Crystallographic root system

How does the root system work?
Roots absorb water and minerals and transport them to stems.
They also anchor and support a plant, and store food.
A root system consists of primary and secondary roots.
Each root is made of dermal, ground, and vascular tissues..
What are the 4 root types?
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras..
What is the root system of a lattice?
Root lattices are orthogonal sums of irreducible lattices which are either the lattice ℤ or the lattices of norm 2 consisting of two infinite families ��n, �� _n and three exceptional lattices ��₆, ��₇, ��₈..
What is the root system of a reflection group?
A root system is a finite set Φ of nonzero vectors in V such that (R1) Φ ∩ Rα = {α, −α} for each α ∈ Φ. (R2) sα(β) ∈ Φ for all α, β ∈ Φ.
Elements of Φ are called roots.
The group W = 〈sα : α ∈ Φ〉 is the reflection group associated to Φ..
What is the root system theory?
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras..
What is the root system theory?
The root system is the descending (growing downwards) portion of the plant axis.
When a seed germinates, radicle is the first organ to come out of it.
It elongates to form primary or the tap root.
It gives off lateral branches (secondary and tertiary roots) and thus forms the root system..
What is the root system?
A root system consists of primary and secondary roots.
Each root is made of dermal, ground, and vascular tissues.
Roots grow in length and width from primary and secondary meristem..
What is the root system?
Root lattices are orthogonal sums of irreducible lattices which are either the lattice ℤ or the lattices of norm 2 consisting of two infinite families ��n, �� _n and three exceptional lattices ��₆, ��₇, ��₈..
A root system is a finite set Φ of nonzero vectors in V such that (R1) Φ ∩ Rα = {α, −α} for each α ∈ Φ. (R2) sα(β) ∈ Φ for all α, β ∈ Φ.
Elements of Φ are called roots.
The group W = 〈sα : α ∈ Φ〉 is the reflection group associated to Φ.
There are two types of root systems: taproots and fibrous roots (also known as adventitious roots; Figure 2.1. 2).

A root system R is called crystallographic if (r^∨,s) is an integer, for any s∈ R,r^∨∈ R^∨ --- these are the root systems considered by Bourbaki. The dimension of the subspace V

In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2; then they call root systems satisfying condition 2 reduced. In this article, all root systems are assumed to be reduced and crystallographic.

Overview

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties

Definitions and examples

As a first example, consider the six vectors in 2-dimensional Euclidean space, R , as shown in the image at the right; call them roots

Positive roots and simple roots

Given a root system we can always choose (in many ways) a set of positive roots

Dual root system, coroots, and integral elements

If Φ is a root system in E, the coroot α of a root α is defined by The set of coroots also forms a root system Φ in E

Classification of root systems by Dynkin diagrams

A root system is irreducible if it cannot be partitioned into the union of two proper subsets , such that for all and

Are Weyl groups the same as crystallographic reflection groups?

The classification of of crystallographic root systems is similar in spirit to the classification of positive definite Coxeter graphs

The resulting Weyl groups are precisely the reflection groups for which all m(α, β) ∈ {2, 3, 4, 6} (when a 6 = β)

So Weyl groups are the same thing as crystallographic reflection groups

What is the best software for crystallographic structure determination?

Acta Crystallogr D Biol Crystallogr 60:2126–2132

(2002) PHENIX: Building new software for automated crystallographic structure determination

Acta Crystallogr D Biol Crystallogr 58:1948–1954

(2011) Fast, scalable generation of high-quality protein multiple sequence alignments using Clustal Omega

Definition A crystallographic root system is a finite set of non zero vectors in Euclidean space

Crystallographic root system

How does the root system work?

What are the 4 root types?

What is the root system of a lattice?

What is the root system of a reflection group?

What is the root system theory?