On the structure of Witt groups and minimal extension conjecture

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. For a braided fusion category $\mathcal{C}$ over $\mathcal{E}$ we give sufficient and necessary conditions that characterize the Witt relation $[\mathcal{C}]=[\mathcal{E}]$. Then we show the Witt group $\mathcal{W}(\mathcal{E})$ is naturally a direct sum of Witt group $\mathcal{W}:=\mathcal{W}(\text{Vec})$ and the group $\text{H}^4(G,\mathbb{K}^\times)$. Consequently, for any non-degenerate fusion category $\mathcal{C}$ over $\mathcal{E}$, there is a positive integer $n$ (e.g. $n=|G|$) such that $\mathcal{C}^{\boxtimes_\mathcal{E}^n}$ admits a minimal extension.

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