Robert's Mistakes

diagram 8: young diagram / young tableau

Sep 08, 2013

In a Young diagram each row has less placeholder squares than the previous. To get a Young tableau we start with a n-square Young diagram and place the numbers from 1 to n into the squares in a particular way: If you step from a given square tor the right or down, the number there has to be bigger than the one in the original box.

Sometimes the top left corner gets the biggest number and right/down has to decrease which, of course, yields the same structure. You can flip a Young diagram along the diagonal to obtain another, usually a different one. And by the way, the french like it upside down.

Alfred Young first used these in 1900 and i'll be trying to give you an idea what for. First we need some basics:

A group is a way to describe symmetries. If you can reverse it and have some identity or zero move you can safely utilize all of group theory. Examples are "addition and zero", "rotations or reflections and the identity", all ways to put a cube in a box, and many many more. If you get finitely many elements the group is called discrete.

The symmetric group S_n (there's one for each natual number n) consists of all possible ways to order the numbers from 1 to n. It is the most general discrete group family and each and every discrete group is present as a subgroup of S_n for some n.

A representation is a model for some structure inside another structure. Like the above paragraph says, every discrete group has a representation in some symmetric group. I hope now you can understand the following from wikipedia:

Young diagrams are in one-to-one correspondence with irreducible representations of the symmetric group over the complex numbers. [...] Many facts about a representation can be deduced from the corresponding diagram.

I'm not going to explain complex numbers here but let me say this: They behave very much like numbers but are 2-dimensional, good at rotation and just beautiful. The above tells us we can find the S_n inside the complex number plane and for each way to do that there is a Young diagram to name it. Further they say:

Young diagrams also parametrize the irreducible polynomial representations of the general linear group GLn (when they have at most n nonempty rows), or the irreducible representations of the special linear group SLn (when they have at most n − 1 nonempty rows), or the irreducible complex representations of the special unitary group SUn (again when they have at most n − 1 nonempty rows).

GL_n is also a group family and again the most general of it's kind and SL_n or SU_n are subgroups of these and are also pretty general beasts. Or to put it differently, much of GL_n's structure is also to be found in SL_n or SU_n.

John Armstrong did a course on representation theory and here's his notes where he writes about Young tableaux:

One popular extension is this: Given a Young tableau, let's allow moving the numbers such that they stay in their respective row and still follow the rules. So we're allowed to change the order horizontally but not vertically. This way you can identify a familiy of tableaux by looking at one member and see where you can go from there. That family is then called a Tabloid.