Hi Doron,
I just rapidly glanced at your most recent addition to ArXiv:
Very nice. In your conclusion about why one should work hard for the
linear algebra proof,
you might want to add that it also shows more, since it is also a
representation theoretic fact.
As you well know, it implies that the associated difference of
characters is a character,
which translates to the special case of Jacobi-trudi:
$$s_{n-k-1,k+1}=3D h_{k+1} h_{n-k-1} - h_k h_{n-k}.$$
I know that you very well know this, but it seems to me
that this is important to publicize. It seems to me that this is one of
the main reasons why it is
worth the effort of looking for an "equivariant" proof.
In a sense, this is in the same spirit as why the Theory of Species is nice. It puts the emphasis on the fact that only natural bijections translate into representation theoretic statements. Moreover, just as for the case of your paper, it becomes natural (and inevitable) in several situations to extend the point of view of Species to "Tensorial Species" (with values in vector spaces, so that we may get truly "equivariant" constructions. More people should get used to considering formal sums of combinatorial objects as being as natural to deal with as individual objects.
I mention this about Species, because I still see often people writing things of the kind: "... although it nicely solves problems, it is not clear that its use is essential for the solution ... I think that they prove this way that entirely miss the point.
Cheers, Francois Bergeron