## The Sign Lemma

As we move towards proving the useful properties of Specht modules, we will find the following collection of results helpful. Through them all, let be a subgroup, and also consider the -invariant inner product on for which the distinct Young tabloids form an orthonormal basis.

First, if , then

where is the alternating sum of the elements of . The proof basically runs the same as when we showed that where has shape .

Next, for any vectors we have

Indeed, we can calculate

where we have used the facts that , and that as runs over a group, so does .

Next, if the swap , then we have the factorization

for some . To see this, consider the subgroup , and pick a transversal. That is, write as a disjoint union:

but then we can write the alternating sum

as we stated.

Finally, if is some tableau with and in the same row, and if the swap , then

Our hypothesis tells us that . We can thus use the above factorization to write

[…] of the Sign Lemma The results we showed last time have a few immediate consequences we will have use […]

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[…] entries in tells us that we must have some pair of and in the same row of . Thus the swap . The sign lemma then tells us that . Since this is true for every summand of , it is true for […]

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[…] we’ve used the sign lemma. So any two polytabloids coming from tableaux in the same column equivalence class are scalar […]

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[…] the other hand, assume in the same column of . Then . But then the sign lemma tells us that is a factor of , and thus […]

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Hi, I have a question, if Specht modules are cyclic, why do we need to find a basis? I know you have just explain that and maybe this is a basic question, but I have troubles with that idea. If one polytabloid generate all Specht module, then what is the point to find a basis?

Comment by Mari | August 28, 2014 |

The module may be generated as an -module by a single element, but not as a vector space.

In fact, that’s exactly what makes the representation theory of (the group algebra of) interesting while the representation theory of a field is boring: the only interesting thing about a field representation (vector space) is its dimension — the number of generators — and each generator behaves the same as every other. -modules have a lot more structure than that.

Comment by John Armstrong | August 28, 2014 |