Representation-directed diamonds
Peter Dräxler
Abstract: A module over a finite-dimensional algebra is called a `diamond'
if it has a simple top and a simple socle.
Using covering theory, the classification of all diamonds for
algebras of finite representation type over algebraically closed fields can be reduced
to representation-directed algebras. The author proves a criterion referring to the positive roots
of the corresponding Tits quadratic form, which makes it easy to check
whether a representation-directed algebra has a faithful diamond. Using an implementation of
this criterion in the CREP program system on representation theory, he is able
to classify all exceptional representation-directed algebras having a faithful
diamond. He obtains a list of 157 algebras up to isomorphism and duality.
The 52 maximal members of this list are presented at the end of this paper.
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