$ (5^* \times 5^*)_{asym}={10}$ in A. Zee's book p.409 versus PDG Sec.114

Question

What is the mathematical or physical way to understand why the 4th and 5th components in the Georgi Galshow SU(5) model has the SU(2) doublet $(1,2,-1/2)$: $$ \begin{pmatrix} \nu\\e \end{pmatrix} $$ with left-handed $\nu$ in the 4th component and $e$ in the 5th component of $5^*$; while in the contrary, the SU(2) doublet $(3,2,1/6)$: $$ \begin{pmatrix} u\\d \end{pmatrix} $$ with the left-handed $u$ in the 5th component (column or row) and $d$ in the 4th component (column or row) of $10$?

My question is that why not the left-handed $u$ in the 4th component (column or row in the anti-symmetrix rank-5 matrix, say in Zee's book p.409 below) and $d$ in the 5th component (column or row in the anti-symmetrix rank-5 matrix, say in Zee's book p.409 below) of $10$?

My understanding is doe to the complex conjugation $$ (5 \times 5)_{asym}={10}^* $$ instead of $$ (5^* \times 5^* )_{asym}={10}. $$ But is it this the case? Would $2^*$ flips the doublet component of 2? $$ 2: \begin{pmatrix} v\\v' \end{pmatrix}\to 2^* \begin{pmatrix} v'\\v \end{pmatrix}? $$

In contrast, we see the PDG writes in a very different manner: https://pdg.lbl.gov/2018/mobile/reviews/pdf/rpp2018-rev-guts-m.pdf

Can we compare the two notations? Notice the contrary locations of $\nu, e, u, d$.

This post imported from StackExchange Physics at 2020-12-01 17:43 (UTC), posted by SE-user annie marie heart