Two Technical Lemmas for the Construction of the Irreps of S_n

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 .

Motivation

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.

A Weird Condition on Tableaux

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.

Motivation

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.

Row and Column Stabilizer

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.

Motivation

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.

The Hook-length Formula

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.

Motivation

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.

Hook-length in a Ferrer’s Diagram

Point of Post: In this post we discuss the notion of hook-length in a Ferrer’s diagram and give a few characterizations of the product of the hook-length over every square in a Ferrer’s diagram in preparation for the hook length formula.

Motivation

In our last post we let slip the deal with looking at the combinatorial objects we have been looking at. In particular, we noted that we will associate to each -frame an irrep of . What we mentioned though about this association was that . Accordingly, it would be great if there was some formula that could compute . In fact, believe it or not there is such a formula. That said, it involves a somewhat strange idea–the hook-length of a square in a Ferrer’s diagram. Intuitively, the hook-length is just just the number of squares to the right of a square, below the square, and for the square itself. The reason the hook-length gets its name is that because if one imagines the hook-length it makes a ‘hook’ (see below) at the square in the sense that it looks like a line starting from the bottom of the column the square in question sits in, extends up to that square, and then makes a right turn and continues to the end of the row. So, after we define the hook-length we find certain characterizations of the product of the hook-length over all squares in a given Ferrer’s diagram since this is what shows up in the formula for .

The Fundamental Result for Tableaux Combinatorics

Point of Post: In this post we prove that sum of where is taken over all -frames is

Motivation

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 .

Relation Between the Number of Standard Young Tableaux on a Frame and the Number of Young Tableaux on the Frame’s Subordinate/Superordinate Frames

Point of Post: In this post we find a relation between the number of standard Young tableaux on a frame and the number of Young tableaux all the subordinate and superordinate frames to .

Motivation

As was stated in our last post we can find a very interesting way to calculate the number, , of standard Young tableaux  with . In this post we actually prove this claim. The intuitive idea is clear, by construction of Young tableaux we see that if is a Young Tableaux such that is a -frame then the number must lie in a bottom right corner of and then fixing in that position  we see that the possible Young tableaux are just the Young tableaux of and thus it makes sense then that is some sort of sum of where is taken over the subordinate frames to . The other theorem which has to do with finding given the values where is taken over the frames superordinate to .

Subordinate and Superordinate Frames

Point of Post: In this post we define the notion of a subordinate frame and superordinate frame and discuss equivalent ways of defining them.

Motivation

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

Partitions, Ferrer’s Diagrams, and Young Tableaux

Point of Post: In this post we discuss the notion of  partitions, Ferrer’s diagrams, and Young Tableaux (and their standard type).

Motivation

There is beautiful interplay between algebra and combinatorics which can be seen in the representation theory of the symmetric group. At the center of this correspondence is the notion of a Ferrer’s diagram and a Young tableau which, as we shall see, will serve to ‘index’ in a very fruitful way the representations of . This really is one of the most beautiful parts of basic finite group representation theory.