Character generating series of symmetric powers (${\sum }_{n\ge 0}{\chi }_{S^{n}V}(g)t^n=\det (I-tg{)}^{-1}$)

For eigenvalues ${\lambda }_1,\dots ,{\lambda }_d$ of a finite-order linear action, its symmetric power characters satisfy ${\sum }_{n\ge 0}{\chi }_{S^{n}V}{t}^n={\prod }_j(1-{\lambda }_{j}t{)}^{-1}$. The denominator is the alternating generating polynomial of its exterior power characters.