p\) in characteristic \(p>2\) has Lowey length at least 3 upon restriction to the copy of \(C_p\) generated by \((1,2,\dots , p)\), and is therefore not quadratic with respect to any maximal rank elementary abelian p-subgroup.

When \(p=2\), the difference is more dramatic. It is no longer true that every maximum rank abelian 2-subgroup of \(S_n\) lies in \(A_n\), in particular \(H_n\) is not a subgroup of \(A_n\). Let \(\tilde{H}_n\) denote the intersection of \(H_n\) and \(A_n\), this has rank one less than \(H_n\). The maximal rank elementary abelian 2-subgroups of \(A_n\) are as follows:

If \(n = 4b\) or \(4b+1\) then up to conjugacy the only maximal rank elementary abelian 2-subgroup inside \(A_n\) is \(K^b\), and it is of rank 2b. If \(n = 4b+2\) or \(4b+3\) then all maximal rank elementary abelian 2-subgroups in \(S_n\) still have maximal rank when intersected with \(A_n\), and up to conjugacy the maximal rank elementary abelian 2-subgroups inside \(A_n\) are of the form:

$$\begin{aligned} \underbrace{K \times K \times \dots \times K}_{m \text { times}} \times \tilde{H}_{n-4m} \ \subset \ \underbrace{A_4 \times A_4 \times \dots \times A_4}_{m \text { times}} \times A_{n-4m} \ \subset \ A_{n} \end{aligned}$$

and these have rank \(2b-1\).

The appropriate modification to Theorem A.2 for alternating groups is the following:

Theorem A.2’. Suppose V is a non-trivial irreducible representation of \(A_n\) with \(n \ge 9\) over a field of characteristic 2 which is quadratic with respect to a maximal rank elementary abelian 2-subgroup H. Then \(n \equiv 2 \text { or } 3\) modulo 4, \(V \cong D^{(n-1,1)}\), and H is conjugate to \(\tilde{H}_n\).

The proof of Theorem A.2 mostly goes through in this case. Some additional care is needed to handle the representations \(D^{\lambda +}\) and \(D^{\lambda -}\) which are not restrictions of irreducible representations of \(S_n\), however one simplifying observation is that since \(D^{\lambda +}\) and \(D^{\lambda -}\) just differ by conjugation by a transposition, they are actually isomorphic to one another upon restriction to a maximal rank elementary abelian 2-subgroup. We will omit the remaining details though.

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Farb, B., Kisin, M. & Wolfson, J. Modular functions and resolvent problems. Math. Ann. 386, 113–150 (2023). https://doi.org/10.1007/s00208-022-02395-8

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