Begin with a finite-dimensional, complex, semi-simple algebra $\mathcal{A}$. Label its irreducible representations using an index set $\Lambda$ and for $\lambda \in \Lambda$ let $X^{\lambda}$ denote the corresponding irreducible character. Select complex parameters $s_\lambda$ and define a symmetric pairing (i.e. trace-form) on $\mathcal{A}$ by
\begin{equation} \langle u ,v \rangle \, := \, \sum_{\lambda \in \Lambda} \, s_\lambda \, X^\lambda \, (uv) \end{equation}
This pairing will be non-degenerate if and only if the parameters $s_\lambda$ are non-vanishing; equivalently the pairing is non-degenerate if and only if the Gram-determinant $G_B = \det \big( \langle b_i, b_j \rangle \big)$ is non-vanishing for any basis choice $\{b_1, \dots, b_N \}$ of the algebra. Now fix a basis $B = \{b_1, \dots, b_N \}$ for $\mathcal{A}$ and view the parameters $s_\lambda$ as indeterminants: In this case the Gram-determinant $G_B$ may be considered an element within the polynomial ring $\Bbb{C}\big[s_\lambda \, \big| \, \lambda \in \Lambda \big]$; moreover, bearing a scalar factor $F$ depending on the choice of basis and the character values of the algebra, $G_B$ must be a monomial owing to my remark about non-degeneracy.
Question: What are the exponents $p_\lambda$ in the monomial expansion $G_B = F \cdot \prod_{\lambda \in \Lambda} \, s_{\lambda}^{p_\lambda}$ ?
Is there a reference where this is explained ?
ines.