# Complex Representations in SU(3)

Good evening,

I am just reading the wonderful book by H. Georgi "Lie Algebras in Particle Physics", and I'm having a problem with one of the proves. While talking about complex representations especially in SU(3) the author states that the conjugate representation of a representation with Dynkin coefficients (n,m) has Dynkin coefficients (m,n). He argues that the highest weight of (n,m) is $n\mu^1+m\mu^2$ while the lowest one is $-n\mu^2-m\mu^1$, where $\mu^i$ are the fundamental weights, while the heighest weight of (m,n) is just $n\mu^2+m\mu^1$, so it is the negative of the lowest weight of (n,m) and from that he follows over a few steps that (m,n) must be the conjugate representation. What I don't get, and what Georgi doesn't really explain, is why $-n\mu^2-m\mu^1$ shall be the lowest weight of (n,m). I worked it out for some examples, but I can't figure out a simple argument for all n,m (as it should be, if Georgi doesn't bother to explain further). Do you have some ideas? Greetings from Heidelberg,

Markus Zetto

• Greetings Markus. The question might be more suitable for physics, but I am not sure about it. Either way notation and context would need be explained for anyone to understand stuff. – mathreadler Nov 8 '17 at 21:28
• I can do that, but I'm not shure where exactly I've been too sloppy. n and m are just some non-negative integers, the Dynkin coefficients of the representation. I don't think that further context is needed other than that we are working in the Lie Algebra of SU(3) (but I actually am more of a physicists, so if you desire more mathematical rigor, I should mention that as the title suggests, the book is also made for physicists) – Intergalakti Nov 8 '17 at 21:45
• Is your question about why the lowest weight has the negative of the coefficients of the highest weight? Or why the complex conjugate representation exchanges the two coefficients? Or both? – ziggurism Nov 8 '17 at 22:16
• Yes, it's about why the lowest weight the negative of the coefficients of the highest weight, but with n and m interchanged. – Intergalakti Nov 9 '17 at 7:53