Piecewise-linear and birational toggling. (English. French summary) Zbl 1394.06005

Einstein, David; Propp, James

Proceedings of the 26th international conference on formal power series and algebraic combinatorics, FPSAC 2014, Chicago, IL, USA, June 29 – July 3, 2014. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 513-524 (2014).

Summary: We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset \(P\) as studied by J. Striker and N. Williams [Eur. J. Comb. 33, No. 8, 1919–1942 (2012; Zbl 1260.06004)]. Piecewise-linear rowmotion relates to R. P. Stanley’s transfer map for order polytopes [Discrete Comput. Geom. 1, 9–23 (1986; Zbl 0595.52008)]; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When \(P=[a] \times [b]\), a reciprocal symmetry property recently proved by D. Grinberg and T. Roby [Electron. J. Comb. 23, No. 1, Research Paper P1.33, 40 p. (2016; Zbl 1338.06003); ibid. 22, No. 3, Research Paper P3.40, 49 p. (2015; Zbl 1339.06001)] implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order \(a+b\). We prove some homomesy results, showing that for certain functions \(f\), the average of \(f\) over each rowmotion/promotion orbit is independent of the orbit chosen.
For the entire collection see [Zbl 1298.05004].

MSC:

06A07	Combinatorics of partially ordered sets
05A19	Combinatorial identities, bijective combinatorics

Keywords:

poset; order ideal; order polytope; rowmotion; promotion; tropicalization