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I know two formulas by the name of Frobenius.

The first one computes the number $$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$ where $G$ is a finite group and $C_1,\dotsc,C_k$ conjugacy classes in $G$. The Frobenius' formula says that $$\mathcal{N}(G;C_1,\dotsc,C_k)=\frac{|C_1|\cdots |C_k|}{|G|}\sum_{\chi}\frac{\chi(C_1)\cdots\chi(C_k)}{\chi(1)^{k-2}},$$ where the sum is over all characters of irreducible representations of $G$.

The other says that, if $\lambda=(\lambda_1,\dotsc,\lambda_\ell)$ and $\mu=(\mu_1,\dotsc,\mu_m)$ are partitions of $n$, the coefficient of $x_1^{\lambda_1+\ell-1}x_2^{\lambda_2+\ell-2}\cdots x_{\ell}^{\lambda_\ell}$ in $$\prod_{1\leq i<j\leq \ell}(x_i-x_j)\prod_{i=1}^m(x_1^{\mu_i}+x_2^{\mu_i}+\cdots+x_\ell^{\mu_i})$$ is the character of $S^\lambda$ (the Speecht module) evaluated at an element of $S_n$ with cycle type $\mu$.

It would be lovely if we could compute the characters of the Speecht modules utilizing the first formula. If it is possible, how could I do it?

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  • $\begingroup$ Usually the first formula for the symmetric group is, viceversa, used for obtaining some information about products of conjugacy classes (and for characters some other formulae are used.) $\endgroup$
    – Fedor Petrov
    Commented Feb 23, 2019 at 22:24
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    $\begingroup$ For any finite group $G$, the numbers $N(G;C_1,C_2,C_3)$ tell us how to multiply conjugacy classes (that is, the sum of the elements of a conjugacy class) in the center of the group algebra $\mathbb{Z}[G]$. There is a unique set (up to sign and order) of primitive orthogonal idempotents in the center of the group algebra over $\mathbb{Z}$. The character values are the coefficients when the class sums are expanded in terms of these idempotents. (The sign is determined by the positivity of the character on the identity element of $G$.) In fact, this is how one proves Frobenius' formula. $\endgroup$
    – Richard Stanley
    Commented Feb 25, 2019 at 0:23
  • $\begingroup$ @RichardStanley could you explain in more details what you said in "The character values are the coefficients when the class sums are expanded in terms of these idempotents" ? I am in my first representation theory course so I don't have much background in it. $\endgroup$
    – Gabriel
    Commented Feb 26, 2019 at 17:29
  • $\begingroup$ @GabrielRibeiro this should be in any introductory text on representations of finite groups, but I don't have access to this at the moment. One online reference is Proposition 3.7 of staff.fnwi.uva.nl/j.v.stokman/Aanvulling2.pdf. $\endgroup$
    – Richard Stanley
    Commented Feb 26, 2019 at 20:28
  • $\begingroup$ @RichardStanley In the last week I tried to understand what you said but without sucess. If it is not too troublesome for you, it would help me a lot to have a complete explanation. $\endgroup$
    – Gabriel
    Commented Mar 11, 2019 at 20:47

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