Reflection groups and Braid groups

## Groups

A group is a set $G$ with a product $G × G → G g1,g2 ↦ g1g2 such that$
1. If ${g}_{1},{g}_{2},{g}_{3}\in G$ then $\left({g}_{1}{g}_{2}\right){g}_{3}={g}_{1}\left({g}_{2}{g}_{3}\right),$
2. There exists $1\in G$ such that
3. If $g\in G$ then there exists ${g}^{-1}\in G$ such that $g⋅g-1 =g-1⋅g =1.$

Example:

## The symmetric group

with product

Examples:

Equivalently, with matrix multiplication $0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 = 0 0 1 0 1 0 1 0 0 ,$ or equivalently,

## The cyclic group

and $ξr ξs = ξr+s and ξ0=ξm=1.$ So

## The groups ${G}_{m,1,n}$

Recall Then with product matrix multiplcation. So ${S}_{n}={G}_{1,1,n}$.

Example: If $m=3$ and $n=3$ $ℤ/3ℤ= 1,ξ,ξ2 with ξ=e2πi/3.$ $G3,1,3= 1 0 0 0 1 0 0 0 1 , 0 ξ 0 1 0 0 0 0 ξ2 , ξ2 0 0 0 ξ2 0 0 0 ξ , 0 0 ξ2 ξ2 0 0 0 1 0 , …$ and the number of elements in ${G}_{3,1,3}$ is $G3,1,3 =3!⋅33= 3⋅2⋅1⋅33= 162.$

## Homomorphisms and kernels

Let $G$ and $G$ be groups. A homomorphism from $G$ to $H$ is a function $ϕ: G → H g ↦ ϕg such that$
1. If ${g}_{1},{g}_{2}\in G$ then $\varphi \left({g}_{1}\right)\varphi \left({g}_{2}\right)=\varphi \left({g}_{1}{g}_{2}\right),$
2. $\varphi \left(1\right)=1,$
3. If $g\in G$ then $\varphi \left({g}^{-1}\right)=\varphi {\left(g\right)}^{-1}.$
The kernel of $\varphi$ is $ker ϕ= g∈G∣ ϕg=1 .$

Example: $ℤ/2ℤ=\left\{1,-1\right\}.$ A homomorphism is given by So

The alternating group is ${A}_{n}=ker \varphi$.

## The groups ${G}_{m,l,n}$

Let $l$ divide $m$. A homomorphism is $ϕ:Gm,1,n →ℤ/lℤ given by ϕg= ∏ non-zero entries gij m/l$ and $Gm,l,n=ker ϕ.$

Example: If $m=6,l=3$ and $n=5$ then $ϕ 0 0 ξ3 0 0 0 ξ4 0 0 0 0 0 0 0 ξ 1 0 0 0 0 0 0 0 ξ2 0 = ξ3⋅ ξ4⋅ ξ⋅ ξ2⋅ 6/3 = ξ10 2 =ξ2=e2πi2/6 =e2πi/3.$ So with product matrix multiplication.

The dihedral group of order $2m$ is ${G}_{m,m,2}$.

## Reflection groups

A reflection is a matrix with exactly one eigenvalue$\ne 1$.

Example: $\left(\begin{array}{ccc}1& 0& 0\\ 0& {\xi }^{2}& 0\\ 0& 0& 1\end{array}\right)$ is a reflection, and
if $m=5$ then $g 0 0 ξ2 0 1 0 ξ3 0 0 g-1 = 1 0 1 0 1 0 1 0 -1 0 0 ξ2 0 1 0 ξ3 0 0 12 0 12 0 1 0 12 0 - 12 = -1 0 0 0 1 0 0 0 1 ,$ and so $0 0 ξ2 0 1 0 ξ3 0 0 is a reflection.$

A reflection group is a group $G$ of matrices generated by reflections.

Example: $S3= 1 0 0 0 1 0 0 0 1 , 0 1 0 1 0 0 0 0 1 ,… ,$ or

has reflections

and every element of ${S}_{3}$ is a product of reflections.

(Shephard-Todd) Except for $34$ special cases, the ${G}_{m,l,n}$ are all finite reflection groups.

## Invariant rings

Let $G$ be a group of matrices. Then $Gacts on ℂn= c1 c2 ⋱ cn ∣ c1, c2, …, cn∈ℂ .$ If $g\in G$ and ${x}_{i}={i}^{th}\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)$ then $gxi= g11 … g1n ⋮ ⋮ ⋮ ⋮ gn1 … gnn 0 ⋮ 0 1 0 ⋮ 0 = g1i g2i ⋮ gni = g1i x1+ g2i x2 +…+ gni xi.$ Then $G$ acts on polynomials $p\in ℂ\left[{x}_{1},\dots ,{x}_{n}\right]$ by $g p1+p2 = gp1+gp2 and g p1p2 = gp1 gp2.$

Example: $0 ξ2 ξ 0 x1=ξx2 and 0 ξ2 ξ 0 x2= ξ2x1,$ and $0 ξ2 ξ 0 3 x12 x2 + 5 x1 x23 = 3 ξx22 ξ2x1 +ξx2 ξ2x13 =3 ξ4 x1 x22 + 5 ξ7 x13 x2.$

The invariant ring of $G$ is

Example: If $m=5$ then $0 ξ2 ξ 0 x15 x25 = ξx25 + ξ2x1 5 = ξ5 x25 + ξ10 x15 = x15 + x25 .$

(Chevalley, Shephard-Todd) $G$ is a finite reflection group if and only if there exist polynomials ${p}_{1},\dots ,{p}_{n}$ such that $ℂ{\left[{x}_{1},\dots ,{x}_{n}\right]}^{G}=ℂ\left[{p}_{1},\dots ,{p}_{n}\right].$

## Fundamental groups

Let $X$ be a topological space with a fixed base point ${x}_{0}$.

A path in $X$ is a continuous map $p: 0,1 with p0=x0.$ So

A loop in $X$ is a path $g:\left[0,1\right]\to X$ with $g\left(0\right)={x}_{0}=g\left(1\right)$

The fundamental group of $X$ is

## Braid groups

The braid group on $n$ strands is with product

Example:

Forgetting whetner crossings are over or under is a homomorphism $ϕ: ℬn→Sn.$

Example:

## Configuration space

The pure braid group is $𝒫n=ker ϕ.$ Let $ℂn= c1,…,cn ∣ c1,…,cn∈ℂ ,$ and let and let $X=ℂn∖ ⋃ i

${𝒫}_{n}={\pi }_{1}\left(X\right)$.

 Proof. A loop in $X$ is a path $x0$ $x0$ $c1$ $c2$ $…$ $cn$ $c1$ $c2$ $…$ $cn$ $t=0$ $t=1$ such that the travelling points ${c}_{1},\dots ,{c}_{n}$ satisfy ${c}_{i}\ne {c}_{j}$ along the path. $\square$

Let $G$ be a reflection goup. For each reflection ${s}_{\alpha }$ in $G$ let $Hα= c1,…,cn ∈ℂn∣ sα c1,…,cn = c1,…,cn .$ Let $X= ℂn ∖ ⋃ reflections Hα .$

Problem: Describe and understand ${\pi }_{1}\left(X\right)$.

## References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)