Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups

Let $p$ be a prime and let $X$ be a finite group with a strongly $p$-embedded subgroup. Recall that a proper subgroup $M$ of $X$ of order divisible by $p$ is strongly $p$-embedded in $X$ if and only if for every $x\in X\setminus M$, $M\cap M^{x}$ is a $p^{\prime}$-group. The most prominent examples of groups with a strongly $p$-embedded subgroup are the rank $1$ groups of Lie type in characteristic $p$. Throughout, we write $m:=m_{p}(X)$ for the $p$-rank of $X$, the largest rank of an elementary abelian subgroup of $X$.

Restrictions on the Jordan block structure of $p$-elements often impose strong constraints on both the representation and the ambient group, and have proved to be an effective tool in the study of modular representations and the $p$-local structure of finite groups. As examples, see [31], [8] and [32].

In this article, we study faithful $\mathbb{F}_{p}X$-modules which are $k$-active, for some $k\leqslant m$. Our main theorem demonstrates that, with a few exceptions, such modules for groups with a strongly $p$-embedded subgroup arise as explicitly described irreducible modules of rank $1$ groups of Lie type in characteristic $p$. Much of the analysis in this paper is devoted to these groups, alongside the alternating groups of degree $2p$, and their associated $\mathbb{F}_{p}$-representations. Consequently, our results also contribute to the representation theory of small-rank groups of Lie type. Since these groups form fundamental building blocks in the analysis of larger quasisimple groups and their representations, a detailed understanding of their $k$-active modular representations is important and therefore of independent interest. The methodology developed to prove our theorems could be generalized to include larger classes of groups, complementing existing research in this area. As an initial step in this direction, in Section 6 we determine the irreducible $\mathbb{F}_{p^{n}}\operatorname{SL}_{3}(p^{n})$-modules for which a unipotent element has at most two non-trivial Jordan blocks (see Proposition 6.2).

Groups with a strongly $p$-embedded subgroup arise naturally in local group theory through what is now called the Alperin–Goldschmidt theorem [2, 10] and as such play a significant role in the classification of the finite simple groups as does, of course, Bender’s fundamental determination of the groups with a strongly $2$-embedded subgroup. This phenomenon is naturally phrased in the language of saturated fusion systems. By the Alperin–Goldschmidt theorem, a saturated fusion system is largely determined by the automorphism groups of its essential subgroups. The key point being that the outer automorphism group $A$ of an essential subgroup $E$ has a strongly $p$-embedded subgroup, and, as $O_{p}(A)=1$, $E/\Phi(E)$ is a faithful $\mathbb{F}_{p}A$-module that under some conditions must be $k$-active for some $k\leqslant m_{p}(S)$. If the essential subgroup is not normal in the group of the fusion system, then the structure of $A$ can often be studied using the theory of failure of factorization modules (see [23]). Failure of factorization modules are $k$-active for some $k\leqslant m$. More importantly, a result of Oliver [25, §2] imposes restrictions on the Jordan block structure of a $p$-element which acts on a weakly closed abelian subgroup in any simple saturated fusion system. An immediate application of our theorems appears in upcoming work of the authors alongside Oliver [13], where reduced fusion systems which have a weakly closed abelian essential subgroup are classified.

We briefly describe some of the most important examples which appear in our main theorems.

Set $X=\operatorname{SL}_{2}(p^{n})$ and $1\leqslant i\leqslant p-1$. Let $V_{i}(p^{n})$ denotes the $\mathbb{F}_{p^{n}}$-space spanned by homogeneous polynomials of degree $i$ in two commuting variables $y$ and $z$. Then $V_{i}(p^{n})$ is made into an $(i+1)$-dimensional $\mathbb{F}_{p^{n}}X$-module by enforcing that

and extending this action to $V_{i}(p^{n})$. Notice that for $\tau\in\mathrm{Aut}(\mathbb{F}_{p^{n}})$, $V_{i}(p^{n})$ is isomorphic to $V_{i}(p^{n})^{\tau}$ as an $\mathbb{F}_{p}X$-module. When we take tensor products $V_{i}(p^{n})\otimes V_{j}(p^{n})$, this tensor product is over $\mathbb{F}_{p^{n}}$.

We call $V_{1}(p^{n})|_{\mathbb{F}_{p}}$ the natural module for $\operatorname{SL}_{2}(p^{n})$, of dimension $2n$. The natural module for $\mathrm{SU}_{3}(p^{n})$ is the restriction of any $3$-dimensional $\mathbb{F}_{p^{2n}}\operatorname{SL}_{3}(p^{2n})$-module to $\mathrm{SU}_{3}(p^{n})$ and then regarded as an $\mathbb{F}_{p}\mathrm{SU}_{3}(p^{n})$-module of dimension $6n$. Finally, the natural module for $\mathrm{Ree}(3^{n})$ is the restriction of any $7$-dimensional $\mathbb{F}_{3^{n}}\mathrm{G}_{2}(3^{n})$-module to $\mathrm{Ree}(3^{n})$, viewed as an $\mathbb{F}_{3}\mathrm{Ree}(3^{n})$-module, of dimension $7n$.

We may now state our main theorem.

For all cases of Theorem 1.2, we also provide explicit Jordan block structure of $W|_{\langle x\rangle}$ when these cases arise in the analysis. This data is conveniently displayed in Table 2 in Appendix B. If (vii) holds, then the possible composition factors of $V|_{L}$ are described in Proposition 2.15 when $p\geqslant 5$. If instead we have that $p=3$ then $L/Z(L)\cong\mathrm{PSL}_{2}(9)$ and the possible composition factors of $V|_{L}$ are described in (i) of Theorem 1.2.

We may extract the following consequence from Theorem 1.2.

Suppose that $x\in X$ has order $p$ and is $k$-active on $V$ for some $k\leqslant m$. Then $x$ centralizes a subspace of $V$ of codimension at most $m(p-1)$. A straightforward argument (see Lemma 4.1) then shows that for $K$ a certain subgroup of $X$ which also has a strongly $p$-embedded subgroup,

Consequently, there exists a non-trivial $K$-composition factor $W$ of $V$ such that $W$ has dimension at most $4m(p-1)$. This means that a central ingredient in the proof of Theorem 1.2 is Theorem 1.4 applied to the pair $(K/C_{K}(W),W)$. When combined with the known classification of low-dimensional representations of quasisimple groups, this theorem significantly sharpens earlier bounds on the minimal dimension of faithful modular representations for groups with a strongly $p$-embedded subgroup (see, for example, [24, Lemma 1.7], [26, Lemma 4.6]). Results of this type have already found repeated applications in group theory, representation theory and the study of fusion systems, and we expect that the strengthened bounds obtained here will prove similarly useful in future work.

We have the following remark concerning case (vii) of Theorem 1.4 where we do not provide an exact description of $O_{p^{\prime}}(X)$.

With some additional analysis of smaller cases, Theorem 1.4 reduces the proof of Theorem 1.2 to the situation where $X$ is a rank $1$ quasisimple group of Lie type in characteristic $p$, where powerful tools such as Steinberg’s Tensor Product Theorem are available.

The main reduction for Theorem 1.2 is given in Proposition 4.3. From there, we initiate an analysis of the various cases which are treated in Proposition 4.4, Proposition 4.5, Proposition 4.6, Proposition 5.3, Proposition 6.5 and Proposition 7.7.

The following corollary may be extracted from Theorem 1.4.

Finally, we note that the classification of the finite simple groups is used sparingly in this work. In Proposition 2.1 we describe the collection $\mathcal{S}$ of finite groups of $p$-rank at least $2$ with a strongly $p$-embedded subgroup, and in Lemma 3.2 we invoke the Schreier property. All results of this paper, including Theorems 1.2 and 1.4, only concern the groups in $\mathcal{S}$. Thus, if readers are primarily concerned with the groups in the class $\mathcal{S}$ and are willing to impose mild additional hypotheses (for example, that $O_{p^{\prime}}(X)$ is solvable), then they may apply the results of this work without appealing to the classification of the finite simple groups.

The structure of the paper is as follows. Section 2 collects various facts about strongly $p$-embedded subgroups, their generation properties and the structure of their representations of small dimensions. We reserve a specific subsection for the groups $\mathrm{Alt}(2p)$, and in particular some facts about their cohomology. These groups often cause us the most difficulty, and consequently, the results we prove about them in Theorem 1.2 and Theorem 1.4 are often weaker when compared with other classes of groups. Section 3 is devoted to the proof of Theorem 1.4, making use of the facts collected in Section 2. This is the most technical section of the paper, where we undertake an intensive analysis of the structure of a minimal counterexample to Theorem 1.4, ultimately showing that such a counterexample does not exist.

In Section 4, we use Theorem 1.4 to reduce the proof of Theorem 1.2 to the case where $X$ is isomorphic to a quasisimple group of Lie type of rank $1$ and in defining characteristic $p$. The remaining sections of the paper then analyze the possible representations of the linear, unitary and Ree groups respectively, which satisfy the hypothesis of Theorem 1.2. We make liberal use of Steinberg’s Tensor Product Theorem, and the Jordan block structure of tensor products and symmetric powers, as surveyed in Appendix A. This paper also makes use of Magma to eliminate small cases in the proofs of various results. The associated code is attached as an ancillary file.

Acknowledgements: We are all grateful to the University of Birmingham and the Heilbronn Institute for Mathematical Research for organizing and funding the workshop “Patterns in Exotic Fusion Systems” where this work was begun. We thank Bob Oliver for his contributions to this project and for providing the motivation to complete this work. We would also like to thank David Craven and Ellen Henke for their input earlier in the project.

The first author is a member of the GNSAGA INdAM research group and kindly acknowledges its support. The fourth author gratefully acknowledges funding from the UK Research Council EPSRC for the project EP/W028794/1. The fifth author is supported by the Heilbronn Institute for Mathematical Research.

In this section, we analyze the generation and representation theory of groups with a strongly $p$-embedded subgroup. We begin by presenting the groups of $p$-rank at least two which contain a strongly $p$-embedded subgroup.

We shall repeatedly use the following observation.