Standard Polytabloids are Independent
Now we’re all set to show that the polytabloids that come from standard tableaux are linearly independent. This is half of showing that they form a basis of our Specht modules. We’ll actually use a lemma that applies to any vector space with an ordered basis . Here indexes some set of basis vectors which has some partial order .
So, let be vectors in , and suppose that for each we can pick some basis vector which shows up with a nonzero coefficient in subject to the following two conditions. First, for each the basis element should be the maximum of all the basis vectors having nonzero coefficients in . Second, the are all distinct.
We should note that the first of these conditions actually places some restrictions on what vectors the can be in the first place. For each one, the collection of basis vectors with nonzero coefficients must have a maximum. That is, there must be some basis vector in the collection which is actually bigger (according to the partial order ) than all the others in the collection. It’s not sufficient for to be maximal, which only means that there is no larger index in the collection. The difference is similar to that between local maxima and a global maximum for a real-valued function.
This distinction should be kept in mind, since now we’re going to shuffle the order of the so that is maximal among the basis elements . That is, none of the other should be bigger than , although some may be incomparable with it. Now I say that cannot have a nonzero coefficient in any other of the . Indeed, if it had a nonzero coefficient in, say, , then by assumption we would have , which contradicts the maximality of . Thus in any linear combination
we must have , since there is no other way to cancel off all the occurrences of . Removing from the collection, we can repeat the reasoning with the remaining vectors until we get down to a single one, which is trivially independent.
So in the case we care about the space is the Young tabloid module , with the basis of Young tabloids having the dominance ordering. In particular, we consider for our the collection of polytabloids where is a standard tableau. In this case, we know that is the maximum of all the tabloids showing up as summands in . And these standard tabloids are all distinct, since they arise from distinct standard tableaux. Thus our lemma shows that not only are the standard polytabloids distinct, they are actually linearly independent vectors in .