2022-10-13
Root systems
A root system $\Phi \subset V$ is a set of vectors in $V$ such that
- $\Phi$ span $V,$
- only $x$ and $-x$ are the scalar multiples of $x$ in $\Phi,$
- $\Phi$ is closed under reflection in the hyperplane defined by $x,$ for every $x \in \Phi,$
- for $x, y \in \Phi,$ $x$ projected onto $y$ is a half-integral multiple of $y.$
The root lattice is the lattice in $V$ which is generated by $\Phi.$ That is, all points which are integer linear combinations of roots.
A simple root is a root that cannot be written as a positive sum of other roots. Each root $\alpha$ defines a hyperplane, where $\pm \alpha$ are normals to this hyperplane. These hyperplanes divides the space into Weyl chambers. By picking a vector $v$ in the inner of a Weyl chamber, we define a half-space. The positive roots $\Phi^+$ are the roots in this half-space and the base (with respect to $\Phi^+$) is the set of simple roots in $\Phi^+.$
Given $\Phi^+,$ the root poset is a graded partial order on $\Phi^+,$ where $\alpha \geq \beta$ if $\alpha - \beta$ is a sum of simple positive roots. The grading is given by the number of simple positive roots needed to express the root as a sum.
The coroot of a root is defined as $\alpha^\vee = 2\alpha/(\alpha,\alpha).$ This defies the dual or inverse root system. A root system and its dual have the same Weyl group.
Classification of simple root systems
The simple root systems are given by the Dynkin diagrams, $A_{n \geq 1},$ $B_{n \geq 2},$ $C_{n \geq 3},$ $D_{n \geq 4},$ $E_6,$ $E_7,$ $E_8,$ $F_4$ and $G_2.$
Weyl groups and Lie groups
| $\text{Root system}$ | $\text{Weyl group}$ | $\text{Lie group}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $A_{n-1}$ | $\symS_n$ | $Sl_{n}(\setC)$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $B_n$ | $\symB_n$ | $SO_{2n+1}(\setC)$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $C_n$ | $\symB_n$ | $Sp_{2n}(\setC)$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $D_n$ | $\symD_n$ | $SO_{2n}(\setC)$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The Weyl group $\symS_n$ is the permutation group, which we can think of as $n \times n$ permutation matrices.
The group $\symB_n$ known as the hyperoctahedral group, which can be seen as the signed permutations — the $n\times n$ permutation matrices, but entries can be $\pm 1$ and not just $1.$ This can be thought of as the group of permutations of type $B$. The group $\symD_n \subset \symB_n$ is the subgroup of the signed permutation matrices where the number of negative entries is even.
The groups $\symS_n,$ $\symB_n$ and $\symD_n$ act on vectors via matrix multiplication. Let $W$ be a reflection group. The ring of polynomials in $\setR[x_1,\dotsc,x_n]$ invariant under $W,$ can be spanned by homogeneous polynomials. The degrees of these polynomials only depend on $W.$ The order of the reflection group is the product of the degrees.
| $\text{Reflection group $W$}$ | $\text{Degrees}$ | $\text{Order}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $A_{n-1}$ | $2,3,\dotsc,n$ | $n!$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $B_n/C_n$ | $2,4,\dotsc,2n$ | $2^n n!$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| $D_n$ | $2,4,\dotsc,2(n-1),n$ | $2^{n-1} n!$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||