Abstract Nonsense

A Contrived Lemma

Point of Post: In this post we prove a lemma which, although entirely complex analysis, really has no purpose except to help prove the hook-length formula.

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Motivation

As the lemma says, this post is about a complex analysis lemma which really is just a simple interest. The proof is a famous one and uses residue theory, although it doesn’t seem like it should…

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The Lemma

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We jump right into it

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Theorem: Let $z_1,\cdots,z_m\in\mathbb{C}$, then

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$\displaystyle \sum_{j=1}^{m}z_j-\frac{m(m-1)}{2}=\sum_{j=1}^{m}z_j\prod_{i\ne j}\left(1+\frac{1}{z_i-z_j}\right)$

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Proof: The idea is simple, define $f:\mathbb{C}\to\mathbb{C}$ by

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$\displaystyle f(z)=z\prod_{i=1}^{m}\left(1+\frac{1}{z_i-z}\right)$

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Evidently we have that $f$ is holomorphic on $\mathbb{C}-\{z_1,\cdots,z_m\}$ and has a simple pole at each of the excluded points. Thus, letting $\displaystyle R=\max_{j\in[m]}|z_j|+1$ we have from Cauchy’s residue theorem that

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$\displaystyle \oint_{|z|=R}f(z)\text{ }dz=2\pi i\sum_{j=1}^{r}\text{Res}\left(f,z_j\right)$

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that said, one can easily compute (using the fact that the poles are simple) that

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$\displaystyle \text{Res}\left(f,z_j\right)=-z_j\prod_{i\ne j}\left(1+\frac{1}{z_i-z_j}\right)$

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that said, we know from basic complex analysis that

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$\displaystyle \oint_{|z|=R}f(z)\text{ }dz=-2\pi i\text{Res}\left(f,\infty\right)=-2\pi i\text{Res}\left( -f\left(\frac{1}{z}\right)\frac{1}{z^2},0\right)$

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That said, we see that

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$\displaystyle -f\left(\frac{1}{z}\right)\frac{1}{z^2}=\frac{-1}{z^3}\prod_{i=1}^{m}\left(1-z-z_i z^2+\text{higher order terms}\right)$

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and so a quick check shows that the coefficient of $z^{-1}$ of this is

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$\displaystyle \sum_{j=1}^{m}z_j-\frac{m(m-1)}{2}$

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comparing our two results for the contour integral proves the theorem. $\blacksquare$

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References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Ahlfors, Lars Valerian. Complex Analysis: an Introduction to the Theory on Analytic Functions of One Complex Variable. New York [u.a.: McGraw-Hill, 2007. Print.