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
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