Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for $A_8$, $D_8$ and $E_8$ for a choice of basis of simple roots and compute their invariants, us...

This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of root systems for a large family of groups generated only by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the pa...

In this article we present a new and not fully employed geometric algebra model. With this model a generalization of the conformal model is achieved. We discuss the geometric objects that can be represented. Furthermore, we show that the Pin group of this geometric algebra corresponds to inversions with respect to axis aligned quadrics. We discuss the construction for the two- and three-dimensional case in detail and give the construction for arbitrary dimension. Key Words: C...

It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [32] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$. Further research on g...

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)= SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the division algebras. The roots themsel...

We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in $O(3)$. We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions.

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of the quadratic space $\mathbb{R}^{3,3}$. We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspecti...

One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgr...

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators o...

In this paper, we show how regular convex 4-polytopes - the analogues of the Platonic solids in four dimensions - can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonne theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arisin...