Browse issues
no. 2
Combinatorial, piecewise-linear, and birational homomesy for products of two chains
Einstein, David1; Propp, James1
1 University of Massachusetts Lowell Department of Mathematical Sciences
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 201-224.

This article illustrates the dynamical concept of homomesy in three kinds of dynamical systems – combinatorial, piecewise-linear, and birational – and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams [16] can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley [14], and then lifted to birational operations on the positive orthant in ℝ |P| and indeed to a dense subset of ℂ |P| . When the poset P is a product of a chain of length a and a chain of length b, these lifted operations have order a+b, and exhibit the homomesy phenomenon: the time-averages of various quantities are the same in all orbits. One important tool is a concrete realization of the conjugacy between rowmotion and promotion found by Striker and Williams; this recombination map allows us to use homomesy for promotion to deduce homomesy for rowmotion.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.139
Classification: 05E18, 06A07
Keywords: Dynamics, homomesy, order ideal, order polytope, piecewise-linear, promotion, recombination, rowmotion, toggle group, tropicalization.

Einstein, David 1; Propp, James 1

1 University of Massachusetts Lowell Department of Mathematical Sciences
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2021__4_2_201_0,
     author = {Einstein, David and Propp, James},
     title = {Combinatorial, piecewise-linear, and birational homomesy for products of two chains},
     journal = {Algebraic Combinatorics},
     pages = {201--224},
     publisher = {MathOA foundation},
     volume = {4},
     number = {2},
     year = {2021},
     doi = {10.5802/alco.139},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.139/}
}
TY  - JOUR
AU  - Einstein, David
AU  - Propp, James
TI  - Combinatorial, piecewise-linear, and birational homomesy for products of two chains
JO  - Algebraic Combinatorics
PY  - 2021
SP  - 201
EP  - 224
VL  - 4
IS  - 2
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.139/
DO  - 10.5802/alco.139
LA  - en
ID  - ALCO_2021__4_2_201_0
ER  - 
%0 Journal Article
%A Einstein, David
%A Propp, James
%T Combinatorial, piecewise-linear, and birational homomesy for products of two chains
%J Algebraic Combinatorics
%D 2021
%P 201-224
%V 4
%N 2
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.139/
%R 10.5802/alco.139
%G en
%F ALCO_2021__4_2_201_0
Einstein, David; Propp, James. Combinatorial, piecewise-linear, and birational homomesy for products of two chains. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 201-224. doi : 10.5802/alco.139. https://alco.centre-mersenne.org/articles/10.5802/alco.139/

[1] Armstrong, Drew; Stump, Christian; Thomas, Hugh A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc., Volume 365 (2013) no. 8, pp. 4121-4151 | DOI | MR | Zbl

[2] Brouwer, Andries; Schrijver, Alexander On the period of an operator, defined on antichains, Math. Centrum report ZW, Volume 24/74 (1974) | Zbl

[3] Cameron, Peter; Fon-Der-Flaass, Dmitry G. Orbits of antichains revisited, European J. Combin., Volume 16 (1995) no. 6, pp. 545-554 | DOI | MR | Zbl

[4] Einstein, David; Farber, Miriam; Gunawan, Emily; Joseph, Michael; Macauley, Matthew; Propp, James; Rubinstein-Salzedo, Simon Noncrossing partitions, toggles, and homomesies, Electron. J. Combin., Volume 23 (2016) no. 3, Paper no. 3.52, 26 pages | DOI | MR | Zbl

[5] Einstein, David; Propp, James Piecewise-linear and birational toggling, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (Discrete Math. Theor. Comput. Sci. Proc., AT), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2014, pp. 513-524 | MR | Zbl

[6] Fon-Der-Flaass, Dmitry G. Orbits of antichains in ranked posets, European J. Combin., Volume 14 (1993) no. 1, pp. 17-22 | DOI | MR | Zbl

[7] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. 3.40, 49 pages | MR | Zbl

[8] Holroyd, Alexander E.; Levine, Lionel; Mészáros, Karola; Peres, Yuval; Propp, James; Wilson, David B. Chip-firing and rotor-routing on directed graphs, In and out of equilibrium. 2 (Progr. Probab.), Volume 60, Birkhäuser, Basel, 2008, pp. 331-364 | DOI | MR | Zbl

[9] Holroyd, Alexander E.; Propp, James Rotor walks and Markov chains, Algorithmic probability and combinatorics (Contemp. Math.), Volume 520, Amer. Math. Soc., Providence, RI, 2010, pp. 105-126 | DOI | MR | Zbl

[10] Kirillov, Anatol N.; Berenstein, Arkady Groups generated by involutions, Gelʼfand–Tsetlin patterns, and combinatorics of Young tableaux, St. Petersburg Math. J., Volume 7 (1996), pp. 77-127 Originally published in Algebra i Analiz 7 (1995), 92–152; also available at http://math.uoregon.edu/~arkadiy/bk1.pdf

[11] MathOverflow Do all subtraction-free identities tropicalize? (2013) (http://mathoverflow.net/questions/127108/do-all-subtraction-free-identities-tropicalize)

[12] Panyushev, Dmitri I. On orbits of antichains of positive roots, European J. Combin., Volume 30 (2009) no. 2, pp. 586-594 | DOI | MR | Zbl

[13] Propp, James; Roby, Tom Homomesy in products of two chains, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. 3.4, 29 pages | MR | Zbl

[14] Stanley, Richard P. Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986) no. 1, pp. 9-23 | DOI | MR | Zbl

[15] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR | Zbl

[16] Striker, Jessica; Williams, Nathan Promotion and rowmotion, European J. Combin., Volume 33 (2012) no. 8, pp. 1919-1942 | DOI | MR | Zbl

[17] Vorland, Corey Homomesy in products of three chains and multidimensional recombination (2017) (https://arxiv.org/abs/1705.02665) | Zbl

Cited by Sources: