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The formula answers: how many tuples $(\sigma_1,\sigma_2,\dots,\sigma_n)$ of elements of a given group $G$ such that (1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class. (2) $\sigma_1\sigma_2...\sigma_n= \text{id}$.

I want to know the name and exact content of this formula. Also, is there any connection between this formula and Burnside counting theorem (orbit-counting theorem)? I also want of a proof of the formula using idempotent (perhaps) and other related theorems.

Thanks in advance.

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This seems to be cross-posted here on MO –  Old John Dec 31 '12 at 13:01

2 Answers 2

up vote 3 down vote accepted

You haven't told the reader what the formula is. I believe I know the formula you mean, which is $$ \left( \frac{|G|^{n-1}}{\prod_{j=1}^{n} |C_{G}(\sigma_{j})|} \right) \sum_{i=1}^{k} \frac{\prod_{j = 1}^{n} \chi_{i}(\sigma_{j})}{\chi_{i}(1)^{n-2}},$$ where the $\chi_{i}$ are all the complex irreducible characters of $G$. This formula was probably known to Burnside- I know no special name for it- it is a special case of a general formula for the product of class sums in the group algebra. The formula can be derived by writing the class sums as linear combinations of the primitive idempotents of $Z(\mathbb{C}G).$ Such linear combinations are easy to multiply since these idempotents are mutually orthogonal. Then one recovers the coefficient of a particular element $g$ in the product by using the fact that $g$ occurs with coefficient $\frac{\chi(1)\overline{\chi(g)}}{|G|}$ in the primitive central idempotent corresponding to the irreducible character $\chi$. The formula for the product of two class sums appears explicitly in Burnside's book, but some of the exercises in the book make it clear that he was aware of the general formula. As far as I know, there is no obvious connection with Burnside's orbit counting formula.

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Thank you . could you give me the name of this formula and suggest some useful reference for the proof ? I wanna see the complete proof . I've tried to google it , but failed to find things that I want. –  user54794 Dec 31 '12 at 13:22

I came across this formula in the book:

MR2036721 (2005b:14068) Lando, Sergei K. ; Zvonkin, Alexander K.

Graphs on surfaces and their applications.

With an appendix by Don B. Zagier. Encyclopaedia of Mathematical Sciences, 141. Low-Dimensional Topology, II.

Springer-Verlag, Berlin, 2004. xvi+455 pp. ISBN: 3-540-00203-0

I don't have this book in front of me but my recollection is that they (or Zagier) attribute this to Frobenius. You should find better references in this book.

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