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2. FGC for tensor categories

Categorical questions related to FGC conjecture for Hopf algebras.


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      Problem 2.1.

      [Hector Pena Pollastri] If $\mathcal{C}$ satisfies the FGC conjecture and $\mathcal{B} \subset \mathcal{Z}(\mathcal{C})$, is it true that the Müger centralizer $\mathcal{Z}_{(2)}(\mathcal{B} \subset \mathcal{Z}(\mathcal{C}))$ also satisfies the FGC conjecture?


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          Problem 2.2.

          [Paul Balmer] Is there a nice class of categories where the FGC conjecture holds? (Something larger than the class of finite tensor categories.)


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              Problem 2.3.

              Does a braided finite tensor category have the tensor product property, that is, $V(A \otimes B) = V(A) \cap V(B)$?
                  See https://arxiv.org/abs/2306.16082 by Bergh-Plavnik-Witherspoon for recent progress.
                1. Remark. It is known to be true for symmetric finite tensor in characteristic $0$.


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                      Problem 2.4.

                      [Paul Balmer] Construct examples that precisely delineate the boundaries of FGC.

                          Cite this as: AimPL: Finite tensor categories: their cohomology and geometry, available at http://aimpl.org/finitetensor.