Past exam of the groups and representations course of the University of Oxford / 2022 / 3 / e

The symmetric square of a direct sum givesNow ${\mathrm{Sym}}^{2}2=3$ and ${\mathrm{Sym}}^{2}3=6$, rather than the singlet and antitriplet arising from exterior squares. ThusThe dimensions are $3+6+6=15$.

By contrast, solution gives the following left-handed Weyl fermion content of $\overline{5}\oplus 10$:These are one Standard Model family, with weak hypercharge $Y=q_c/6$. The $15$ contains $Q$ but lacks $L$, $d^c$, $u^c$ and $e^c$; instead it contains an SU(2) triplet and an SU(3) sextet. Equality of total dimensions therefore does not give the same representation content, and $15$ cannot supply the family.