Timeline for Does $SU(N)$ have pseudo-real representation?
Current License: CC BY-SA 4.0
21 events
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| Feb 26, 2019 at 9:55 | answer | added | Mikhail Borovoi | timeline score: 11 | |
| Feb 25, 2019 at 19:31 | comment | added | Mikhail Borovoi | (cont.) Condition (1) means that the numerical labels of $\lambda$ at the vertices of the Dynkin diagram are symmetric with respect to $\sigma$. If (1) is satisfied, then (2) means that the numeric label at the vertex $2m+1$ is odd. See Onishchik and Vinberg, "Lie Groups and Algebraic Groups", Exercise 4.3.13. | |
| Feb 25, 2019 at 18:47 | comment | added | Mikhail Borovoi | I think that ${\rm SU}(N)$ has a non-real pseudo-real irreducible representation if and only if $N=4m+2$ for some natural number $m$. For a representation $\rho$ one can take the irreducible complex representation with highest weight $\lambda$ satisfying the following conditions: (1) $\lambda$ is $\sigma$-invariant, where $\sigma$ is the nontrivial automorphism of the Dynkin diagram $A_{N-1}$, and (2) $\lambda(z_1)=-1$, where $z_1$ a generator of the center $Z({\rm SU}(N))$. For example, $\rho=\Lambda^{2m+1}(\mathbb{C}^N)$. | |
| Feb 24, 2019 at 23:09 | comment | added | YCor | @TomGoodwillie anyway, even if physicists are more specific the definition as currently given sounds natural (and would sound ugly if one artificially adds assumptions such as irreducible or non-real). I think there's no need for another synonym to "quaternionic" for irreducible representations :) | |
| Feb 24, 2019 at 23:07 | history | edited | YCor | CC BY-SA 4.0 |
added 12 characters in body
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| Feb 24, 2019 at 13:33 | comment | added | Tom Goodwillie | Nowhere does the question say "irreducible", but I suspect that that is what is meant. I don't know whether physicists use the term "pseudo-real" in speaking of representations that are not necessarily irreducible. | |
| Feb 24, 2019 at 10:00 | comment | added | YCor | @DimaPasechnik of course. This was my point in my previous comment: even if $C=\bar{D}^{-1}D$ for some $D$ then it forces being real. Anyway, pseudo-real was multiply confusing, since it was defined as "if X then..." which is not a way to define X, and being non-real is not part of the definition of pseudo real in the way the question was initially stated, so it doesn't need to be incorporated in the definition of pseudo-real. Eventually I entirely rewrote the question. | |
| Feb 24, 2019 at 9:55 | history | edited | YCor | CC BY-SA 4.0 |
completely written as every single line was confusingly written
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| Feb 24, 2019 at 8:41 | answer | added | Dima Pasechnik | timeline score: 5 | |
| Feb 24, 2019 at 7:05 | comment | added | Dima Pasechnik | Requiring $C$ not to be identity is not enough, as certainly you do not want $C=-I$. | |
| Feb 24, 2019 at 3:28 | comment | added | user34104 | @ Marcel Bischoff @Tom Gzoodwillie $SU(2)$ is a special case. In general $N$ dimensional representation of $SU(N)$ is not pseudo real.It would be nice to know a schematic construction of the pseudo real representation for all $SU(N)$, optimistically with minimal dimension. | |
| Feb 24, 2019 at 3:22 | comment | added | Marcel Bischoff | As pointed out above, the two-dimensional irreducible representation of SU(2) is pseudo-real. | |
| Feb 24, 2019 at 2:30 | comment | added | Aaron Bergman | Irreducible pseudoreal reps are the quaternionic reps. In physics, people classify representations as real, pseudoreal and complex instead of real, complex and quaterniomic. | |
| Feb 24, 2019 at 2:06 | comment | added | Tom Goodwillie | The irreducible $2$-dimensional representation of $SU(2)$ is pseudo-real but not real, if I understand what you mean by these terms. | |
| Feb 24, 2019 at 1:38 | comment | added | YCor | No, it doesn't... anyway I understand what you're asking. | |
| Feb 24, 2019 at 1:36 | comment | added | Learner | Thanks for the comment! I edited the question. Here, I meant a non-real pseudo-real representation as LSpice was saying. It means that $C$ can not be an identity matrix. | |
| Feb 24, 2019 at 1:34 | history | edited | Learner | CC BY-SA 4.0 |
added 35 characters in body
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| Feb 24, 2019 at 1:29 | comment | added | YCor | @LSpice thanks! indeed the question is asked 2 times before an additional sentence amends it :) also I guess that "real" means conjugate to a real-valued rep. | |
| Feb 24, 2019 at 1:10 | comment | added | LSpice | @YCor, although I agree that it would be nice to clarify whether the second paragraph is a definition, the first paragraph does say (although the title doesn't) that the author wants a non-real pseudo-real representation. | |
| Feb 24, 2019 at 1:04 | comment | added | YCor | Would you say what a pseudo-real representation is? Is the sentence "If $g$..." a definition? If so, every real representation, or conjugate thereof, is pseudo-real, so what is the question? | |
| Feb 24, 2019 at 0:53 | history | asked | Learner | CC BY-SA 4.0 |