(Redirected from
SU(3)XSU(2)XU(1))
In mathematics and mathematical physics, the Lie group SU(3) × SU(2) × U(1) is the formulation of the Standard Model as a gauge theory with the gauge group SU(3) × SU(2) × U(1) or SU(3) × SU(2) × U(1)/Z6 with a couple of fermion fields and a Higgs field, which is a 
and/or a 
. SU(3) describes quantum chromodynamics, SU(2) describes the weak interaction* and U(1) describes hypercharge.
*Technically speaking, the Z and W bosons are described by a field which is really a linear combination of SU(2) and U(1). See electroweak.
There are three families of fermions, each consisting of the representations, 
(q for left-handed quark), 
(dc for the left-handed anti d-quark), 
(uc for the left handed up antiquark), (l for the left handed leptons), (1,1)1(ec for the left-handed positron) and (1,1)0(νc for the left-handed antineutrino, which is now known to exist. See Neutrino oscillation.).
The Higgs field acquires a VEV, resulting in a spontaneous symmetry breaking from ![[SU(2)\times U(1)]/\mathbb{Z}_2](a/../upload/math/65be98fb4c9ca83feef92e6bb3855ace.png)
or 
to U(1)em.
Of course, calling the representations things like is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, it is standard among high energy physicists.
Since the homotopy group
![\pi_2\left(\frac{[SU(2)\times U(1)]/\mathbb{Z}_2}{U(1)_{em}}\right)=0](a/../upload/math/72f557c4295f6ffe0e95fd249430fedd.png)
this model predicts no monopoles associated with the electroweak breaking scale. See Hooft-Polyakov monopole.
The Yukawa couplings of the scalar Higgs fields with the fermions produces the fermion masses after the Higgs field acquires a VEV.
See also grand unified theory.