Renner The orbit structure of 2-reducible algebraic monoids

In this paper we study the orbit structure of semisimple algebraic monoids with exactly two nonzero minimal G×G-orbits. The case of one minimal orbit was solved earlier by the authors. The key notion for reductive monoids is the type map λ, which is the monoid notion of the Dynkin diagram. It is the ultimate combinatorial invariant of a reductive monoid. To calculate λ, we associate with each 2-reducible monoid M, certain invariants (I+, I−) and (∆+,∆−). These invariants are not entirely independent, but can be regarded as the minimal information needed to determine the much sought after type map of M. We obtain a combinatorial recipe for the cross-section lattice and the type map in terms of (I+, I−) and (∆+,∆−). We end the discussion with several carefully chosen examples. The first one illus-trates how the Cartan matrix can be involved in calculating (∆+,∆−) from (I+, I−) and the positive vertices of the polytope of M