= Solution
The <trivial representation> and <sign of a permutation> give
$$
R_0(\sigma)=1,\qquad R_1(\sigma)=\operatorname{sgn}(\sigma).
$$
They are <one-dimensional representations>, hence <irreducible representations>. Their <characters>, in the <conjugacy class> order of <2/a/solution>, are
$$
\chi_0=(1,1,1,1,1),\qquad
\chi_1=(1,-1,1,1,-1).
$$
These are the only <one-dimensional representations>. All transpositions are conjugate, so they have the same scalar image $t$ in any <one-dimensional representation>; since a transposition squares to the identity, $t=\pm1$. Transpositions generate the <symmetric group>, fixing the whole <representation> once $t$ is chosen.
By the <sum of squares of irreducible representation dimensions>, the three remaining dimensions satisfy $d_2^2+d_3^2+d_4^2=24-1-1=22$, with each $d_i\ge2$. No $d_i$ can be at least $4$, since $16+4+4>22$. Thus the only possible dimensions are
$$
\boxed{d_2=2,\qquad d_3=d_4=3.}
$$
Back to article page
