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On the Orthogonal Projections

Jiarui Fei

TL;DR

This work develops a rigorous framework for projecting representations onto orthogonal subcategories defined by a rigid presentation $e$ via the left adjoint projection $L_e$, and extends it to a modified projection $L_e^\pm$ in the quiver-with-potential setting. It establishes bijections between ${\rm rep}(e^\perp)$ and ${\rm rep}({\lfloor e \rfloor})$, introduces $e$-regularity to control functorial behaviour, and connects these constructions to Bongartz completions and Schur reductions. For quivers with potentials, it proves an equivalence ${\rm rep}(e^\perp) \simeq {\rm mod}({_e(Q,\mathcal{S})})$ when complements are extended-reachable, and provides mutation formulas for positive/negative complements and the associated $C$-matrices, enabling an algorithm to obtain the projected QP. The paper further develops modified projections that preserve general presentations and links these projections to stabilization functors in cluster algebra theory, yielding practical tools for projecting seeds and analyzing F-polynomials on facets. Overall, it offers a comprehensive toolkit for orthogonal projection in representation theory and cluster algebra contexts, including explicit mutation rules, algorithmic prescriptions, and structural equivalences for quivers with potentials.

Abstract

For any ${\rm E}$-rigid presentation $e$, we construct an orthogonal projection functor to ${\rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${\rm rep}(e^\perp)$ and presentations compatible with $e$. For quivers with potentials, we show that ${\rm rep}(e^\perp)$ forms a module category of another quiver with potential. We derive mutation formulas for the $δ$-vectors of positive and negative complements and the dimension vectors of simple modules in ${\rm rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

On the Orthogonal Projections

TL;DR

This work develops a rigorous framework for projecting representations onto orthogonal subcategories defined by a rigid presentation via the left adjoint projection , and extends it to a modified projection in the quiver-with-potential setting. It establishes bijections between and , introduces -regularity to control functorial behaviour, and connects these constructions to Bongartz completions and Schur reductions. For quivers with potentials, it proves an equivalence when complements are extended-reachable, and provides mutation formulas for positive/negative complements and the associated -matrices, enabling an algorithm to obtain the projected QP. The paper further develops modified projections that preserve general presentations and links these projections to stabilization functors in cluster algebra theory, yielding practical tools for projecting seeds and analyzing F-polynomials on facets. Overall, it offers a comprehensive toolkit for orthogonal projection in representation theory and cluster algebra contexts, including explicit mutation rules, algorithmic prescriptions, and structural equivalences for quivers with potentials.

Abstract

For any -rigid presentation , we construct an orthogonal projection functor to left adjoint to the natural embedding. We establish a bijection between presentations in and presentations compatible with . For quivers with potentials, we show that forms a module category of another quiver with potential. We derive mutation formulas for the -vectors of positive and negative complements and the dimension vectors of simple modules in , enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

Paper Structure

This paper contains 31 sections, 67 theorems, 107 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Work of Geigle-Lenzing, Schofield, and Jasso
  3. Projections from Rigid Presentations
  4. The Bijection between ${rep}(e^\perp)$ and ${rep}({{ \lfloor}{e} {\rfloor}})$
  5. The Case of Quivers with Potentials
  6. Organization
  7. Notations
  8. Preliminary
  9. The Category of Presentations
  10. General Presentations
  11. Rigid Presentations
  12. The $\mathop{\mathrm{E}}\nolimits$-truncating Functors and Complements
  13. Torsion Theory
  14. The Orthogonal Projection Functor
  15. Stable and Schur Reductions
  16. ...and 16 more sections

Key Result

Theorem 1.1

The functor $L_e: \mathop{\mathrm{rep}}\nolimits(A)\to \mathop{\mathrm{rep}}\nolimits(e^\perp)$ is left adjoint to the inclusion functor $\mathop{\mathrm{rep}}\nolimits(e^\perp) \to \mathop{\mathrm{rep}}\nolimits(A)$.

Theorems & Definitions (145)

  • Theorem 1.1: Theorem \ref{['T:L_e']}
  • Corollary 1.2: Corollaries \ref{['C:equi']} and \ref{['C:bimod']}
  • Theorem 1.3: Theorems \ref{['T:bijection']} and \ref{['T:bijection_space']}
  • Theorem 1.4: Theorem \ref{['T:perpQP']}
  • Theorem 1.5: Theorem \ref{['T:mucomp']}
  • Theorem 1.6: Theorem \ref{['T:musimple']}
  • Theorem 1.8: Theorem \ref{['T:Lstab']}
  • Theorem 1.9: Theorem \ref{['T:L_e^-']}
  • Corollary 1.10: Corollary \ref{['C:faces']}
  • Lemma 2.1
  • ...and 135 more