Past exam of the groups and representations course of the University of Oxford / 2018 / 1 / e / Solution

The stated coordinate substitution splits the polynomial basis into two three-cycles:The identity fixes all six basis vectors; neither nonidentity element fixes any. A permutation representation has character equal to the number of fixed basis vectors, henceThe representation multiplicity of each irreducible character is thereforeAn explicit eigenbasis confirms the result. For $k=0,1,2$, putThen $\tilde{R}(1)f_k={\alpha }^k{f}_k$ and $\tilde{R}(1)h_k={\alpha }^k{h}_k$. Thus each three-cycle carries one copy of every irreducible representation. These signs use precisely the source's substitution action; taking an inverse substitution would interchange ${\rho }_1$ and ${\rho }_2$, leaving the total decomposition unchanged.