Symmetric Rep(Z/2) has singular Hopf-link matrix

The projectors $(1+T)\mathrm{/}2$ and $(1-T)\mathrm{/}2$ split every representation as $V=V_{+}\oplus V_{-}$, with $T=+1$ on $V_{+}$ and $T=-1$ on $V_{-}$. Choosing finite bases decomposes these spaces into copies of the one-dimensional trivial representation $1$ and sign representation $\epsilon$. They are nonisomorphic simples, their endomorphism rings are $\mathbb{C}$, and $\epsilon \otimes \epsilon \cong 1$ since $(-1{)}^2=1$. Thus the category is finite semisimple with simple unit; ordinary duality and flip make it symmetric ribbon with the stated spherical structure. Contraction of the ordinary evaluation and coevaluation is the ordinary linear trace, so both simple dimensions are one. By [F2], for any simple labels $i,j$, $s_{ij}=\mathrm{tr}(1_{i\otimes j})=d_i{d}_j=1$. In the ordered basis $(1,\epsilon )$ this gives $s=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)\mathrm{.}$

The two rows coincide, and $\det (s)=1\cdot 1-1\cdot 1=0$. Therefore $s$ is not invertible and the category is not modular. This shows why nondegeneracy must be a hypothesis or separately proved.