[lpetrich]

Consider groups of matrices under matrix multiplication. Associativity: automatic. Identity: the identity matrix. Inverses: matrix inverses -- the matrices must be nonsingular.


A group homomorphism is some function f of a group's elements that acts as a group and that satisfies f(a)*f(b) = f(a*b).

Its kernel is all group elements that map onto the identity of the new group.


A function that takes determinants of matrices is a homomorphism of a matrix group, with its kernel being all matrices with determinant 1.
det(A)*det(B) = det(A.B)

For a group of some subset of complex numbers under multiplication, like nth roots of unity, f(a) = an for integer n is a homomorphism.


Groups may be realized as sets of matrices, D(a) for element a. A matrix group can be realized with a different set of matrices, for instance. Such matrices are called representations, and those that cannot be reduced to combinations of representations irreducible ones or irreps.

More formally, consider some X that satisfies X.D(a) = D(a).X for all a in the group. If every such X is a multiple of the identity matrix, then the D's are an irrep.


There is a trivial rep for every group, the identity rep: D(a) = 1.

For abelian groups, all their irreps have dimension 1.

For nonabelian groups, at least one irrep has a dimension more than 1.

A rep of a group can be a rep of any of its quotient groups.

For group G,
a*G = (permutation of G)

So every group can be realized as a permutation group. The permutations form a representation:
D(a)cb = 1 if c = a*b, 0 otherwise

This is the "regular representation", and it is irreducible only for the identity group. It contains n copies of every irrep, where n is the irrep's dimension.


I'll now make a table of the irreps of a few groups.
Z(2):
e, a
1, 1
1, -1

Z(4):
e, a, a2, a3
1, 1, 1, 1
1, i, -1, -i
1, -1, 1,-1
1,-i, -1, i

Z(2)*Z(2) -- the 4-group or Viergruppe
e, a, b, a*b
1, 1, 1, 1
1, 1, -1, -1
1, -1, 1, -1
1, -1, -1, 1