I know and understand from standard gauge theory the fundamental (and anti-fundamental) and the adjoint representations. This is standard when considering for example QCD or say Seiberg-Witten theory to name two examples.

Still, in more advanced studies one can encounter various other types of representations. For example in quiver gauge theories one encounters the bifundamental representations.

I have encountered in various papers some of the following representations: pseudo-real representation, tensor representation, matter representation, twisted representation.

All this seems quite confusing to me since, at least in mathematics text books I have not encountered something similar. So, is there a reference or a guide to all possible types of representations?

I mean what is the difference of a real and pseudo-real and also complex? Is matter the quartenionic representation? Are fundamental and adjoint also split in real and pseudo-reals?

Finally, would you recommend some text where a lost soul could read about these?


I strongly suggest "Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists (Cambridge Monographs on Mathematical Physics)" by Jurgen Fuchs.

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  • $\begingroup$ I will check it out, I was not aware of this text. Would it contain all the above? $\endgroup$ – Gorbz May 16 '17 at 13:32
  • $\begingroup$ Detailed discussion of reality properties of representations plus many side applications related to theoretical physics. Maybe less systematic than math books but imho very good for theoretical physics. $\endgroup$ – Matteo Beccaria May 16 '17 at 13:34
  • $\begingroup$ Thank you. I see the table of contents. Seems very concise. But, I would also like a small "classification" (not in the mathematical sense), a list of all such representations summarized or so. Would you be aware of anything like this? $\endgroup$ – Gorbz May 16 '17 at 13:36
  • $\begingroup$ Some terms ( real , pseudo real etc ) are standard , other less. For instance matter representation refers to the representation of the gauge group carried by matter fields in physical sense. $\endgroup$ – Matteo Beccaria May 16 '17 at 13:39
  • $\begingroup$ Yes, I know that matter representation refers to the matter content but in many cases it is a quaternionic representation. But matter is represented by spinors, e.g. quarks, which are in the spin representation (over flat space). If we put it in a compact manifold then requirement of spin c structure makes it quaternionic I think. But I would like to have all this somewhere well explained :) $\endgroup$ – Gorbz May 16 '17 at 15:11

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