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 Incoming links: Character restriction norm bound

By the Maschke theorem, the restricted representation is a direct sum of irreducible

H

ψ_{i}

d_{i} = ⟨ χ ↓_{H}, ψ_{i} ⟩_{H}

. Therefore

χ ↓_{H} = \sum_{i} d_{i} ψ_{i}, d_{i} \in Z_{\geq 0} .

Using the preceding character inner product identity and the irreducibility of

χ

\sum_{i} d_{i}^{2} = \frac{1}{∣ H ∣} \sum_{h \in H} ∣ χ (h) ∣^{2} \leq \frac{1}{∣ H ∣} \sum_{g \in G} ∣ χ (g) ∣^{2} = \frac{∣ G ∣}{∣ H ∣} .

[G : H] = 2

, the nonzero sum of squares of integer multiplicities is either one or two. Hence the restriction is irreducible, or is the sum of two distinct irreducible characters, each with multiplicity one. This proves the requested index-two character restriction dichotomy directly; normality of an index-two subgroup is a general group theory fact.

\sum_{i = 1}^{r} a_{i}^{2} = ⟨ χ ↓_{H}, χ ↓_{H} ⟩_{H} = \frac{1}{∣ H ∣} \sum_{h \in H} ∣ χ (h) ∣^{2} .

Since

χ

G

is one, and hence

\sum_{i} a_{i}^{2} \leq \frac{1}{∣ H ∣} \sum_{g \in G} ∣ χ (g) ∣^{2} = \frac{∣ G ∣}{∣ H ∣} = [G : H] .

The omitted summands are nonnegative, so the character restriction norm bound is an equality precisely when

χ (g) = 0 for every g \in G ∖ H .

⟨ χ ↓_{K}, χ ↓_{K} ⟩_{K}

is a positive integer, because it is the sum of the squares of the multiplicities of the irreducible constituents. The character restriction norm bound and

[G : K] = 3

make it at most three. It is therefore $1$ , $2$ , or $3$ .

All three values occur:

In $G = S_{3}$ with $K = ⟨(12)⟩$ , the trivial character remains irreducible on $K$ , giving norm $1$ .
For the same pair, the two-dimensional standard representation of the symmetric group restricts to the sum of the trivial and nontrivial linear characters of $K$ , giving norm $1^{2} + 1^{2} = 2$ .
In $G = A_{4}$ with its normal Klein four-group $K = V_{4}$ , the three-dimensional irreducible representation of the alternating group on four letters restricts to the sum of the three nontrivial linear characters of $V_{4}$ , giving norm $1^{2} + 1^{2} + 1^{2} = 3$ .