Overgroups of exterior powers of an elementary group.
Levels

The present paper is devoted to the solution of the following general problem.

The conjectural answer, the standard overgroup description, in a general case can be formulated as follows. For any overgroup $H$ of the elementary group there exists a net of ideals $\mathbb{A}$ of the ring $R$ such that

where $\operatorname{E}_{N}(R,\mathbb{A})$ is a relative elementary subgroup for the net $\mathbb{A}$.

In the special case of a trivial net, i. e., $\mathbb{A}=\{A\}$ consists of one ideal $A$ of the initial ring $R$, overgroups of the group $\operatorname{E}_{G}(\Phi,R)$ can be parametrized by the ideal $A$ of the ring $R$:

where $\operatorname{E}_{N}(R,A)$ equals $\operatorname{E}_{N}(A)^{\operatorname{E}_{N}(R)}$ by definition.

Based on the classification of finite simple groups in 1984, Michael Aschbacher proved the Subgroup structure theorem [1]. It states that every maximal subgroup of a finite classical group either falls into one of the eight explicitly described classes $\mathcal{C}_{1}$–$\mathcal{C}_{8}$, or is an ‘‘almost’’ simple group in an irreducible representation (class $\mathcal{S}$). In the recent past, many experts studied overgroups of groups from the Aschbacher classes for some special cases of fields. For finite fields and algebraically closed fields maximality of subgroups was obtained by Peter Kleidman and Martin Liebeck, see [2, 3]. Oliver King, Roger Dye, and Shang Zhi Li proved maximality of groups from Aschbacher classes for arbitrary fields or described its overgroups in cases where they are not maximal, see [4, 5, 6, 7, 8, 9, 10, 11, 12]. We recommend the surveys [13, 14, 15], which contain necessary preliminaries, complete history, and known results about the initial problem.

In the present paper, we consider the case of the $m$-th fundamental representation of a simply connected group of type $A_{n-1}$, i. e., the scheme $G_{\rho}(\Phi,\mathord{\hbox to6.45831pt{\hrulefill}}\,)$ equals a [Zariski] closure of the affine group scheme $\operatorname{SL}_{n}(\mathord{\hbox to6.45831pt{\hrulefill}}\,)$ in the representation with the highest weight $\varpi_{m}$. In our case the extended Chevalley group scheme coincides with the $m$-th fundamental representation of the general linear group scheme $\operatorname{GL}_{n}(\mathord{\hbox to6.45831pt{\hrulefill}}\,)$. This case corresponds to the Aschbacher class ${\mathcal{S}}$ consisting of almost simple groups in certain absolutely irreducible representations. Morally, the paper is a continuation of a series of papers by the St. Petersburg school on subgroups in classical groups over a commutative ring, see [16, 17, 13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

We deal only with the case of a trivial net, i. e., a net consists of only one ideal $A$. As shown below (Propositions 1 and 16), it imposes a constraint $n\geqslant 3m$, we proceed with this restriction. In this case the general answer has the following form. Let $N=\binom{n}{m}$ and $H$ be a subgroup in $\operatorname{GL}_{N}(R)$ containing $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R)$. Then there exists a unique maximal ideal $A\trianglelefteqslant R$ such that

The present paper is the first part in the serial study of the problem. We construct a level and calculate a normalizer of connected (i. e., perfect) intermediate subgroups. Further, it is necessary to construct invariant forms for $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{SL}_{n}(R)$ and calculate a normalizer of $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R)$. Finally, we will extract an elementary transvection from an intermediate subgroup $H$. These steps are enough to solve the problem completely, see [29].

There are separate results for special cases of the ring $R$ such as a finite field $K$ or an algebraically closed field. For finite fields Bruce Cooperstein proved maximality of the normalizer $N_{G}\bigl{(}\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!2}}$}}}\operatorname{E}_{n}(K)\bigr{)}$ in $\operatorname{GL}_{N}(K)$ [30]. For algebraically closed fields a description of overgroups of $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(K)$ follows from the classical results about maximal subgroups of classical algebraic groups, for instance, see [31].

The present paper is organized as follows. In the next Section we formulate main results of the paper. In Section 2 we set up the notation. Section 3 contains all complete proofs, for instance, in §3.1 we have considered an important special case — a level computation for exterior squares of elementary groups. In §3.2–§3.4 we develop a technique for an arbitrary general exterior power. Finally, a level reduction for exterior powers is proved in §3.5.

Acknowledgment. We would like to express our sincere gratitude to our scientific adviser Nikolai Vavilov for formulating the problem and for a constant support, without which this paper would never have been written. The authors are grateful to Alexei Stepanov for carefully reading our original manuscript and for numerous remarks and corrections. Also, we would like to thank an anonymous referee for bringing our attention to the paper [32].

Fundamental representations of the general linear group $\operatorname{GL}_{n}$, as well as of the special linear group $\operatorname{SL}_{n}$, are the ones with the highest weights $\varpi_{m}=\underbrace{(1,\dots,1)}_{m}$ for $m=1,\dots,n$. The representation with the highest weight $\varpi_{n}$ degenerates for the group $\operatorname{SL}_{n}$. The explicit description of these representations uses exterior powers of the standard representation.

In detail, for a commutative ring $R$ by $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}R^{n}$ we denote an $m$-th exterior power of the free module $R^{n}$. We consider the following natural transformation, an exterior power,

which extends the action of the group $\operatorname{GL}_{n}(R)$ from $R^{n}$ to $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}R^{n}$.

An elementary group $\operatorname{E}_{n}(R)$ is a subgroup of the group of points $\operatorname{GL}_{n}(R)$, so its exterior power $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R)$ is a well defined subgroup of the group of points $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{GL}_{n}(R)$. A more user–friendly description of the elementary group $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R)$ will be presented in Subsections 3.1 and 3.2.

Let $H$ be an arbitrary overgroup of an elementary group $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R)$:

For any unequal weights $I,J\in\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}[n]$, which are indices for matrix entries of elements from $\operatorname{GL}_{\binom{n}{m}}(R)$, by $A_{I,J}$ we denote the following set

It turns out these sets are ideals that coincide for any pair of unequal weights $I\neq J$.

In the case $\frac{n}{3}\leqslant m\leqslant n$ description of overgroups cannot be done by a parametrization only by a single ideal. Moreover, as it could be seen from further calculations we need up to $m$ ideals in some cases for a complete parametrization of overgroups. There are a lot of nontrivial relationships between the ideals. So even the notion of a relative elementary group is far more complicated and depends on a Chevalley group (for instance, see [28]), let alone formulations of the Main Theorems. The authors work in this direction and hope this problem would be solved in the near future. In a general case this [partially ordered] set of ideals forms a net of ideals (due to Zenon Borevich; for a definition see [33], for further progress in the direction of subgroup classification see [34, 35]).

Back to the case $n\geqslant 3m$, the set $A:=A_{I,J}$ is called a level of an overgroup $H$. The description of overgroups goes as follows.

The left–hand side subgroup is denoted by $\operatorname{E}\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R,A)$. We note that this group is perfect (Lemma 22). Motivated by the expected relations $(*)$, we present an alternative description of the normalizer $N_{\operatorname{GL}_{N}(R)}\bigl{(}\operatorname{E}\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}\operatorname{E}_{n}(R,A)\bigr{)}$.

For this we introduce the canonical projection $\rho_{A}:R\longrightarrow R/A$ mapping $\lambda\in R$ to $\bar{\lambda}=\lambda+A\in R/A$. Applying the projection to all entries of a matrix, we get the reduction homomorphism

Eventually, we have the following explicit congruence description.

Our notation for the most part is fairly standard in Chevalley group theory. We recall all necessary notion below for the purpose of self-containment.

First, let $G$ be a group. By a commutator of two elements we always mean the left-normed commutator $[x,y]=xyx^{-1}y^{-1}$, where $x,y\in G$. Multiple commutators are also left-normed; in particular, $[x,y,z]=[[x,y],z]$. By ${}^{x}y=xyx^{-1}$ we denote the left conjugates of $y$ by $x$. Similarly, by $y^{x}=x^{-1}yx$ we denote the right conjugates of $y$ by $x$. In the sequel, we will use the Hall–Witt identity:

For a subset $X\subseteq G$, we denote by $\langle X\rangle$ a subgroup it generates. The notation $H\leqslant G$ means that $H$ is a subgroup in $G$, while the notation $H\trianglelefteqslant G$ means that $H$ is a normal subgroup in $G$. For $H\leqslant G$, we denote by $\langle X\rangle^{H}$ the smallest subgroup in $G$ containing $X$ and normalized by $H$. For two groups $F,H\leqslant G$, we denote by $[F,H]$ their mutual commutator: $[F,H]=\langle[f,g]\text{ for }f\in F,h\in H\rangle.$

Also we need some elementary ring theory notation. Let $R$ be an associative ring with $1$. By default, it is assumed to be commutative. By an ideal $I$ of a ring $R$ we understand the two-sided ideal and this is denoted by $I\trianglelefteqslant R$. As usual, $R^{*}$ denotes a multiplicative group of a ring $R$. A multiplicative group of matrices over a ring $R$ is called a general linear group and is denoted by $\operatorname{GL}_{n}(R)=\operatorname{M}_{n}(R)^{*}$. A special linear group $\operatorname{SL}_{n}(R)$ is a subgroup of $\operatorname{GL}_{n}(R)$ consisting of matrices of determinant $1$. By $a_{i,j}$ we denote an entry of a matrix $a$ at the position $(i,j)$, where $1\leqslant i,j\leqslant n$. Further, $e$ denotes the identity matrix and $e_{i,j}$ denotes the standard matrix unit, i. e., the matrix that has $1$ at the position $(i,j)$ and zeros elsewhere. For entries of the inverse matrix we will use the standard notation $a_{i,j}^{\prime}:=(a^{-1})_{i,j}$.

By $t_{i,j}(\xi)$ we denote an elementary transvection, i. e., a matrix of the form $t_{i,j}(\xi)=e+\xi e_{i,j}$, $1\leqslant i\neq j\leqslant n$, $\xi\in R$. Hereinafter, we use (without any references) standard relations [36] among elementary transvections such as

1. (1)

   additivity:

   $$t_{i,j}(\xi)t_{i,j}(\zeta)=t_{i,j}(\xi+\zeta).$$

2. (2)

   the Chevalley commutator formula:

   $$[t_{i,j}(\xi),t_{h,k}(\zeta)]=\begin{cases}e,&\text{ if }j\neq h,i\neq k,\\ t_{i,k}(\xi\zeta),&\text{ if }j=h,i\neq k,\\ t_{h,j}(-\zeta\xi),&\text{ if }j\neq h,i=k.\end{cases}$$

A subgroup $\operatorname{E}_{n}(R)\leqslant\operatorname{GL}_{n}(R)$ generated by all elementary transvections is called an $($absolute$)$ elementary group:

Now define a normal subgroup of $\operatorname{E}_{n}(R)$, which plays a crucial role in calculating the level of intermediate subgroups. Let $I$ be an ideal in $R$. Consider a subgroup $\operatorname{E}_{n}(R,I)$ generated by all elementary transvections of level $I$, i. e., $\operatorname{E}_{n}(R,I)$ is a normal closure of $\operatorname{E}_{n}(I)$ in $\operatorname{E}_{n}(R)$. This group is called an $($relative$)$ elementary group of level $I$:

It is well known (due to Andrei Suslin [37]) that the elementary group is normal in the general linear group $\operatorname{GL}_{n}(R)$ for $n\geqslant 3$. The normality is crucial for further considerations, so hereafter we suppose that $n\geqslant 3$. Furthermore, the relative elementary group $\operatorname{E}_{n}(R,I)$ is normal in $\operatorname{GL}_{n}(R)$ if $n\geqslant 3$. This fact, first proved in [37], is cited as Suslin’s theorem. Moreover, if $n\geqslant 3$, then the group $\operatorname{E}_{n}(R,I)$ is generated by transvections of the form $z_{i,j}(\xi,\zeta)=t_{j,i}(\zeta)t_{i,j}(\xi)t_{j,i}(-\zeta)$, $1\leqslant i\neq j\leqslant n$, $\xi\in I$, $\zeta\in R$. This fact was proved by Leonid Vaserstein and Andrey Suslin [38] and, in the context of Chevalley groups, by Jacques Tits [39].

By $[n]$ we denote the set $\{1,2,\dots,n\}$ and by $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}[n]$ we denote an exterior power of the set $[n]$. Elements of $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}[n]$ are ordered subsets $I\subseteq[n]$ of cardinality $m$ without repeating entries:

We use the lexicographic order on $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}[n]$ by default: $12\dots(m-1)m<12\dots(m-1)(m+1)<\dots$

Usually, we write an index $I=\{i_{j}\}_{j=1}^{m}$ in the ascending order, $i_{1}<i_{2}<\dots<i_{m}$. Sign $\operatorname{sgn}(I)$ of the index $I=(i_{1},\dots,i_{m})$ equals the sign of the permutation mapping $(i_{1},\dots,i_{m})$ to the same set in the ascending order. For example, $\operatorname{sgn}(1234)=\operatorname{sgn}(1342)=+1$, but $\operatorname{sgn}(1324)=\operatorname{sgn}(4123)=-1$.

Finally, let $n\geqslant 3$ and $m\leqslant n$. By $N$ we denote the binomial coefficient $\binom{n}{m}$. In the sequel, we denote an elementary transvection in $\operatorname{E}_{N}(R)$ by $t_{I,J}(\xi)$ for $I,J\in\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!m}}$}}}[n]$ and $\xi\in R$. For instance, the transvection $t_{12,13}(\xi)$ equals the matrix with $1$’s on the diagonal and $\xi$ in the position $(12,13)$.

We consider a case of an exterior square of a group scheme $\operatorname{GL}_{n}$ at first. We have two reasons for this way of presentation. Firstly, proofs of statements in a general case belong to the type of technically overloaded statements. At the same time, simpler proofs in the basic case present all ideas necessary for a general case. In particular, for $n=4$ Nikolai Vavilov and Victor Petrov completed the standard description of overgroups¹¹1The restriction of the exterior square map $\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!2}}$}}}:\operatorname{GL}_{4}(R)\longrightarrow\operatorname{GL}_{6}(R)$ to the group $\operatorname{E}_{4}(R)$ is an isomorphism onto the elementary orthogonal group $\operatorname{EO}_{6}(R)$ [18].. Secondly, for exterior squares there are several important results that cannot be obtained for the exterior cube or other powers, see [40, 41, 42]. For example, in [41] the author construct a transvection $T\in\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!2}}$}}}\operatorname{E}_{n}(R)$ such that it stabilizes an arbitrary column of a matrix $g$ in $\operatorname{GL}_{\binom{n}{2}}(R)$. And there are no such transvections for other exterior powers.