Combinatorial, piecewise-linear, and birational homomesy for products of two chains

Abstract

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.

Birational Rowmotion and the Octahedron Recurrence

“…Combinatorial rowmotion permutes order ideals and so can also be thought of as a permutation of the vertices of O(P ). As is the case with the transfer map, there is a natural piecewise-linearization of this bijection from [4].…”

“…This lifting process is called detropicalization. (See [3,4,17] for more detailed discussion.) The functions resulting from detropicalization are generally subtraction-free and therefore well-defined on positive labelings of P .…”

“…for any linear extension L of P . From the birational setting, we can obtain the piecewise linear analogue via a valuation: see, for instance, [3,4].…”

“…This action is well studied in the dynamical algebraic combinatorics literature; for background on rowmotion, see [23]. On certain classes of posets (triangles, skeletal posets, rectangles, root posets, and others), rowmotion has a surprisingly small period, and it also sometimes exhibits other interesting phenomena such as homomesy and cyclic sieving: see [4,13,16,19,24].…”

“…Reinterpreting these toggles as acting on lattice points in R n , one can lift toggles and hence rowmotion to piecewise-linear maps. One can then lift these further to the birational realm by replacing max with addition, addition with multiplication, and subtraction with division [3,4]. Surprisingly, many results from the combinatorial level remain true on the birational level.…”