The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper.
It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here):

Theorem: Let $x \in {\small \rm Irr}(\mathcal{C})$ with ${\small \rm FPdim} \ x = 2$. Then at least one of the following holds:

  • $G[x] \neq 1.$
  • $\mathcal{C}$ has a fusion subcategory $\mathcal{D}$ of type $[[1,2],[2,1],[3,2]]$, such that $x \not\in {\small \rm Irr}(\mathcal{D})$ which has an invertible object $g$ of order $2$ such that $gx \neq x$.
  • $\mathcal{C}$ has a fusion subcategory of type $[[1,3],[3,1]]$ or $[[1,1],[3,2],[4,1],[5,1]]$.

Question: Is this statement also true for integral fusion rings?

Remark: The basic definitions for a fusion ring are written in this post.

Observation: Consider an integral fusion ring $\mathcal{F}$ of type $[[1,n],[2,m], \dots]$, with $n, m \ge 1$, then by some arguments of the proof of Theorem 11 of Nichols-Richmond paper (arguments which work automatically in the fusion ring setup), we deduce that $n>1$ or $\mathcal{F}$ admits a fusion subring of type $[[1,1],[3,2],[4,1],[5,1]]$ (and nothing else because $\mathcal{F}$ is finite dimensional).
Corollary: A simple integral fusion ring can't have a basic element $h$ with $d(h) = 2$ (except $\mathcal{G}_2$).

  • $\begingroup$ Yes, according to Sonia Natale (by email). $\endgroup$ – Sebastien Palcoux Aug 5 '16 at 5:43

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