mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the

identity transformation Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...

, has a finite (multiplicative)

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

, and fixes a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

. The concept of pseudoreflection generalizes the concepts of reflection and complex reflection and is simply called reflection by some mathematicians. It plays an important role in Invariant theory of finite groups, including the Chevalley-Shephard-Todd theorem.

Formal definition

Suppose that ''V'' is vector space over a field ''K'', whose dimension is a finite number ''n''. A pseudoreflection is an invertible linear transformation

g: V\to V

such that the order of ''g'' is finite and the fixed subspace

V^g = \

of all vectors in ''V'' fixed by ''g'' has dimension ''n-1''.

Eigenvalues

A pseudoreflection ''g'' has an eigenvalue 1 of multiplicity ''n-1'' and another eigenvalue ''r'' of multiplicity 1. Since ''g'' has finite order, the eigenvalue ''r'' must be a root of unity in the field ''K''. It is possible that ''r'' = 1 (see Transvections).

Diagonalizable pseudoreflections

Let ''p'' be the characteristic of the field ''K''. If the order of ''g'' is coprime to ''p'' then ''g'' is diagonalizable and represented by a diagonal matrix diag(1, ... , 1, ''r'' ) =

\begin
  1  &  0  &  0  & \cdots & 0      \\
  0  &  1  &  0  & \cdots & 0      \\
  \vdots & \vdots & \ddots & \vdots \\ 
  0  &  0  &  \cdots & 1  & 0      \\
  0  &  0  &  0  & \cdots & r      \\
\end

where ''r'' is a root of unity not equal to 1. This includes the case when ''K'' is a field of characteristic zero, such as the field of real numbers and the field of complex numbers. A diagonalizable pseudoreflection is sometimes called a semisimple reflection.

Real reflections

When ''K'' is the field of real numbers, a pseudoreflection has matrix form diag(1, ... , 1, -1). A pseudoreflection with such matrix form is called a real reflection. If the space on which this transformation acts admits a symmetric bilinear form so that

orthogonality In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

of vectors can be defined, then the transformation is a true reflection.

Complex reflections

When ''K'' is the field of complex numbers, a pseudoreflection is called a complex reflection, which can be represented by a diagonal matrix diag(1, ... , 1, r) where r is a complex root of unity unequal to 1.

Transvections

If the pseudoreflection ''g'' is not diagonalizable then ''r'' = 1 and ''g'' has Jordan normal form

\begin
  1  &  0  &  0  & \cdots & 0      \\
  0  &  1  &  0  & \cdots & 0      \\
\vdots & \vdots & \ddots & \vdots & \vdots \\ 
  0  &  0  &  \cdots & 1  & 1      \\
  0  &  0  &  0  & \cdots & 1      \\
\end

In such case ''g'' is called a transvection. A pseudoreflection ''g'' is a transvection if and only if the characteristic ''p'' of the field ''K'' is positive and the order of ''g'' is ''p''. Transvections are useful in the study of finite geometries and the classification of their groups of motions. ''(Reprint of the 1957 original; A Wiley-Interscience Publication)''

References

{{Reflist Functions and mappings