Determinant of the character table of a finite group $G$

This is an exercise from the book "Groups and Representations" by Alperin & Bell. This quantity is well defined upto a sign. By column orthogonality relations, its squared norm is $\displaystyle\prod_{\substack{g}} |C_G(g)|$, where g runs over the representatives of the conjugacy classes. If the group is cyclic, the determinant is just a Vandermonde determinant.

I wonder if there is a nice explanation for an arbitrary group.

Solution 1:

Let $A$ be the character table as a square matrix. In other words $A_{ij}=\chi_i(g_j)$, where $\chi_i$ the distinct irreducible characters, and $g_j$ are representatives of conjugacy classes. Let $A^H$ be the conjugate transpose. Let us compute the matrix product $B:=A^HA$. At position $(i,j)$ we get $$ B_{ij}=\sum_k A^H_{ik}A_{kj}=\sum_k \overline{A_{ki}}A_{kj}=\sum_k\overline{\chi_k(g_i)}\chi_k(g_j)=|C_G(g_j)|\delta_{ij} $$ by the second orthogonality relation. Therefore $B$ is diagonal, and $$ \det B=\prod_j|C_G(g_j)|. $$ So by the multiplicativity of determinant we also have $$ \det(A^H)\det(A)=\det B=\prod_j|C_G(g_j)|. $$ Let us study the relation between $A^H$ and the transpose $A^T$. Let us define a permutation $s$ of the conjugacy classes by the mapping $[g]\mapsto [g^{-1}]$. It is obviously a product of disjoint 2-cycles, and its fixed points are exactly the conjugacy classes stable under taking the inverse element. Let us denote by $\ell$ the number of orbits of size two. If we denote by $\tilde{A}$ the matrix that we get from $A$ by permuting the columns according to $s$, then the general fact $\chi(g^{-1})=\overline{\chi(g)}$ allows us to identify $A^H$ as $\tilde{A}^T$. Clearly $$ \det\tilde{A}=(-1)^{\ell}\det A,$$ so we get the equation $$ (-1)^{\ell}(\det A)^2=\det B=\prod_j|C_G(g_j)|. $$ The sign of $\det A$ will always remain ambiguous, because we have no natural ordering neither for the conjugacy classes nor for the characters, so all we can say is that $$ \det A=\pm i^{\ell}\sqrt{\prod_j|C_G(g_j)|}. $$