# Euclidean Reflection Groups _ Kane's text

• Math Amateur
In summary, Kane defines reflections in Euclidean space E as linear operators with specific rules for their action on hyperplanes and vectors. The two properties of reflections, (1) and (2), follow from this definition. Property (1) states that the reflection of a vector x in the hyperplane Hα is x minus twice the projection of x onto α. Property (2) states that reflections are orthogonal, meaning their action preserves the inner product of two vectors. These properties can be shown using the geometric interpretation of reflections and the fact that they are uniquely determined by their action on Hα and α.
Math Amateur
Gold Member
MHB
I am reading Kane - Reflection Groups and Invariant Theory and need help with two of the properties of reflections stated on page 7

(see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7)

On page 6 Kane mentions he is working in $\ell$ dimensional Euclidean space ie $E = R^{\ell}$ where $R^{\ell}$ has the usual inner product (x,y).

In defining reflections with respect to vectors Kane writes:

" Given $0 \ne \alpha \in E$ let $H_{\alpha}\subset E$ be the hyperplane

$H_{\alpha} = \{ x | (x, \alpha ) = 0 \}$

We then define the reflection $s_{\alpha} : E \longrightarrow E$ by the rules

$s_{\alpha} \cdot x = x$ if $x \in H_{\alpha}$

$s_{\alpha} \cdot \alpha = - \alpha$ "

Then Kane states that the following two properties follow:

(1) $s_{\alpha} \cdot x = x - [2 ( x, \alpha) / (\alpha, \alpha)] \alpha$ for all $x \in E$

(2) $s_{\alpha}$ is orthogonal, ie $( s_{\alpha} \cdot x , s_{\alpha} \cdot y ) = (x,y)$ for all $x, y \in E$

I would appreciate help to show (1) and (2) above.

Peter

Note that ##H_\alpha## is codim 1 in E with perp spanned by ##\alpha##. So any linear operator defined on E is completely determined by its action on ##H_\alpha## and ##\alpha##. In particular, there is a unique linear operator satisfying the defining conditions for ##s_\alpha##. Since the mapping defined by the RHS of (1) is linear and satisfies these conditions as well, it must be ##x \mapsto s_\alpha x##.

Of course what's going on geometrically is that ##s_\alpha## is simply reflection in the hyperplane ##H_\alpha##. Try this out in ##R^2##, with ##H_\alpha## a line through the origin that is perpendicular to ##\alpha##: what is the formula for reflecting about ##H_\alpha##?

As for (2), it suffices to check that it holds in the 3 cases: (a) ##x,y \in H_\alpha##; (b) ##x\in H_\alpha## and ##y=\alpha##; and (c) ##x=y=\alpha##. Or you could use (1).

## 1. What is a Euclidean Reflection Group?

A Euclidean Reflection Group is a mathematical concept that describes a group of geometric transformations (such as reflections, rotations, and translations) that preserve the Euclidean geometry of an object. In other words, these groups consist of all the possible ways that we can reflect an object in a Euclidean space (such as a plane or a line) while still maintaining its shape and size.

## 2. Who is Kane and what is their text about?

Kane refers to the mathematician and physicist Gordon Kane. His text, "Reflection Groups and Invariant Theory," is a comprehensive guide to Euclidean Reflection Groups and their applications in mathematics and physics. It covers topics such as group theory, symmetry, and algebraic geometry.

## 3. How do Euclidean Reflection Groups relate to real-world objects?

Euclidean Reflection Groups have many real-world applications. For example, they can be used to study the symmetries of crystals, the shapes of molecules, and the structure of mathematical lattices. They are also used in computer graphics, robotics, and computer vision to analyze and manipulate images.

## 4. What is the significance of Euclidean Reflection Groups in mathematics?

Euclidean Reflection Groups are important in mathematics because they provide a framework for studying symmetries and transformations. They have connections to many areas of mathematics, including geometry, algebra, and topology. They also have applications in other fields, such as physics and computer science, making them a valuable tool in interdisciplinary research.

## 5. Are there any limitations to Euclidean Reflection Groups?

While Euclidean Reflection Groups are a powerful tool, they do have some limitations. For example, they only apply to objects in Euclidean space, and they do not account for non-Euclidean geometries. Additionally, they may not be able to fully describe the symmetries of complex structures, such as fractals. However, they are still a fundamental concept in mathematics and have many practical uses.

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