Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory
Abstract
Let P be a finite connected poset and let ΛP be the opposite endomorphism algebra of the direct sum of all interval representations of P over a field. Via projectivization, this algebra governs resolutions relative to interval-decomposable representations, which arise naturally in persistence theory. We first show that ΛP carries a Reedy algebra structure in the sense of Dalezios--Šťov\'ıček. Its Reedy degree is given by the cardinality of the indexing interval, and the induced quasi-hereditary order is given by reverse interval cardinality. With respect to the resulting quasi-hereditary structure, we give a concrete combinatorial description of the standard modules and construct explicit projective resolutions of these modules. Using these resolutions, we reduce the calculation of standard--simple Ext groups to the reduced cohomology of simplicial complexes determined by the interval combinatorics. Order reversal gives the corresponding simple--costandard formula. Building on these calculations, we determine all simple--simple Ext groups. These groups are one-dimensional in a unique degree when the corresponding pair of intervals is saturated, and vanish otherwise. As a consequence, we obtain an exact combinatorial formula for the global dimension of ΛP, which in particular shows that it is independent of the coefficient field. As an application, for the m by ℓ grid Gm,ℓ with m≥ℓ≥2, we give the explicit formula gldimΛGm,ℓ=min{2ℓ,m+ℓ−2}. This also gives an explicit formula for the interval-resolution global dimension of these grids, settling the corresponding grid conjectures of Asashiba--Escolar--Nakashima--Yoshiwaki and determining the stable value and the precise stabilization threshold.
Related
BibTeX
@misc{aoki2026interval,
title = {{Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory}},
author = {Toshitaka Aoki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15927},
eprint = {2609.15927},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15927},
}
RIS
TY - UNPB
TI - Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory
AU - Toshitaka Aoki
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15927
UR - https://arxiv.org/abs/2609.15927
AB - Let P be a finite connected poset and let Λ_P be the opposite endomorphism algebra of the direct sum of all interval representations of P over a field. Via projectivization, this algebra governs resolutions relative to interval-decomposable representations, which arise naturally in persistence theory. We first show that Λ_P carries a Reedy algebra structure in the sense of Dalezios--Šťov\'ıček. Its Reedy degree is given by the cardinality of the indexing interval, and the induced quasi-hereditary order is given by reverse interval cardinality. With respect to the resulting quasi-hereditary structure, we give a concrete combinatorial description of the standard modules and construct explicit projective resolutions of these modules. Using these resolutions, we reduce the calculation of standard--simple Ext groups to the reduced cohomology of simplicial complexes determined by the interval combinatorics. Order reversal gives the corresponding simple--costandard formula. Building on these calculations, we determine all simple--simple Ext groups. These groups are one-dimensional in a unique degree when the corresponding pair of intervals is saturated, and vanish otherwise. As a consequence, we obtain an exact combinatorial formula for the global dimension of Λ_P, which in particular shows that it is independent of the coefficient field. As an application, for the m by ℓ grid G_m,ℓ with m≥ℓ≥2, we give the explicit formula gldimΛ_G_m,ℓ=min{2ℓ,m+ℓ−2}. This also gives an explicit formula for the interval-resolution global dimension of these grids, settling the corresponding grid conjectures of Asashiba--Escolar--Nakashima--Yoshiwaki and determining the stable value and the precise stabilization threshold.
ER -