## Kostka Numbers

Now we’ve finished our proof that the intertwinors coming from semistandard tableauxspan the space of all intertwinors from the Specht module to the Young tabloid module . We also know that they’re linearly independent, and so they form a basis of the space of intertwinors — one for each semistandard generalized tableau.

Since the Specht modules are irreducible, we know that the dimension of this space is the multiplicity of in . And the dimension, of course, is the number of basis elements, which is the number of semistandard generalized tableaux of shape and content . This number we call the “Kostka number” . We’ve seen that there is a decomposition

Now we know that the Kostka numbers give these multiplicities, so we can write

We saw before that when , the multiplicity is one. In terms of the Kostka numbers, this tells us that . Is this true? Well, the only way to fit entries with value , with value , and so on into a semistandard tableau of shape is to put all the entries on the th row.

In fact, we can extend the direct sum by removing the restriction on :

This is because when we have . Indeed, we must eventually have , and so we can't fit all the entries with values through on the first rows of . We must at the very least have a repeated entry in some column, if not a descent. There are thus no semistandard generalized tableaux with shape and content in this case.