M Theory Lesson 214
The Young operator for a tableau of $n$ boxes is an element of the group algebra for the permutation group $S_{n}$. For $S_{3}$ it is described as follows. Let $R$ be the sum of all permutations which fix the rows. Similarly, $C$ is the sum of all permutations that fix the columns, with a sign for the parity of the permutation. 
In each case, $Y$ is the product $RC$. It will be a multiple of a primitive idempotent for the group algebra of $S_{3}$. The normalisation factors are $\sqrt{2}$ and $\sqrt{3}$, up to $i$.
In each case, $Y$ is the product $RC$. It will be a multiple of a primitive idempotent for the group algebra of $S_{3}$. The normalisation factors are $\sqrt{2}$ and $\sqrt{3}$, up to $i$.
posted by Kea | 7:47 PM
7 Comments:
This seems very useful but I don't understand it. Why don't you write a blog post on it that is about 10x as long?
For example, what the heck does "sum of all permutation" mean? How do you add two permutations? Or are you talking about the group function as addition instead of multiplication?
To make this understandable, you need to give explicit examples, explicit calculations, and, in general, it needs to be about 10x as long as it is.
Sorry, Carl. The group algebra is just 'linear' combinations, say over Z or C, of the elements of the group.
Hi Kea,
I love Young's diagrams. I used to understand them inside out, and then, twenty years afterwards... It is fading, fading away. Maybe I do need some more text in this post! Can I ask you to explain the basic rules for combining them, over again ? For instance, decomposing 27 in 1+8+8+10.
Cheers,
T.
Actually, let me ask you if you would be willing to write a guest post on Young's diagrams, explaining them to somebody who has not even ever heard of group theory, and giving the example of the SU(3) of flavor. I can help out with that part if you want... We could do it two-handedly and publish it on both sites. But my first offer is for you to write a guest post on "Fun with Young's Diagrams" on my blog....
xxx
T.
Yes, Tommaso, I'd love to write a guest post for you. But you might have to wait a couple of days until I find some time, since I am busy waitressing at present.
That's great. I'll diligently wait. Your customers paid for what they get, I am not... They must come first in this darn dollar-driven society!
Cheers,
T.
I join to the petition of Dorigo; I would like to read a pretty explanation of SU(3) flavor 1+8+8+10, and the differences (parity, other symmetries) between the terms of the sum.
Slightly on this topic, I am thinking to rename my generation-crossing "SU(5) flavor" to "SU(5) odor". It does not taste exactly as a justification for three generations, but it smells as if there is one such explanation cooking there. Thus I ask, has anybody already done this joke of "SU(n) odor group"? documented where?
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