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For $G$ a finite group, the sign representation is the one-dimensional representation $\pi : G \to \mathbb{C}$ with $\pi(g)$ the sign of the permutation given by the action of $g$ on $G$ by left multiplication.

Question: how to generalize the sign representation to semisimple finite dimensional Hopf algebras?

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    $\begingroup$ 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. $\endgroup$
    – darij grinberg
    Commented Feb 17, 2016 at 20:24
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    $\begingroup$ 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? $\endgroup$
    – David E Speyer
    Commented Feb 17, 2016 at 20:40
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    $\begingroup$ 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 $\endgroup$
    – Denis Nardin
    Commented Feb 17, 2016 at 22:15
  • $\begingroup$ @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). $\endgroup$
    – David Hill
    Commented Feb 18, 2016 at 0:42
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    $\begingroup$ 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. $\endgroup$
    – Ehud Meir
    Commented Feb 23, 2016 at 15:22

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