Some Results on the Schur Index of a Representation of a Finite Group

1970 ◽  
Vol 22 (3) ◽  
pp. 626-640 ◽  
Author(s):  
Charles Ford
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Complex Field ◽  
Complex Vector Space ◽  
Linear Transformations ◽  
Complex Vector ◽  
Finite Dimensional ◽  
A Finite Group

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.

scholarly journals On quasi-permutation representations of finite groups

1994 ◽  
Vol 36 (3) ◽  
pp. 301-308 ◽  
Author(s):  
J. M. Burns ◽  
B. Goldsmith ◽  
B. Hartley ◽  
R. Sandling
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Minimal Degree ◽  
Permutation Representation ◽  
Permutation Groups ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Group Situation ◽  
A Finite Group

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

scholarly journals Linear groups analogous to permutation groups

1963 ◽  
Vol 3 (2) ◽  
pp. 180-184 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Linear Group ◽  
Singular Matrices ◽  
A Finite Group ◽  
Complex Coefficients ◽  
Finite Linear

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

scholarly journals The minimal degree of a faithful quasi-permutation representation of an abelian group

1997 ◽  
Vol 39 (1) ◽  
pp. 51-57 ◽  
Author(s):  
Houshang Behravesh
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Linear Group ◽  
Singular Matrices ◽  
A Finite Group ◽  
Complex Coefficients ◽  
Finite Linear

Let G be a finite linear group of degree n; that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients). We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [5].

Reflection Subquotients of Unitary Reflection Groups

1999 ◽  
Vol 51 (6) ◽  
pp. 1175-1193 ◽  
Author(s):  
G. I. Lehrer ◽  
T. A. Springer
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Vector Fields ◽  
Reflection Group ◽  
Complex Vector Space ◽  
Reflection Groups ◽  
Complex Vector ◽  
Invariant Vector ◽  
A Finite Group ◽  
Invariant Vector Fields

AbstractLet G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset gG, a subquotient of G which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in G of certain elements of the coset. A criterion is also given in terms of the invariant degrees of G for an integer to be regular for G. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

Linear Symmetry Classes

1976 ◽  
Vol 28 (6) ◽  
pp. 1311-1319 ◽  
Author(s):  
L. J. Cummings ◽  
R. W. Robinson
Keyword(s):  
Vector Space ◽  
Permutation Group ◽  
Cyclic Group ◽  
Symmetry Class ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Linear Character ◽  
Finite Permutation Group ◽  
Symmetry Classes ◽  
Finite Dimensional

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.

EQUIVARIANT HOLOMORPHIC HERMITIAN PRINCIPAL BUNDLES OVER A EUCLIDEAN SPACE

2013 ◽  
Vol 05 (03) ◽  
pp. 345-360
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Euclidean Space ◽  
Vector Space ◽  
Algebraic Group ◽  
Inner Product ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Principal Bundles ◽  
Isomorphism Classes ◽  
Finite Dimensional

Let V be a finite dimensional complex vector space equipped with an inner product. Let G denote the group of all affine automorphisms of V preserving the metric defined by the inner product. Let H be a connected reductive affine algebraic group defined over ℂ. We give an explicit classification of the isomorphism classes of G-equivariant holomorphic hermitian principal H-bundles over V.

scholarly journals Classification of regular dicritical foliations

2016 ◽  
Vol 37 (5) ◽  
pp. 1443-1479 ◽  
Author(s):  
GABRIEL CALSAMIGLIA ◽  
YOHANN GENZMER
Keyword(s):  
Vector Space ◽  
Normal Forms ◽  
Blow Up ◽  
The Other ◽  
Holomorphic Foliations ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Dimensional ◽  
Birational Transformation

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

scholarly journals On the Block of Defect Zero

1971 ◽  
Vol 44 ◽  
pp. 57-59 ◽  
Author(s):  
Yukio Tsushima
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Prime Divisor ◽  
Residue Class ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Residue Class Field ◽  
A Finite Group

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

scholarly journals The index of elliptic operators over V-manifolds

1981 ◽  
Vol 84 ◽  
pp. 135-157 ◽  
Author(s):  
Tetsuro Kawasaki
Keyword(s):  
Finite Group ◽  
Differential Operator ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Elliptic Operators ◽  
Complex Vector ◽  
Pseudo Differential Operator ◽  
Finite Dimensional ◽  
The Difference ◽  
A Finite Group

Let M be a compact smooth manifold and let G be a finite group acting smoothly on M. Let E and F be smooth G-equivariant complex vector bundles over M and let be a G-invariant elliptic pseudo-differential operator. Then the kernel and the cokernel of the operator P are finite-dimensional representations of G. The difference of the characters of these representations is an element of the representation ring R(G) of G and is called the G-index of the operator P.

scholarly journals A short note on the noncoprime regular module problem

2020 ◽  
Vol 72 (11) ◽  
pp. 1589-1592
Author(s):  
G. Ercan ◽  
Ş. Güloğlu
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Semidirect Product ◽  
Short Note ◽  
Linear Transformations ◽  
Special Configuration ◽  
Regular Module ◽  
A Finite Group

UDC 512.5 Considering a special configuration in which a finite group acts by automorphisms on а finite group and the semidirect product acts on the vector space by linear transformations, we discuss the existence of a regular -module in

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