Abstract Nonsense

Crushing one theorem at a time

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.

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May 12, 2011 - Posted by | Analysis | , ,

1 Comment »

  1. […] This is the big theorem that we discussed in our last post that will give us, using the hook-lengths of a frame, the number of standard Young tableaux with that frame. Consequently, as was previously mentioned this will also give us the degree of the irrep for . The idea of the proof is simple, we induct on the size of the frames (how many blocks it contains) and then use the relation between the number of standard Young tableaux on a frame and the number of standard Young tableaux on the subordinate frames to use our induction hypothesis in which we will use our so-called contrived lemma. […]

    Pingback by The Hook-length Formula « Abstract Nonsense | May 14, 2011 | Reply


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