Here the definition of complete symmetric polynomial $h_{k}$ and elementary symmetric polynomial $e_{k}$ are:

$$ e_{k}=\sum_{1\le i_1<\cdots <i_k\le n}x_{i_1}\cdots x_{i_k}, h_{k}=\sum_{1\le i_1\le \cdots \le i_k\le n}x_{i_1}\cdots x_{i_k} $$

I know that they are "dual" to each other in the symmetric function ring $\Lambda_{k}$, for the map $e_{k}\rightarrow h_{k}$ is an involution of the ring. But this does not explain some other beautiful dual relationship to me.

For example in Pieri's formula we have $$ s_{\lambda}e_{k}=\sum_{\mu\in \lambda\otimes 1^{k}}s_{\mu}, s_{\lambda}h_{k}=\sum_{\mu\in \lambda\otimes k}s_{\mu} $$ And in the reverse side using Kostka numbers we have $$ h_{\mu}=\sum K_{\lambda \mu}s_{\lambda}, e_{\mu}=\sum_{\lambda}K_{\lambda \mu}s_{\lambda^{*}} $$ The Jacobi-Trudi formula claim that for $|\lambda|\le n$, we have $$ s_{\lambda}=\det(h_{\lambda_i-i+j})_{1\le i,j\le n},s_{\lambda^{*}}=\det(e_{\lambda_i-i+j})_{1\le i,j\le n} $$

To me, all these suggests that there some deeper relation underlying these dualizing relationships. The formulas are so stunningly beautiful that they cannot come from mere coincidence in computation. In particular if we consider their action on Schur polynomials using the Tableaux, we can visualize the dual relationship. I want to ask, is there any deep reason behind these dual relationships? The definition itself seems to reveal very little and I felt very puzzled by the unexpected beauty.