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## Main Question or Discussion Point

I was reading this thread on "representation and field theory" under

physics > quatnum physics

https://www.physicsforums.com/showthread.php?t=91764

which got me thinking...and perhaps there are experts out there who can clarify these for me:

(A.1) In particle physics we often speak of something as being a singlet, doublet, triplet...etc. in SU(3), SU(2) or the

(A.2) eg. we put leptons into doublets, typically in a 2x1 vector... so I guess we are assuming the 2x2 matrix representation of SU(2)? My question is does the

OR

does the fact that there are "two"(or three or one) items to be juggled with dictate whether something is called a "doublet" (or triplet or singlet) respectively?

(A.3) Related to this is the issue of irreducibility of tensors/representations. Do we always choose irreducible rep for putting in our particle contents? ie. say in SU(3), there are

(A.4) finally, when we introduce the Higgs doublet to create a mass term which is invariant under the group action, I noticed that in that context, we often say that we need a doublet Higgs because 2x2=1+3 contains a singlet! So are we now talking about combining two reps in such a way that we end up not having juggled anything? Or is there anything deeper than that? Doesn't a direct product of 2 and 2 give us a "bigger" space? or only the irreducible bit (related to A.3) matters?

I've got a feeling that I may have mixed up a lot of concenpts (irreducible representations, irreducible tensors, representation space, dim of rep and dim of matrix...etc.), pls feel free to correct me where applicable. Thanks in advance.

physics > quatnum physics

https://www.physicsforums.com/showthread.php?t=91764

which got me thinking...and perhaps there are experts out there who can clarify these for me:

(A.1) In particle physics we often speak of something as being a singlet, doublet, triplet...etc. in SU(3), SU(2) or the

**5**in SU(5)... or whatever. What do they actually mean?(A.2) eg. we put leptons into doublets, typically in a 2x1 vector... so I guess we are assuming the 2x2 matrix representation of SU(2)? My question is does the

*dimensionality*of the vector space/tensor/representation (not sure which one I should be talking about) got anything to do with whether we call something a singlet, doublet, triplet... etc?OR

does the fact that there are "two"(or three or one) items to be juggled with dictate whether something is called a "doublet" (or triplet or singlet) respectively?

(A.3) Related to this is the issue of irreducibility of tensors/representations. Do we always choose irreducible rep for putting in our particle contents? ie. say in SU(3), there are

**1**,**3**,**6**,**8**irreducible reps etc, does that mean one cannot have a SU(3) doublet (there is no**2**rep)? If not, how can this be written down?(A.4) finally, when we introduce the Higgs doublet to create a mass term which is invariant under the group action, I noticed that in that context, we often say that we need a doublet Higgs because 2x2=1+3 contains a singlet! So are we now talking about combining two reps in such a way that we end up not having juggled anything? Or is there anything deeper than that? Doesn't a direct product of 2 and 2 give us a "bigger" space? or only the irreducible bit (related to A.3) matters?

I've got a feeling that I may have mixed up a lot of concenpts (irreducible representations, irreducible tensors, representation space, dim of rep and dim of matrix...etc.), pls feel free to correct me where applicable. Thanks in advance.