Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Eigenvalues of Coxeter Elements

In course

MATH-335: Coxeter groups

Study groups generated by reflections ...

Description

This lecture covers the eigenvalues of a Coxeter element and the uniqueness of the Coxeter plane associated with it. It explains how any permutation among the Coxeter generators can be achieved through cyclic permutations and transpositions. The lecture also discusses the invariance of the Coxeter number and the absence of eigenvalue 1 for a Coxeter element. Additionally, it explores the relationship between the Coxeter element and its matrix, as well as the decomposition of eigenspaces. The properties of Coxeter elements, their actions as rotations in unique planes, and the projection of root systems onto the Coxeter plane are also examined.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Anna Lachowska

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number h of an irreducible root system. A Coxeter element is a product of all simple reflections.

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

Coxeter notation

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram.

Coxeter–Dynkin diagram

In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge), that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees.

Coxeter group

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 .

Related lectures (41)

Conformal Transformations: Part 1

Covers the topic of conformal transformations, including translations, dilations, rotations, and the conformal algebra.

Group Theory: Basics

Covers the basics of group theory, including sets, applications, and examples like permutations and rotations.

Geometric Transformations in R2 and R3: Visualizations and Rotations

Covers visualizations in orthonormal bases in R2, focusing on rotations and reflections.

Orbital Angular Momentum

Explores orbital angular momentum in quantum mechanics, including rotation operators and eigenvalues.

Rotations: Angular Momentum