Reflection Subquotients of Unitary Reflection Groups

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.

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On quasi-permutation representations of finite groups

Glasgow Mathematical Journal ◽

10.1017/s0017089500030901 ◽

1994 ◽

Vol 36 (3) ◽

pp. 301-308 ◽

Cited By ~ 5

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.

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Linear groups analogous to permutation groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700027920 ◽

1963 ◽

Vol 3 (2) ◽

pp. 180-184 ◽

Cited By ~ 5

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.

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Normal forms of invariant vector fields under a finite group action

Publicacions Matemàtiques ◽

10.5565/publmat_37193_05 ◽

1993 ◽

Vol 37 ◽

pp. 75-82

Author(s):

F. Sánchez-Bringas

Keyword(s):

Finite Group ◽

Group Action ◽

Vector Fields ◽

Normal Forms ◽

Finite Group Action ◽

Invariant Vector ◽

A Finite Group ◽

Invariant Vector Fields

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-069-3 ◽

1970 ◽

Vol 22 (3) ◽

pp. 626-640 ◽

Cited By ~ 7

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.

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

Glasgow Mathematical Journal ◽

10.1017/s0017089500031906 ◽

1997 ◽

Vol 39 (1) ◽

pp. 51-57 ◽

Cited By ~ 5

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].

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Vector fields on certain quotients of complex Stiefel manifolds

Mathematica Slovaca ◽

10.2478/s12175-013-0142-1 ◽

2013 ◽

Vol 63 (4) ◽

Cited By ~ 2

Author(s):

Shilpa Gondhali ◽

Parameswaran Sankaran

Keyword(s):

Vector Space ◽

Vector Fields ◽

Scalar Multiplication ◽

Complex Vector Space ◽

Complex Vector ◽

Differentiable Structure ◽

Stiefel Manifolds ◽

Cyclic Groups ◽

Tangent Bundles ◽

Mod P

AbstractWe consider quotients of complex Stiefel manifolds by finite cyclic groups whose action is induced by the scalar multiplication on the corresponding complex vector space. We obtain a description of their tangent bundles, compute their mod p cohomology and obtain estimates for their span (with respect to their standard differentiable structure). We compute the Pontrjagin and Stiefel-Whitney classes of these manifolds and give applications to their stable parallelizability.

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The Hessian map in the invariant theory of reflection groups

Nagoya Mathematical Journal ◽

10.1017/s0027763000002737 ◽

1988 ◽

Vol 109 ◽

pp. 1-21 ◽

Cited By ~ 5

Author(s):

Peter Orlik ◽

Louis Solomon

Keyword(s):

Vector Space ◽

Invariant Theory ◽

Polynomial Functions ◽

Complex Vector Space ◽

Dual Basis ◽

Reflection Groups ◽

Complex Vector ◽

Module Homomorphism

Let V be a complex vector space of dimension l. Let S be the C-algebra of polynomial functions on V. Let Ders be the S-module of derivations of S and let Ωs = Homs (Ders, S) be the dual S-module of differential 1-forms. Let {ei} be a basis for V and let {xi} be the dual basis for V. Then {Di = ∂/∂xi and {dxi} are bases for Ders and Ωs as S-modules. If f ∈ S, define a map Hess (f): Ders → Ωs byThen Hess (f) is an S-module homomorphism which does not depend on the choice of basis for V.

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