For a group $G$ there is a well-defined map $\operatorname{Irr}(G) \to \operatorname{Lin}(G)$ which sends $\chi \mapsto \det \chi$, where $\det \chi$ the linear character of $G$ given by taking the determinant of the representation affording $\chi$.


In general, is there a good way to go about computing $\det \chi$ without having to construct a representation affording $\chi$, or are there some conditions on $G$ or $\chi$ under which this can be done easily?



Computing determinants of characters
If you know $\chi$ then you can write down $\det \chi$ using [Newton's identities](http://en.wikipedia.org/wiki/Newton%27s_identities). This is simply the observation that one can express the determinant of a matrix $\rho(g)$ in terms of traces of powers $\rho(g)^k=\rho(g^k)$ of that matrix.


