Abstract Nonsense

Crushing one theorem at a time

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 S_n.

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Motivation

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We are at the penultimate post before carrying through with our long-ago promised goal of constructing the irreps of S_n in a way for which they are naturally labeled by n-frames. In this post we just need to prove two technical lemmas before this.

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May 23, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , | Leave a comment

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 S_n associated to each n-frame.

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Motivation

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So, enough being cryptic. I promised that we will create a bijection \text{Frame}_n\to\widehat{S_n} in such a way that \deg\rho^{(\mathcal{F})}=f_{\text{st}}\left(\mathcal{F}\right)–it’s about time I explained roughly how. So, in our last post we created this interesting function E:\text{Tab}\left(\mathcal{F}\right)\to\mathbb{C}\left[S_n\right]. Our main goal to the construction is to show that up to normalization E\left(\mathcal{T}\right) 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 \mathcal{T} and another tableau \mathcal{T}'.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 n-frames are different.

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May 22, 2011 Posted by | Algebra, Algebraic Combinatorics | , , , , , , , , | 1 Comment

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.

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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 \mathcal{T} two certain subsets of S_n 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.

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May 20, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , | 2 Comments

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.

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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 \rho^{(\mathcal{F})} for S_n. 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.

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May 14, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , , , , | Leave a comment

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.

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Motivation

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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 n-frame \mathcal{F} an irrep \rho^{(\mathcal{F})} of S_n. What we mentioned though about this association was that \deg\rho^{(\mathcal{F})}=f_{\text{st}}\left(\mathcal{F}\right). Accordingly, it would be great if there was some formula that could compute f_{\text{st}}\left(\mathcal{F}\right). 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 1 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 \text{f}_{\text{st}}\left(\mathcal{F}\right).

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May 12, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , , , | 1 Comment

The Fundamental Result for Tableaux Combinatorics


Point of Post: In this post we prove that sum of f\left(\mathcal{F}\right)^2 where \mathcal{F} is taken over all n-frames is n!

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Motivation

The ultimate goal of this brief journey into combinatorics land is that we will eventually show that there is a map \left\{n\text{-frames}\right\}\to\widehat{S_n}. But, the fact that there exists a correspondence is obvious since we know that \#\left(\widehat{S_n}\right)=p(n)=\left\{n\text{-frames}\right\}. What is interesting is that we are able to correspond an element \mathcal{F}\in\left\{n\text{-frames}\right\} an element of \widehat{S_n} 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 \rho^{(\mathcal{F})} is the irrep corresponding to \mathcal{F}\in\left\{n\text{-frames}\right\} then \deg\rho^{(\mathcal{F})}=f_{\text{st}}\left(\mathcal{F}\right)–the number of standard Young tableaux on \mathcal{F}. 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 n-frames \mathcal{F} with f_{\text{st}}\left(\mathcal{F}\right)^2 is n!.

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May 12, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , , , , , | 3 Comments

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 \mathcal{F} and the number of Young tableaux all the subordinate and superordinate frames to \mathcal{F}.

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Motivation

As was stated in our last post we can find a very interesting way to calculate the number, f_{\text{st}}\left(\mathcal{F}\right), of standard Young tableaux \mathcal{T} with \text{Frame}\left(\mathcal{T}\right)=\mathcal{F}. In this post we actually prove this claim. The intuitive idea is clear, by construction of Young tableaux we see that if \mathcal{T} is a Young Tableaux such that \text{Frame}\left(\mathcal{T}\right) is a n-frame then the number n must lie in a bottom right corner of \text{Frame}\left(\mathcal{T}\right) and then fixing n in that position b  we see that the possible Young tableaux are just the Young tableaux of \mathcal{F}-b and thus it makes sense then that f_{\text{st}}\left(\mathcal{F}\right) is some sort of sum of \text{f}_{\text{st}}\left(\mathcal{G}\right) where \mathcal{G} is taken over the subordinate frames to \mathcal{F}. The other theorem which has to do with finding f_{\text{st}}\left(\mathcal{F}\right) given the values f_{\text{st}}\left(\mathcal{H}\right) where \mathcal{H} is taken over the frames superordinate to \mathcal{F}.

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May 11, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , , , , | 3 Comments

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.

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Motivation

It’s clear that in our definition of n-frames that sitting inside each n-frame \mathcal{F} is a lot of n-1-frames which can be gotten simply by removing a single box from \mathcal{F}. These n-1-frames ‘sitting’ inside \mathcal{F} shall be what we call the n-1-frames ‘subordinate’ to \mathcal{F}. Of course, there is a dual notion where given an n-frame \mathcal{F} we see that \mathcal{F} sits subordinately inside a lot of n+1-frames \mathcal{G}, we shall say in this case that \mathcal{G} is ‘superordinate’ to \mathcal{F}. Said slightly differently the n+1-frames superordinate to \mathcal{F} are the n+1-frames which can be obtained from \mathcal{F} by adding a single box to \mathcal{F}. The interesting thing is that given f_{\text{st}}\left(\mathcal{G}\right) (the number of standard Young tableaux) for each \mathcal{G} subordinate to \mathcal{F} we can calculate f_{\text{st}}\left(\mathcal{F}\right) and dually given f_{\text{st}}\left(\mathcal{G}\right) for all n+1-frames \mathcal{G} superordinate to \mathcal{F} we can calculate f_{\text{st}}\left(\mathcal{F}\right). That will be the topic of our next post

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May 11, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , , , , | 2 Comments

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).

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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 S_n. This really is one of the most beautiful parts of basic finite group representation theory.

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May 10, 2011 Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , , , , | 7 Comments

   

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