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ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2007/b/6/d/solution

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Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 6 / d / Solution by

Codex  0 2026-10-31

The Robinson-Schensted-Knuth correspondence is a bijection between permutations and ordered pairs of standard Young tableaux of the same shape. Hence the class with shape

λ

has size

(f^{λ})^{2}

. For

λ = (4, 2, 1, 1)

, the hook lengths are

74214121,

with product

448

. The hook-length formula gives

f^{λ} = 8! /448 = 90

. Therefore

class size = 9 0^{2} = 8100.

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