A note on multiplicity-free permutation characters

Involutions acting on representations

“…From Proposition 4, since ε ι (π) = 1 for every irreducible π of G, we have (1) from Clifford's theorem, since χ i (1) = χ i | N (1). Since the union of the G-conjugate orbits of the characters of N is all of the characters of N , and…”

“…There is a much less complicated proof of the multiplicity free part of this statement given by Balmaceda [1] using the double coset method of obtaining a Gelfand pair. The next proposition shows that we can still conclude information about the twisted indicators from this proof.…”