Modules of Generalized Young Tableaux
We can obviously create vector spaces out of generalized Young tableaux. Given the collection of tableaux of shape and content , we get the vector space . We want to turn this into an -module.
First, given any tabloid of shape , we can product a (generalized) tableau by defining to be the number of the row in that contains the entry . As an example, consider the tabloid
This gives us the function , , and . If and we use the usual reference tableau , this gives us the generalized tabloid
The shape of is obviously , and it’s easy to see that the content is exactly . Indeed, there are entries in with the value , just as there are entries in the first row of .
It should also be clear that this correspondence is a bijection. That is, given any generalized tableau of shape and content we can get a tabloid of shape by turning into a function and then putting on row of if .
That means that the basis of generalized tableaux of the vector space is in bijection with the basis of -tabloids of the vector space . And this space carries an action of — the linear extension of the action on tabloids. We want to pull this action across the bijection we just set up to get an action on .
On the one hand, this is as easy as saying it: if corresponds to , we define to be the generalized tableau corresponding to and we’re done. To be a bit more explicit, we define by considering it as a function and setting
So, for example, if
then we can calculate
Even more explicitly, if
then we calculate
We should be clear about a major distinction here: the permutation acts on the entries in — replacing by — but it acts on the places in — moving to the position of .
If we write the correspondence as , then for to be an intertwinor we need . This forces
and so this explicit action is forced on us.
The really interesting thing is that when we use this action on the generalized tableaux in , we always get a module , no matter what shape we start with.