Conjugacy Classes in Algebraic Monoids

Putcha, Mohan S.

doi:10.2307/2000682

Cited by 5 publications

(6 citation statements)

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“…A one-to-one correspondence between conjugacy classes in a symplectic rook monoid and (symplectic) partitions is established in [1], with precise formulas for calculating the number of conjugacy classes and formulas for computing the order of each class given. For a reductive monoid M , Putcha [24] showed that there exist affine subsets M 1 , M 2 , ..., M k such that every element of M is conjugate to an element of some M i . Furthermore, he gave a necessary and sufficient condition for two elements in M i to be conjugate.…”

Section: The ∼-Conjugacy Classesmentioning

confidence: 99%

Conjugacy classes of Renner monoids

Cao

2013

Journal of Algebra

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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).

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Section: The ∼-Conjugacy Classesmentioning

confidence: 99%

Conjugacy classes of Renner monoids

Cao

2013

Journal of Algebra

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“…The Renner monoid of M, denoted by R, is a finite semigroup which parametrizes the B ×B-orbits in M, [18]. The nilpotent variety of M, denoted by M nil , is the subvariety consisting of all nilpotent elements of M. It is studied by Putcha in a series of papers, [12,14,16,17]. Unlike M, the nilpotent variety does not decompose into B×B-orbits.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Semigroups and Two-sided Weak Orders

Can¹

2019

Preprint

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“…For integrity of the paper, we will briefly describe this algorithm in the next section. We refer the reader who is interested in other conjugacies in semigroup theory to [5,6,8,9,10,16,17], and those who are interested in conjugacy classes in semisimple algebraic groups and finite groups of Lie type to [2,3] and [7].…”

Section: Introductionmentioning

confidence: 99%

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

Li¹,

Li²

2013

Preprint

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Conjugacy Classes in Algebraic Monoids

Cited by 5 publications

References 1 publication

Conjugacy classes of Renner monoids

Conjugacy classes of Renner monoids

Asymptotic Semigroups and Two-sided Weak Orders

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

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