Questions: Representations of Symmetric Groups

Question 1

The number of non-isomorphic irreducible representations of S₅ is:

• A. 5
• B. 7
• C. 10
• D. 120

Answer: (B)

The number of irreducible representations equals the number of conjugacy classes, which for Sₙ equals the number of partitions of n. The partitions of 5 are: (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), (1,1,1,1,1) — exactly 7 partitions. So S₅ has 7 irreducible representations.

Question 2

Which partition of n corresponds to the trivial representation of Sₙ?

• A. (1, 1, ..., 1) — the partition into all 1s
• B. (n) — the single-row partition
• C. (n−1, 1)
• D. The answer depends on n

Answer: (B)

The partition (n) — a single row of n boxes — corresponds to the trivial representation, where every permutation acts as the identity. The partition (1,1,...,1) — a single column of n boxes — corresponds to the sign representation, where each permutation acts as multiplication by its sign. The partition (n−1,1) gives the (n−1)-dimensional standard representation.

Question 3

Conjugacy classes of Sₙ are determined by cycle type, and partitions of n parametrize both conjugacy classes and irreducible representations. Is this bijection between the two sets 'natural' in any canonical sense?

Model answer: No. While both sets are indexed by partitions of n (giving the same count), the correspondence is not canonical — there is no single 'natural' bijection that works uniformly for all n. The labeling of irreducible representations by partitions uses the construction of Specht modules, which involves specific combinatorial choices.

This is a subtle point. The equality |{conjugacy classes}| = |{irreducibles}| holds for any finite group (not just symmetric groups), but for Sₙ both sets happen to be naturally parametrized by partitions of n, creating a tempting but misleading identification. The conjugacy class of cycle type λ and the irreducible representation labeled by λ are related, but the relationship is indirect — it goes through the character table rather than a canonical bijection.

Question 4

The dimension of the irreducible representation of Sₙ corresponding to partition λ equals the number of standard Young tableaux of shape λ.

• T. True
• F. False

Answer: True

A standard Young tableau of shape λ is a filling of the Young diagram of λ with the numbers 1, ..., n such that entries increase along each row and down each column. The hook length formula gives an explicit count: dim(Sλ) = n! / ∏ h(□), where the product runs over all boxes and h(□) is the hook length. For S₃ and λ = (2,1), the two standard tableaux are [[1,2],[3]] and [[1,3],[2]], confirming dim = 2.