Guest post: Marni D. Sheppeard – Fun With Young Diagrams August 12, 2008

Posted by dorigo in physics, science.
Tags: , ,

Marni (shown in a recent photo on the right) is a friend, a blogger, and a researcher in category theory. She does not need to be introduced to most readers of this site, since she has been contributing (with her nickname Kea) to many discussions with meaningful remarks, as well as with a good measure of scepticism about the Higgs boson (which she calls a “fairy field”…), and a guest post on category theory. Upon reading one of the latest lessons on M-theory in her blog, I was reminded I had loved Young’s diagrams in my student days, after having been forced to study them for my undergrad course of Theoretical Physics. I also realized I had since forgotten most of their niceties. Her post was synthetic, so I asked her to produce a better introductory explanation for this site. Please find it below!

FUN WITH YOUNG’S DIAGRAMS

No mathematical knowledge is assumed for this post.

As the mathematician Gian-Carlo Rota once said, combinatorics is an honest subject. He meant that, unlike in many other branches of mathematics, in combinatorics there is no possibility of fudging a definition to make things half work, because either you have figured out how to count the marbles in the bag, or you haven’t. Today we are going to look at ways to count using stacks of boxes. These ideas, due to Alfred Young, go back to 1900 and were applied a few years later by Frobenius to count the number of non reducible ways that one could linearly realise (ie. represent using simple algebra) the permutations of N objects. Let us illustrate the general idea using three objects, which we will call 1, 2 and 3. First, draw all possible ways to neatly stack three boxes against a wall (see figure below).

How many ways are there to fill the tall stack with the numbers 1, 2 and 3? Once we choose a number for the bottom box, there are only two numbers left to choose from for the next box and then one remaining for the top box. In other words there are 6 = 1 x 2 x 3 ways to fill the stack. In general there will be N! = 1 x 2 x 3 x … x N ways to fill a vertical stack of N boxes. Note that what we really did was put all possible orderings, ie. permutations, of the numbers 1, 2 and 3 into the boxes. Similarly, the other stacks can also be labelled with the set of 6 permutations of three objects. How would we count the number of stacks themselves, for any N?

In order to do some impressive mathematics, it is helpful to consider the set of permutations not as a list of orderings of 1, 2 and 3, but rather as a set of 6 operations that can be done on the numbers 1, 2 and 3 to reorder them. Operations are more powerful than elements of lists, because they can be combined to form new operations. But in this case, the operations can be labelled by the orderings themselves. Imagine to start with that the boxes were numbered (123). By definition, the operation (312) puts the label 3 where the label 1 was, the label 1 where there was a 2, and the label 2 where there was a 3. Observe that the combination (312)(231) is the same as the operation (123), since after doing (312) we put the second label into the first box, but this is now a 1. In other words, it is always possible to get back to the original order (123) by rearranging the labels with a second operation.
The beauty of Young’s diagrams is that a given numbered stack gets turned into some concrete algebra using the operations just described. Each diagram gives us two subsets of the permutations: the operations that leave the rows unaltered, and the operations that leave the columns unaltered. Let us now choose the central stack, with the labels 1 and 3 on the bottom and the label 2 above the box 1. For the chosen
diagram, the first set of operations is {(123),(321)} and the second is {(123),(213)}. Note that the remaining operations (312) and (231) are unused here, because they shift numbers between rows or columns. The so called Young symmetrizer for a given diagram is an operation in a bigger algebra, which mixes ordinary numbers and all possible combinations of permutation operations. In our example, the symmetrizer is written as a product

[(123) + (321)][(123) – (213)]

We can write down such an expression for any numbered stack. The rule to remember is that we should put a minus sign in front of operations inside the right bracket whenever the operation involves an odd number of basic two object swaps. A tricky but rewarding homework problem is to verify that if we multiply a symmetrizer by itself, we get the same symmetrizer back again, at least up to a numerical factor. But the truly amazing thing is, it turns out that the different symmetrizers, which are all distinct, cleverly label the ways of linearly representing the permutations. This works for any number N, not just 3.

The property of a symmetrizer that makes it useful for physics is the way that it multiplies itself to get itself back again. A more familiar example of an operation with this property is the operation of creating a shadow on the ground, since the shadow of the shadow is really just the shadow again. In quantum physics, the operation of measurement behaves this way.

(Series to be continued later at Arcadian Functor).

1. Kea - August 12, 2008

Thanks for making it so neat and friendly! I’m surprised Lubos has nothing to say.

2. Alejandro Rivero - August 12, 2008

You forget to mention (perhaps Kea will?) that a typical use of Young diagrams is to work out SU(n) representations; as such it is included in the pdg, here: http://pdg.lbl.gov/2008/reviews/youngrpp.pdf
Actually, the recipe for the product of representations is a bit obscure; I was forced to look at it last week –to evaluate the products of the SU(5) fundamental and antifundamental, 5×5, (-5)x(-5) and 5x(-5) as a basis for 3 generations of particles– and I felt not any hint of happiness, fun or enjoyment.

3. Stefan - August 12, 2008

Oh,

Young diagrams are really fun – some very nice recreational math!
But I guess that’s quite geeky 😉

Cheers, Stefan

4. carlbrannen - August 12, 2008

Tommaso,

Whatever you did to her to get her to write this post, please do it again. This was only a few cm over my head on first read and I think I could figure it out if I read it over and over a few dozen times.

I constantly end up rediscovering things previously Marni wrote, except I have to rediscover them the hard way.

5. dorigo - August 14, 2008

Hi Alejandro,

the point was indeed to continue this with a discussion of SU(N), but it will have to wait… I am presently overburdened with a project.

Cheers,
T.

6. dorigo - August 14, 2008

Hi Stephan,

well, what can I say, I am one. In some twisted, broad sense 😉

Carl, take a book. Get on top of it with your feet. Then you should be able to be at the correct height to understand it 😉

Cheers,
T.

7. Luboš Motl - August 18, 2008

Dear Kea #1,
I was in Tunisia for 1 week.

But not sure what to say. Unlike Stefan #3, I don’t think that Young diagrams are “recreational maths”. They’re a key tool to deal with representations of classical groups which became key concepts in physics a century ago.

Even very complicated Young diagrams are important in physics and e.g. in the AdS/CFT correspondent, they represent large configurations of many spherical branes of various types.

Almost every physicist who knows what he’s doing has written a paper using composition rules for complicated Young diagrams, even super-Young diagrams, see e.g. last 2 pages of

Very colorful. 😉

Best
Lubos

8. Kea - August 19, 2008

Why thank you for the interesting and colourful reference. Perhaps we will get around to blogging about such Young diagrams.

9. Creptio - September 23, 2008