Example 15.5.1 (Restriction and induction between and ). Let with irreducibles (dimensions 1, 1, 3, 3, 2), and with irreducibles (dimensions 1, 1, 2). We compute restrictions using character tables (restricting the character of a representation of to elements of ):
and (trivial and sign restrict to trivial and sign).
. Indeed, the permutation representation restricts as (the 4th coordinate is fixed by ), so .
(since and ).
. The representation factors through , and as a representation of this quotient is the standard representation .
By Frobenius reciprocity, we can read off the induced representations:
contains each with multiplicity 1, giving (dim ).
(dim ).
(dim ).
Exercise 15.5.1 (Exercises).
Let . Show that where is the regular representation of and is viewed as a vector space with trivial -action on the second factor.
Let for . Show that , where is the trivial representation of , is isomorphic to (the trivial plus the sign representation of ).
Using Frobenius reciprocity and the character table of , compute (where is the standard representation of , embedded as the subgroup fixing 5).
Prove that the character formula for induction can alternatively be written as where are coset representatives.
(Transitivity of induction) If , show .
Show solution
(1), a direct sum of copies of , with permuting the summands. This is exactly where the -action is on the factor.
(2). Since , this is 2-dimensional with basis indexed by the two cosets. Even permutations fix both; odd permutations swap them. The invariant subspace , and the anti-invariant .
(3) By Frobenius reciprocity, the multiplicity of an -irreducible in equals . We restrict each -irreducible to (the subgroup fixing 5) by evaluating its character on elements of cycle types . The inner product with (class sizes 1, 6, 3, 8, 6) gives: So (dimensions ).
(4) The sum counts each valid conjugate exactly once per coset. In the formula , replacing by for : the condition becomes , which holds iff . The values of give (class function), contributing times the same value. Dividing by recovers the coset-representative sum.
(5) Write coset representatives for by combining those for and : if represents and represents , then represents . The direct sum decompositions are compatible: .