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Chevalley-Monk formulas for bow varieties

Till Wehrhan

TL;DR

This work extends the Chevalley–Monk framework to the broad class of bow varieties by giving an explicit localization-friendly formula for multiplying first Chern classes of tautological bundles with stable envelopes. The authors develop a robust toolkit, combining tie diagrams, binary contingency tables, and Hanany–Witten transitions, to express localization coefficients as skein-like sums over crossings and simple moves. They prove the antidominant-case formula first in separated diagrams and then lift it to arbitrary chambers via HW-transitions and symmetric-group actions, using an equivariant resolution theorem to relate bow-variety data to cotangent bundles of partial flag varieties. The results provide a versatile, diagrammatic computation method for stable-envelopes-based multiplication, with potential applications to geometric representation theory and 3d mirror-symmetry-inspired structures. Overall, the paper unifies bow varieties with classical flag-variety calculus, delivering concrete combinatorial rules for their equivariant Chern-class actions.

Abstract

We prove a formula for the multiplication of equivariant first Chern classes of tautological bundles of type A bow varieties with respect to the stable envelope basis. This formula naturally generalizes the classical Chevalley-Monk formula and can be formulated in terms of creating crossings of skein type diagrams that label the stable envelope basis.

Chevalley-Monk formulas for bow varieties

TL;DR

This work extends the Chevalley–Monk framework to the broad class of bow varieties by giving an explicit localization-friendly formula for multiplying first Chern classes of tautological bundles with stable envelopes. The authors develop a robust toolkit, combining tie diagrams, binary contingency tables, and Hanany–Witten transitions, to express localization coefficients as skein-like sums over crossings and simple moves. They prove the antidominant-case formula first in separated diagrams and then lift it to arbitrary chambers via HW-transitions and symmetric-group actions, using an equivariant resolution theorem to relate bow-variety data to cotangent bundles of partial flag varieties. The results provide a versatile, diagrammatic computation method for stable-envelopes-based multiplication, with potential applications to geometric representation theory and 3d mirror-symmetry-inspired structures. Overall, the paper unifies bow varieties with classical flag-variety calculus, delivering concrete combinatorial rules for their equivariant Chern-class actions.

Abstract

We prove a formula for the multiplication of equivariant first Chern classes of tautological bundles of type A bow varieties with respect to the stable envelope basis. This formula naturally generalizes the classical Chevalley-Monk formula and can be formulated in terms of creating crossings of skein type diagrams that label the stable envelope basis.
Paper Structure (51 sections, 52 theorems, 213 equations, 3 figures)

This paper contains 51 sections, 52 theorems, 213 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bow varieties
  3. Brane diagrams
  4. Construction of bow varieties
  5. Torus actions
  6. Tie diagrams
  7. Torus fixed points
  8. Binary contingency tables
  9. Localization in equivariant cohomology
  10. Tautological bundles
  11. Hanany--Witten transition
  12. Cotangent bundles of partial flag varieties as bow varieties
  13. Realization via parabolic subgroups
  14. Bow variety realization
  15. Torus action and tautological bundles
  16. ...and 36 more sections

Key Result

Theorem 1

Let $\mathcal{C}(\mathcal{D})$ be a bow variety and $(\mathrm{Stab}(x_D))_D$ a fixed choice of stable basis and $\xi_i$ a tautological bundle. Then, we have where $h$ is the equivariant parameter corresponding to the scaling of the symplectic form, $\mathrm{SM}_{D,i}$ a certain set of tie diagrams which are obtained from $D$ by resolving one crossing and $\operatorname{sgn}(D,D')$ a explicitly co

Figures (3)

  • Figure 1: Relative positions of simple moves.
  • Figure 2: Illustration of simple moves for general brane diagrams.
  • Figure 3: Example of simple moves for general brane diagrams.

Theorems & Definitions (125)

  • Theorem : Multiplication formula
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 115 more