Exceptional modules for Serre's converse semisimplicity theorems for symmetric and exterior powers in positive characteristic

Let $G$ be a group and $V$ a finite-dimensional representation of $G$ over a field of characteristic $p > 0$. Serre's 1997 "converse theorems" (with an appendix of counterexamples by Walter Feit) determine when semisimplicity of a tensor construction on $V$ — a tensor product $V \otimes W$, a tensor power, an exterior power $\Lambda^m V$, or a symmetric power $\mathrm{Sym}^m V$ — forces $V$ itself to be semisimple: the implication holds outside explicit exceptional classes of $\dim V$ modulo $p$, and Feit's constructions realize most of those classes by counterexamples. One component of the source's open scope, Serre's conjecture extending the $\Lambda^m$ and $\mathrm{Sym}^m$ criteria to symmetric rigid tensor categories, was proved by Gelaki in 2010; his counterexamples, however, live in super-tensor categories, not in categories of group representations. The surviving problem, open since 1997, is to complete the classification for actual group representations: whether a non-semisimple representation with semisimple symmetric square exists at the primes not covered by Feit's construction (for example $p = 7$); the minimal dimension of such counterexamples; exactly which even dimensions admit characteristic-2 exterior-square counterexamples; and whether any converse criterion survives for symmetric powers in the small-characteristic regime $p \le m$.

Background

By a representation of a group $G$ over a field $k$ we mean a finite-dimensional $k$-vector space $V$ together with a linear action of $G$, that is, a module over the group algebra $kG$. A representation is called semisimple (or completely reducible) if it decomposes as a direct sum of irreducible subrepresentations. Maschke's theorem guarantees semisimplicity for every representation of a finite group when the characteristic of $k$ is 0 or prime to the group order; when the characteristic $p > 0$ divides the group order, non-semisimple representations exist, and their study is modular representation theory. Throughout, $k$ is an algebraically closed field of characteristic $p > 0$ unless stated otherwise.

Several standard constructions build new representations from $V$: the tensor product $V \otimes W$; the tensor powers $V^{\otimes m}$; and the exterior powers $\Lambda^m V$ and symmetric powers $\mathrm{Sym}^m V$, which are the quotients of $V^{\otimes m}$ obtained by imposing, respectively, antisymmetry ($v \otimes w = -w \otimes v$) and symmetry ($v \otimes w = w \otimes v$); for $m = 2$ these are the familiar second exterior and symmetric squares. A classical theorem of Chevalley states that over a field of characteristic 0 the tensor product of semisimple representations is semisimple; Serre (1994) proved that the same conclusion holds in characteristic $p$ provided $\dim V + \dim W < p - 2$. The tensor-product converse is now standard graduate-textbook material (see, e.g., Kowalski's ETH lecture notes).

The source of the present problem is Serre's paper "Semisimplicity and Tensor Products of Group Representations: Converse Theorems" (Journal of Algebra 194 (1997) 496–520, with an appendix by Walter Feit), which studied the reverse implication: if a tensor construction on $V$ is semisimple, must $V$ be semisimple? Serre proved: (i) if $V \otimes W$ is semisimple for some $W$ with $\dim W \not\equiv 0 \pmod{p}$, then $V$ is semisimple; (ii) if some tensor power $V^{\otimes m}$, $m \ge 1$, is semisimple, then $V$ is semisimple; (iii) if $\Lambda^m V$ is semisimple and $\dim V \not\equiv 2, 3, \ldots, m \pmod{p}$, then $V$ is semisimple; and (iv) if $\mathrm{Sym}^m V$ is semisimple with $p > m$, and $\dim V \not\equiv -2, -3, \ldots, -m \pmod{p}$, then $V$ is semisimple. In characteristic 2 the symmetric-square criterion is unconditional: $\mathrm{Sym}^2 V$ semisimple forces $V$ semisimple. The excluded dimension congruence classes in (iii) and (iv) are the exceptional classes.

The exceptional classes are largely genuine. A construction in Section 7 of the paper (a semidirect product of a cyclic group of order $p$ with a finite abelian group on which it acts freely; the special case $i = 2$ of Proposition 7.3.1) realizes, for odd $p$, every dimension $n \equiv 2 \pmod{p}$ by a non-semisimple $V$ with $\Lambda^2 V$ semisimple, and Feit's appendix (Theorem A1) shows that whenever $p$ divides $2^m + 1$ for odd $p$ there is such a $V$ of dimension $2^{m+1} \equiv -2 \pmod{p}$ with $\mathrm{Sym}^2 V$ semisimple. In characteristic 2, Feit constructed (Theorem A2), for $G = \mathrm{SL}(2, q)$ with $q \equiv 3 \pmod{8}$ a prime power, a non-semisimple module of dimension $(q+1)/2$ with semisimple exterior square. The symmetric-square constructions all rest on the cyclotomic factorization of $2^m + 1$: at a prime such as $p = 7$ or $p = 23$, where no power of 2 is congruent to $-1$ modulo $p$ (equivalently, the multiplicative order of 2 modulo $p$ is odd), Feit's method produces nothing.

Serre also conjectured (p. 511 of the paper) that criteria (iii) and (iv) extend from representations of groups to symmetric rigid tensor categories: abelian $k$-linear categories equipped with a tensor product, duals, and a symmetry isomorphism $V \otimes W \cong W \otimes V$ satisfying the familiar axioms of vector-space duality. The category $\mathrm{Rep}(G)$ of finite-dimensional representations of a group is the basic example, but a general symmetric rigid tensor category need not admit a forgetful functor to vector spaces. Gelaki (2010) proved the conjecture: over a field of characteristic 0 or characteristic greater than $n$, for every Schur functor $S_\lambda$ of degree $n$ (the functorial constructions on tensor powers, such as $\Lambda^m$ and $\mathrm{Sym}^m$, indexed by partitions $\lambda$ of $n$), semisimplicity of $S_\lambda V$ together with $\dim V$ outside an explicit finite exceptional set forces $V$ semisimple, and every exceptional dimension is realized by a counterexample in some symmetric rigid tensor category. Gelaki's counterexamples are built in super-tensor categories — categories graded by $\mathbb{Z}/2$ in which $v \otimes w = -w \otimes v$ for odd homogeneous elements — which are not categories of group representations, and his theorem retains the hypothesis that the characteristic exceed $n$. The categorical conjecture is therefore resolved while the finite-group questions are untouched; moreover, Gelaki's result shows that any proof of nonexistence in the group case must use group-specific structure (the forgetful functor), since the exceptional classes are always realized categorically.

The converse theorems also motivated Serre's notion of $G$-complete reducibility for subgroups of a reductive algebraic group $G$: a subgroup $H$ is $G$-completely reducible if whenever $H$ lies in a parabolic subgroup of $G$ it lies in a Levi factor of it. This theory, developed in Serre's 2005 Bourbaki survey and studied intensively since, translates semisimplicity questions about modules into structural questions about subgroups. McNinch (2000) settled the forward direction for exterior powers (when $V$ semisimple implies $\Lambda^m V$ semisimple), and Himstedt–Symonds (2014) gave algorithms decomposing exterior and symmetric powers of modules over cyclic 2-groups, a computational toolkit for the characteristic-2 questions. The exterior-square question in its subgroup-theoretic form is posed as an archetypal open problem in the 2022–2023 survey of Litterick–Stewart–Thomas. None of these later works resolves the group-case exceptional-dimension questions that follow.

Problem Statement

Objective: complete the classification of exceptional dimensions for Serre's converse semisimplicity theorems for symmetric and exterior powers of group representations — that is, determine, for each characteristic $p$ and each $m$, exactly which dimensions admit a non-semisimple module $V$ with $\mathrm{Sym}^m V$ or $\Lambda^m V$ semisimple, and determine the minimal such dimensions. Throughout, $k$ is a field of characteristic $p > 0$ and $V$ is a finite-dimensional $kG$-module for a group $G$; the known results and counterexamples are those of Serre's theorems and Feit's appendix summarized in the Background. The specific open questions, all posed by or left open in the 1997 paper and its 1998 Moursund-lecture summary, are:

(1) Symmetric square at uncovered primes. For a prime $p$ such that no power of 2 is congruent to $-1$ modulo $p$ (equivalently, the multiplicative order of 2 modulo $p$ is odd; e.g. $p = 7, 23$): does there exist a group $G$ and a non-semisimple $kG$-module $V$ with $\mathrm{Sym}^2 V$ semisimple? Any such $V$ necessarily has $\dim V \equiv -2 \pmod{p}$. Serre stated in 1998 that at $p = 7$ no example was known; either a construction or a proof of nonexistence answers the question.

(2) Minimal dimensions. For odd primes $p$ for which Feit's construction applies (i.e. $p$ divides $2^m + 1$ for some $m$): determine the minimal dimension of a non-semisimple $V$ with $\mathrm{Sym}^2 V$ semisimple. The dimension $2^{m+1} = 2p - 2$ is attained when $p = 2^m + 1$ is a Fermat prime; the general minimal dimension is unknown.

(3) Characteristic-2 exterior square. In characteristic 2, determine exactly the set of even dimensions $n$ for which there exists a non-semisimple module $V$ of dimension $n$ with $\Lambda^2 V$ semisimple. The known examples have dimensions $(q+1)/2$ for $G = \mathrm{SL}(2, q)$ with $q \equiv 3 \pmod{8}$ a prime power — all $\equiv 2 \pmod{4}$, so dimensions divisible by 4, for instance, are unaccounted for.

(4) Small-characteristic symmetric powers. For $\mathrm{Sym}^m$ with $0 < p \le m$ and $(p, m) \ne (2, 2)$ (the case $p = m = 2$ is resolved unconditionally in the source): does $\mathrm{Sym}^m V$ semisimple imply $V$ semisimple under any congruence hypothesis on $\dim V$ modulo $p$? The source's Theorem 5.3.1 requires $p > m$ — its proof uses the duality between symmetric powers and divided powers, which fails when $p \le m$ — and this regime is not covered.

(5) Base fields. The same questions over base fields that are not algebraically closed; the source's constructions use algebraic closedness, and the general case is not discussed there.

A complete answer for a stated family (e.g. $\mathrm{Sym}^2$ at all odd $p$, or $\Lambda^2$ in characteristic 2) consists of the exact exceptional set of dimensions, together with explicit constructions attaining each realized dimension and proofs of nonexistence for the excluded ones; the minimal-dimension question is answered by the same data. The sub-questions may be answered independently: a partial answer (a single prime, a bounded range of dimensions, a specific small-characteristic pair $(p, m)$) resolves the corresponding sub-question.

References

1. Jean-Pierre Serre and Walter Feit, "Semisimplicity and Tensor Products of Group Representations: Converse Theorems", Journal of Algebra 194 (1997), no. 2, 496–520. DOI: 10.1006/jabr.1996.6929. URL: https://doi.org/10.1006/jabr.1996.6929.
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9. Michael Bate, Sebastian Herpel, Benjamin Martin, and Gerhard Röhrle, "G-complete reducibility and semisimple modules", Bulletin of the London Mathematical Society 43 (2011), no. 6, 1069–1078. DOI: 10.1112/blms/bdr042; arXiv:1011.2012. URL: https://arxiv.org/abs/1011.2012
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11. Alastair J. Litterick, David I. Stewart, and Adam R. Thomas, "Complete reducibility and subgroups of exceptional algebraic groups" (survey; to appear in the proceedings of Groups St Andrews 2022 in Newcastle), arXiv:2209.11310. URL: https://arxiv.org/abs/2209.11310
12. E. Kowalski, "Representation theory", lecture notes, ETH Zürich, Spring Semester 2011 (version of February 17, 2017). URL: https://people.math.ethz.ch/~kowalski/representation-theory.pdf