Quantum Bayesian Networks

August 6, 2014

Young Diagram Crib Sheet

Filed under: Uncategorized — rrtucci @ 6:23 am

In my next paper and patent, I apply group representation theory to quantum computing. (At least I’m quite convinced that I do…I’ve been wrong many times before). An important tool in group representation theory is Young diagrams. Here is a little crib sheet on some of the things that I know about Young diagrams. This crib sheet might be of use as a refreshener to those who already know about Young diagrams. If you don’t know the difference between a Young diagram and an old diagram, you better skip this blog post.

Mind you, I am a physicist, so my ramblings about mathematics should be taken with a big grain of salt.

Want to draw a Young diagram or tableau in your research paper? No problem. The LaTex gods have foreseen this eventuality and provided us with several packages for drawing Young diagrams and tableaux with the utmost of ease. I used the packages “youngtab” and “xy” to draw the figures below. In fact, here is the latex source code that I used.

Let S_n= Symmetric group on n letters.|S_n|=n!

Let \lambda = partition of numbers 1,2,…,n = a distinct Young diagram = a distinct conjugacy class

Let N_{stab}(\lambda)= number of standard tableau of Young diagram \lambda.

Let \tau^\lambda_j be the j th standard Young tableau of \lambda,
where j=1,2,\ldots, N_{stab}(\lambda).

Let dim(\tau^\lambda_j) be the dimension of the S_n irrep labelled by \tau^\lambda_j. Then

\forall j, dim(\tau^\lambda_j)=N_{stab}(\lambda).

n! =|S_n|=\sum_\lambda \sum_{j=1}^{N_{stab}(\lambda)}  \underbrace{dim(\tau^\lambda_j)}_{  N_{stab}(\lambda)}=  \sum_\lambda N^2_{stab}(\lambda)

For example, for S_4,

4!=24 =1 + 3^2 + 2^2 + 3^2 + 1

Each \clubsuit in the figure below represent one basis element for a total of 24.

Each distinct Young diagram corresponds to a distinct, inequivalent irrep of S_n. Each Young tableaux with the same Young diagram corresponds to a copy of the same irrep of S_n.

young-s4

A Young Tree (a sapling) of standard tabelaux. I think that I shall never see A poem lovely as a tree.

A Young Tree (a sapling) of standard tableaux.

I think that I shall never see
A poem lovely as a tree.

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