 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2008/b/6/c/solution

The two partition diagrams are transposes, so their characteristic-zero Specht modules differ by tensoring with the sign representation. A transposition has sign minus one. Consequently

χ^{(2, 1^{n - 2})} ((i, j)) = - χ^{(n - 1, 1)} ((i, j)) = 3 - n .

The middle character is

n - 3

, since the standard representation is the point-permutation representation minus its trivial line, and a transposition fixes

n - 2

points.

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