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Isotropic cuspidal functions in the Hall algebra of a quiver

Lucien Hennecart

TL;DR

The paper delivers an explicit linear form whose kernel equals the cuspidal subspace in the regular part of the Hall algebra of an affine quiver, enabling a concrete description of isotropic cuspidal functions and their affine-support property. It introduces a degree-preserving permutation action on the Hall algebra, yielding elementary proofs of two Berenstein–Greenstein conjectures and revealing deep symmetry in the regular tube structure. By connecting regular cuspidal functions of affine quivers to those of the Kronecker quiver, the work unifies several classically separate counts (Kac, indecomposables) and provides normalization and dimension-comparison results across Jordan, cyclic, and affine cases. The findings imply Bozec–Schiffmann positivity for absolutely cuspidal polynomials in isotropic dimensions and advance the understanding of primitive elements in Hall algebras, with potential implications for the realization of quantum groups. Overall, the work offers a complete framework for isotropic cuspidal functions in quiver Hall algebras and demonstrates powerful symmetry-based methods for longstanding conjectures.

Abstract

From the structure of the category of representations of an affine cycle-free quiver, we determine an explicit linear form on the space of regular cuspidal functions over a finite field: its kernel is exactly the space of cuspidal functions. Moreover, we show that any isotropic cuspidal dimension has an affine support. Brought together, this two results give an explicit description of isotropic cuspidal functions of any quiver. The main theorem together with an appropriate action of some permutation group on the Hall algebra provides a new elementary proof of two conjectures of Berenstein and Greenstein previously proved by Deng and Ruan. We also prove a statement giving non-obvious constraints on the support of the comultiplication of a cuspidal regular function allowing us to connect both mentioned conjectures of Berenstein and Greenstein. Our results imply the positivity conjecture of Bozec and Schiffmann concerning absolutely cuspidal polynomials in isotropic dimensions.

Isotropic cuspidal functions in the Hall algebra of a quiver

TL;DR

The paper delivers an explicit linear form whose kernel equals the cuspidal subspace in the regular part of the Hall algebra of an affine quiver, enabling a concrete description of isotropic cuspidal functions and their affine-support property. It introduces a degree-preserving permutation action on the Hall algebra, yielding elementary proofs of two Berenstein–Greenstein conjectures and revealing deep symmetry in the regular tube structure. By connecting regular cuspidal functions of affine quivers to those of the Kronecker quiver, the work unifies several classically separate counts (Kac, indecomposables) and provides normalization and dimension-comparison results across Jordan, cyclic, and affine cases. The findings imply Bozec–Schiffmann positivity for absolutely cuspidal polynomials in isotropic dimensions and advance the understanding of primitive elements in Hall algebras, with potential implications for the realization of quantum groups. Overall, the work offers a complete framework for isotropic cuspidal functions in quiver Hall algebras and demonstrates powerful symmetry-based methods for longstanding conjectures.

Abstract

From the structure of the category of representations of an affine cycle-free quiver, we determine an explicit linear form on the space of regular cuspidal functions over a finite field: its kernel is exactly the space of cuspidal functions. Moreover, we show that any isotropic cuspidal dimension has an affine support. Brought together, this two results give an explicit description of isotropic cuspidal functions of any quiver. The main theorem together with an appropriate action of some permutation group on the Hall algebra provides a new elementary proof of two conjectures of Berenstein and Greenstein previously proved by Deng and Ruan. We also prove a statement giving non-obvious constraints on the support of the comultiplication of a cuspidal regular function allowing us to connect both mentioned conjectures of Berenstein and Greenstein. Our results imply the positivity conjecture of Bozec and Schiffmann concerning absolutely cuspidal polynomials in isotropic dimensions.

Paper Structure

This paper contains 56 sections, 43 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main results
  3. Cuspidal functions as the kernel of a linear form
  4. An action of a permutation group on the Hall algebra
  5. Acknowledgements
  6. Structure of the category of representations of affine quivers over a finite field
  7. Notation and recollections on quiver representations
  8. The Euler form
  9. Dualization
  10. The structure of the category of representations of an affine quiver
  11. Cyclic and Jordan quivers
  12. Decomposition in blocks of $\mathop{\mathrm{\textup{Rep}}}\nolimits_{C_n}(k)$
  13. Acyclic affine quivers
  14. Identification of the tubes with the help of the Kronecker quiver
  15. Representations of the Kronecker quiver
  16. ...and 41 more sections

Key Result

Theorem 1.1

The kernel of the linear form is $\mathop{\mathrm{\mathbf{H}}}\nolimits_{Q,\mathop{\mathrm{\mathbf{F}}}\nolimits_q}^{\mathop{\mathrm{\textup{cusp}}}\nolimits}[r\delta]$.

Figures (1)

  • Figure 1: Non-homogeneous tubes of affine quivers and their period SchiffmannHall

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5: Ringel, MR774589
  • Remark 2.6
  • Theorem 2.7
  • ...and 68 more