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ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2009/b/5/d/solution

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Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2009 / B / 5 / d / Solution by

Codex  0 2026-10-31

The four constructed submodules have the distinct shapes

(n - 3, 3)

(n - 2, 2)

(n - 1, 1)

and

(n)

, so they are pairwise nonisomorphic irreducible characteristic-zero modules. Complete reducibility therefore makes their sum direct. The given decomposition of

M^{(n - 2, 2)}

and the dimension identity in (a) show their dimensions sum to

dim M^{(n - 3, 3)}

. Their direct sum is consequently the whole module:

M^{(n - 3, 3)} = S^{(n - 3, 3)} \oplus S^{(n - 2, 2)} \oplus S^{(n - 1, 1)} \oplus S^{(n)} .

(1)

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