Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra

2009/04/30 by Boris Feigin, Alexander Tsymbaliuk · 1 citation
Mathematics · #math.RT #math.AG

paper · pdf · doi:10.1215/21562261-1424875

published as Kyoto J. Math. 51 (2011), no. 4, 831-854 · some typos fixed, the last 2 subsections added, 20 pages

arxiv created 2013/05/29 · arxiv updated 2019/02/12

Abstract

In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C². We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part h_ii>0 of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h₁,h₂,...].

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