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The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.
We prove some combinatorial conjectures extending those proposed in [13, 14]. The proof uses a vertex operator due to Nekrasov, Okounkov, and the first author [4] to obtain a gluing formula for the relevant generating series, essentially reducing the
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly
In arXiv:0907.3784, we introduced a variant of non-commutative Donaldson-Thomas theory in a combinatorial way, which is related with topological vertex by a wall-crossing phenomenon. In this paper, we (1) provide an alternative definition in a geomet
We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representation
In this paper, we explore a canonical connection between the algebra of $q$-difference operators $widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $mathcal{A}$ of the Lie algebra $mathfrak{gl}_{infty}$