Reflection groups and Braid groups

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 G×G G g1,g2 g1g2 such that
  1. If g1,g2,g3G then g1g2 g3= g1 g2g3,
  2. There exists 1G such that if  gG  then  1g=g1=g,
  3. If gG then there exists g-1G such that gg-1 =g-1g =1.

Example:

with product { , , , , , , , }
with product g1g2= g1 g2 so that = .

The symmetric group

Sn= graphs with  n  top vertices and  n  bottom vertices such that each top vertex is connected to exactly one bottom vertex with product

with product g1g2= g1 g2

Examples:

S3= { , , , , , , , }
S2= { , } and S1= { }

Equivalently, Sn= n×n matrices with (a) exactly one nonzero entry in each row and each column (b) the nonzero entries are 1. 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,

Complex numbers

= x+yi x,y with i2=-1, = ereiθ r, θ [0,2π) 0 .

x θ 1 iy x+iy ereiθ eiθ

The cyclic group

/m = mth roots of unity = ggn=1 = 1,ξ,ξ2 ,,ξm-1 with ξ=e2πi/2m, and ξr ξs = ξr+s and ξ0=ξm=1. So

/5= 1 ξ4 ξ3 ξ2 ξ=e2πi/5

The groups Gm,1,n

Recall Sn= n×n matrices with (a) exactly one nonzero entry in each row and each column (b) the nonzero entries are 1. Then Gm,1,n= n×n matrices with (a) exactly one nonzero entry in each row and each column (b) the nonzero entries are in /m with product matrix multiplcation. So Sn=G1,1,n.

Example: If m=3 and n=3 /3= 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 G3,1,3 is G3,1,3 =3!33= 32133= 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 g1,g2G then ϕg1 ϕg2= ϕg1g2,
  2. ϕ1=1,
  3. If gG then ϕg-1= ϕg-1.
The kernel of ϕ is kerϕ= gG ϕg=1 .

Example: /2= 1,-1 . A homomorphism is given by ϕ: Sn /2 by ϕg= -1 #  of crossings in g . So

ϕ ( ) =-15=-1.

The alternating group is An=kerϕ.

The groups Gm,l,n

Let l divide m. A homomorphism is ϕ:Gm,1,n /l given by ϕg= non-zero entries gij m/l and Gm,l,n=kerϕ.

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 Gm,l,n= n×n matrices with (a) exactly one nonzero entry in each row and each column (b) the nonzero entries are in /m (c)  non-zero entries gij m/l =1 with product matrix multiplication.

The dihedral group of order 2m is Gm,m,2.

Reflection groups

A reflection is a matrix with exactly one eigenvalue1.

Example: 1 0 0 0 ξ2 0 0 0 1 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

S3= { , , , , , , , }

has reflections

, ,

and every element of S3 is a product of reflections.

(Shephard-Todd) Except for 34 special cases, the Gm,l,n are all finite reflection groups.

Invariant rings

Let G be a group of matrices. Then Gacts on n= c1 c2 cn c1, c2, , cn . If gG and xi=ith 0 0 1 0 0 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 x1,,xn 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 x1,,xn G = p x1,,xn gp=p  for all gG .

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 p1,,pn such that x1,,xn G = p1,,pn .

Fundamental groups

Let X be a topological space with a fixed base point x0.

x 0 =

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

x 0 = [0,1]= and x2 is a path.

A loop in X is a path g:[0,1]X with g0=x0=g1

x 0 =

The fundamental group of X is π1 X= loops in X with product g1g2 t = g1t, if 0t 12 g2t, if  12 t1.

Braid groups

The braid group on n strands is n= string diagrams with n top vertices and n  bottom vertices and each top vertex is connected to exactly one bottom vertex with product

with product g1g2= g1 g2

Example:

= ( ) ( )

Forgetting whetner crossings are over or under is a homomorphism ϕ: nSn.

Example:

ϕ =

Configuration space

The pure braid group is 𝒫n=kerϕ. Let n= c1,,cn c1,,cn , and let Hij= c1,,cn c1,,cn for 1j<in, and let X=n i<j Hij .

𝒫n=π1X.

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 c1,,cn satisfy cicj along the path.

Let G be a reflection goup. For each reflection sα in G let Hα= c1,,cn n sα c1,,cn = c1,,cn . Let X= n reflections Hα .

Problem: Describe and understand π1X.

References

[Bou] N. Bourbaki, Groupes et Alg├Ębres 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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