Conjugacy Classes Of Representations Research Articles

Classifying finite monomial linear groups of prime degree in characteristic zero

Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in MAGMA.

A decomposition of the group algebraof a hyperoctahedral group

The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafe asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.

The irreducible characters of the alternating Hecke algebras

This paper computes the irreducible characters of the alternating Hecke algebras, which are deformations of the group algebras of the alternating groups. More precisely, we compute the values of the irreducible characters of the semisimple alternating Hecke algebras on a set of elements indexed by minimal length conjugacy class representatives and we show that these character values determine the irreducible characters completely. As an application, we determine a splitting field for the alternating Hecke algebras in the semisimple case.

A Note on Fine Graphs and Homological Isoperimetric Inequalities

Abstract In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected 2-complex X with a linear homological isoperimetric inequality, a bound on the length of attachingmaps of 2-cells, and finitely many 2-cells adjacent to any edge must have a fine 1-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity and show that a group G is hyperbolic relative to a collection of subgroups P if and only if G acts cocompactly with ûnite edge stabilizers on a connected 2-dimensional cell complex with a linear homological isoperimetric inequality and P is a collection of representatives of conjugacy classes of vertex stabilizers.

Borel subalgebras of the Witt algebra

Let $\mathbb{F}$ be an algebraically closed field of characteristic p > 3, and $\mathfrak{g}$ be the Witt algebra over $\mathbb{F}$ . Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ . An explicit description of $\mathcal{N}$ is given, so that the conjugacy classes of Borel subalgebras of $\mathfrak{g}$ under the automorphism group of $\mathfrak{g}$ are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in g. The representatives of conjugacy classes of Borel subalgebras, i.e., the so-called standard Borel subalgebras, are precisely given.

Conjugacy classes of torsion in GL_n(Z)

The problem of integral similarity of block-triangular matrices over the ring of integers is connected to that of finding representatives of the classes of an equivalence relation on general integer matrices. A complete list of representatives of conjugacy classes of torsion in the 4 × 4 general linear group over ring of integers is given. There are 45 distinct such classes and each torsion element has order of 1, 2, 3, 4, 5, 6, 8, 10 or 12.

Computing conjugacy classes of elements in matrix groups

This article describes a setup that – given a composition tree – provides functionality for calculation in finite matrix groups using the Trivial-Fitting approach that has been used successfully for permutation groups. It treats the composition tree as a black-box object. It thus is applicable to other classes of groups for which a composition tree can be obtained. As an example, we consider an effective algorithm for determining conjugacy class representatives.

Invariants of 2 × 2 matrices, irreducible $${\hbox{SL}(2, \mathbb{C})}$$ characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of G=SL(2,C) on the cartesian product G^n by simultaneous conjugation on each factor, in terms of the corresponding invariant functions, and derive from it a simple criterion for irreducibility of representations of finitely generated groups into G. We also obtain analogous results for the action of SL(2,C) on the vector space of n-tuples of 2 by 2 complex matrices. For a free group F_n of rank n, we show how to generically reconstruct the 2^{n-2} conjugacy classes of representations F_n -> G from their values under the map T_n : G^n = Hom(F_n,G) -> C^{3n-3} considered in [M], defined by certain 3n-3 traces of words of length one and two.

Computing conjugacy class representatives in permutation groups

We describe a significantly improved algorithm for computing the conjugacy classes of a finite permutation group with trivial soluble radical. We rely on existing methods for groups that are almost simple, and we are concerned here only with the reduction of the general case to the almost simple case.

Conjugacy Class Representatives in the Monster Group

AbstractThe paper describes a procedure for determining (up to algebraic conjugacy) the conjugacy class in which any element of the Monster lies, using computer constructions of representations of the Monster in characteristics 2 and 7. This procedure has been used to calculate explicit representatives for each conjugacy class.

Maximal subalgebras of simple modular Lie algebras

The maximal graded subalgebras of the restricted simple Lie algebras of Cartan type W n , S n , H n , and K n are studied. The notion of an R-subalgebra and that of an S-subalgebra are introduced for maximal subalgebras. All maximal R-subalgebras are described completely. The number of conjugacy classes, representatives of conjugacy classes and the dimensions of all R-subalgebras are found. An invariant description of graded S-subalgebras is also obtained.

Isometry classes of codes arising from sets of points in the projective plane

Consider the natural action of PGL 3( q) on the projective plane PG 2( q) over a finite field GF( q). In this paper we split a set of representatives of conjugacy classes of PGL 3( q) into a disjoint union of subfamilies gathering the elements that have the same cycle type as a permutation of the points of PG 2( q). Also, we count the number of elements in each subfamily. This allows us to obtain formulas for the number of orbits of the action of PGL 3( q) on different sets of n-subsets and n-multisubsets of PG 2( q). As an application we obtain explicit formulas for the number of isometry classes of different families of codes of dimension three.

The Solvable Primitive Permutation Groups of Degree at Most 6560

AbstractThe authors present an algorithm to construct conjugacy class representatives of the solvable primitive subgroups of Sd for a given degree d. Using this method, they determine the solvable primitive permutation groups of degree at most 6560 (that is, 38 – 1), up to conjugacy.

Conjugacy Class Representatives in Fischer's Baby Monster

AbstractA set conjugacy class representatives is given in this paper for the elements in Fischer's baby monster simple group, up to inversion.

Isometry classes of codes arising from sets of points in the projective plane

Consider the natural action of PGL 3 (q) on the projective plane PG 2 (q) over a finite field GF(q). In this paper we split a set of representatives of conjugacy classes of PGL 3 (q) into a disjoint union of subfamilies gathering the elements that have the same cycle type as a permutation of the points of PG 2 (q). Also, we count the number of elements in each subfamily. This allows us to obtain formulas for the number of orbits of the action of PGL 3 (q) on different sets of n-subsets and n-multisubsets of PG 2 (q). As an application we obtain explicit formulas for the number of isometry classes of different families of codes of dimension three.

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let Π be the fundamental group of a closed orientable surface of genus g ≥ 1, and let R(Π, G)/G be the space of conjugacy classes of representations of Π into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map φ¯ on R(Π, G)/G and investigate whether the fibres of φ¯ are isotropic with respect to the natural symplectic structure on R(Π, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of φ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of φ¯ are also isotropic, but we give an example to show that in general they are not.