Editing Grand Unified Theory (section)

=== SO(10) ===
{{main|SO(10)}}

[[File:E6GUT.svg|200px|right|thumb|The pattern of [[weak isospin]], W, weaker isospin, W′, strong g3 and g8, and baryon minus lepton, B, charges for particles in the [[SO(10)]] Grand Unified Theory, rotated to show the embedding in [[E6 (mathematics)|E<sub>6</sub>]].]]

The next simple Lie group which contains the standard model is
: <math>\rm SO(10)\supset SU(5)\supset SU(3)\times SU(2)\times U(1) .</math>

Here, the unification of matter is even more complete, since the [[group representation|irreducible]] [[spinor]] [[group representation|representation]] {{math|'''16'''}} contains both the {{math|{{overline|'''5'''}}}} and {{math|'''10'''}} of {{math|SU(5)}} and a right-handed neutrino, and thus the complete particle content of one generation of the extended [[standard model]] with [[Neutrino|neutrino masses]]. This is already the largest [[simple group]] that achieves the unification of matter in a scheme involving only the already known matter particles (apart from the [[Higgs mechanism|Higgs sector]]).

Since different standard model fermions are grouped together in larger representations, GUTs specifically predict relations among the fermion masses, such as between the electron and the [[down quark]], the [[muon]] and the [[strange quark]], and the [[tau lepton]] and the [[bottom quark]] for {{math|SU(5)}} and {{math|SO(10)}}. Some of these mass relations hold approximately, but most don't (see [[Georgi-Jarlskog mass relation]]).

The boson matrix for {{math|SO(10)}} is found by taking the {{math|15 × 15}} matrix from the {{math|'''10''' + '''5'''}} representation of {{math|SU(5)}} and adding an extra row and column for the right-handed neutrino. The bosons are found by adding a partner to each of the 20 charged bosons (2 right-handed W bosons, 6 massive charged gluons and 12 X/Y type bosons) and adding an extra heavy neutral Z-boson to make 5 neutral bosons in total. The boson matrix will have a boson or its new partner in each row and column. These pairs combine to create the familiar 16D Dirac [[spinor]] matrices of {{math|SO(10)}}.