Table 2 Character table for the one-dimensional irreps of \(D_{n}\). For n even, all four irreps are valid, while for n odd, only the first two rows apply. In discussing \(D_{3}\), we will commonly refer to these one-dimensional irreps as the trivial representation, denoted by , and the σ-representation
\(r^{k}\) | \(sr^{k}\) | |
|---|---|---|
\(\chi_{1}(g)\) | 1 | 1 |
\(\chi_{2}(g)\) | 1 | -1 |
\(\chi_{3}(g)\) | \((-1)^{k}\) | \((-1)^{k}\) |
\(\chi_{4}(g)\) | \((-1)^{k}\) | \((-1)^{k+1}\) |