Unit forms occur in the representation theory of algebras, partially ordered sets and related areas. They appear as Tits forms and Euler forms. Their weak positivity resp. non-negativity frequently characterizes finite and tame representation types. Their positive roots correspond to indecomposable representations.
Let ε(n) be the vector in n with co-ordinates of which are all 1. We show how all weakly positive unit forms can be reconstructed from those unit forms such that ε(n) is a root and such that there is no radical vector μ ≠ 0, where ϵ(n) ± μ is a positive root as well. We call unit forms with these properties ‘good thin forms’. Our main result is the complete classification of all good thin unit forms, which rests on the purely combinatorial construction of all these forms in few variables by using a computer.
