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I am reading the construction of the Specht module from James's book. The Specht module of a symmetric group corresponding to a partition $\lambda$ is spanned by all polytabloids $e_{t}$ associated with a tableau $t$ of shape $\lambda$.

There is an another description for the Specht module of symmetric group in Chapter 3, Section 2 of [A. Mathas, Iwahori–Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 15, Amer. Math. Soc., Providence, RI, 1999; MR1711316].

I would like to understand how the bases are equivalent.

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    $\begingroup$ Disclaimer : I have never fully understand the approach of Jucys-Murphy elements. But my understanding is the following : in James' book, we construct Specht modules having a basis in standard tableaux, and we derive the branching rule from them ; whereas in the other approach, we construct a basis using the branching rule, and show that it corresponds to standard tableaux. The following notes make links between these two approachs : pi.math.cornell.edu/~dmehrle/notes/partiii/…. $\endgroup$
    – eti902
    Commented Oct 26 at 14:45
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    $\begingroup$ Could you give a summary of the construction from Mathas's book? $\endgroup$
    – LSpice
    Commented Oct 27 at 19:32
  • $\begingroup$ By Proposition 3.22 of Mathas's book, the Specht module $S^{\lambda}$ is generated as a free module with the basis $\{m_{t}: t \text{ is a standard tableau of shape } \lambda\}$. This is coming from the cell basis of a symmetric group. So, my question is how $m_t$ is related with $e_t$. $\endgroup$
    – noone
    Commented Oct 28 at 4:34
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    $\begingroup$ If it is coming from the cell basis, then he gives you a rule to write $\sigma \cdot m_t$ ($\sigma \in \mathfrak{S}_n$) in the given basis. It which should be equivalent to the rule used to write $\sigma \cdot e_t = e_{\sigma \cdot t}$ in the standard basis. The latter is given by Garnir elements, so you should see if Mathas' construction is equivalent to them. $\endgroup$
    – eti902
    Commented Oct 28 at 15:32

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