The four electric and magnetic sectors of $D({\mathbb{Z}}_2)$

The group ${\mathbb{Z}}_2$ is abelian, so its two conjugacy classes are the singletons $\{e\}$ and $\{s\}$, and each centralizer is ${\mathbb{Z}}_2$. A group representation is determined by $T=\pi (s)$ with $T^2=I$. The operators $P_{+}=(I+T)\mathrm{/}2$ and $P_{-}=(I-T)\mathrm{/}2$ satisfy $P_{+}^2=P_{+}$, $P_{-}^2=P_{-}$, $P_{+}{P}_{-}=0$, and $P_{+}+P_{-}=I$ by $T^2=I$; they commute with $T$, so their ranges are invariant. Irreducibility forces one range to be all of $V$ and the other zero, giving $T=I$ or $T=-I$. A scalar action on a complex vector space of dimension greater than one leaves every line invariant, so irreducibility forces dimension one. Conversely each of these two scalar actions defines a character because $(\pm 1{)}^2=1$. Thus there are exactly the two characters ${\chi }_{+}$ and ${\chi }_{-}$ for each flux label.

By the definition of ${\chi }_{-}$, its value on the nontrivial flux is ${\chi }_{-}(s)=-1$.

The coordinate cycles $p_1,q_1$ specified in [F3] have $I=1$. Applying [F2] to their closed strings gives commutation sign $(-1{)}^1=-1$, equal to ${\chi }_{-}(s)$. With no crossings, $I=0$ and the string sign is $+1={\chi }_{-}(e)$. The claim is only equality of these finite algebraic signs; it asserts no general braid representation or physical realization.