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→Gg1,g2↦g1g2such that
If g1,g2,g3∈G then
g1g2g3=g1g2g3,
There exists 1∈G such that
ifg∈Gthen1⋅g=g⋅1=g,
If g∈G then there exists g-1∈G such that
g⋅g-1=g-1⋅g=1.
Example:
The symmetric group
Sn=graphs withntop vertices andnbottom vertices such thateach top vertex is connected to exactly one bottom vertex
with product
Examples:
Equivalently,
Sn=n×nmatrices with(a) exactly one nonzero entry in each row and each column(b) the nonzero entries are 1.
with matrix multiplication
010100001010001100=001010100,
or equivalently,
Complex numbers
ℂ=x+yi∣x,y∈ℝwithi2=-1,=ereiθ∣r∈ℝ,θ∈[0,2π)∪0.
The cyclic group
ℤ/mℤ=mthroots of unity=g∈ℂ∣gn=1=1,ξ,ξ2,…,ξm-1withξ=e2πi/2m,
and
ξrξs=ξr+sandξ0=ξm=1.
So
The groups Gm,1,n
Recall
Sn=n×nmatrices with(a) exactly one nonzero entry in each row and each column(b) the nonzero entries are 1.
Then
Gm,1,n=n×nmatrices with(a) exactly one nonzero entry in each row and each column(b) the nonzero entries are inℤ/mℤ
with product matrix multiplcation. So
Sn=G1,1,n.
Example: If m=3 and n=3ℤ/3ℤ=1,ξ,ξ2withξ=e2πi/3.G3,1,3=100010001,0ξ010000ξ2,ξ2000ξ2000ξ,00ξ2ξ200010,…
and the number of elements in G3,1,3 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→Hg↦ϕgsuch that
If g1,g2∈G then
ϕg1ϕg2=ϕg1g2,
ϕ1=1,
If g∈G then
ϕg-1=ϕg-1.
The kernel of ϕ is
kerϕ=g∈G∣ϕg=1.
Example:ℤ/2ℤ=1,-1.
A homomorphism is given by
ϕ:Sn→ℤ/2ℤbyϕg=-1#of crossings ing.
So
The alternating group is An=kerϕ.
The groups Gm,l,n
Let l divide m. A homomorphism is
ϕ:Gm,1,n→ℤ/lℤgiven byϕg=∏non-zeroentriesgijm/l
and
Gm,l,n=kerϕ.
Example: If m=6,l=3 and n=5 then
ϕ00ξ3000ξ40000000ξ10000000ξ20=ξ3⋅ξ4⋅ξ⋅ξ2⋅6/3=ξ102=ξ2=e2πi2/6=e2πi/3.
So
Gm,l,n=n×nmatrices with(a) exactly one nonzero entry in each row and each column(b) the nonzero entries are inℤ/mℤ(c)∏non-zeroentriesgijm/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 eigenvalue≠1.
Example:1000ξ20001
is a reflection, and
if m=5 then
g00ξ2010ξ300g-1=10101010-100ξ2010ξ30012012010120-12=-100010001,
and so
00ξ2010ξ300is a reflection.
A reflection group is a group G of matrices generated by reflections.
Example:S3=100010001,010100001,…,
or
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 onℂn=c1c2⋱cn∣c1,c2,…,cn∈ℂ.
If g∈G and
xi=ith0⋮010⋮0
then
gxi=g11…g1n⋮⋮⋮⋮gn1…gnn0⋮010⋮0=g1ig2i⋮gni=g1ix1+g2ix2+…+gnixi.
Then G acts on polynomials
p∈ℂx1,…,xn
by
gp1+p2=gp1+gp2andgp1p2=gp1gp2.
Example:0ξ2ξ0x1=ξx2and0ξ2ξ0x2=ξ2x1,
and
0ξ2ξ03x12x2+5x1x23=3ξx22ξ2x1+ξx2ξ2x13=3ξ4x1x22+5ξ7x13x2.
The invariant ring of G is
ℂx1,…,xnG=p∈ℂx1,…,xn∣gp=pfor allg∈G.
Example:
If m=5 then
0ξ2ξ0x15x25=ξx25+ξ2x15=ξ5x25+ξ10x15=x15+x25.
(Chevalley, Shephard-Todd) G is a finite reflection group if and only if there exist polynomials
p1,…,pn
such that
ℂx1,…,xnG=ℂp1,…,pn.
Fundamental groups
Let X be a topological space with a fixed base point x0.
A path in X is a continuous map
p:0,1withp0=x0.
So
A loop in X is a path
g:[0,1]→X
with
g0=x0=g1
The fundamental group of X is
π1X=loops inXwith productg1g2t=g1t,if0≤t≤12g2t,if12≤t≤1.
Braid groups
The braid group on n strands is
ℬn=string diagrams withntop vertices andnbottom vertices andeach top vertex is connected to exactly one bottom vertex
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=kerϕ.
Let
ℂn=c1,…,cn∣c1,…,cn∈ℂ,
and let
Hij=c1,…,cn∣c1,…,cn∈ℂfor1≤j<i≤n,
and let
X=ℂn∖⋃i<jHij.
𝒫n=π1X.
Proof.
A loop in X is a path
such that the travelling points c1,…,cn satisfy
ci≠cj along the path.
□
Let G be a reflection goup. For each reflection sα in G let
Hα=c1,…,cn∈ℂn∣sαc1,…,cn=c1,…,cn.
Let
X=ℂn∖⋃reflectionsHα.
Problem:
Describe and understand π1X.
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)