For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries:
$$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq s,t\leq k} $$
where
$$a_{rst} = \frac{1}{|C_t|} |\{(x,y) \in C_r\times C_s\ |\ xy\in C_t\}|$$
Has anyone ever heard of them? I've been trying to find more Information on these matrices. I even checked then collected works of William Burnside.
They have very interesting properties, mainly they are simulatiously diaganalizable by a unitary matrix $V$. Calling the column vectors $v_s$ and defining:
$$ \chi_s (g) = \sqrt{|G| } \sum_{j=1}^k\frac{v_{sj}}{\sqrt{|C_j|}} \delta_{C_j}(g)\ 1\leq s\leq k $$
where g is an element of $G$ and the delta is $1$ for $g\in C_j$ and else zero. then the matrix:
$$\big(\chi_s(C_t)\big)_{1\leq s,t\leq k}$$
is exactly the character table of all irreducible representations of $G$.
This is the basic theory behind the Dixon-Schneieder algorithm for computing the character table of finite groups, which is used by GAP and Magma. The original implementations used floating point approximation for complex numbers, but the modern methods do all of the calculations in a finite field ${\mathbb F}_p$ for some moderately large prime number $p$, and then lift the character values (which ar esums of roots of unity) to the algebraic numbers.