# Hook-length formula [closed]

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 $$\lambda$$ $$\chi_{2^{d/2}}^{\lambda}?$$

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

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• what does $\chi_a^b$ mean ? – mercio Sep 21 '18 at 0:43
• Character of symmetric group. – GGT Sep 21 '18 at 2:00
• but characters are not integers. – mercio Sep 21 '18 at 5:33
• So what it has to do with my question? – GGT Sep 21 '18 at 6:07
• 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 $\dim V_\lambda$. – hardmath Sep 21 '18 at 16:22