# Constructing a centrally primitive idempotent in the group algebra of the symmetric group

Consider the group algebra of the symmetric group $$\mathbb{C} S_k$$. Given some Young tableau $$T$$ of shape $$\lambda$$, let $$a_{\lambda,T}$$ and $$b_{\lambda,T}$$ be the row symmetrizer and column antisymmetrizer of the tableau respectively.

It is known that the Young symmetrizer $$c_{\lambda,T} = a_{\lambda,T} b_{\lambda,T}$$ is proportional to an idempotent. That is, $$c_{\lambda,T}^2 = m_\lambda c_{\lambda,T}$$ with $$m_\lambda \in \mathbb{R}$$.

Using character theory, one can show that the element $$\omega_\lambda = \sum_{\pi \in S_k} \pi c_{\lambda,T} \pi^{-1}$$ is proportional to a centrally primitive idempotent (see e.g. Proposition 2 in the notes by Graham Gill, representation theory of the symmetric group: basic elements). It therefore projects onto the isotypic component associated to $$\lambda$$.

Is there a more straightforward way (i.e. one that doesn't use character theory) to show that $$\omega_\lambda$$ is proportional to an idempotent, that is to show that $$\omega_\lambda^2 = n_\lambda \omega_\lambda$$ with $$n_\lambda \in \mathbb{R}$$?

Alternatively, is there a (published) reference for the construction of $$\omega_\lambda$$ by such averaging operation?

edit: I feel one should be able to take advantage of the averaging operation / Reynolds operator $$\alpha \mapsto \sum_{g \in G} g \alpha g^{-1}$$. I don't quite know how to however.

• I think (to help out representation theorists like me who follow Harish-Chandra's dictum that "characters tell all") it might be helpful to specify what tools are allowable as more straightforward than character theory. Commented Sep 9, 2019 at 18:57
• I was hoping for some trick involving Young tableaux and/or the Reynolds operator, as the derivation in Graham Gill's notes relies on Theorem 4 and Lemma 2 which aren't that straightforward either (to me, I'm a physicist.) Maybe I should adapt my question a bit to alternative ask for a (published) reference for that construction of $\omega_\lambda$. Commented Sep 10, 2019 at 10:30
• Answered on math.stackexchange. (math.stackexchange.com/questions/3345754/…) Commented Sep 10, 2019 at 12:26