  We prove a theorem classifying the equivariant $K$-theoretic pushforwards of
the product of arbitrary Schur functors applied to the tautological bundle on
the moduli space of framed rank $r$ torsion-free sheaves on $\mathbb{P}^2$, and
its dual. This is done by deriving a formula for similar coefficients on
Grassmannian varieties, and by thinking of the moduli space as a class in the
$K$-theory of the Grassmannian, in analogy with the construction of the Hilbert
scheme when the rank is one. Our motivations stem from some vertex operator
calculus studied recently by Nekrasov, Okounkov, and the author when the rank
is one, with applications to four-dimensional gauge theory.

Algebraic Geometry