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

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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? $$ 2: \begin{pmatrix} v\\v' \end{pmatrix}\to 2^* \begin{pmatrix} v'\\v \end{pmatrix}? $$

enter image descri+ption here

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

enter image description here

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
asked Jul 26, 2020 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]

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