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no. 6
Torsors and Tilings from Toric Toggling
Defant, Colin1; Joseph, Michael2; Macauley, Matthew3; McDonough, Alex4
1 Harvard University Department of Mathematics Cambridge MA 02139 (USA)
2 Dalton State College Department of Technology and Mathematics Dalton GA 30720 (USA)
3 Clemson University School of Mathematical and Statistical Sciences Clemson SC 29631 (USA)
4 University of Oregon Department of Mathematics Eugene OR 97403 (USA)
Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1695-1729.

Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a path graph. We investigate a “toric” analogue of this work by analyzing the dynamics arising from toggling independent sets of a cycle graph. Each orbit in the dynamical system can be encoded via a grid of 0s and 1s; two commuting bijections on the set of 1s in this grid produce torsors for what we call the infinite snake group and the finite ouroboros groups. By studying related covering maps, we deduce precise combinatorial properties of the orbits. Because the snake and ouroboros groups are abelian, they define tilings of cylinders and tori by parallelograms, which we also characterize. Many of the ideas developed here should be adaptable both to other toggle actions in combinatorics and to other cellular automata.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.386
Classification: 05E18
Mots-clés : dynamical algebraic combinatorics, independent set, toggle group, snake group, snakes on a plane, ouroboros on a torus, torsor

Defant, Colin 1; Joseph, Michael 2; Macauley, Matthew 3; McDonough, Alex 4

1 Harvard University Department of Mathematics Cambridge MA 02139 (USA)
2 Dalton State College Department of Technology and Mathematics Dalton GA 30720 (USA)
3 Clemson University School of Mathematical and Statistical Sciences Clemson SC 29631 (USA)
4 University of Oregon Department of Mathematics Eugene OR 97403 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Torsors and {Tilings} from {Toric} {Toggling}},
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Defant, Colin; Joseph, Michael; Macauley, Matthew; McDonough, Alex. Torsors and Tilings from Toric Toggling. Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1695-1729. doi : 10.5802/alco.386. https://alco.centre-mersenne.org/articles/10.5802/alco.386/

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