Reflection groups and Braid groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 15 November 2010

Groups

A group is a set $G$ with a product $$\begin{array}{ccc}G\hspace{0.17em}\times \hspace{0.17em}G& \to & G\\ \left(g_1,g_2\right)& \mapsto & g_1{g}_2\end{array}\text{such that}$$

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 $$\text{if} g\in G \text{then} 1\cdot g=g\cdot 1=g,$$
3. If $g\in G$ then there exists $g^{-1}\in G$ such that $$g\cdot g^{-1}=g^{-1}\cdot g=1.$$

Example:

The symmetric group

$$S_n=\left\{\begin{array}{c}\text{graphs with} n \text{top vertices and} n \text{bottom vertices such that}\\ \text{each top vertex is connected to exactly one bottom vertex}\end{array}\right\}$$ with product

Examples:

Equivalently, $$S_n=\left\{\begin{array}{l}n\hspace{0.17em}\times \hspace{0.17em}n \text{matrices with}\\ \text{(a) exactly one nonzero entry in each row and each column}\\ \text{(b) the nonzero entries are 1.}\end{array}\right\}$$ with matrix multiplication $$\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right),$$ or equivalently,

Complex numbers

$$\begin{array}{rcl}ℂ& =& \left\{x+yi\mid x,y\in ℝ\right\}\text{with} i^2=-1,\\ & =& \left\{e^r{e}^{i\hspace{0.17em}\theta }\mid r\in ℝ,\theta \in [0,2\pi )\right\}\cup \left\{0\right\}.\end{array}$$

The cyclic group

$$\begin{array}{rcl}\mathbb{Z}/m\mathbb{Z}& =& \left\{m^{th} \text{roots of unity}\right\}\\ & =& \left\{g\in ℂ\mid g^n=1\right\}\\ & =& \left\{1,\xi ,{\xi }^2,\dots ,{\xi }^{m-1}\right\}\text{with}\xi =e^{2\pi i/2m},\end{array}$$ and $${\xi }^r{\xi }^s={\xi }^{r+s}\text{and}{\xi }^0={\xi }^m=1.$$ So

The groups $G_{m,1,n}$

Recall $$S_n=\left\{\begin{array}{l}n\hspace{0.17em}\times \hspace{0.17em}n \text{matrices with}\\ \text{(a) exactly one nonzero entry in each row and each column}\\ \text{(b) the nonzero entries are 1.}\end{array}\right\}$$ Then $$G_{m,1,n}=\left\{\begin{array}{l}n\hspace{0.17em}\times \hspace{0.17em}n \text{matrices with}\\ \text{(a) exactly one nonzero entry in each row and each column}\\ \text{(b) the nonzero entries are in} \mathbb{Z}/m\mathbb{Z}\end{array}\right\}$$ with product matrix multiplcation. So $S_n=G_{1,1,n}$.

Example: If $m=3$ and $n=3$ $$\mathbb{Z}/3\mathbb{Z}=\left\{1,\xi ,{\xi }^2\right\}\text{with}\xi =e^{2\pi i/3}.$$ $$G_{3,1,3}=\left\{\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& \xi & 0\\ 1& 0& 0\\ 0& 0& {\xi }^2\end{array}\right),\left(\begin{array}{ccc}{\xi }^2& 0& 0\\ 0& {\xi }^2& 0\\ 0& 0& \xi \end{array}\right),\left(\begin{array}{ccc}0& 0& {\xi }^2\\ {\xi }^2& 0& 0\\ 0& 1& 0\end{array}\right),\dots \right\}$$ and the number of elements in $G_{3,1,3}$ is $$\left|G_{3,1,3}\right|=3!\cdot 3^3=3\cdot 2\cdot 1\cdot 3^3=162.$$

Homomorphisms and kernels

Let $G$ and $G$ be groups. A homomorphism from $G$ to $H$ is a function $$\begin{array}{cccc}\varphi :& G& \to & H\\ & g& \mapsto & \varphi \left(g\right)\end{array}\text{such that}$$

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 $$ker\hspace{0.17em}\varphi =\left\{g\in G\mid \varphi \left(g\right)=1\right\}.$$

Example: $\mathbb{Z}/2\mathbb{Z}=\left\{1,-1\right\}.$ A homomorphism is given by $$\varphi :S_n\to \mathbb{Z}/2\mathbb{Z}\text{by}\varphi \left(g\right)={\left(-1\right)}^{\left(\# \text{of crossings in} g\right)}.$$ So

The alternating group is $A_n=ker\hspace{0.17em}\varphi$.

The groups $G_{m,l,n}$

Let $l$ divide $m$. A homomorphism is $$\varphi :G_{m,1,n}\to \mathbb{Z}/l\mathbb{Z}\text{given by}\varphi \left(g\right)={\left(\prod _{\begin{array}{c}\text{non-zero}\\ \text{entries}\end{array}}{g}_{ij}\right)}^{m/l}$$ and $$G_{m,l,n}=ker\hspace{0.17em}\varphi .$$

Example: If $m=6,l=3$ and $n=5$ then $$\varphi \hspace{0.17em}\left(\begin{array}{ccccc}0& 0& {\xi }^3& 0& 0\\ 0& {\xi }^4& 0& 0& 0\\ 0& 0& 0& 0& \xi \\ 1& 0& 0& 0& 0\\ 0& 0& 0& {\xi }^2& 0\end{array}\right)={\left({\xi }^3\cdot {\xi }^4\cdot \xi \cdot {\xi }^2\cdot \right)}^{6/3}={\left({\xi }^{10}\right)}^2={\xi }^2=e^{2\pi i2/6}=e^{2\pi i/3}.$$ So $$\begin{array}{}{G}_{m,l,n}=\left\{\begin{array}{l}n\hspace{0.17em}\times \hspace{0.17em}n \text{matrices with}\\ \text{(a) exactly one nonzero entry in each row and each column}\\ \text{(b) the nonzero entries are in} \mathbb{Z}/m\mathbb{Z}\\ \text{(c)} {\left(\prod _{\begin{array}{c}\text{non-zero}\\ \text{entries}\end{array}}{g}_{ij}\right)}^{m/l}=1\end{array}\right\}\end{array}$$ 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 $$g\hspace{0.17em}\left(\begin{array}{ccc}0& 0& {\xi }^2\\ 0& 1& 0\\ {\xi }^3& 0& 0\end{array}\right)\hspace{0.17em}{g}^{-1}=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& -1\end{array}\right)\hspace{0.17em}\left(\begin{array}{ccc}0& 0& {\xi }^2\\ 0& 1& 0\\ {\xi }^3& 0& 0\end{array}\right)\hspace{0.17em}\left(\begin{array}{ccc}\frac{1}{2}& 0& \frac{1}{2}\\ 0& 1& 0\\ \frac{1}{2}& 0& -\frac{1}{2}\end{array}\right)=\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),$$ and so $$\left(\begin{array}{ccc}0& 0& {\xi }^2\\ 0& 1& 0\\ {\xi }^3& 0& 0\end{array}\right)\text{is a reflection.}$$

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

Example: $$S_3=\left\{\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right),\dots \right\},$$ 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 $$G\text{acts on}{ℂ}^n=\left\{\hspace{0.17em}\left(\begin{array}{c}{c}_1\\ c_2\\ \ddots \\ c_n\end{array}\right)\mid c_1,c_2,\dots ,c_n\in ℂ\right\}.$$ If $g\in G$ and $x_i=i^{th}\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)$ then $$g{x}_i=\left(\begin{array}{ccc}{g}_{11}& \dots & g_{1n}\\ ⋮& & ⋮\\ ⋮& & ⋮\\ g_{n1}& \dots & g_{nn}\end{array}\right)\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)=\left(\begin{array}{c}{g}_{1i}\\ g_{2i}\\ ⋮\\ g_{ni}\end{array}\right)=g_{1i}{x}_1+g_{2i}{x}_2+\dots +g_{ni}{x}_i.$$ Then $G$ acts on polynomials $p\in ℂ\left[x_1,\dots ,x_n\right]$ by $$g\left(p_1+p_2\right)=g{p}_1+g{p}_2\text{and}g\left(p_1{p}_2\right)=\left(g{p}_1\right)\left(g{p}_2\right).$$

Example: $$\left(\begin{array}{cc}0& {\xi }^2\\ \xi & 0\end{array}\right)\hspace{0.17em}{x}_1=\xi x_2\text{and}\left(\begin{array}{cc}0& {\xi }^2\\ \xi & 0\end{array}\right)\hspace{0.17em}{x}_2={\xi }^2{x}_1,$$ and $$\left(\begin{array}{cc}0& {\xi }^2\\ \xi & 0\end{array}\right)\left(3x_1^2{x}_2+5x_1{x}_2^3\right)=3{\left(\xi x_2\right)}^2\left({\xi }^2{x}_1\right)+\left(\xi x_2\right){\left({\xi }^2{x}_1\right)}^3=3{\xi }^4{x}_1{x}_2^2+5{\xi }^7{x}_1^3{x}_2.$$

The invariant ring of $G$ is $$ℂ{\left[x_1,\dots ,x_n\right]}^G=\left\{p\in ℂ\left[x_1,\dots ,x_n\right]\mid gp=p \text{for all} g\in G\right\}.$$

Example: If $m=5$ then $$\left(\begin{array}{cc}0& {\xi }^2\\ \xi & 0\end{array}\right)\left(x_1^5{x}_2^5\right)={\left(\xi x_2\right)}^5+{\left({\xi }^2{x}_1\right)}^5={\xi }^5{x}_2^5+{\xi }^{10}{x}_1^5=x_1^5+x_2^5.$$

(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 $$p:\left[0,1\right]\text{with}p\left(0\right)=x_0.$$ So

A loop in $X$ is a path $g:[0,1]\to X$ with $g\left(0\right)=x_0=g\left(1\right)$

The fundamental group of $X$ is $${\pi }_1\left(X\right)=\left\{\text{loops in} X\right\}\text{with product}\left(g_1{g}_2\right)\left(t\right)=\left\{\begin{array}{ll}{g}_1\left(t\right),& \text{if} 0\le t\le \frac{1}{2}\\ g_2\left(t\right),& \text{if} \frac{1}{2}\le t\le 1.\end{array}\right.$$

Braid groups

The braid group on $n$ strands is $${ℬ}_n=\left\{\begin{array}{c}\text{string diagrams with} n \text{top vertices and} n \text{bottom vertices and}\\ \text{each top vertex is connected to exactly one bottom vertex}\end{array}\right\}$$ with product

Example:

Forgetting whetner crossings are over or under is a homomorphism $$\varphi :{ℬ}_n\to S_n.$$

Example:

Configuration space

The pure braid group is $${𝒫}_n=ker\hspace{0.17em}\varphi .$$ Let $${ℂ}^n=\left\{\left(c_1,\dots ,c_n\right)\mid c_1,\dots ,c_n\in ℂ\right\},$$ and let $$H_{ij}=\left\{\left(c_1,\dots ,c_n\right)\mid c_1,\dots ,c_n\in ℂ\right\}\text{for} 1\le j<i\le n,$$ and let $$X={ℂ}^n\setminus \left(\bigcup _{i<j}{H}_{ij}\right).$$

${𝒫}_n={\pi }_1\left(X\right)$.

		Proof.
	A loop in $X$ is a path 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 $$H_{\alpha }=\left\{\left(c_1,\dots ,c_n\right)\in {ℂ}^n\mid s_{\alpha }\left(c_1,\dots ,c_n\right)=\left(c_1,\dots ,c_n\right)\right\}.$$ Let $$X={ℂ}^n\setminus \left(\bigcup _{reflections}{H}_{\alpha }\right).$$

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)

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