A question about the Tannaka-Krein reconstruction of finite groups

Question

In Chapter 15 Section 15.2.1 of Quantum Groups and Noncommutative Geometry, 2nd edition, the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional complex representations $\text{rep}_{\mathbb{C}}(G)$.

The answer depends on how we consider $\text{rep}_{\mathbb{C}}(G)$. Actually we cannot reconstruct $G$ if we only consider $\text{rep}_{\mathbb{C}}(G)$ as an abelian category or monoidal category, but we can reconstruct $G$ if we consider $\text{rep}_{\mathbb{C}}(G)$ as a symmetric monoidal category.

The proof outlined in the book is to consider the forgetful functor $$ F: \text{rep}_{\mathbb{C}}(G)\to \text{vect}_{\mathbb{C}} $$ to the symmetric monoidal category of finite dimensional $\mathbb{C}$-vector spaces. Then we show there is an isomorphism $$ G\to \text{Aut}^{\otimes}(F) $$ where $\text{Aut}^{\otimes}(F)$ denotes the group of natural isomorphisms of $F$ which are compatible with the tensor product on $\text{rep}_{\mathbb{C}}(G)$.

I want to see the impact $\text{Aut}^{\otimes}(F)$ when we add the symmetric structure on the monoidal categories. But it seems to me that there is no difference between the symmetric monoidal case and merely monoidal case. For example, symmetric means the following diagram commutes for $\alpha\in \text{Aut}^{\otimes}(F)$ $$ \require{AMScd} \begin{CD} F(V\otimes W)@>F(S_{V,W})>> F(W\otimes V)\\ @V \alpha_{V\otimes W} VV @VV \alpha_{W\otimes V} V\\ F(V\otimes W) @>>F(S_{W,V})> F(W\otimes V) \end{CD} $$ where $S_{V,W}$ is the symmetry functor in $\text{rep}_{\mathbb{C}}(G)$.

However since $F$ is the forgetful functor we automatically have $$ F(S_{V,W})\cong S_{F(V),F(W)} $$ and since $\alpha$ preserve the monoidal structure we have $$ \alpha_{V\otimes W}\cong \alpha_V\otimes \alpha_W. $$ Therefore since we have the commutative diagram $$ \begin{CD} F(V)\otimes F(W)@>S_{F(V),F(W)}>> F(W)\otimes F(V)\\ @V \alpha_{V} \otimes\alpha_{W} VV \circlearrowright @VV \alpha_{W}\otimes\alpha_{V} V\\ F(V)\otimes F(W) @>>S_{F(W),F(V)}> F(W)\otimes F(V) \end{CD} $$ the first diagram also commute. In conclusion there is no difference of $\text{Aut}^{\otimes}(F)$ whether we consider $\text{rep}_{\mathbb{C}}(G)$ as a symmetric monoidal category or just a monoidal category.

What did I do wrong in this argument?

Answer 1

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces. See Theorem 5.3.12 of the Tensor Categories. This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$. However, for the Hopf algebra one you look at all endomorphisms instead of tensor isomorphisms. If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as H-mod because H is not spanned by its grouplike elements.

Answer 2

Although you already have one sufficient answer, I thought I would add here that Etingof and Gelaki, "Isocategorical groups" considered exactly this question. For finite groups, they gave a complete characterization of when two non-isomorphic finite groups can have the same monoidal categories $rep(G)$ (ignoring the symmetric structure). One consequence of their characterization is that all finite groups of odd order, or twice odd order, and all simple finite groups are determined even by $rep(G)$ considered only as a monoidal category.