The irreducible representation of $S_4$ corresponding to the partition $2+2$ is two-dimensional and unfaithful. Are there other unfaithful irreps of $S_n$ with dimension greater than 1?
2026-04-07 09:44:42.1775555082
Unfaithful irreps of $S_n$ with dimension greater than one
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The kernel of a representation is a normal subgroup, and of course the representation is the result of lifting the representation of the quotient. Since we know the only normal subgroups of $S_n$ are $1,A_n,S_n$ (except $n=4$ having $V\lhd S_4$ too) and the quotients $S_n/A_n\cong C_2$ is abelian, the only case left is $S_4/V\cong S_3$ which has only one irrep of degree >1. So there are no other non-faithful irreps of $S_n$ of degree >1.