3
$\begingroup$

(I asked the following question at StackExchange but received no answer.)

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \otimes V^*$ for each (representative of isomorphism class of) simple object of $C$.

I want to show that $D=\mathrm{Rep}(G/Z(G))$.

It seems that this problem is claimed in Tensor Categories By Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik without a complete proof.

I appreciate any help.

$\endgroup$
3
  • $\begingroup$ This depends on the field of coefficients. For example, it is false for $G=\mathbb Z/3$ over $\mathbb Q$. $\endgroup$ Jul 2 '16 at 17:02
  • $\begingroup$ @BenWieland How about when the base field is algebraically closed and characteristic zero? $\endgroup$
    – user46846
    Jul 2 '16 at 18:01
  • $\begingroup$ This even works for compact groups using Doplicher Roberts duality. $\endgroup$ Jul 2 '16 at 23:24
4
$\begingroup$

I'll work over $\mathbb{C}$ below for simplicity, although it can be replaced by an algebraically closed field of characteristic $0$.

Assuming that "fusion subcategory generated by" means what I think it means (tensor products, direct sums, direct summands), here is a general observation.

Lemma: The fusion subcategory of $\text{Rep}(G)$ generated by some reps $V_i$ is $\text{Rep}(G/N)$ where $N$ is the intersection of the kernels of $G$ acting on each $V_i$.

This follows in turn from the standard fact that $\text{Rep}(H)$ is generated as a fusion category (in the above sense) by any faithful representation of $H$; see, for example, this MO question. From here it suffices to show that the intersection of the kernels of $G$ acting on $V \otimes V^{\ast}$ for each irrep $V$ is $Z(G)$.

Recall that the regular representation $\mathbb{C}[G]$ decomposes, as a representation of $G \times G$ (acting on the left and right), as $\bigoplus_V V \boxtimes V^{\ast}$ where $\boxtimes$ denotes the external tensor product. If we restrict to the diagonal copy of $G$ in $G \times G$, we get that the permutation representation coming from $G$ acting on itself by conjugation decomposes as $\bigoplus_V V \otimes V^{\ast}$ where $\otimes$ is now the ordinary tensor product of representations. The kernel of this permutation representation is clearly $Z(G)$, as desired.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy