Editing Georgi–Glashow model (section)

==Construction==
[[File:SU(5) representation of fermions.png|thumb|Schematic representation of fermions and bosons in {{math|SU(5)}} GUT showing {{nowrap|'''5''' + '''10'''}} split in the multiplets. Row for {{math|'''1'''}} (the [[sterile neutrino]] singlet) is omitted, but would likewise be isolated. Neutral bosons (photon, Z-boson, and neutral gluons) are not shown but occupy the diagonal entries of the matrix in complex superpositions.]]
{{Expand section|date=July 2019}}

SU(5) acts on <math>\mathbb{C}^5</math> and hence on its [[exterior algebra]] <math>\wedge\mathbb{C}^5</math>.  Choosing a <math>\mathbb{C}^2\oplus\mathbb{C}^3</math> splitting restricts SU(5) to {{math|S(U(2)×U(3))}}, yielding matrices of the form

:<math>\begin{matrix}
\phi: & U(1)\times SU(2)\times SU(3) & \longrightarrow & S(U(2)\times U(3)) \subset SU(5) \\
& (\alpha, g, h) & \longmapsto &
\begin{pmatrix}
\alpha^3 g & 0\\
0 & \alpha^{-2}h
\end{pmatrix}\\
\end{matrix}</math>

with [[kernel (algebra)|kernel]] <math>\{(\alpha, \alpha^{-3} \mathrm{Id}_2, \alpha^2 \mathrm{Id}_3) | \alpha \in \mathbb C , \alpha ^6 = 1 \}\cong \mathbb Z_6</math>, hence isomorphic to the [[Standard Model]]'s true [[gauge group]] <math>SU(3)\times SU(2)\times U(1)/\mathbb{Z}_6</math>. For the zeroth power <math>{\textstyle\bigwedge}^0\mathbb{C}^5</math>, this acts trivially to match a left-handed [[neutrino]], <math>\mathbb{C}_0\otimes\mathbb{C}\otimes\mathbb{C}</math>.  For the first exterior power <math>{\textstyle\bigwedge}^1\mathbb{C}^5 \cong \mathbb{C}^5</math>, the Standard Model's group action preserves the splitting <math>\mathbb{C}^5 \cong \mathbb{C}^2\oplus\mathbb{C}^3</math>.  The <math>\mathbb{C}^2</math> transforms trivially in {{math|SU(3)}}, as a doublet in {{math|SU(2)}}, and under the {{math|Y {{=}} {{sfrac|1|2}}}} representation of {{math|U(1)}} (as [[weak hypercharge]] is conventionally normalized as {{math|α<sup>3</sup> {{=}} α<sup>6Y</sup>}}); this matches a right-handed anti-[[lepton]], <math>\mathbb{C}_{\frac 1 2}\otimes\mathbb{C}^{2*}\otimes\mathbb{C}</math> (as <math>\mathbb{C}^{2}\cong\mathbb{C}^{2*}</math> in SU(2)).  The <math>\mathbb{C}^3</math> transforms as a triplet in SU(3), a singlet in SU(2), and under the Y = &minus;{{sfrac|1|3}} representation of U(1) (as {{math|α<sup>&minus;2</sup> {{=}} α<sup>6Y</sup>}}); this matches a right-handed [[down quark]], <math>\mathbb{C}_{-\frac 1 3}\otimes\mathbb{C}\otimes\mathbb{C}^3</math>.

The second power <math>{\textstyle\bigwedge}^2\mathbb{C}^5</math> is obtained via the formula <math>{\textstyle\bigwedge}^2(V\oplus W)={\textstyle\bigwedge}^2 V^2 \oplus (V\otimes W) \oplus {\textstyle\bigwedge}^2 W^2</math>.  As SU(5) preserves the canonical volume form of <math>\mathbb{C}^5</math>, [[Hodge dual]]s give the upper three powers by <math>{\textstyle\bigwedge}^p\mathbb{C}^5\cong({\textstyle\bigwedge}^{5-p}\mathbb{C}^5)^*</math>.  Thus the Standard Model's representation {{math|''F'' ⊕ ''F*''}} of one [[Generation (particle physics)|generation]] of [[fermion]]s and antifermions lies within <math>\wedge\mathbb{C}^5</math>.

Similar motivations apply to the [[Pati–Salam model|Pati–Salam]] model, and to [[SO(10)]], E6, and other supergroups of SU(5).