Point of Post: In this post we prove two technical lemmas in relation to the row and column stabilizer functions which will ultimately help us construct the irreps of .
We are at the penultimate post before carrying through with our long-ago promised goal of constructing the irreps of in a way for which they are naturally labeled by -frames. In this post we just need to prove two technical lemmas before this.
Point of Post: In this post we discuss an interesting property between two tableaux which will ultimately help us construct the irreps of associated to each -frame.
So, enough being cryptic. I promised that we will create a bijection in such a way that –it’s about time I explained roughly how. So, in our last post we created this interesting function . Our main goal to the construction is to show that up to normalization is a minimal projection from where we shall get our corresponded irrep. In the journey to prove this we will need a strange, un-motivated concept which has to do with the relationship between the rows of one tableau and another tableau .Luckily, the motivation and usefulness will become apparent shortly. That said, we can at least give a glance of why anyone would even care about this condition. In particular, we shall use this condition to prove that the irreps associated to two different -frames are different.
Point of Post: In this post we define the notion of the column and row stabilizers for a Young tableaux and some standard results. Of course we’ll have to talk about the appropriate action first.
We now start to move away from pure combinatorics we’ve been engaging in and start to prepare for the representation theory that lies ahead. But, before we get into the pure rep theory we need to start with a mix of algebra and combinatorics to start. Roughly, in this post we define for each tableau two certain subsets of that ‘stabilizes’ it in a particular interesting way. We then consider certain sums and products in the group algebra associated to these two certain subsets.
Point of Post: In this post we derive the hook-length which well tell us, given a frame, the number of standard Young tableaux that have that frame.
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.
Point of Post: In this post we prove that sum of where is taken over all -frames is
The ultimate goal of this brief journey into combinatorics land is that we will eventually show that there is a map . But, the fact that there exists a correspondence is obvious since we know that . What is interesting is that we are able to correspond an element an element of in a meaningful way. What precisely I mean by ‘interesting’ I will wait to say, but probably the most useful part of it is that if is the irrep corresponding to then –the number of standard Young tableaux on . In this post we prove a result which is not only integral in proving this fact but is consistent with this hypothesis, namely that the sum over all -frames with is .
Point of Post: In this post we define the notion of a subordinate frame and superordinate frame and discuss equivalent ways of defining them.
It’s clear that in our definition of -frames that sitting inside each -frame is a lot of -frames which can be gotten simply by removing a single box from . These -frames ‘sitting’ inside shall be what we call the -frames ‘subordinate’ to . Of course, there is a dual notion where given an -frame we see that sits subordinately inside a lot of -frames , we shall say in this case that is ‘superordinate’ to . Said slightly differently the -frames superordinate to are the -frames which can be obtained from by adding a single box to . The interesting thing is that given (the number of standard Young tableaux) for each subordinate to we can calculate and dually given for all -frames superordinate to we can calculate . That will be the topic of our next post
Point of Post: In this post we discuss the conjugacy class structure of the symmetric group .
Often times knowing the conjugacy classes of a group gives you much information about the group. In particular, we have seen that the number of irreducible characters of a group is equal to the number of conjugacy classes of . That said, it is often difficult without getting one’s hands dirty to find all the conjugacy classes of a group, and moreover finding the number of elements in each such class. It is an interesting fact that for one of the most important groups (for example, the ‘comfort theorem’ given by Cayley) has easily findable conjugacy classes, and even explicitly computable sized classes–I am of course talking about the symmetric group. So, in this post we shall classify the conjugacy classes of and find their order.