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Enumerating m-clusters using exceptional sequences

Kiyoshi Igusa

Abstract

We give a bijection between ordered $m$-clusters and (complete) $m$-exceptional sequences, a concept that we introduce for this purpose. This holds for all hereditary artin algebras. This extends the bijection in the $m = 1$ case shown in arXiv:1706.02041.

Enumerating m-clusters using exceptional sequences

Abstract

We give a bijection between ordered -clusters and (complete) -exceptional sequences, a concept that we introduce for this purpose. This holds for all hereditary artin algebras. This extends the bijection in the case shown in arXiv:1706.02041.
Paper Structure (15 sections, 20 theorems, 70 equations)

This paper contains 15 sections, 20 theorems, 70 equations.

Table of Contents

  1. Basic definitions
  2. Relative projective terms
  3. $m$-exceptional sequences
  4. Recursive formula
  5. Proof of main theorem
  6. Statement of main theorem
  7. The key lemma
  8. Outline of proof of key lemma
  9. The bijection $\alpha_j:{\mathcal{A}}_j\to{\mathcal{A}}_{j-1}'$
  10. The bijection $\beta_j:{\mathcal{B}}_j\to {\mathcal{B}}_{j}'$
  11. Compatibility
  12. More properties of $m$-exceptional sequences
  13. Relation to Buan-Reiten-Thomas 9-BRT
  14. Tropical duality
  15. Mutation formula for $\tilde{c}$-vectors

Key Result

Theorem 1.2.2

Let $\Lambda$ be a finite dimensional hereditary algebra over any field and $k,m\ge0$. Then there is a 1-1 correspondence between (isomorphism classes of) ordered $k$-tuples of pairwise compatible objects in the $m$-cluster category of $\Lambda$ and (isomorphism classes of) $m$-exceptional sequences

Theorems & Definitions (43)

  • Definition 1.2.1
  • Theorem 1.2.2
  • Proposition 1.2.3
  • proof
  • Proposition 1.2.4
  • Lemma 1.3.1
  • Proposition 1.3.2
  • proof
  • Corollary 1.3.3
  • Theorem 2.1.1
  • ...and 33 more