## 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