I am doing a project on group association schemes, in particular looking at the structure constant $$p_{KL}^M = \#\{(x, y, xy) : x \in K, y\in L, xy \in M\}$$ where $K, L$ and $M$ are conjugacy classes.

I have been given a formula to help me, but cannot get hold of it anymore! It has this expression on the right-hand side: $$ \sum_{\chi \in \DeclareMathOperator{\irr}{Irr} \irr(G)} \frac{\chi(x) \chi(y)}{\chi(xy)}$$ Does anyone recognise this?


closed as off-topic by user10534, Carlo Beenakker, Andrés E. Caicedo, Yemon Choi, Felipe Voloch Aug 29 '13 at 23:00

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There is a well-known formula for these structure constants using character theory. It has a slightly different form to the one you write down (which doesn't look right to me), and you can find it as Exercise (3.9) of Isaacs' Character Theory of Finite Groups.

I have an e-version of this book so, if you want a copy, email me. For the record, it's one of my favourite maths books ever.


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