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Is there a formula somewhere in the literature for the action of the generators $T_1,\ldots,T_{n-1}$ of the Iwahori-Hecke algebra on the standard basis of its Specht modules? It is well-known that the coefficients of these matrices are elements of $\mathbb{Z}[q,q^{-1}]$ but I am unsure if there is a closed formula (there certainly is for the seminormal form, which I thought was derived from this integral form).

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There is no closed form formula. There is a straightening algorithm which you will find in text books under "Garnir relations".

The integral form can also be constructed from the seminormal form. The change of basis matrix is constructed using the Yang-Baxter equation (with spectral parameter). This was the subject of the MO question 66602.

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