## 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 .

*The Result*

The idea of the proof is simple. We’ll induct on (the in the ‘sum over -frames) and evaluate a certain double sum two ways: one way will tell us that the double actual reduces to the single sum in question and the other way will make the double sum amenable to applying the dimension hypothesis. We’ll heavily use our two facts about the sum over subordinate and superordinate frames and the standard tableaux on them. So, let’s get to it:

*Remark: *For notational convenience let

**Theorem: ***For any it is true that*

**Proof: **We proceed by induction on . Indeed, the result is true for since the only -frame is and so

So, assume the result is true for and consider the sum

Writing it firstly as

and using our previous result about summing over subordinate frames we find that

That said, a little thought shows that we may reverse this sum and write it as

But, by our previous result on summing over superordinates we may rewrite the right side of the above as

and applying our induction hypothesis we see that this is equal to . Thus,

and so the induction is complete.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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