= Solution
The specified <linear maps> form a <permutation representation>, since
$$
R(\sigma)R(\tau)e_i=e_{\sigma(\tau(i))}=R(\sigma\tau)e_i.
$$
The <trace> of a <permutation matrix> counts fixed coordinate <basis vectors>. On the identity, three-cycles and <transpositions>, respectively, this gives $\chi_R=(3,0,1)$. Comparing with <2/b/solution>,
$$
\boxed{\chi_R=\chi_{\mathbf1}+\chi_D,\qquad R\cong\mathbf1\oplus D.}
$$
For an explicit <direct sum>,
$$
\mathbb C^3=\mathbb C(e_1+e_2+e_3)\oplus
\{z_1e_1+z_2e_2+z_3e_3:z_1+z_2+z_3=0\}.
$$
Both summands are <invariant subspaces> under coordinate <permutations>. The first is fixed pointwise and hence gives the <trivial representation>; the second has <character> $(2,-1,0)$ and hence gives $D$. The <sign representation> has multiplicity zero.
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