Recently I have been studying the representation type of algebras (tame-wild dichotomy). A question that I feel is probably well known but that I cannot answer is the following:
Can I define tame algebras to be those algebras for which any representation may be written (after choosing a basis) in a form "only zeros and ones, and a free parameter lambda"? In other words, no scalars of the base field apart from lambda.
Assume of course that the base field is algebraically closed.
In all the examples I know (tame quivers, all the tame algebras that I have seen for which a classification exists) this is true.
What if I relax the hypotheses to "defined over F(lambda)", where F is the prime field, and lambda the scalar? Maybe it is true if we allow rational functions in lambda?
If this is not true, I would love to see a counterexample, so that I can rest in peace with this question.