Conjugacy Classes in Algebraic Monoids II

1994 ◽  
Vol 46 (3) ◽  
pp. 648-661 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Unit Group ◽  
Conjugacy Classes ◽  
Reductive Groups ◽  
Bijective Correspondence ◽  
Twisted Conjugacy ◽  
Algebraic Monoid ◽  
Algebraic Monoids

AbstractLet M be a connected linear algebraic monoid with zero and a reductive unit group. We show that there exist reductive groups G1,..., Gt, each with an automorphism, such that the conjugacy classes of M are in a natural bijective correspondence with the twisted conjugacy classes of Gi, i = 1,..., t.

CARTAN SUBMONOIDS OF ALGEBRAIC MONOIDS

1995 ◽  
Vol 05 (03) ◽  
pp. 367-377 ◽  
Author(s):  
WENXUE HUANG
Keyword(s):  
Maximal Torus ◽  
Unit Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Completely Regular ◽  
Algebraic Monoid ◽  
Algebraic Monoids

Let M be an irreducible linear algebraic monoid defined over an algebraically closed field K with idempotent set E(M), T a maximal torus of the unit group G of M. We call CM(T)c a Cartan submonoid of M. The following are proved: (1) If M is reductive with zero or completely regular, then CM(T) is irreducible and regular and [Formula: see text]; (2) If M is regular, then M is solvable iff NM(CM(T))=CM(T), in which case, CM(T) is irreducible and regular; (3) If M is regular, then [Formula: see text].

Affinely spanned quasi-stochastic algebraic monoids

2017 ◽  
Vol 27 (08) ◽  
pp. 1061-1072
Author(s):  
W. Huang ◽  
J. Li
Keyword(s):  
Matrix Element ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Unit Group ◽  
Initial Study ◽  
Zariski Closure ◽  
Closed Field ◽  
Basic Class ◽  
Algebraic Monoid ◽  
Algebraic Monoids

A linear algebraic monoid over an algebraically closed field [Formula: see text] of characteristic zero is called (row) quasi-stochastic if each row of each matrix element is of sum one. Any linear algebraic monoid over [Formula: see text] can be embedded as an algebraic submonoid of the maximum affinely spanned quasi-stochastic monoid of some degree [Formula: see text]. The affinely spanned quasi-stochastic algebraic monoids form a basic class of quasi-stochastic algebraic monoids. An initial study of structure of affinely spanned quasi-stochastic algebraic monoids is conducted. Among other things, it is proved that the Zariski closure of a parabolic subgroup of the unit group of an affinely spanned quasi-stochastic algebraic monoid is affinely spanned.

scholarly journals Products of idempotents in algebraic monoids

2006 ◽  
Vol 80 (2) ◽  
pp. 193-203 ◽  
Author(s):  
Mohan S. Putcha
Keyword(s):  
Sufficient Conditions ◽  
Unit Group ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Monoid ◽  
Necessary And Sufficient ◽  
Algebraic Monoids

AbstractLet M be a reductive algebraic monoid with zero and unit group G. We obtain a description of the submonoid generated by the idempotents of M. In particular, we find necessary and sufficient conditions for M\G to be idempotent generated.

scholarly journals Automorphisms of higher rank lamplighter groups

2015 ◽  
Vol 25 (08) ◽  
pp. 1275-1299 ◽  
Author(s):  
Melanie Stein ◽  
Jennifer Taback ◽  
Peter Wong
Keyword(s):  
Cayley Graph ◽  
Infinite Number ◽  
Conjugacy Classes ◽  
Generating Set ◽  
Twisted Conjugacy ◽  
Higher Rank ◽  
Lamplighter Groups

Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].

Twisted conjugacy classes in saturated weakly branch groups

2008 ◽  
Vol 134 (1) ◽  
pp. 61-73 ◽  
Author(s):  
Alexander Fel’shtyn ◽  
Yuriy Leonov ◽  
Evgenij Troitsky
Keyword(s):  
Conjugacy Classes ◽  
Twisted Conjugacy

scholarly journals Geometry of Reidemeister classes and twisted Burnside theorem

2008 ◽  
Vol 2 (3) ◽  
pp. 463-506 ◽  
Author(s):  
Alexander Fel'shtyn ◽  
Evgenij Troitsky
Keyword(s):  
Conjugacy Classes ◽  
Type I ◽  
Finitely Generated ◽  
Dual Object ◽  
Finitely Generated Groups ◽  
Noncommutative Version ◽  
Twisted Conjugacy ◽  
Unitary Dual ◽  
Expository Paper ◽  
Definition Of

AbstractThe purpose of the present mostly expository paper (based mainly on [17, 18, 40, 16, 11]) is to present the current state of the following conjecture of A. Fel'shtyn and R. Hill [13], which is a generalization of the classical Burnside theorem.Let G be a countable discrete group, φ one of its automorphisms, R(φ) the number of φ-conjugacy (or twisted conjugacy) classes, and S(φ) = #Fix the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other.This conjecture plays a important role in the theory of twisted conjugacy classes (see [26], [10]) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Noncommutative Harmonic Analysis.First we prove this conjecture for finitely generated groups of type I and discuss its applications.After that we discuss an important example of an automorphism of a type II1 group which disproves the original formulation of the conjecture.Then we prove a version of the conjecture for a wide class of groups, including almost polycyclic groups (in particular, finitely generated groups of polynomial growth). In this formulation the role of an appropriate dual object plays the finite-dimensional part of the unitary dual. Some counter-examples are discussed.Then we begin a discussion of the general case (which also needs new definition of the dual object) and prove the weak twisted Burnside theorem for general countable discrete groups. For this purpose we prove a noncommutative version of Riesz-Markov-Kakutani representation theorem.Finally we explain why the Reidemeister numbers are always infinite for Baumslag-Solitar groups.

scholarly journals Twisted Conjugacy Classes in Chevalley Groups

2015 ◽  
Vol 53 (6) ◽  
pp. 481-501 ◽  
Author(s):  
T. R. Nasybullov
Keyword(s):  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Twisted Conjugacy

scholarly journals Stable base change for spherical functions

1987 ◽  
Vol 106 ◽  
pp. 121-142 ◽  
Author(s):  
Yuval Z. Flicker
Keyword(s):  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Base Change ◽  
Conjugacy Classes ◽  
Natural Homomorphism ◽  
Convolution Algebra ◽  
Orbital Integral ◽  
Twisted Conjugacy ◽  
Complex Valued

Let E/F be an unramified cyclic extension of local non-archimedean fields, G a connected reductive group over F, K(F) (resp. K(E)) a hyper-special maximal compact subgroup of G(F) (resp. G(E)), and H(F) (resp. H(E)) the Hecke convolution algebra of compactly-supported complex-valued K(F) (resp. G(E))-biinvariant functions on G(F) (resp. G(E)). Then the theory of the Satake transform defines (see § 2) a natural homomorphism H(E) → H(F), θ→f. There is a norm map N from the set of stable twisted conjugacy classes in G(E) to the set of stable conjugacy classes in G(F); it is an injection (see [Ko]). Let Ω‱(x, f) denote the stable orbital integral of f in H(F) at the class x, and Ω‱(y, θ) the stable twisted orbital integral of θ in H(E) at the class y.

Sign in / Sign up

Export Citation Format

Share Document