Cluster Algebras and Semi-invariant Rings I. Triple Flags

Jiarui Fei

Cluster Algebras and Semi-invariant Rings I. Triple Flags

Commutative Algebra 2021-12-01 v3 Combinatorics Rings and Algebras Representation Theory

Authors: Jiarui Fei

Abstract

We prove that each semi-invariant ring of the complete triple flag of length $n$ is an upper cluster algebra associated to an ice hive quiver. We find a rational polyhedral cone ${\sf G}_n$ such that the generic cluster character maps its lattice points onto a basis of the cluster algebra. As an application, we use the cluster algebra structure to find a special minimal set of generators for these semi-invariant rings when $n$ is small.

Keywords

cluster algebras semi-invariant rings triple flags ice hive quiver generic cluster character minimal generators

Cite

@article{arxiv.1411.4693,
  title  = {Cluster Algebras and Semi-invariant Rings I. Triple Flags},
  author = {Jiarui Fei},
  journal= {arXiv preprint arXiv:1411.4693},
  year   = {2021}
}

Comments

Text overlap with arXiv:1210.1888 by other authors. v2. notation revised, typos corrected. v3. a shortened version. remove a wrong citation on local acyclicity, simplify the proof of Lemma 8.9

Cluster Algebras and Semi-invariant Rings I. Triple Flags

Abstract

Keywords

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