Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 19F (Representation Theory)

Let $\varphi =(1+\sqrt{5})/2$ and $\bar{\varphi }=(1-\sqrt{5})/2$. In class order $(1,2A,3A,5A,5B)$, character orthogonality giveswith class sizes $1,15,20,12,12$. The rows are orthonormal and their squared degrees sum to $60$, so the table is complete.

A three-dimensional representation of $S_5$ would decompose into irreducibles. Since $S_5$ has irreducible degrees $1,1,4,4,5,5,6$, only one-dimensional constituents could occur, whose restrictions to $A_5$ are trivial. It cannot restrict to either irreducible $3$.

Its character is the number of fixed points, and averaging it gives the number of orbits. Complementation $\Sigma \mapsto X\setminus \Sigma$ is a $G$-equivariant bijection ${\Gamma }_n\to {\Gamma }_{d-n}$.

The exterior-power assertion is false. For $n=2$ and $g=(12)(34)\in A_5$, $g$ fixes two $2$-subsets, so ${\chi }_{\mathbb{C}[{\Gamma }_2]}(g)=2$. The natural permutation character has $\chi (g)=1$ and $\chi (g^2)=5$, henceThe representations are not isomorphic.