Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character.

Consider the following combinatorial property of $\Lambda$: for all triple $(j,k,\ell)$ $$\sum_i \frac{\lambda_{i,j}\lambda_{i,k}\lambda_{i,\ell}}{\lambda_{i,1}} \ge 0.$$ It is a consequence of a more general result involving subfactor planar algebra and fusion category (see here Corollary 7.5, see also this answer).

*Question*: Is this combinatorial property already known to finite group theorists?

If yes: What is a reference?

If no: Is there a group theoretical *elementary* proof?

*In any case*: Are there other properties of the same kind?

To avoid any misunderstanding, let us see one example. Take $G=A_5$, its character table is:

$$\left[ \begin{matrix}
1&1&1&1&1 \\
3&-1&0&\frac{1+\sqrt{5}}{2}&\frac{1-\sqrt{5}}{2} \\
3&-1&0&\frac{1-\sqrt{5}}{2}&\frac{1+\sqrt{5}}{2} \\
4&0&1&-1&-1 \\
5&1&-1&0&0
\end{matrix} \right] $$
Take for example $(j,k,\ell) = (2,4,5)$, then $\sum_i \frac{\lambda_{i,j}\lambda_{i,k}\lambda_{i,\ell}}{\lambda_{i,1}} = \frac{5}{3} \ge 0$.