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 .
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.
We dive right into it:
Lemma 1: Let be a fixed -tableaux and suppose that cannot be written as where and (the row and column stabilizers and the product is the product in not in itself). Then, there exists some and such that is odd, and .
Proof: Note that since cannot be written as by assumption we have by a previous theorem that . Thus, there exists which are in the same row of and the same column of . We have by definition that and so consider , we have by a previous theorem that . Moreover, it’s clear that and so is odd. To finish we note that
from where there the conclusion follows.
Lemma 2: Let be a fixed -tableau and be such that for all and it’s true that . Then, there exists some such that .
Proof: We know by definition that there exists for each such that
But, this then implies by assumption that for each and . But, in particular we see that this implies that . But by the previous lemma if we choose with odd then and so . It thus follows that most of the terms in the above sum fall out and we are left with
the conclusion follows.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996.
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