Timeline for Generalization of the sign representation to Hopf algebras
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 23, 2016 at 15:22 | comment | added | Ehud Meir | It seems that the only groups in which this representation will not be trivial are groups $G$ in which the 2-Sylow subgroup is cyclic, isn't it? In any case, a general finite dimensional semisimple Hopf algebra will not necessarily have a 1-dimensional nontrivial representation, and we cannot talk about determinants in representations of a noncocommutative Hopf algebra. | |
| Feb 18, 2016 at 13:23 | comment | added | Denis Nardin | @SébastienPalcoux Because I think it is needed to pass the module structure on $V^{\otimes r} $ to the quotient $\Lambda^r V $. I haven't checked all details but you need something in order to behave well with respect to the $S_r $ action | |
| Feb 18, 2016 at 5:36 | comment | added | Sebastien Palcoux | @DenisNardin: why do you need to assume cocommutativity? | |
| Feb 18, 2016 at 0:42 | comment | added | David Hill | @DenisNardin In the group case the determinant is $\pm 1$. This is not going to carry over in general. It would be helpful if to see a list of properties that one might want to see in such a sign representation (besides giving the sign rep. in the group case). | |
| Feb 17, 2016 at 22:15 | comment | added | Denis Nardin | I might be confused but isn't the representation defined in the OP just the determinant of the regular representation? I think that this can be generalized to all cocommutative finite-dimensional Hopf algebras | |
| Feb 17, 2016 at 20:40 | comment | added | David E Speyer | This definition of the sign representation doesn't give what I would expect to be the desired answer for $S_n$, when $n \geq 4$. Specifically, the sign of the permutation of $g \cdot$ on $S_n$ is always $1$ for $n \geq 4$ and any $g \in S_n$. Is that what you wanted your definition to do? | |
| Feb 17, 2016 at 20:24 | comment | added | darij grinberg | If the Hopf algebra is cocommutative, then there is Larson's character (see Section 7.1 of Sweedler's "Hopf algebras" book). Not sure about the general situation. | |
| Feb 17, 2016 at 20:12 | history | asked | Sebastien Palcoux | CC BY-SA 3.0 |