Conjugacy classes of Renner monoids

Abstract

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 the Munn conjugacy on a rook monoid to any Renner monoid and show that the Munn conjugacy coincides with the 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 the Munn conjugacy classes in R.Proof. Write I = I(σ), J = J(σ), and K = J(τ ). Then τ = e K w 1 e J and σ = e J we I for some w 1 , w ∈ W . It follows that τ e J σ = e K w 1 e J we I = τ σ. So p(τ )p(σ) = µ − K τ µ J µ − J σµ I = µ − K τ e J σµ I = µ − K τ σµ I = p(τ σ).

The 1882 Conjugacy Classes of the First Basic Renner Monoid of Type E_6

“…The main result of this paper is the list of all conjugacy classes of the first basic Renner monoid of type E 6 (see Definition 2.1). This list is a result of an extensive computation on a Dell desktop computer OPTIPLEX 745 with a total physical memory 2,048.00 MB (about 1 week of computing time in May, 2012) using the method described in [12]. The code of the computation is written in GAP, a computer software system for algebraic computations [4].…”

In this paper we find all conjugacy classes of the first basic Renner monoid of type E6. They are listed in the tables of the paper.

“…The main result of this paper is the list of all conjugacy classes of the first basic Renner monoid of type E 6 (Definition 2.1). This list is a result of an extensive computation on a Dell desktop computer OPTIPLEX 745 with a total physical memory 2,048.00 MB (about 1 week of computing time in May, 2012) using the method described in [1]. The code of the computation is written in GAP, a computer software system for algebraic computations [2].…”

“…4,2,3,1,5,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3 ] [ 2, 4, 2, 3, 1, 5, 4, 3, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5 ] [ 2, 4, 2, 5, 4, 2, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2,6, 5, 4, 3 ] [ 2, 4, 2, 3, 5, 4, 2, 3, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5 ] [ 2, 4, 2, 3, 1, 4, 3, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, 5, 6 ] [ 2, 4, 3, 5, 4, 2, 3, 1, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 24, 3, 1, 5, 4, 2, 3, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5 ] [ 2, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 6 ] [ 2, 4, 2, 3, 1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2 ] [ 2, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2 ] [ 2, 4, 2, 3, 1, 4, 3, 5, 4, 3, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5 ] [ 2, 3, 1, 4, 3, 1, 5, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 54, 3, 1, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5 ] [ 2, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6 ] [ 2, 4, 2, 3, 1, 5, 4, 2, 3, 1, 4, 5, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2 ] [ 2, 3, 1, 4, 3, 1, 5, 4, 2, 3, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5 ]…”

“…[1,3,1,4,3]=s 1 s 3 s 1 s 4 s 3 e 1 and [ ]=e 1 . Tables 4-10 shows conjugacy classes associated with e 2 to e 8 respectively.…”

Computation in Linear Algebraic Monoids

“…The conjugacy classes of any Renner monoid were investigated in [24]. One of its main results is that the conjugacy of two elements in the monoid is described by using the action of the centralizer of idempotents on certain parabolic subgroups of the Weyl group.…”