From: Dave Rusin Date: Tue, 14 Apr 1998 23:17:48 -0500 (CDT) To: stromme@mi.uib.no Subject: Re: Commutative matrix algebras Newsgroups: sci.math.research In article you write: >I'm looking for references for the classification of n-dimensional >commutative subalgebras (with 1) of the algebra of nxn matrices over the >complex numbers C (or any commutative ring, if possible), up to >conjugation or abstract algebra isomorphism. Am I missing something here? If e_1, ... are the minimal idempotents, then you can diagonalize simultaneously so that each is diagonal with just a few 1's on the diagonal; the algebra A is then the block sum of the sub-matrix algebras Ae_i. But each of these is then equivalent to an algebra consisting of all "striped" upper-triangular matrices (M_ij depends only on j-i). dave ============================================================================== Date: Wed, 15 Apr 1998 10:18:53 -0500 (CDT) From: Stein Stromme To: Dave Rusin Subject: Re: Commutative matrix algebras Newsgroups: sci.math.research [Dave Rusin] | In article you write: | >I'm looking for references for the classification of n-dimensional | >commutative subalgebras (with 1) of the algebra of nxn matrices over the | >complex numbers C (or any commutative ring, if possible), up to | >conjugation or abstract algebra isomorphism. | | Am I missing something here? If e_1, ... are the minimal idempotents, | then you can diagonalize simultaneously so that each is diagonal with | just a few 1's on the diagonal; the algebra A is then the block sum of | the sub-matrix algebras Ae_i. But each of these is then equivalent to | an algebra consisting of all "striped" upper-triangular matrices | (M_ij depends only on j-i). I think there are more than those, the simplest being a 0 b 0 a c 0 0 a The striped algebras are just the ones of the form k[x]/x^n. -- Stein A. Str\o mme, Dept. of Math., Univ. of Chicago, Tel: (773) 702-5754 ============================================================================== From: Dave Rusin Date: Wed, 15 Apr 1998 10:37:44 -0500 (CDT) To: stromme@mi.uib.no Subject: Re: Commutative matrix algebras Yes, of course you're right. I'm not quite sure what I was thinking. As another poster pointed out, the problem may be insoluble. There's some sort of result to the effect that the elementary-abelian subgroups of GL(n,F_p) of rank k are unclassifiable for fixed k larger than 3 (?), as n is allowed to vary. Of course you are asking for k=n, so maybe there is a solution in that case. If you state this as a mod-p question, then >Stein A. Str\o mme, Dept. of Math., Univ. of Chicago, Tel: (773) 702-5754 maybe you can get a quick answer out of Alperin! dave ============================================================================== From: stromme@mi.uib.no (Stein A. Stromme) Newsgroups: sci.math.research Subject: Re: Commutative matrix algebras Date: 16 Apr 1998 05:02:42 +0200 Thanks for the feedback. I found a paper by G. Mazzola (Comment. Math. Helvetici. 55 (1980) 267--293) that convinced me that the problem is indeed hard. Stein -- Stein A. Str\o mme -- Matematisk institutt, Universitetet i Bergen epost: stromme@mi.uib.no telefon: 5558 4825 telefax: 5558 9672