# Character formula for $S_n$ and $GL(V)$

In a set of lecture notes I'm reading, we consider representations of the symmetric group $S_n$ via treatment of Young tableaux, partitions of $n$ etc. (in what I believe is the standard approach) - likewise, representations of the general linear group $GL(V)$. In the notes, the following is stated for $S_n$:

For partition $\lambda = (\lambda_1,\ldots,\lambda_m) \in \mathbb{Z}^m$, $\alpha$ a conjugacy class in $S_n$ with cycle type $n^{\alpha_n}(n-1)^{\alpha_{n-1}}\ldots 1^{\alpha_1}$, we define $\psi_\lambda(\alpha)$ to be the coefficient of the monomial $x_1^{\lambda_1} \ldots x_m^{\lambda_m}$ in $s_1^{\alpha_1}\ldots s_n^{\alpha_n}$, where $s_i = x_1^i + x_2^i + \ldots x_m^i$. We then define $w_\lambda(\alpha) = \sum \limits_{\pi \in S_n} \varepsilon_\pi \, \psi_{(l_{\pi(1)}+1-m,\ldots,l_{\pi(m)})}(\alpha)$, where $l_i = \lambda_i+m-i$ (these are introduced for notational convenience).

We go to prove that $w_\lambda = \operatorname{char}(\mathbb{C}S_nh_{\lambda})$, where $h_\lambda$ is the Young symmetriser, and also prove the "Character formula":

$s_1^{\alpha_1}\ldots s_n^{\alpha_n}\,|\,x^{m-1},x^{m-2},\ldots,1\,|\, = \sum \limits_{\lambda \in \Lambda(n,m)} w_{\lambda}(\alpha)\, | \,x^{l_1},\ldots, x^{l_m}\, |\, \,\,(*)$

where $|A,B,\ldots,\circ|$ represents the determinant of the matrix with columns $A,\,B$ etc (for example the LHS has first column $(x_1^{m-1},x_2^{m-1},\ldots,x_m^{m-1})^\top$), and $\Lambda(n,m)$ is the set of partitions of $n$ with at most $m$ parts.

We use this relation later in combination with Schur-Weyl duality to deduce that for $\xi \in GL(V)$ with eigenvalues $x_1,\ldots,x_m$, the character $\phi_\lambda$ of a certain module $D_{(\lambda_1,\ldots,\lambda_m)}$ is given by

$\phi_\lambda(\xi) = \frac{|x^{l_1},\ldots,x^{l_m}|}{|x^{m-1},\ldots,x^1,1|}$ (apologies if you can't read that) - this is labelled "Weyl's character formula".

I won't go into detail about how we define the $D_\lambda$ - it's not complicated but I'm not sure it's necessary - but what I want to know is, is there actually any benefit to knowing $(*)$ for the character of $S_n$, rather than just knowing the definition of the $\psi_\lambda$ as coefficients of the monomial in $s_1^{\alpha_1}\ldots s_n^{\alpha_n}$? It seems like $(*)$ is almost surely more complicated to calculate anything with in the first place, so what (if any) is the use? Or do you think it is purely stated with the intention of getting a more useful result in $GL(V)$? Does $(*)$ have any mathematical merit in and of itself?

Obviously you won't know all of the content of my notes but any thoughts at all would be greatly appreciated. If I need to clarify anything (not sure how standard this all is) then please just ask.

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