TITLE: A new combinatorial property for the character table of a finite group?
QUESTION [21 upvotes]: 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 $\sum_i \frac{\lambda_{i,j}\lambda_{i,k}\lambda_{i,\ell}}{\lambda_{i,1}} = \frac{5}{3} \ge 0$.

REPLY [12 votes]: This is indeed well-known in the character theory literature, and goes back to Frobenius and Burnside. What you are calculating is a positive rational multiple of a class algebra constant, and class algebra constants are clearly non-negative.
Using the notation in David Speyer's comment, it is well-known, and derived in most representation theory texts that 
$\frac{|G|}{|C_{G}(f)| |C_{G}(g)|} \sum_{\chi} \frac{\chi(f)\chi(g)\chi(h)}{\chi(1)}$ is the number of times 
$h^{-1}$ is expressible as a product of a conjugate of $f$ and a conjugate of $g$.
  The character theory formula is easily derived from the expressions of the class sums as linear combinations of primitive central idempotents of the group algebra $\mathbb{C}G$, and can be found in many texts .