Ordered sets with the standardizing property and straightening laws for algebras of invariants

Abstract

In Math. Z. ( 176 (1981), 359–374) I explicitly determined the invariants of a certain class of unipotent group actions, and obtained a positive partial answer to Hilbert's 14th problem for nonreductive groups. The class of groups for which the method worked remained quite obscure. Theorem (4.2) of the present paper gives a precise description of the cases where the algebras of invariants are spanned by standard bitableaux, hence have a straightening law. The unipotent groups in question (“radizielle Untergruppen” of GL n ) correspond, up to conjugation, to finite (partially) ordered sets. The promised description is done by properties of the ordered sets that are easy to test. This is another example where combinatorial methods are important for the theory of invariants.