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

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

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

Conjugacy Classes on the Symmetric Group


Point of Post: In this post we discuss the conjugacy class structure of the symmetric group S_n.

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Motivation

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 G is equal to the number of conjugacy classes of G. 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 S_n and find their order.

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May 10, 2011 Posted by | Algebra, Group Theory | , , , , , | 1 Comment