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Nakajima's creation operators and the Kirwan map

Jakub Koncki, Magdalena Zielenkiewicz

TL;DR

The paper tackles the problem of expressing Nakajima's creation operators on the Hilbert scheme of points in the plane through the Kirwan map, bridging cohomology and K-theory. It develops a localization-based framework on the nested Hilbert scheme to push forward tautological classes and uses Adams–Riemann–Roch to relate K-theoretic computations to cohomology, ultimately giving explicit formulas for the action of $\mathfrak q_1$ on power-sum and Chern-class bases and extending to all $\mathfrak q_m$. The results provide a concrete description of how Nakajima’s operators act on the Chern classes of the tautological bundle via the Kirwan map, and they extend to K-theory through the Ding–Iohara/elliptic Hall algebra perspective. This work connects geometric representation theory with equivariant localization, enhancing computational tools for tautological classes and their algebraic structures on Hilbert schemes.

Abstract

We consider the Hilbert scheme of points in the affine complex plane. We find explicit formulas for the Nakajima's creation operators and their K-theoretic counterparts in terms of the Kirwan map. We obtain a description of the action of Nakajima's creation operators on the Chern classes of the tautological bundle.

Nakajima's creation operators and the Kirwan map

TL;DR

The paper tackles the problem of expressing Nakajima's creation operators on the Hilbert scheme of points in the plane through the Kirwan map, bridging cohomology and K-theory. It develops a localization-based framework on the nested Hilbert scheme to push forward tautological classes and uses Adams–Riemann–Roch to relate K-theoretic computations to cohomology, ultimately giving explicit formulas for the action of on power-sum and Chern-class bases and extending to all . The results provide a concrete description of how Nakajima’s operators act on the Chern classes of the tautological bundle via the Kirwan map, and they extend to K-theory through the Ding–Iohara/elliptic Hall algebra perspective. This work connects geometric representation theory with equivariant localization, enhancing computational tools for tautological classes and their algebraic structures on Hilbert schemes.

Abstract

We consider the Hilbert scheme of points in the affine complex plane. We find explicit formulas for the Nakajima's creation operators and their K-theoretic counterparts in terms of the Kirwan map. We obtain a description of the action of Nakajima's creation operators on the Chern classes of the tautological bundle.
Paper Structure (23 sections, 48 theorems, 189 equations)

This paper contains 23 sections, 48 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements:
  3. Partitions, Young diagrams and symmetric functions
  4. Equivariant K-theory
  5. Hilbert scheme of points in a complex plane
  6. Definition and torus action
  7. Equivariant cohomology and K-theory
  8. Kirwan map
  9. Nested Hilbert Scheme
  10. Definition and torus action
  11. Restriction to a one-dimensional subtorus
  12. Tangent space
  13. Tautological line bundle
  14. Pushforward in the equivariant K-theory
  15. Nakajima operators
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\lambda=(\lambda_1,\dots,\lambda_l)$ be a sequence of nonnegative integers and $m$ a positive integer. For a subset $A\subseteq\{1,\dots,l\}$, let $\lambda_A$ be the sequence obtained from $\lambda$ by removing indices corresponding to elements of $A$ and $l(A):=\sum_{i\in A}\lambda_i\,.$ Then In particular for a nonegative integer $k\ge 0$ we have

Theorems & Definitions (96)

  • Theorem : \ref{['tw:qm']}
  • Proposition : \ref{['pro:qK']}
  • Theorem : \ref{['tw:push']} and \ref{['tw:pushH']}
  • Corollary
  • Remark 2.1
  • Definition 2.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3: Tho and CG
  • Proposition 3.4
  • ...and 86 more