> < ^ Date: Fri, 11 Jun 1999 11:55:05 +0200 (CEST)
> ^ From: Juergen Mueller <juergen.mueller@math.rwth-aachen.de >
< ^ Subject: Re: algebras over cyclotomics

Dear GAP-Forum,

In addition to Alexander's remarks on Frank Quinn's question, I would
like to present a few ideas, how Frank Quinn's problem could at least
in principle be solved using tools available in GAP.

As far as I understand, we have given a f.d. commutative semisimple
algebra A over the rationals Q. We are looking for a basis for
A\otimes L, where L is a splitting field for A, reflecting its
Wedderburn decomposition into one-dimensional two-sided ideals, and
hence giving its character table, if this is wanted. As we are in a
commutative situation, we do not have to worry about Schur indices,
and we have a uniquely determined minimal splitting field L for A,
which additionally is galois/Q.

We use a generating system for A, which we now think of as given as a
matrix algebra; otherwise, one at first has to compute e.g. the
regular representation. Then the simultaneous generalised eigenspaces
of the generators, found by linear algebra and factorisation of their
characteristic polynomials, give the rational Wedderburn decomposition of A.

Now we are reduced to a simple commutative algebra over Q, which hence
is isomorphic to its splitting field L. Hence A has a primitive
element, i.e., an element having an irreducible characteristic
polynomial, which is `easily' found by a random search. Its minimum =
characteristic polynomial gives us L; and if this indeed is a
cyclotomic field, we can now switch to linear algebra over L, to find
the homogeneous components over L. Otherwise, we could of course also
do linear algebra, but L would not be so easily realizable within GAP.

This reminds me to the Dixon-Schneider algorithm, where A=Z(K[G]).
Here a reduction into a suitably chosen finite field F is done, to
ease computations. But a criterion is needed to ensure a unique lift
back to characteristic zero, which is found using the group G. One
might think of a generalisation of these ideas to the more general
situation described above.

Best, J"urgen.

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