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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?

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If you know $\chi$ then you can write down $\det \chi$ using Newton's 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.

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    $\begingroup$ Strictly speaking, knowing $\chi$ in the sense of knowing the function $\chi : G \to \mathbb{C}$ is not enough: you also need to know enough of the multiplication table of $G$ to take powers. $\endgroup$
    – Qiaochu Yuan
    Nov 27, 2012 at 2:33

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