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 ndimensional
>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 submatrix algebras Ae_i. But each of these is then equivalent to
an algebra consisting of all "striped" uppertriangular matrices
(M_ij depends only on ji).
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 ndimensional
 >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 submatrix algebras Ae_i. But each of these is then equivalent to
 an algebra consisting of all "striped" uppertriangular matrices
 (M_ij depends only on ji).
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) 7025754
==============================================================================
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 elementaryabelian 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 modp question, then
>Stein A. Str\o mme, Dept. of Math., Univ. of Chicago, Tel: (773) 7025754
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)
267293) 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