# $(5^* \times 5^*)_{asym}={10}$ 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? $$2: \begin{pmatrix} v\\v' \end{pmatrix}\to 2^* \begin{pmatrix} v'\\v \end{pmatrix}?$$ 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
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