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Here is a paragraph with some statements about the Gauge Symmetry Breaking from Georgi's book Lie Algebras in Particle Physics 2nd ed -- From Isospin to Unified Theories (Georgi, 1999) p.285.

Georgi wrote:

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His claim is too quick. Can some experts explain which symmetry breaking pattern he is thinking of? In particular, he uses three Higgs fields in three representations:

  • 24 should be the adjoint representation of SU(5).

  • 5 should be the fundamental representation of SU(5).

  • 45 should be EITHER the adjoint representation of SO(10) or some (what kind?) representation of SU(5).

question: How do we use representation 24, 5, and 45 to break from $SO(10)$ to $SU(5)$ to $SU(3) \times SU(2) \times U(1)$?

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    $\begingroup$ It's all in Slansky, equation (3.4) and Table 43 but you must learn the language... The WP article on the SO(10) GUT: the antisym 45 of SO(10) breaks it down to SU(5) GG and contains the 24 of GG for further breaking to the SM. I think the higgs doublet of the SM is coming out of the 5 of SU(5) , etc... I'm not sure why you'd care about the 45 of SU(5), though--its YTableau, 4column & 2column is in Table 28. Really, there are no easy summaries... $\endgroup$ – Cosmas Zachos Jul 25 '20 at 15:25
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    $\begingroup$ Answering my own question of the 45 of SU(5). Georgi and Jarlskog, PhysLett B86 (1979) 297 use it for the Yukawas of SU(5) which yield better mass relations. It has a Young tableau of a 4-story column and a 2-story column next to it. Lots of them in the 120 of SO(10). $\endgroup$ – Cosmas Zachos Jul 25 '20 at 20:36
  • $\begingroup$ MANY THANKS FOR PRECIOUS COMMENTS - <3 thanks ~ $\endgroup$ – annie marie heart Jul 25 '20 at 22:19
  • $\begingroup$ any comments math.stackexchange.com/questions/3777000/… ? thanks! $\endgroup$ – annie marie heart Aug 1 '20 at 20:18
  • $\begingroup$ Frankly, emailing Kephart should produce a superior answer....! There can be a self-dual and an anti-self-dual one... $\endgroup$ – Cosmas Zachos Aug 1 '20 at 20:27

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