Representation-theoretical reasons for positivity of product of two Schubert polynomials?

Question

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when expressed in terms of the basis of Schubert polynomials. I know such a reason exists for Schur polynomials, but as far as I know the only known proofs of this fact for Schubert polynomials use algebraic geometry. Is there some purely representation-theoretic proof I've never heard of?

Answer 1

I doubt it. Presumably that's a typo, and it should read "geometric reasons". The usual proof identifies the coefficients of the product of two Schubert polynomials on the basis of Schubert polynomials as intersection numbers of transverse subvarieties, which are therefore non-negative integers. It has been generalised in various directions by increasingly difficult and technical variations on this theme: by Michel Brion to K-theory, by Bill Graham to equivariant cohomology, and by Dave Anderson, Ezra Miller and me to equivariant K-theory. There is presumably a quantum equivariant K-theory analog as well (as yet unproved).

My previous work in Schubert calculus notwithstanding, I'm mostly a representation theorist, so I suspect I would have known if someone had identified Schubert structure constants with some kind of composition multiplicities or dimensions of hom spaces (though I could be wrong!). Because of the relation to Demazure operators perhaps a connection to characters can be made, and positivity proved that way---if you can do this, let me know!