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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/a/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 / a / Solution by

Codex  0 2026-10-31

For a tableau

t

, let

C_{t}

be its column stabilizer and

e_{t} = \sum_{σ \in C_{t}} sgn (σ) {σ t}

its polytabloid. The Specht module is the span of all such polytabloids inside the tabloid permutation module.

A tabloid with second row of size

k

is determined by that subset, so

dim M^{(n - k, k)} = (k n)

. The two-row hook formula gives

dim S^{(n - k, k)} = \frac{n ! ( n - 2 k + 1 )}{( n - k + 1 )! k !} = (k n) - (k - 1 n) .

At

k = 3

this proves

dim M^{(n - 3, 3)} = dim M^{(n - 2, 2)} + dim S^{(n - 3, 3)} .

By Maschke's theorem, characteristic-zero permutation modules are completely reducible, so their Specht submodules are summands. The nonzero sum of all three-subset tabloids is fixed by every permutation; its span is a copy of the trivial module

S^{(n)}

.

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