2.4 The Standard Model Representation
We are now in a position to put the entire Standard Model together in a single
picture, much as we combined the isospin and hypercharge
into the electroweak gauge group,
, in
Section 2.3.3. We then tensored the hypercharge
representations with the isospin representations to get the
electroweak representations.
Now let us take this process one step further, by bringing in a factor of
, for the color symmetry, and tensoring the representations of
with the representations of . Doing this, we get the
Standard Model. The Standard Model has this gauge group:
The fundamental fermions described by the Standard Model combine to form
representations of this group. We know what these are, and describe all of
them in Table 1.
Table:
Fundamental fermions as representations of
The Standard Model Representation 
Name 
Symbol 
rep 



Lefthanded leptons 





Lefthanded quarks 





Righthanded neutrino 





Righthanded electron 





Righthanded up quarks 





Righthanded down quarks 






All of the representations of
in the lefthand column are irreducible,
since they are made by tensoring irreps of this group's three factors,
, and . This is a general
fact: if is an irrep of , and is an irrep of , then
is an irrep of . Moreover, all irreps of arise in this
way.
On the other hand, if we take the direct sum of all these irreps,
we get a reducible representation containing all the firstgeneration fermions
in the Standard Model. We call the fermion representation.
If we take the dual of , we get a representation describing all the
antifermions in the first generation. And taking the direct sum of these
spaces:
we get a representation of
that we will call the Standard Model
representation. It contains all the firstgeneration elementary particles in
the Standard Model. It does not contain the gauge bosons or the mysterious
Higgs.
The fermions living in the Standard Model representation interact by exchanging
gauge bosons that live in the complexified adjoint representation of
. We
have already met all of these, and we collect them in
Table 2.
Table 2:
Gauge bosons
Gauge Bosons 
Force 
Gauge Boson 
Symbol 



Electromagnetism 
Photon 




Weak force 
and bosons 
, and 



Strong force 
Gluons 





Of all the particles and antiparticles in , exactly two of
them are fixed by the action of
. These are the righthanded neutrino
and its antiparticle
both of which are trivial representations of
; they thus do not
participate in any forces mediated by the gauge bosons of the Standard
Model. They might interact with the Higgs boson, but very little about
righthanded neutrinos is known with certainty at this time.
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