χ_{ρ} (g) = Tr ρ (g)

; it is irreducible if the representation is irreducible. The character inner product is

(α, β) = \frac{1}{∣ G ∣} \sum_{g \in G} \overline{α (g)} β (g) .

Distinct irreducible characters are orthogonal and each has norm one. For a permutation representation, the diagonal entry of the permutation matrix at a basis vector is one exactly when that vector is fixed. Its trace therefore counts fixed points.

Let

r

be rotation through

π /2

. Let

s_{x}, s_{y}

be reflections in the coordinate axes, and

s_{d}, s_{d^{'}}

reflections in the two diagonals. The eight values are

g χ_{Ω} (g) e 4 r 0 r^{2} 0 r^{3} 0 s_{x} 0 s_{y} 0 s_{d} 2 s_{d^{'}} 2.

(2)

Only diagonal reflections fix vertices, two each. Hence

(1, χ_{Ω}) = \frac{4 + 2 + 2}{8} = 1, (χ_{Ω}, χ_{Ω}) = \frac{16 + 4 + 4}{8} = 3.

(3)

Complete reducibility and character orthogonality imply that the squared multiplicities sum to three. The trivial constituent has multiplicity one, so the remaining two units come from two distinct nontrivial constituents, each with multiplicity one. Their degrees add to

4 - 1 = 3

, so they are two and one. Thus

χ_{Ω} = 1 + χ + η, χ (1) = 2, η (1) = 1.

(1)

The two-dimensional constituent is the complexification of the geometric action on the plane. Its trace is

2

on the identity,

- 2

on the half-turn, zero on the quarter-turns, and zero on every reflection. Its character norm is

(4 + 4) /8 = 1

, proving it irreducible. Alternatively, the span of the two coordinate functions on the vertex set is a two-dimensional invariant subspace of the permutation module with this geometric action. This identifies the requested representation rather than merely its degree.