Discussion Overview
The discussion revolves around the time reversal properties of specific irreducible representations (irreps) in the context of group theory, particularly focusing on the Pnnm(58) group and the Herring test. Participants explore theoretical implications and results related to the representations $\Gamma_{3}^+$ and $\Gamma_{4}^+$, as well as the conditions under which these representations may be time reversal degenerate.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- One participant references J. O. Dimmock's work suggesting that $\Gamma_{3}^+$ and $\Gamma_{4}^+$ should be time reversal degenerate, implying that $\Sigma\chi\{R^2\}$ should equal 0.
- Another participant points out that the irreps of the double group #58 are being considered, noting that the $\gamma^+$ and $\gamma^-$ for 1-4 are irreps of the simple group and are one-dimensional real representations, which cannot be complex, leading to a conclusion that Herring must equal +1.
- A participant acknowledges the consensus among several researchers that $\Gamma_{3}^+$ and $\Gamma_{4}^+$ should be time reversal degenerate, expressing skepticism about the possibility that all these researchers could be incorrect.
- Another participant asserts that for any real irrep, Herring must equal +1, emphasizing that one-dimensional irrep characters correspond to matrices, thus reinforcing the claim that $\Gamma_{3}$ and $\Gamma_{4}$ are real and cannot yield Herring = 0.
Areas of Agreement / Disagreement
Participants express differing views on the time reversal properties of the representations in question. While some assert that $\Gamma_{3}^+$ and $\Gamma_{4}^+$ cannot be time reversal degenerate, others reference established literature suggesting otherwise. The discussion remains unresolved with competing perspectives on the validity of the Herring test results.
Contextual Notes
Participants have not reached consensus on the interpretation of the Herring test results or the implications of the irreps' properties, highlighting potential limitations in their assumptions regarding time reversal degeneracy.