Timeline for What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Kac number")?
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| Oct 11, 2022 at 23:18 | comment | added | Kimyeong Lee | For a single BPS instanton in the gauge theory of the group $G$ on $R^4$, the number of zero modes is $h^\vee_G$, the dual Coxeter number. When the theory is considered on $R^3\times S^1$, the instanton gets fractionalized to $h^\vee_G$ magnetic monopoles. After the twisted compactification of the theory, the number of magnetic monopoles $h^\vee_G$ remain intact. | |
| Feb 23, 2019 at 3:27 | comment | added | Theo Johnson-Freyd | The fact that "dual" and "affine" do not commute always --- "surprises" isn't the right word, "confronts"? "reminds"? --- me. In addition to the $B_\ell$ example, there are my favourites, the two non-simply-laced exceptional groups $G_2$ and $F_4$, which are self-dual but with affine algebras that are not self-dual. | |
| May 24, 2010 at 12:34 | comment | added | Jim Humphreys | The term comes up in the context of root systems, Dynkin diagrams, Weyl groups (crystallographic reflection groups), but doesn't come from Coxeter's more general study of the order and the eigenvalues of Coxeter elements. I don't think Coxeter would have seen any group-theoretic meaning for "dual Coxeter number". The later characterization of Coxeter numbers which Kac starts with was proved by Kostant after observations by A. Shapiro. Anyway, the question of renaming is moot by now (I recall Borel's belated attempt to ascribe the "Bruhat ordering" more correctly to Chevalley). | |
| May 23, 2010 at 22:33 | history | answered | Peter Tingley | CC BY-SA 2.5 |