Is determinant useful in representation of group

Question

I am learning representation theory of finite group, the trace of matrix is used to define character, which is very useful. the determinant of a matrix is also independent of choice of basis, is determinant of matrix useful in representation of group?

Answer 1

This is certainly a good observation and is applicable to character theory: if $\chi$ is a complex character of a finite group $G$, define $det(\chi): G \rightarrow \mathbb{C}$ as follows. Choose a representation $\mathfrak{X}$ affording $\chi$ and put $det(\chi)( g ) = det(\mathfrak{X})$. Then $det(\chi)$ is a uniquely defined linear complex character of $G$. This character is used in essential ways in many theorems, notably in relation to the order $o(\chi)$, of $det(\chi)$ now seen as an element of the group of linear characters of $G$. See for example the book Character Theory of Finite Groups by Marty Isaacs.