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Community Bot
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Since you only care about the completely reducible case, I'll assume $K=\mathbb{C}$.

The easiest way that I know of to construct irreducible $S_n$ representations is a special case of the construction herehere. In particular, symmetric group is a quotient of the (degenerate) affine Hecke algebra appearing there (obtained by setting $x_1=0$). If you take $\mu=0$ in the link, then the skew shape $\lambda/\mu$ is just $\lambda$ and the module that is constructed is the irreducible representation $S_\lambda$.

This recovers the character formula $s_\lambda=\sum_T x^T$, where the sum is over all standard tableaux of shape $\lambda$ (as described in Macdonald's text, for example).

Since you only care about the completely reducible case, I'll assume $K=\mathbb{C}$.

The easiest way that I know of to construct irreducible $S_n$ representations is a special case of the construction here. In particular, symmetric group is a quotient of the (degenerate) affine Hecke algebra appearing there (obtained by setting $x_1=0$). If you take $\mu=0$ in the link, then the skew shape $\lambda/\mu$ is just $\lambda$ and the module that is constructed is the irreducible representation $S_\lambda$.

This recovers the character formula $s_\lambda=\sum_T x^T$, where the sum is over all standard tableaux of shape $\lambda$ (as described in Macdonald's text, for example).

Since you only care about the completely reducible case, I'll assume $K=\mathbb{C}$.

The easiest way that I know of to construct irreducible $S_n$ representations is a special case of the construction here. In particular, symmetric group is a quotient of the (degenerate) affine Hecke algebra appearing there (obtained by setting $x_1=0$). If you take $\mu=0$ in the link, then the skew shape $\lambda/\mu$ is just $\lambda$ and the module that is constructed is the irreducible representation $S_\lambda$.

This recovers the character formula $s_\lambda=\sum_T x^T$, where the sum is over all standard tableaux of shape $\lambda$ (as described in Macdonald's text, for example).

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David Hill
David Hill
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Since you only care about the completely reducible case, I'll assume $K=\mathbb{C}$.

The easiest way that I know of to construct irreducible $S_n$ representations is a special case of the construction here. In particular, symmetric group is a quotient of the (degenerate) affine Hecke algebra appearing there (obtained by setting $x_1=0$). If you take $\mu=0$ in the link, then the skew shape $\lambda/\mu$ is just $\lambda$ and the module that is constructed is the irreducible representation $S_\lambda$.

This recovers the character formula $s_\lambda=\sum_T x^T$, where the sum is over all standard tableaux of shape $\lambda$ (as described in Macdonald's text, for example).