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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 ?

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    $\begingroup$ The first construction you list isn't in symmetric polynomials, and the second construction you list isn't a construction of an irreducible representation. $\endgroup$
    – Ben Webster
    Aug 9, 2015 at 12:06
  • $\begingroup$ Yes, the polynmials $E_{\{T\}}$ are not symmetric (I never claimed them to be), but, for example, if we take their sum (varying $\pi$ over the Young subgroup then we get a symmetric polynomial associated to $\lambda$; is their any relation between them and $h_\lambda$'s, do they form a basis like $h_\lambda$ ? $\endgroup$ Aug 9, 2015 at 12:17
  • $\begingroup$ The second construction can be found in Fulton's 'Young Tableau' (section 7.3, page 91). $\endgroup$ Aug 9, 2015 at 12:21
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    $\begingroup$ I know that construction very well, but it's not a construction of an irreducible representation. It's a way of understanding their Grothendieck group. $\endgroup$
    – Ben Webster
    Aug 9, 2015 at 12:30
  • $\begingroup$ Yes, you are right; may be I should rephrase the first line of the question. My main queries are the last two questions. Are they not interesting, or may be does not have a nice answer? $\endgroup$ Aug 9, 2015 at 12:47

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