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The character of a representation is its trace. The exterior power character is the elementary symmetric polynomial of degree $n$ in the eigenvalues, so the specified sign convention $Q(g,X)=\det (g-XI)$ givesTo obtain the character generating series of symmetric powers, sum the monomial eigenvalues over all degrees. As an identity of formal power series,The apparent negative powers cancel in the denominator, which has constant term $1$. Equivalently, this series is the reciprocal of ${\sum }_{j=0}^d(-1{)}^j{\chi }_{{\Lambda }^{j}V}(g)t^j$. Comparing coefficients gives ${\sum }_{j=0}^{\min (d,n)}(-1{)}^j{\chi }_{{\Lambda }^{j}V}(g){\chi }_{S^{n-j}V}(g)=0$ for $n>0$, an explicit recurrence computing the symmetric-power characters from $Q$.