Intertwinors from Generalized Tableaux
Given any generalized Young tableau with shape and content , we can construct an intertwinor . Actually, we’ll actually go from to , but since we’ve seen that this is isomorphic to , it’s good enough. Anyway, first, we have to define the row-equivalence class and column-equivalence class . These are the same as for regular tableaux.
So, let be our reference tableau and let be the associated tabloid. We define
Continuing our example, with
we we define
Now, we extend in the only way possible. The module is cyclic, meaning that it can be generated by a single element and the action of . In fact, any single tabloid will do as a generator, and in particular generates .
So, any other module element in is of the form for some . And so if is to be an intertwinor we must define
Remember here that acts on generalized tableaux by shuffling the entries by place, not by value. Thus in our example we find
Now it shouldn’t be a surprise that since so much of our construction to this point has depended on an aribtrary choice of a reference tableau , the linear combination of generalized tableaux on the right doesn’t quite seem like it comes from the tabloid on the left. But this is okay. Just relax and go with it.