The Schur functions are symmetric functions which appear in several different contexts:

1. The characters of the irreducible representations for the symmetric group.
2. The characters of the polynomial irreducible representations for the general linear group or unitary groups.
3. The cohomology classes of the Schubert cycles in the Grassmannian.
4. The images of the basis elements in the fermionic Fock space under the boson-fermion correspondence.
5. The orthonormal basis of the ring of symmetric functions with respect to a certain scalar product.

There are surely many more examples (that I would love to learn about!).

I know that the Schur-Weyl duality relates the representation theory of the symmetric group with the representation theory of the general linear group, so these two can be related. There are also several different papers addressing the relation of the Schubert calculus and the representations of the general linear group. I know less about the fermionic Fock space.

However, as far as I understand, many of these results hold only "in type A" and when their analogues are formulated for other types, different generalizations appear. For example, in the symplectic Schur-Weyl duality, the general linear group is replaced by the symplectic group, and the symmetric group is replaced by the Brauer group; but the characters of these two objects are no longer the same. Similarly, the connection between the cohomology ring of the Grassmannians in other Lie types seem to be different from the representation theory of the corresponding Lie objects.

Question: Is it a coincidence that the Schur functions appear in these independent contexts?

The Schur functions are symmetric functions which appear in several different contexts:

1. The characters of the irreducible representations for the symmetric group (under the characteristic isometry).
2. The characters of the polynomial irreducible representations for the general linear group or unitary groups.
3. The cohomology classes of the Schubert cycles in the Grassmannian.
4. The images of the basis elements in the fermionic Fock space under the boson-fermion correspondence.
5. The orthonormal basis of the ring of symmetric functions with respect to a certain scalar product (satisfying certain desirable properties).

There are surely many more examples (that I would love to learn about!).

I know that the Schur-Weyl duality relates the representation theory of the symmetric group with the representation theory of the general linear group, so these two can be related. There are also several different papers addressing the relation of the Schubert calculus and the representations of the general linear group. I know less about the fermionic Fock space.

However, as far as I understand, many of these results hold only "in type A" and when their analogues are formulated for other types, different generalizations appear. For example, in the symplectic Schur-Weyl duality, the general linear group is replaced by the symplectic group, and the symmetric group is replaced by the Brauer group; but the characters of these two objects are no longer the same. Similarly, the connection between the cohomology ring of the Grassmannians in other Lie types seem to be different from the representation theory of the corresponding Lie objects.

Question: Is it a coincidence that the Schur functions appear in these independent contexts?

I want to understand the rigidity of the Schur functions in these examples when modified. For instance, the equivariant cohomology of the Grassmannian yields the factorial Schur functions. Are there analogous concepts in the representation theory of the general linear group? The representation theory of the super Lie algebra for the general linear group gives the supersymmetric Schur functions. Is there an analogue in the cohomology of the Grassmannian? Or is the appearance of Schur functions in these contexts coincidental, without any generalizations having a corresponding analogue?