Let X be a finite set with the action of a group G. Show that
the multiplicity of the trivial representation in C[X] is |G\X|,
the number of G-orbits in X.
Use the previous exercise, together with character theory to prove
Burnside’s lemma:
Here Xg denotes the set of points x ∈ X such that g ⋅ x = x.
Let X, Y and Z be three G-sets. Given functions k1 : Y × Z → C
and k2 : X × Y → C, what is the function k : X × Z → C such
that
The dihedral group D2n acts on the set V of 2n vertices
of the regular n-gon. What is |G\(V × V )|? What about
|G\(V × V × V )|?
Each group G acts on itself by left multiplication. What
are the relative positions of pairs of elements in G for this
action?
Let Xk denote the set of all subsets of order k in {1,…,n}.
Show that the representations C[Xk] and C[Xn-k] of Sn are
isomorphic.
