The cross section lattices and Renner monoids of the odd special orthogonal algebraic monoids

Abstract

Let MSO n (n is odd) be the special orthogonal algebraic monoid and M n the monoid of all n × n matrices over an algebraically closed field. We will explicitly determine the cross section lattices Λ and the Renner monoids R of MSO n by using admissible subsets (see Definition 3.1) and the Weyl group. It turns out that Λ is a sublattice of the cross section lattice of M n and that R is a submonoid of the Renner monoid M n . Also, we obtain some interesting properties of the submonoid (MSO n ) e = {y ∈ MSO n | ye = ey = e} of MSO n where e is an idempotent in MSO n .