Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 2 / b

Solution

Repeated use of the induction branching rule gives

S^{(2)} ↑_{S_{2}}^{S_{4}} ≅ S^{(4)} \oplus 2 S^{(3, 1)} \oplus S^{(2, 2)} \oplus S^{(2, 1, 1)},

while

S^{(1^{3})} ↑_{S_{3}}^{S_{4}} ≅ S^{(2, 1, 1)} \oplus S^{(1^{4})} .

1, (12), (12) (34), (123), (1234)

, their characters are respectively

(12, 2, 0, 0, 0) and (4, - 2, 0, 1, 0) .

(48, - 4, 0, 0, 0)

. Taking inner products with the five irreducible characters of

S_{4}

gives multiplicities

1, 5, 4, 7, 3

V ≅ S^{(4)} \oplus 5 S^{(3, 1)} \oplus 4 S^{(2, 2)} \oplus 7 S^{(2, 1, 1)} \oplus 3 S^{(1^{4})} .

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