Representation theory of finite groups: Difference between revisions

Let the finite group in question be G. All the linear reps in this article are finite dimensional.

Schur's lemma:

If ρ is a rep of G, then all linear transformations mapping the rep to itself which are invariant under the action of G are multiples of the identity if and only if the rep is irreducible.

Invariant here means, if you think about it, the linear transformation commutes with ρ(g) for all g in G. Suppose we have an invariant linear transformation A which is NOT a multiple of the identity. Let λ be one of its eigenvalues. Then, A-λ1 is also invariant. Look at its kernel. It can't be zero dimensional since its determinant is zero, but it can't be the entire vector space either because A is NOT a multiple of the identity. The kernel is obviously a proper subrep. Hence, the rep is reducible.

If we have a subrep of a subrep, then the quotient space is another rep.

Group algebra: Let C[G], the group algebra of G be the associative algebra generated the elements of g subject to g₁g₂ (algebra product)=g₁g₂ (group product). It is a |G| dimensional algebra.

C[G] is an algebra representation of G. (See Peter-Weyl for topological groups).

C[G] as a vector space is a rep of G×G. More specifically, if g₁ and g₂ are elements of G and h is an element of C[G] corresponding to the element h of G,

(g₁,g₂)[h]=g₁h g^-1₂.

In particular, C[G] (as a linear rep) decomposes into the direct sum of n copies of each irrep from each equivalence class where n is the dimension of the irrep.

If ρ is an n dimensional irrep of G, then the subspace of C[G] consisting of the elements $\sum _{g\in G}<x,\rho (g)[y]>g$ for all y in the irrep and x in the dual irrep is a n×n dimensional irrep of C[G] as a rep of G×G, which of course would be n direct sum copies of equivalent n dimensional irreps of G.

There is a mapping from G to the complex numbers for each rep called the character given by the trace of the linear transformation upon the rep generated by the element of G in question. All elements of G belonging to the same conjugacy class have the same character.

Tr[ρ(ghg^-1)]=Tr[ρ(g)ρ(h)ρ(g)^-1]=Tr[ρ(h)] (cyclic property of the trace)

The number of conjugacy classes of G is the same as the number of inequivalent irreps of G.

The characters are orthogonal to each other.