Multiple cluster algebra structure for triple crossing diagram maps I: theoretical framework
Abstract
We introduce triple crossing diagram (TCD) maps, which encode projective configurations of points and lines, as a unified framework for constructions arising in various areas of geometry, such as discrete differential geometry, discrete geometric dynamics and hyperbolic geometry. We define two types of local moves for TCD maps, one of which is governed by the discrete Schwarzian KP (dSKP) equation, and establish their multi-dimensional consistency. We construct two distinct cluster structures on the space of TCD maps, called projective and affine cluster structures, and show that they are related via an operation called section. This framework organizes and unifies a wide range of examples, including Q-nets, Darboux maps, line complexes, T-graphs, t-embeddings, triangulations and geometric discrete integrable systems such as the pentagram map and cross-ratio dynamics, which are further developed in the companion paper [AGGR25] and in [AGR25].
Related
- Decentralized Safe Path Following for Multiple Quadrotors on Intersecting Paths with Theoretical Guarantees
- A taxonomy of reliability indices for cluster-level constructs measured by multiple individual-level indicators.
- Generalized Functional ANOVA: A Complete Theoretical Framework
- Cluster randomized crossover trials with very few clusters but multiple periods: which analyses for continuous outcomes should be used?
- Learning environment at home and primary school dropout status in Bangladesh: An analysis of multiple indicator cluster survey
BibTeX
@misc{affolter2026multiple,
title = {{Multiple cluster algebra structure for triple crossing diagram maps I: theoretical framework}},
author = {Niklas Affolter and Terrence George and Max Glick and Sanjay Ramassamy},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05762246v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiple cluster algebra structure for triple crossing diagram maps I: theoretical framework
AU - Niklas Affolter
AU - Terrence George
AU - Max Glick
AU - Sanjay Ramassamy
PY - 2026
DA - 2026/09/24
PB - HAL
UR - https://hal.science/hal-05762246v1
AB - We introduce triple crossing diagram (TCD) maps, which encode projective configurations of points and lines, as a unified framework for constructions arising in various areas of geometry, such as discrete differential geometry, discrete geometric dynamics and hyperbolic geometry. We define two types of local moves for TCD maps, one of which is governed by the discrete Schwarzian KP (dSKP) equation, and establish their multi-dimensional consistency. We construct two distinct cluster structures on the space of TCD maps, called projective and affine cluster structures, and show that they are related via an operation called section. This framework organizes and unifies a wide range of examples, including Q-nets, Darboux maps, line complexes, T-graphs, t-embeddings, triangulations and geometric discrete integrable systems such as the pentagram map and cross-ratio dynamics, which are further developed in the companion paper [AGGR25] and in [AGR25].
ER -