SU6 tensor branchings to SU5 ($6\downarrow =5\oplus 1$)

For the embedding $U\mapsto \mathrm{diag}(1,U)$, the defining SU(6) representation restricts to the defining SU(5) representation plus a trivial representation. Taking dual representations, symmetric squares, exterior squares and traceless endomorphisms gives $\overline{6}\downarrow =\overline{5}\oplus 1$, $21\downarrow =15\oplus 5\oplus 1$, $15\downarrow =10\oplus 5$, and $35\downarrow =24\oplus 5\oplus \overline{5}\oplus 1$. The fixed diagonal direction in the Adjoint representation is $\mathrm{diag}(-5,1,1,1,1,1)$. Dimensions label representations of different groups on the two sides: the fifteen-dimensional SU(6) representation is alternating, while the fifteen-dimensional SU(5) representation is symmetric.