Conjugacy classes of Renner monoids

In this paper we describe conjugacy classes of a Renner monoid R with unit group W, the Weyl group. We show that every element in R is conjugate to an element ue where u∈W and e is an idempotent in a cross section lattice. Denote by W(e) and W⁎(e) the centralizer and stabilizer of e∈Λ in W, respectively. Let W(e) act by conjugation on the set of left cosets of W⁎(e) in W. We find that ue and ve (u,v∈W) are conjugate if and only if uW⁎(e) and vW⁎(e) are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of R and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of R, and describe in detail the conjugacy classes of the Renner monoid of some J-irreducible monoids.We then generalize Munn conjugacy on a rook monoid to any Renner monoid and show that Munn conjugacy coincides with semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of R over an algebraically closed field of characteristic zero equals the number of Munn conjugacy classes in R.