 Solution

ID: past-exam-of-the-introduction-to-representation-theory-course-of-the-university-of-oxford/2023/2/d/solution

The given permutations satisfy

r^{4} = s^{2} = 1

and

srs = r^{- 1}

. The eight elements

r^{j}, r^{j} s

for

0 \leq j < 4

are distinct:

s \in / ⟨ r ⟩

, whose only element of order two is

r^{2} = (13) (24)

. Thus

H

is the order-eight dihedral group, and we may use the matrices

R = σ (r) = (01 - 1 0), S = σ (s) = (10 0 - 1) .

They obey the same relations and describe a quarter-turn and a reflection of a square.

R

i

and

- i

, with respective eigenlines spanned by

(1, - i)

and

(1, i)

. Since

S

interchanges these eigenlines, neither is an invariant subspace for

H

. Any proper nonzero subrepresentation of this two-dimensional complex representation would be a line invariant under both

R

and

S

, and an

R

-invariant line must be one of those eigenlines. None qualifies, so $W$ is irreducible.

For the calculation in part (e), its character is

χ_{W} (1) = 2, χ_{W} (r^{2}) = - 2, χ_{W} (r) = χ_{W} (r^{3}) = χ_{W} (r^{j} s) = 0 (0 \leq j < 4) .

Indeed

R^{2} = - I

, while a quarter-turn or a reflection has trace zero. The character inner product

(2^{2} + (- 2)^{2}) /8 = 1

provides a second irreducibility check.

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