I will describe two ways to associate irreducible representations of $S_n$ with polynomials inside the ring of symmetric polynomials and I want to know if there is any connection between the two.
$\textit{via Specht polynomials}:$ For a partition $\lambda\vdash n$, let $\{T\}$ be a $\lambda $-tabloid. Define the following polynomial
$$E_{\{T\}}:=\prod\limits_{i=1}^{d}x_i^{\left(\text{number of the row in which }i\text{ lies in the tableau }T\right)-1}.$$
Then $\{E_{\pi\{T\}} : \pi\in S_n/S_{\lambda_1}\times\cdots\times S_{\lambda_k}\} $ is a $\mathbb{C}$-basis for $U_\lambda:=\text{Ind}^{S_n}_{S_{\lambda_1}\times\cdots\times S_{\lambda_k}} (1)$. And the following polynomials $$F_T=\prod\limits_{i < j;\ i\text{ and }j\text{ lie in the same column of }T}\left(x_i-x_j\right)$$
spans the irreducible representation $V_\lambda$ of $S_n$. Now, we can, for example, take the sum of $E_{\pi\{T\}}$'s and get the symmetric polynomial associated to $U_\lambda$ and similarly get symmetric polynomial associated to $V_\lambda$.
$\textit{via Schur polynomials}:$ For a partition $\lambda\vdash n$, let $h_\lambda$ be the complete symmetric polynomial and $s_\lambda$ be the Schur polynomial. Define the map $\phi$ from the set of all symmetric polynomials to the Grothendieck group of representations of $S_n$ by $\phi (h_\lambda )=U_\lambda $. Then $\phi$ is an isomorphism and $\phi (s_\lambda )=V_\lambda$.
$\textbf{Questions}$:
Is there any relationship between $h_\lambda$ and $E_{\pi\{T\}}$'s and between $s_\lambda$ and $F_T$'s ?
Can we construct a basis of the ring of symmetric polynomials using $E_{\pi\{T\}}$'s ?