Conjugacy classes of reductive groups defined over local commutative rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:55:15Z http://mathoverflow.net/feeds/question/11277 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11277/conjugacy-classes-of-reductive-groups-defined-over-local-commutative-rings Conjugacy classes of reductive groups defined over local commutative rings Vinoth 2010-01-10T01:31:07Z 2010-04-07T11:33:12Z <p><strong>Background:</strong> I'm trying a problem on representations of reductive groups over various finite rings towards which this is very relevant (what I want to do is a very specialized case of this problem, and I want to know what background theory has been done for this situation in the literature and what is known about this). In characteristic $0$ over an algebraically closed field, and over finite fields, classifying conjugacy classes in reductive groups over field is a very well-known and well-studied problem (sometimes in preparation for studying representations of these). </p> <p><strong>Question</strong>: Let $R$ be a local commutative ring, either in characteristic $0$, algebraically closed, or in characteristic $p$ (algebraically closed OR finite field). If you want to complicate matters, and have an answer for non-commutative rings as well, I would be happy to see it, but I think the problem is non-trivial enough as is - and as far as I know, only $GL$ can be easily defined over non-commutative rings). <em>Edit:</em> Also specify that $R$ is an algebra over its residue field, and has an identity (which I believe is necessary to make the following argument work; again if you don't need this restriction feel free to not use it). </p> <p>Let $G$ be a reductive group (if you want, feel free to restrict to just the classical groups, $GL$, $SL$, $Sp$, and $SO$) defined over $R$. <em>What can be said about classifying conjugacy classes in $G$?</em> What is clear is the Levi decomposition of $G$, as the semi-direct product of the reductive group defined over the residue field of $R$, and $N$, the set of all matrices that are congruent entry-wise to the identity matrix, modulo the maximal ideal of $R$ (the latter is the normal subgroup). Using the semi-direct product, one can say something implicit about the conjugacy classes; first by studying the conjugation action of the reductive group on the unipotent algebraic group $N$, then studying the conjugacy classes in $N$, then extending this to the whole group. </p> <p>Are there any special cases of this problem that have been studied in the literature? Is there something more than can be said in general (further to what I have said above about the semi-direct product)?. </p> http://mathoverflow.net/questions/11277/conjugacy-classes-of-reductive-groups-defined-over-local-commutative-rings/11284#11284 Answer by Ben Webster for Conjugacy classes of reductive groups defined over local commutative rings Ben Webster 2010-01-10T02:39:25Z 2010-01-10T02:39:25Z <p>I don't have a complete answer or a reference, but I have a principle:</p> <p>(*) a conjugacy class in GL(n) is the same as an isomorphism class of n-dimensional representation of $\mathbb{Z}$.</p> <p>Similarly, one in SO(n) is an isomorphism class of n-dimensional representation with symmetric form, etc.</p> <p>So, over an algebraically closed field, the Jordan decomposition says that indecomposable representations of $\mathbb{Z}$ are a single Jordan block, for example.</p> http://mathoverflow.net/questions/11277/conjugacy-classes-of-reductive-groups-defined-over-local-commutative-rings/20618#20618 Answer by A Stasinski for Conjugacy classes of reductive groups defined over local commutative rings A Stasinski 2010-04-07T11:17:44Z 2010-04-07T11:33:12Z <p>The short answer is that over local PIRs which are not fields, all the conjugacy classes are known explicitly for $\mathrm{SL}_2$, $\mathrm{GL}_2$, and $\mathrm{GL}_3$, <em>some</em> can be explicitly described for $\mathrm{GL}_n$, and in general the problem is hopelessly complex (in the sense that it is <a href="http://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild" rel="nofollow">wild</a>).</p> <p>The longer answer is as follows. The cases $\mathrm{SL}_2$ and $\mathrm{GL}_2$ over finite local PIRs have been known for a long time, and can easily be worked out directly. The case of $\mathrm{GL}_2$ and $\mathrm{GL}_3$ over arbitrary local PIRs has been treated more recently by Avni, Onn, Prasad and Vaserstein in <a href="http://arxiv.org/abs/0708.1608" rel="nofollow">Similarity classes of 3x3 matrices over a local principal ideal ring</a>. </p> <p>Trying to tackle the $\mathrm{GL}_4$ case, one runs into the problem of classifying pairs of $2\times2$ matrices over the residue field, under simultaneous conjugation. More generally, the problem of describing the conjugacy classes in <code>$\mathrm{GL}_{4n}$</code>, for all $n$ contains the matrix pair problem over the residue field. This is a wild classification problem (see the above link) and hence one cannot expect a useful explicit classification in general. I think (but I have not checked it carefully) that if you do not assume that the ring is a PIR, you run into the matrix pair problem already for the groups <code>$\mathrm{GL}_{2n}$</code>.</p> <p>On the other hand, despite the lack of a general canonical form for matrices over local rings, there is a canonical form for a large subset of matrices, notably the <em>regular</em> ones. Regular elements were defined by Hill (for matrices over local PIRs) in <em>Regular Elements and Regular Characters of</em> $\mathrm{GL}_n(\mathcal{O})$, and are characterised by the property of being conjugate to their respective companion matrix, i.e., the canonical form consists of a single companion matrix block. One can of course generalise this to a canonical form with several companion matrix blocks. This is discussed in detail in Guralnick, <em>Similarity of Matrices over Local Rings</em>. Over fields this is the familiar rational canonical form.</p>