= Solution
On a one-dimensional <representation>, $s_i$ acts by a nonzero scalar $\varepsilon_i$. Its square is one. The braid relation becomes $\varepsilon_i^2\varepsilon_{i+1}=\varepsilon_{i+1}^2\varepsilon_i$, and hence $\varepsilon_{i+1}=\varepsilon_i$. Thus all adjacent transpositions act by one common scalar $\varepsilon$ with $\varepsilon^2=1$.
In characteristic different from two, $\varepsilon=1$ or $-1$. Since adjacent transpositions generate the group, these choices give exactly the <trivial representation> and <sign representation>. For $n\le1$ they coincide, leaving only the <trivial representation>.
Back to article page
