Orders Of The Renner Monoids And Orbit Structure Of (J,σ)-irreducible Monoids Of Type D₄～3

Let M be a reductive monoid with unit group G. Let B(?)G be a Borel subgroup,T(?)B be a maximal torus, and W=N_G(T)/T be the Weyl group. Let (?) be theZariski closure of N_G(T) in M. Then R=(?)/T is an finit inverse monoid with unitgroup W. Let K be an algebraic closed field, and its' multiplicative group be K^*. Let G bea simple algebraic group, andρ: G→GL(V) be an rational irreducible representationwith finite kernel. Then M(ρ)=(?) is a J-irreducible monoid. Let M be areductive monoid with zero, and suppose thatσ:M→M is a bijective morphism. Amonoid M is called (J,σ)-irreducible ifσis transitive on the set of minimal G×G orbitsof M\{0}.In this paper, we give an alternate proof to the formula for the orders of Rennermonoids of reductive monoids, and apply this formula to find the orders of Renner monidsof J-irreducible monoids M(ρ)=(?), whereρis the adjoint representation of G.At last, we describe the orbit structure of (J,σ)-irreducible monoids of type D₄³.