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Let $\lambda=(\lambda_1,\ldots,\lambda_n)$ be a partition of $d$. Then hook length formula gives us

$$dim(\lambda)=\chi_{1^d}^{\lambda}=\frac{d!}{\prod h_{\lambda}(i,j)}$$ where $\chi_{a}^{b}$ denote the character of symmetric group.

Say $d$ is even,does there exists hook-length formula for all $\lambda$ $$\chi_{2^{d/2}}^{\lambda}? $$

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closed as unclear what you're asking by mercio, Christopher, Paul Frost, hardmath, Lord Shark the Unknown Sep 21 '18 at 16:35

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    $\begingroup$ what does $\chi_a^b$ mean ? $\endgroup$ – mercio Sep 21 '18 at 0:43
  • $\begingroup$ Character of symmetric group. $\endgroup$ – GGT Sep 21 '18 at 2:00
  • $\begingroup$ but characters are not integers. $\endgroup$ – mercio Sep 21 '18 at 5:33
  • $\begingroup$ So what it has to do with my question? $\endgroup$ – GGT Sep 21 '18 at 6:07
  • $\begingroup$ There is some machinery in the hook-length formula that you are skipping over. One associates to integer partition $\lambda$ of $d$ a complex irreducible representation $V_\lambda$, and $\dim \lambda$ is then understood as an abbreviated form of $\dim V_\lambda$. $\endgroup$ – hardmath Sep 21 '18 at 16:22