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.
[…] First let’s mention a few more general results about Kostka numbers. […]
Pingback by More Kostka Numbers « The Unapologetic Mathematician | February 18, 2011 |
Your math is far beyond my comprehension, but I love reading this blog daily.