Varieties of chain complexes and mixed dimer covers

Abstract

A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type A which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a 2 × n grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type A roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.

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BibTeX RIS

BibTeX

@misc{anderson2026varieties,
  title = {{Varieties of chain complexes and mixed dimer covers}},
  author = {Portia X. Anderson and Esther Banaian and Melanie J. Ferreri and Nicholas Mayers and Shiyun Wang and Alexander N. Wilson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00202},
  eprint = {2609.00202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00202},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Varieties of chain complexes and mixed dimer covers
AU  - Portia X. Anderson
AU  - Esther Banaian
AU  - Melanie J. Ferreri
AU  - Nicholas Mayers
AU  - Shiyun Wang
AU  - Alexander N. Wilson
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00202
UR  - https://arxiv.org/abs/2609.00202
AB  - A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type A which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a 2 × n grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type A roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.
ER  -

Abstract

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