On bounded elementary generation for $SL_n$ over polynomial rings

TL;DR

This paper proves that for polynomial rings over finite fields, the special linear group SL_n can be expressed as a bounded product of elementary matrices when n is at least 3.

Contribution

It establishes the bounded elementary generation property for SL_n over polynomial rings over finite fields, extending known results to this setting.

Findings

SL_n(F[X]) is boundedly generated by elementary matrices for n ≥ 3

Provides new insights into the structure of linear groups over polynomial rings

Extends bounded generation results to polynomial rings over finite fields

Abstract

Let $F[X]$ be the polynomial ring over a finite field $F$. It is shown that, for $n\geq 3$, the special linear group $SL_n(F[X])$ is boundedly generated by the elementary matrices.

Equations

$$\left(\begin{array}{ccc}a& b& \\ c& d& \\ & & I_{n-2}\end{array}\right)$$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{F}[X]).$$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\sim \left(\begin{array}{cc}a& b^{\prime }\\ c^{\prime }& d^{\prime }\end{array}\right)$$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{F}[X])$$

$$\delta (b):=\frac{q^{\mathrm{deg}(b)}-1}{q-1},\delta (c):=\frac{q^{\mathrm{deg}(c)}-1}{q-1}$$

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=X{Y}^{-1},$$

$$X={\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{x\delta (b)},Y={\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{y\delta (c)}.$$

$$†$$

$$\left(\begin{array}{cc}X{Y}^{-1}& \\ & 1\end{array}\right)=\left(\begin{array}{cc}X& \\ & 1\end{array}\right)\left(\begin{array}{cc}{Y}^{-1}& \\ & 1\end{array}\right)\sim \left(\begin{array}{cc}D(u)& \\ & -1\end{array}\right)\left(\begin{array}{cc}D(u^{-1})& \\ & -1\end{array}\right)=I_3$$

$${\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}^{y\delta (c)}=e{I}_2+f\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}e+fa& fb\\ fc& e+fd\end{array}\right)$$

$$\left(\begin{array}{cc}a& b\\ sc& d\end{array}\right)\in {\mathrm{SL}}_2(A)$$

$$\left(\begin{array}{ccc}a& b& \\ sc& d& \\ & & 1\end{array}\right)\sim \left(\begin{array}{ccc}\pm a& -sb& \\ c& \mp d& \\ & & -1\end{array}\right)$$

$$s=s_1{s}_2,a=k_1{s}_1+1=k_2{s}_2-1.$$

$$\left(\begin{array}{ccc}a& b& 0\\ sc& d& 0\\ 0& 0& 1\end{array}\right)\sim \left(\begin{array}{ccc}a& b& 0\\ sc& d& 0\\ s_1& 0& 1\end{array}\right)\sim \left(\begin{array}{ccc}1& b& -k_1\\ 0& d& -s_{2}c\\ s_1& 0& 1\end{array}\right)\sim \left(\begin{array}{ccc}1& b& -k_1\\ 0& d& -s_{2}c\\ 0& -s_{1}b& a\end{array}\right)$$

$$\left(\begin{array}{ccc}1& b& -k_1\\ 0& d& -s_{2}c\\ 0& -s_{1}b& a\end{array}\right)\sim \left(\begin{array}{ccc}1& 0& s_2\\ 0& d& -s_{2}c\\ 0& -s_{1}b& a\end{array}\right)$$

$$\left(\begin{array}{ccc}1& 0& s_2\\ 0& d& -s_{2}c\\ 0& -s_{1}b& a\end{array}\right)$$

$$\sim \left(\begin{array}{ccc}-a& -sb& 0\\ c& d& 0\\ -k_2& -s_{1}b& -1\end{array}\right)\sim \left(\begin{array}{ccc}-a& -sb& 0\\ c& d& 0\\ 0& 0& -1\end{array}\right)$$

$$\left(\begin{array}{ccc}e+fa& fb& \\ fc& e+fd& \\ & & 1\end{array}\right)\sim \left(\begin{array}{ccc}-(e+fa)& -f^{2}b& \\ c& e+fd& \\ & & -1\end{array}\right)\sim \left(\begin{array}{ccc}-u& \dots & \\ c& u^{-1}& \\ & & -1\end{array}\right)$$

Full text

1. Introduction

The special linear group $\mathrm{SL}_{n}(\mathbb{Z})$ is generated by the elementary matrices, that is, matrices which differ from the identity by at most one non-zero off-diagonal entry. Far more remarkable is the following fact:

Theorem 1** (Carter - Keller [6]).**

Let $n\geq 3$. Then $\mathrm{SL}_{n}(\mathbb{Z})$ is boundedly generated by the elementary matrices.

This means that, for some positive integer $\nu_{n}$, every matrix in $\mathrm{SL}_{n}(\mathbb{Z})$ is a product of at most $\nu_{n}$ elementary matrices. The Carter - Keller theorem provides, in fact, the explicit bound $\nu_{n}=\tfrac{1}{2}(3n^{2}-n)+36$. See [2] for a variation on the Carter - Keller argument, with a slightly worse bound. In [5], Carter and Keller extend their argument to rings of integers in algebraic number fields.

A different approach to bounded elementary generation for $\mathrm{SL}_{n}$ over rings of integers, based on unpublished work of Carter, Keller, and Paige, can be found in [12]. The novelty is the use of model-theoretic ideas. Unlike the original Carter - Keller approach, this is a non-explicit argument. One proves the existence of a bound on the number of elementary matrices needed to express a matrix in $\mathrm{SL}_{n}$.

Elementary generation of $\mathrm{SL}_{n}$ also holds for polynomial rings over fields. However, bounded elementary generation may fail. In [8] van der Kallen shows, by means of algebraic K-theory, that $\mathrm{SL}_{n}(\mathbb{C}[X])$, $n\geq 2$, is not boundedly generated by the elementary matrices. From an arithmetical viewpoint, however, the closest relative of $\mathbb{Z}$ is a polynomial ring over a finite field. The purpose of this note is to show the following analogue of Theorem 1, which appears to be new (cf., e.g., [11, p.523]).

Theorem 2**.**

Let $\mathbb{F}$ be a finite field, and let $n\geq 3$. Then $\mathrm{SL}_{n}(\mathbb{F}[X])$ is boundedly generated by the elementary matrices.

The proof is an adaptation of the Carter - Keller argument. Here is one technical difference. A crucial role in [6], and also in [2], is played by a ‘power lemma’ [6, Lemma 1] whose origins lie in properties of the so-called Mennicke symbols. We use instead a simple ‘swindle lemma’, Lemma 4 herein. A version of this swindle was used in [9, §2.3]. The proof of Theorem 2 yields the explicit bound $\nu_{n}=\tfrac{1}{2}(3n^{2}-n)+29$. As this bound does not depend on the size of the finite field $\mathbb{F}$, it follows that Theorem 2 holds, more generally, whenever $\mathbb{F}$ is an algebraic extension of a finite field.

One cannot take $n=2$ in Theorem 2: $\mathrm{SL}_{2}(\mathbb{F}[X])$ is not boundedly generated by the elementary matrices. This fact, and the reason behind it, are analogous to what happens for $\mathrm{SL}_{2}(\mathbb{Z})$. The principal congruence subgroup of $\mathrm{SL}_{2}(\mathbb{F}[X])$ corresponding to the ideal $(X)$, in other words the kernel of the surjective homomorphism $\mathrm{SL}_{2}(\mathbb{F}[X])\to\mathrm{SL}_{2}(\mathbb{F})$ given by the evaluation $X=0$, has a free product structure.

2. Proof of Theorem 2

Throughout, an elementary operation will be called, simply, a move. We allow the degenerate move of multiplying by the identity matrix. We write $\sim$ for the equivalence relation of being connected by a finite number of moves.

2.1. Reduction to a framed $\mathrm{SL}_{2}$ matrix

The first step is to reduce a matrix in $\mathrm{SL}_{n}$, $n\geq 3$, to a matrix of the following form:

[TABLE]

This is a standard reduction which works over any principal domain $A$. The general concept underpinning this procedure is Bass’s stable range [3]. For the sake of completeness, let us sketch the argument for $n=3$. Let $(u,v,w)$ be the last row of a matrix in $\mathrm{SL}_{3}(A)$. Thus, $u$, $v$, and $w$ are relatively prime, and we may assume that either $u$ or $v$ is non-zero. A suitable move takes us to a matrix whose last row is $(u^{\prime},v^{\prime},w)$, and such that $u^{\prime}$ and $v^{\prime}$ are relatively prime. The key fact here is that, if $\gcd(u,v,w)=1$ and $u$ is non-zero, then $\gcd(u,v+tw)=1$ for some $t\in A$. (An explicit choice for $t$ is the product of all primes dividing $u$ but not $v$. More precisely, we take one prime per associate class. We set $t=1$ if there are no such primes.) Now $w-1$ is a combination of $u^{\prime}$ and $v^{\prime}$, so two moves turn $w$ into $1$. Four additional moves clear the last row and the last column. In summary, we have reached a framed $\mathrm{SL}_{2}$ matrix, as desired, in $7$ moves. More generally, this argument reduces an $\mathrm{SL}_{n}$ matrix to a framed $\mathrm{SL}_{2}$ matrix in $\tfrac{1}{2}(3n^{2}-n)-5$ moves.

The remainder of the argument is devoted to showing that $34$ moves are sufficient in order to reduce, in $\mathrm{SL}_{3}$, a framed $\mathrm{SL}_{2}$ matrix to the identity. For the purposes of the next step, we assume that $a\neq 0$; otherwise, the reduction is trivial and quick, in only $3$ moves.

2.2. A convenient anti-diagonal

The second step will use the following analogue of Dirichlet’s theorem on primes in arithmetic progressions.

Theorem 3** (Kornblum - Artin).**

If $a,b\in\mathbb{F}[X]$ are relatively prime and $a\neq 0$, then there are infinitely many primes congruent to $b$ mod $a$. Furthermore, such a prime can have arbitrary degree, provided the degree is sufficiently large.

The first part is due to Kornblum (1919). The second part is a sharpening due to Artin (1921). See [10, Chapter 4] for a modern treatment.

Consider a matrix

[TABLE]

As $a$ and $b$ are relatively prime, the first part of Theorem 3 ensures that there is a prime $b^{\prime}\in\mathbb{F}[X]$ satisfying $b^{\prime}\equiv b$ mod $a$. Similarly, there is a prime $c^{\prime}\in\mathbb{F}[X]$ satisfying $c^{\prime}\equiv c$ mod $a$. Thus

[TABLE]

in $2$ moves. Furthermore, we can assume that $b^{\prime}$ and $c^{\prime}$ have relatively prime degrees: once $b^{\prime}$ has been chosen, we use the second part of Theorem 3 to pick $c^{\prime}$ of suitable degree.

2.3. The main step

Let

[TABLE]

be a matrix enjoying the property granted by the previous step: the anti-diagonal entries $b$ and $c$ are prime, with relatively prime degrees.

Let $q$ denote the number of elements in $\mathbb{F}$. Then the integers

[TABLE]

are relatively prime, as well. Let $x$ and $y$ be positive integers satisfying $x\delta(b)-y\delta(c)=1$. We write

[TABLE]

where

[TABLE]

We aim to reduce $X$ and $Y$ independently in $\mathrm{SL}_{3}$. More precisely, we will show that

[TABLE]

in $14$ moves, for some $u\in\mathbb{F}^{*}$. The same will hold for $Y^{-1}$ in place of $Y$, and $u^{-1}$ in place of $u$, by inverting. It also holds for $X$ in place of $Y$, by interchanging $b$ and $c$, and then transposing, with respect to some other unit $v\in\mathbb{F}^{*}$. However, $D(v)\sim D(u)$ in $\mathrm{SL}_{2}$, in $4$ additional moves. We can then deduce that

[TABLE]

in $14+4+14=32$ moves. Along the way, we are using the fact that diagonal matrices normalize the elementary matrices. Taking into account the second step, we conclude that $34$ moves are sufficient in order to reduce a framed $\mathrm{SL}_{2}$ matrix to the identity.

Let us turn to proving $(\dagger)$. By the Cayley - Hamilton theorem, there are $e,f\in\mathbb{F}[X]$ such that:

[TABLE]

Modulo $c$, the above matrices become upper triangular. So $e+fa\equiv a^{y\delta(c)}$ mod $c$. On the other hand, $a^{\delta(c)}$ mod $c$ is in $\mathbb{F}^{*}$. This follows by viewing the finite field $\mathbb{F}[X]/(c)$ as an extension of $\mathbb{F}$ of degree $\deg(c)$. Thus $e+fa\equiv u\in\mathbb{F}^{*}$ mod $c$. A similar argument applies to the lower diagonal entry. Keeping in mind that the determinant is $1$, we find that $e+fd\equiv u^{-1}\in\mathbb{F}^{*}$ mod $c$. At this point, we would like to replace the lower entry, $fc$, by $c$ so as to be able to perform reductions.

These considerations motivate the following lemma. Roughly speaking, it provides a way of swindling factors across the diagonal.

Lemma 4**.**

Let $A$ be a principal domain, and let

[TABLE]

where $a\equiv d$ mod $s$. Then

[TABLE]

in $11$ moves.

Proof.

The degenerate case $s=0$ is easily seen to hold, so let us assume that $s\neq 0$. The hypotheses imply that $a^{2}\equiv ad\equiv 1$ mod $s$. So there are $s_{1},s_{2}$ and $k_{1},k_{2}$ in $A$ such that

[TABLE]

Now

[TABLE]

by a column move, two row moves, and one more row move. We have basically swindled $s_{1}$ across the diagonal, and we now go for $s_{2}$. Firstly,

[TABLE]

by two column moves. Next,

[TABLE]

by two row moves, another row move, and two column moves. Overall, we have performed $11$ moves, as claimed.

For the other choice of signs on the diagonal, one could ‘pivot’ around $d$ instead of $a$. Alternatively, start from the above choice of signs, invert both matrices, interchange $a$ and $d$, and switch the signs of $b$ and $c$. ∎

Applying the above lemma, we obtain

[TABLE]

in $11+2=13$ moves. Taking the determinant reveals that the missing entry of the last matrix is [math]. One additional move, bringing the total to $14$, clears out the entry $c$. This completes the argument for $(\dagger)$, and so for Theorem 2 as well.

3. Further remarks

3.1.

The notion of bounded generation is commonly used for the property that a group is a product of finitely many cyclic subgroups. For $\mathrm{SL}_{n}(\mathbb{Z})$, $n\geq 3$, bounded cyclic generation follows from bounded elementary generation. This is no longer the case over $\mathbb{F}[X]$. In fact, bounded cyclic generation fails for $\mathrm{SL}_{n}(\mathbb{F}[X])$, $n\geq 3$. The idea that bounded cyclic generation is essentially a characteristic [math] phenomenon, is crystallized by the following result from [1]: if a linear group in positive characteristic enjoys bounded cyclic generation, then the group is virtually abelian.

3.2.

Lemma 4 can also be used over $A=\mathbb{Z}$. In this case, it leads to a simplification of the original Carter - Keller argument for $\mathrm{SL}_{n}(\mathbb{Z})$, and to the better bound $\nu_{n}=\tfrac{1}{2}(3n^{2}-n)+25$. The improved bound is irrelevant from an asymptotic perspective, but it becomes interesting in the case $n=3$. The question, which seems to us quite appealing, is how many elementary operations are needed to reduce a matrix in $\mathrm{SL}_{3}(\mathbb{Z})$ to the identity? Carter and Keller have shown that $48$ operations are sufficient. We have reduced this number to $37$. We challenge the reader to reduce this bound even further.

3.3.

There is an interesting issue of effectiveness in using the Kornblum - Artin theorem. The usual Dirichlet theorem, used in [6], is made effective by a result of Linnik and its modern improvements (see, for instance, [7]). Theorem 3 is also effective, since the Riemann hypothesis in the function field context is already known.

See [4] for further instances of Dirichlet-type theorems over polynomial rings.

Acknowledgements. I would like to thank Dave Witte Morris for thoughtful comments, and for pointing out Theorem 3 and reference [10]. I am also grateful to B. Sury for many valuable remarks–notably, 3.1 herein.