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

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 can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley, and then lifted to birational operations on the positive orthant in $\mathbb{R}^{|P|}$ and indeed to a dense subset of $\mathbb{C}^{|P|}$. 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. We also show that Stanley's transfer map between the order polytope and the chain polytope arises as the tropicalization of an analogous map in the bilinear realm.


Introduction
Numerous authors (specifically Brouwer and Schrijver [BS74], Cameron and fon der Flaass [Fon93][CF95], Panyushev [Pan08], and Striker and Williams [SW12]) have studied an operation ρ on the set of order ideals of a poset P that, following Striker and Williams, we call rowmotion. In exploring the properties of rowmotion, Striker and Williams also introduced and studied a closely related operation π they call promotion on account of its ties with promotion of Young tableaux, which depends on the choice of an rc embedding (a particular kind of embedding of P into the poset N × N). Here we focus on a very particular case, where P is of the form [a] × [b] and the rc embedding sends (i, j) ∈ P to (j − i, i + j − 2) ∈ Z 2 , and we explore how the cardinality of an order ideal I ∈ J(P ) behaves as one iterates rowmotion and promotion.
It has been shown [PR13] that the order of ρ acting on J([a] × [b]) is n = a + b. Propp and Roby furthermore showed that the average of |I| as I varies over a ρ-orbit in J(P ) is ab/2, regardless of which orbit one takes. That is: The notion of looking at the average of a quantity over an orbit was an outgrowth of the second author's work on chip-firing and rotor-routing [H+08], [HP10]; see in particular Proposition 3 of [PR13]. Further inspiration came from conjectures of Panyushev [Pan08] (later proved by Armstrong, Stump, and Thomas [AST11]).
This article presents a new proof of Theorem 1 which, although less direct than the Propp-Roby proof, indicates that the "constant-averages-over-orbits phenomenon" (also called "homomesy") applies not just for actions on order ideals and antichains but also for dynamical systems of a different character. Specifically, we define (continuous) piecewise-linear maps from the order polytope of P to itself (piecewise-linear rowmotion and promotion) that satisfy homomesy, and birational maps from a dense open subset of C ab to itself (birational rowmotion and promotion) that satisfy a multiplicative version of homomesy. We give other examples of functions that exhibit homomesy, and in the case of combinatorial rowmotion and promotion, we are able to show that, in a The plan of the article is as follows. In section 2, after introducing needed preliminaries and notation, including the general definition of (additive) homomesy and its specialization to the case of functions with periodic behavior, we review some of the background on the rowmotion and promotion operations ρ, π : J(P ) → J(P ). We then define (in section 3) piecewise-linear maps ρ P , π P : R |P | → R |P | and show that ρ P and π P specialize to ρ and π if one restricts attention to the vertices of O(P ). Changing variables, we obtain slightly different piecewise-linear maps ρ H , π H : R |P | → R |P | that are homogeneous versions of ρ P , π P . Then we show (in section 4) how ρ H , π H can in turn each be viewed as a tropicalization of a birational map from a dense open subset U of C ab to itself; we call the elements of U P -arrays. In section 5, we digress to give an alternative characterization of birational rowmotion in terms of Stanley's transfer map between the order polytope and chain polytope of a poset [Sta86]. In section 6, we construct two actions of (birational) Theorem 4 ⇓ (tropicalization) (piecewise-linear) Theorem 3 ⇓ (change of variables) (piecewise-linear) Theorem 2 ⇓ (specialization) (combinatorial) Theorem 1 Figure 1: Implications between combinatorial, piecewise-linear, and birational results about homomesy.
S n on U. The first (due to Darij Grinberg [?]) gives invariants for birational rowmotion; the second gives invariants for birational promotion, and also yields a proof of Theorem 4: for v = (v 1 , . . . , v ab ) ∈ U, the product of the coordinates of v associated with elements of the file S (denoted here by |v| S ) has the property that |v| S |π(v)| S |π 2 (v)| S · · · |π n−1 (v)| S = 1. In section 7, we prove a fundamental pairing property of birational rowmotion, and use it to show that birational rowmotion is of order a + b. In section 8, building upon Theorem 5.4 of [SW12], we demonstrate a concrete operation on P -arrays, which we call recombination, that gives an equivariant bijection between birational rowmotion and promotion; the resulting bijection between rowmotion orbits and promotion orbits lets us deduce that birational promotion, like birational rowmotion, is of order a + b, and furthermore lets us deduce that Theorem 4 holds when promotion is replaced by rowmotion. In section 9, we use tropicalization to deduce from Theorem 4 a piecewise-linaer analogue (Theorem 3) that by a linear change of variables yields the homomesy result for the action of promotion on O(P ) (Theorem 2). This last result then yields homomesy for the action of promotion on J(P ) (Theorem 1). Figure 1 shows the structure of this chain of deductions schematically (omitting the use of recombination for passing back and forth between rowmotion and promotion). In section 10 we use the pairing property of section 7 to prove that the function that sends f to f (x)f (x ′ ) (with x = (i, j) and x ′ = (a+1−i, b+1−j)) is multiplicatively homomesic. In section 11, we show that all of the homomesies obtained thus far give all homomesies for the action of rowmotion and promotion on J(P ), if one restricts to functions spanned by the indicator functions 1 x (I). In section 12, we point out directions for future work.
The philosophy of lifting combinatorial actions to piecewise-linear actions and thence to birational actions (called "geometric actions" by some authors, as in the phrase "geometric Robinson-Schensted-Knuth") is not original, and in particular Kirillov and Berenstein's work on operations on Gelfand-Tsetlin patterns [KB95] has some parallels with our constructions. Computer experiments suggest that the homomesy phenomenon and the implicational structure given in Figure 1 are not unique to the poset [a] × [b] but apply to many other posets, notably root posets and minuscule posets. For more background of homomesy, including several examples different in nature from the ones considered here but philosophically similar, see [PR13]. The authors are grateful to Arkady Berenstein, Darij Grinberg, Tom Roby, and Jessica Striker for helpful conversations.

Homomesy
Given a set X, an operation T : X → X, and a function F from X to a field K of characteristic 0, we say that F is homomesic relative to (or under the action of) T , or that the triple (X, T, F ) exhibits homomesy, if, for all x ∈ X, the long-term average equals some constant c (independent of x). We also say in this situation that the function F (which in this context we will sometimes call a functional on X) is c-mesic relative to the map T . In the case where the sequence F (T k (x)) (k = 0, 1, 2, . . . ) is periodic, it is easy to show that the limit is equal to the average of F (T k (x)) over one period, and this will be the situation throughout this article, except in one paragraph in the final section. The article [PR13] gives examples of combinatorial situations in which homomesy holds, and we have found that such situations are easy to find experimentally once one starts to look for them, though proving theorems that support the experimental results is another matter.

Posets and toggling
We assume readers are familiar with the definition of a finite poset (P, ≤), as for instance given in Ch. 3 of [Sta11]. For the most part, we are studying the We write x ⋖ y ("x is covered by y") or equivalently y ⋗ x ("y covers x") when x < y and no z ∈ P satisfies x < z < y. We call S ⊆ P a filter (or upset or dual order ideal) of P when x ∈ S and y ≥ x imply y ∈ S. We call S ⊆ P an order ideal (or downset) of P when x ∈ S and y ≤ x imply y ∈ S. We call S ⊆ P an antichain when x, y ∈ S and x = y imply that x and y are incomparable (i.e., neither x ≤ y nor y ≤ x). The set of filters, order ideals, and antichains of P are denoted by F (P ), J(P ), and A(P ), respectively.
There are natural bijections α 1 : J(P ) → F (P ), α 2 : F (P ) → A(P ), and α 3 : A(P ) → J(P ) given by the following recipes: (1) for I ∈ J(P ), let α 1 (I) be the complement P \ I; (2) for F ∈ F (P ), let α 2 (F ) be the set of minimal elements of F (i.e., the set of x ∈ F such that y < x implies y ∈ F ); and (3) for A ∈ A(P ), let α 3 (A) be the downward saturation of A (i.e., the set of y ∈ P such that y ≤ x for some x ∈ A). The composition ρ := α 3 • α 2 • α 1 : J(P ) → J(P ) is not the identity map; e.g., it sends the "full" order ideal (P itself) to the empty order ideal (∅).
Cameron and fon der Flaass [CF95] gave an alternative characterization of ρ. Given x ∈ P and I ∈ J(P ), let τ x (I) ("I toggled at x" in Striker and Williams' terminology) denote the set I △ {x} if this set is in J(P ) and I otherwise. Equivalently, τ x (I) is I unless y ∈ I for all y ⋖ x and y ∈ I for all y ⋗ x, in which case τ x (I) is I △ {x}. (We will sometimes say "Toggling x turns I into τ x (I)".) Clearly τ x is an involution. It is also easy to show that τ x and τ y commute unless x ⋖ y or x ⋗ y. If x 1 , x 2 , . . . , x |P | is any linear extension of P (that is, a listing of the elements of P such that x i < x j implies i < j), then the composition τ x 1 • τ x 2 • · · · • τ x |P | coincides with ρ. In the case where the poset P is ranked (that is, where the elements can be partitioned into ranks 0, 1, 2, . . . such that all minimal elements of P belong to rank 0 and such that x⋖y implies that the rank of x is 1 less than the rank of y), one natural way to linearly extend P is to list the elements by rank, starting with rank 0 and working upward. Given the right-to-left order of composition of τ x 1 • τ x 2 • · · · • τ x |P | , this corresponds to toggling the top rank first, then the next-to-top rank, and so on, lastly toggling the bottom rank.
Note that when x, y belong to the same rank of P , the toggle operations τ x and τ y commute, so even without availing ourselves of the theorem of Cameron and fon der Flaass, we can see that this composite operation on J(P ) is well-defined. Striker and Williams, in their theory of rc posets, use the term "row" as a synonym for "rank", and they refer to ρ as rowmotion.
For example, let P = [2] × [2], and write (1, 1), (2, 1), (1, 2), (2, 2) as w, x, y, z for short, with w < x < z and w < y < z, and with the rc embedding shown. admits an embeding in the plane that maps (i, j) to the point (j − i, i + j − 2). This sends the minimal element (1, 1) to the origin and sends all the poset-elements of rank m (= i + j − 2) to the horizontal line at height m above the origin, We refer to elements of P that lie on a common vertical line as belonging to the same file. In particular, we say (i, j) belongs to the (j − i + a)th 1 file of P . See Figure 2. Note that if x and y belong to the same file, the toggle operations τ x and τ y commute, since neither of x, y can cover the other. Thus the composite operation of toggling the elements of P from left to right is well-defined; Striker and Williams call this operation promotion, and we denote it by π.
3 Piecewise-linear fiber-toggling Given a poset P = {x 1 , . . . , x n }, let R P denote the set of functions f : P → R; we can represent such an f as a P -array (or array for short) in which the values of f (x) (for all x ∈ P ) are arranged according to the rc embedding of P in the plane. We will sometimes identify R P with R |P | , associating , and the associated partitions of P into ranks and files. Diagonal edges are associated with the covering relation in P .
. . , f (x n )). LetP denote the augmented poset obtained from P by adding two extra elements0 and1 (which we sometimes denote by x 0 and x n+1 ) satisfying0 < x <1 for all x ∈ P . The order polytope O(P ) ⊂ R P (see [Sta86]) is the set of vectors (f (x 1 ), . . . ,f (x n )) arising from functionsf :P → R that satisfyf (0) = 0 andf (1) = 1 and are order-preserving (x ≤ y in P impliesf (x) ≤f (y) in R). In some cases it is better to work with the augmented vector (f ( In either case we have a convex compact polytope. Given a convex compact polytope K in R n (we are only concerned with the case K = O(P ) here but the definition makes sense more generally), we define the piecewise-linear toggle operation φ i (1 ≤ i ≤ n) as the unique map from K to itself whose action on the 1-dimensional fibers of K in the ith coordinate direction is the linear map that switches the two endpoints of the fiber. That is, where the real numbers L and R are respectively the left and right endpoints of the set {t ∈ R : each toggle operation is an involution. Similar involutions were studied by Kirillov and Berenstein [KB95] in the context of Gelfand-Tsetlin triangles. Indeed, one can view their action as an instance of our framework, where instead of looking at the rectangle posets [a]×[b] one looks at the triangle posets with elements (1 ≤ i ≤ j ≤ n − 1); Kirillov and Berenstein use the term "elementary transformations" in their Definition 0.1, whereas we use the term "fibertoggles".
In the case where K is the order polytope of P , and a particular element x ∈ P has been indexed as x i , we write φ i as φ x . The L and R that appear in (1) are given by and Using these definitions of L and R, 3 we see that equation (1) defines an involution on all of R P , not just O(P ). It is easy to show that φ x and φ y commute unless x ⋖ y or x ⋗ y. These piecewise-linear toggle operations φ x are analogous to the combinatorial toggle operations τ x (and indeed φ x generalizes τ x in a sense to be made precise below), so it is natural to define piecewise-linear rowmotion We will see that ρ P and π P are of order n. Specifically, we will show as a corollary of Theorem ?? that ρ P is of order n, and by then applying recombination (see ??) we will conclude that π P is of order n as well.
The vertices of O(P ) are precisely the 0,1-valued functions f on P with the property that x ≤ y in P implies f (x) ≤ f (y) in {0, 1}. That is, they are precisely the indicator functions of filters. Filters are in bijection with order ideals by way of the complementation map, so the vertices of O(P ) are in bijection with the elements of lattice J(P ). Each toggle operation acts as a permutation on the vertices of O(P ). Indeed, if we think of each vertex O(P ) as determining a cut of the poset P into an upset (filter) S up and a downset (order ideal) S down (the pre-image of 1 and 0, respectively, under the order-preserving map from P to {0, 1}), then the effect of the toggle , unless this would violate the property that S up must remain an upset and S down must remain a downset. In particular, we can see that when our point v ∈ O(P ) is a vertex associated with the cut (S up , S down ), the effect of φ x on S down is just toggling the order ideal S down at the element x ∈ P .
It is not hard to show that each toggle operation preserves the quantity min{f (y) − f (x) : x, y ∈P , x ⋖ y}. Therefore ρ P and π P preserve this quantity as well.
Define |v| as the sum of the entries of v (since all the entries are nonnegative, this is the L 1 -norm of v). Then we will show: Theorem 1 is just the special case of this theorem where v is a vertex of O(P ).
One can generalize the above construction by allowing f (0) and f (1) to have fixed values in R other than 0 and 1 respectively. The nicest choice is to take f (0) = f (1) = 0. In this case, we write the promotion operation as π H (and likewise we write the homogeneous piecewise-linear rowmotion operation as ρ H ). The maps π P and π H are closely related. Suppose that P has r + 1 ranks, numbered 0 (bottom) through r (top). Given an arbitrary f in R P (considered as a map fromP to R that sends0 to 0 and1 to 1), define f (x) = f (x) − m/r where x belongs to rank m. Then f sends0 and1 to 0, and each function fromP to R that sends0 and1 to 0 arises as f for a unique f in O(P ). Furthermore, the map f → f commutes with promotion. 4 In this way it can be seen that Theorem 2 is equivalent to the following claim: is just a chain with n = ab elements, then our piecewise-linear maps are all linear, and the effect of the map φ i (1 ≤ i ≤ n) is just to swap the ith and i+1st elements of the difference-vector , a vector of length n + 1 whose entries sum to 0. Consequently π H is just a cyclic shift of a vector whose entries sum to 0, and the claim of Theorem 3 follows.
It might be possible to prove Theorem 2 by figuring out how the map π P dissects the order polytope into pieces and re-arranges them via affine maps. Likewise, it might be possible to prove Theorem 3 by giving a precise analysis of the piecewise-linear structure of the map π H . However, we will take a different approach, proving the result in the piecewise-linear setting by proving it in the birational setting and then tropicalizing.

Birational toggling
The definition of the toggling operation involves only addition, subtraction, min, and max. As a result of this one can define birational transformations on (R + ) P that have some formal resemblance to the toggle operations on O(P ). This transfer makes use of a dictionary in which 0, addition, subtraction, max, and min are respectively replaced by 1, multiplication, division, addition, and parallel addition (defined below), resulting in a subtractionfree rational expression. 5 Parallel addition can be expressed in terms of the other operations, but taking a symmetrical view of the two forms of addition turns out to be fruitful. Indeed, in setting up the correspondence we have a choice to make: by "series-parallel duality", one could equally well use a dictionary that switches the roles of addition and parallel addition. We hope the choice that we have made here will prove to be at least as convenient as the other option would have been.
For x, y satisfying x + y = 0, we define the parallel sum of x and y as x y = xy/(x+y). In the case where x, y and x+y are all nonzero, xy/(x+y) is equal to 1/( 1 x + 1 y ), which clarifies the choice of notation and terminology: if two electrical resistors of resistance x and y are connected in parallel, the resulting circuit element has an effective resistance of x y. Note that if x and y are in R + , then x + y and x y are in R + as well. Also note that is commutative and associative, so that a compound parallel sum x y z · · · is well-defined; it is equal to product xyz · · · divided by the sum of all products that omit exactly one of the variables, and in the case where x, y, z, . . . are all positive, it can also be written as 1/( 1 x + 1 y + 1 z + · · · ). We thus have the reciprocity relation The identity (x y)(x + y) = xy (5) also plays an important role.
Recall formulas (1), (2) and (3) above. Instead of taking the maximum of the x j 's satisfying x j ⋖ x i , we can take their ordinary (or "series") sum, and instead of taking the minimum of the x j 's satisfying x j ⋗x i , we can take their parallel sum. Proceeding formally, given a non-empty set S = {s 1 , s 2 , . . . }, let + S denote s 1 + s 2 + · · · and S denote s 1 s 2 · · · . Then for and We call the maps φ i given by (6) birational toggle operations, as opposed to the piecewise-linear toggle operations treated in the previous section. 6 As the 0th and n + 1st coordinates of v aren't affected by any of the toggle operations, we can just omit those coordinates, reducing our toggle operations to actions on (R + ) P . Since LR/(LR/v i ) = v i , each birational toggle operation is an involution on the orthant (R + ) P . The birational toggle operations are analogous to the piecewise-linear toggle operations (in a sense to be made precise below), so it is natural to define birational rowmotion ρ B : (R + ) P → (R + ) P as the composite operation accomplished by toggling from top to bottom, and to define birational promotion π B : (R + ) P → (R + ) P as the composite operation accomplished by toggling from left to right. It is not hard to show that each birational toggle operation preserves the quantity + {f (x)/f (y) : x, y ∈P , x ⋖ y} (or if one prefers the reciprocal quantity {f (y)/f (x) : x, y ∈P , x⋖y}). Therefore ρ B and π B also preserve this quantity.
Continuing our running example P = [2] × [2] = {w, x, y, z}, let v = (1, 2, 3, 4) ∈ R P , corresponding to the positive function f that maps w, x, y, z to 1, 2, 3, 4, respectively. Under the action of φ z , φ y , φ x , and φ w , the vector v = (1, 2, 3, 4) gets successively mapped to (1, 2, 3, 5 4 ), (1, 2, 5 12 , 5 4 ), (1, 5 8 , 5 12 , 5 4 ), and ( 1 4 , 5 8 , 5 12 , 5 4 ) = ρ B (v). We can check that the quantity + {f (x)/f (y) : x, y ∈P , x ⋖ y} retains the value 85 12 throughout the process. For most of the purposes of this article, it would suffice to take π B to be a map from (R + ) P to itself. However, π B can be extended to a map from a dense open subset of R P to itself, and indeed, from a dense open subset U of C P to itself. All expressions we consider are well-defined on the open orthant (R + ) P , and all the theorems we prove amount to identities that are valid when all variables lie in this orthant; this implies that the identities hold outside of some singular variety in C P . Identifying the singular subvariety on which π B (or one of its powers) is undefined seems like an interesting question, but it is one that we leave to others. Alternatively, Tom Roby has pointed out that one can replace R + by a ring of rational functions in formal indeterminates indexed by the elements of P , thereby avoiding the singularity issue (once one checks that the rational functions in question can be expressed as ratios of polynomials with positive coefficients). Experiments in Sage conducted by Darij Grinberg have taken this formal approach, and there is strong evidence there for the conjecture that for P = [a] × [b], the operation π B is of order n.
We focus on the "multiplicatively homogeneous" case f (0) = f (1) = 1, since (just as in the piecewise-linear setting) no generality is lost. at least if we restrict to vectors v in the positive orthant, so that rth roots are globally well-defined, where r + 1 is the number of ranks of P . Given an arbitrary f :P → R + , let α = f (0) 1/r and ω = f (1) 1/r , and define f by f (x) = f (x)/α r−m ω m for x belonging to rank m. Then f sends0 and1 to 1, and for any choice of α, ω ∈ R + , each function fromP to R + that sends0 and1 to 1 arises as f for a unique f :P → R + sending0 to α and1 to ω. Furthermore, the map f → f commutes with promotion.
Define |v| to be the product of the entries of v. As in the piecewise-linear case, our proof-method will enable us to demonstrate (multiplicative) homomesy of the action even allowing for the possibility that the orbits are infinite. That is, even though we cannot yet show that the sequence v, π B (v), π 2 B (v), . . . is periodic, we will show that the sequence |v|, |π B (v)|, |π 2 B (v)|, . . . , is periodic with period n. Indeed, we will prove more: Theorem 4. For every v in (R + ) P corresponding to an f :P → R + with In other words, the geometric mean of the values of |π B (v)| as v traces out an orbit in (R + ) P is equal to 1 for every orbit. Theorem 4 also applies to a dense open subset of R P , and indeed to a dense open subset of C P , but the paraphrase in terms of geometric means does not hold in general since z → z 1/N is not single-valued on C.

Birational promotion and Stanley's transfer map
Although most of our work with rowmotion treats it as a composition of |P | toggles (from the top to the bottom of P ), we noted in subsection 2.2 that ρ can also be defined as a composition of three operations α 1 , α 2 , α 3 . 7 Here we point out that this alternative definition can be lifted to the piecewise-linear setting and indeed to the birational setting. For the piecewise-linear setting, we first recall Richard Stanley's definition of the chain polytope C(P ) of a poset P [Sta86]. A chain in a poset P is a totally ordered subset of P , and a maximal chain in a poset P is a chain that is not a proper subset of any other chain. If the poset P is ranked, then the maximal chains in P are precisely those chains that contain an element of every rank. The chain polytope of a poset P is the set of maps from P to [0, 1] such that for every chain C in P , Just as the vertices of the order polytope of P correspond to the indicator functions of filters of P , the vertices of the chain polytope of P correspond 8 to the indicator functions of antichains in P .
(12) Note that the third of these operations is just the map Ψ "turned upside down". It is not hard to check that (12) can be replaced by the recursive definition which turns out to be the form most suitable for lifting to the birational setting. It readily follows from Theorem 5 below (by tropicalization) that ρ P = α 1 • α 3 • α 2 . Moreover, if we make α 1 homogeneous by redefining α 1 (f ) so that (α 1 (f ))(x) is −f (x) instead of 1 − f (x), then the composition of the three maps gives ρ H .
Theorem 5. For any poset P , the birational rowmotion operation ρ B coincides with the composition α 1 • α 3 • α 2 , where α 1 , α 2 , α 3 : R P → R P are defined by ( Proof. We wish to prove that (α 1 α 3 α 2 f )(x) = (ρ B f )(x) for all x in P . We prove the claim by backward induction in the poset P , i.e., from top to bottom. The base case x =1 is trivial (both sides of the equation equal 1), so we need only prove the induction step. Assume that every y ∈ x + satisfies the induction hypothesis. We have from equation (15) ( Next, applying α 3 to α 2 f using (16) we have where the last line follows from (4) and (14). Now applying α 1 to both sides we get But by the induction hypothesis, (α 1 α 3 α 2 f )(y) for y ∈ x + is just (ρ B f )(y), so we get

Symmetric group actions
Move this remark someplace else: It should be noted while the leftmost file and top rank each consist of just a single element of P , the two associated toggle-operations on J(P ), viewed as permutations of J(P ), do not have the same cycle-structure: the former acts nontrivially on many elements of J(P ) while the latter acts nontrivially on only two of them.

Rank-toggling and rowmotion
This subsection needs to be written.

File-toggling and promotion
Recall that P = [a] × [b] can be partitioned into files numbered 1 through n − 1 from left to right. Given f :P → R + with f (0) = f (1) = 1, let p i (1 ≤ i ≤ n − 1) be the product of the numbers f (x) with x belonging to the ith file of P , let p 0 = p n = 1, and for 1 ≤ i ≤ n let q i = p i /p i−1 . Call q 1 , . . . , q n the quotient sequence associated with f , and denote it by Q(f ). This is analogous to the difference sequence introduced in [PR13]. Note that the product q 1 · · · q n telescopes to p n /p 0 = 1. For i between 1 and n − 1, let φ * i be the product of the commuting involutions φ x for all x belonging to the ith file. Lastly, given a sequence of n numbers w = (w 1 , . . . , w n ), and given 1 ≤ i ≤ n − 1, define σ i (w) = (w 1 , . . . , w i−1 , w i+1 , w i , w i+2 , . . . , w n ); that is, σ i switches the ith and i + 1st entries of w.
Lemma 6. For all 1 ≤ i ≤ n − 1, That is, toggling the ith file of f swaps the ith and i + 1st entries of the quotient sequence of f . Proof.
We have p ′ j = p j for all j = i (since only the values associated with elements of P of the ith file are affected by φ i ), so we have q ′ j = p ′ i /p ′ i−1 = p i /p i−1 = q j for all j other than i and i + 1. The product q ′ 1 · · · q ′ n telescopes to 1 as before. The lemma asserts that q ′ i = q i+1 and q ′ i+1 = q i . It suffices to prove just one of the two assertions, since we know ahead of time that q ′ i q ′ i+1 = q i q i+1 . Expressed in terms of the p j 's, the assertion q ′ i = q i+1 amounts to the claim p ′ i /p ′ i−1 = p i+1 /p i , or equivalently At this point we write p i p ′ i as the product of f (x)f ′ (x) as x varies over the ith file of P . For each x in the ith file of P , {f (y) : y ⋗ x} (with w, y ∈P ). Now we note a key property of the structure of P = [a] × [b]: if x + and x − are vertically adjacent elements of a given file, with x + above x − , the w's that contribute to L x + are precisely the y's that contribute to R x − . So, when we take the product of f (x)f ′ (x) = L x R x over all x in the ith file, the factors L x + and R x − combine to give w f (w) where w varies over the elements satisfying x − ⋖ w ⋖ x + (here we are using the identity (5)). The only factors that do not combine in this way are L x where x is the bottom element of the ith file and R x where x is the top element of the ith file. Both of these factors can be written in the form f (z) where z is a single element ofP belonging to either the i − 1st or i + 1st file. By examining cases, it is easy to check that every element of the i − 1st file or i + 1st file makes a single multiplicative contribution, so that p i p ′ i is the product of f (z) as z varies over the union of the i − 1st and i + 1st files of P . But this product is precisely p i−1 p i+1 . So we have proved that p i p ′ i = p i−1 p i+1 , which concludes the proof.
Perhaps the rest of this subsection should be moved elsewhere.
Recalling that π B is the composition φ * n−1 • · · · • φ * 1 , we have: Proof of Theorem 4. For each i between 1 and n−1, view q i as a function of f . Corollary 7 tells us that the values q i (π 0 B f ), q i (π 1 B f ), q i (π 2 B f ), . . . q i (π n−1 B f ) are respectively equal to the numbers q i (f ), q i+1 (f ), . . . , q i−1 (f ), which multiply to 1. Therefore q i (viewed as a function of f ) is multiplicatively homomesic under the action of π B (with average value 1 on all orbits), for all 1 ≤ i ≤ n−1. Therefore the same is true of p 1 = q 1 , p 2 = q 1 q 2 , p 3 = q 1 q 2 q 3 , etc. Hence p 1 p 2 · · · p n−1 = |f | is also multiplicatively 1-homomesic, as claimed.
The preceding argument does more than confirm Theorem 4. If we define |v| i to be the quantity p i (f ) (where as earlier f and v are two different symbols for the same mathematical object), the argument tells us that the n − 1 functions v → |v| i have the property that (The original theorem asserts this just for the product |v| 1 |v| 2 · · · |v| n−1 = |v|.) Also note that, although the theorem asserts homomesy only for v's in (R + ) P , the same holds for every v in a dense open subset of R P , whose complement consists of points for which the orbit v, π B (v), . . . , π n−1 B (v) is not well-defined because of some denominator vanishing. The same is true for As a final remark, it should be mentioned that although the operations φ * i generate an S n -action on the Q-vectors associated with arrays f , they do not generate an S n -action on the arrays f themselves; for instance, in the case P = [2] × [2], (φ * 1 φ * 2 ) 3 does not send (.3, .4, .5, .7) to itself. 10

Pairing
This section needs to be written.

Recombination
We have explained rowmotion and promotion as toggling by ranks (from top to bottom) and by files (from left to right). A different way to understand these operations is by toggling fibers 11 of [a] × [b], that is, sets of the form [a]×{j} or {i}× [b]. We refer to fibers as being positive or negative according to whether their slope in the rc embedding is +1 or −1.
We illustrate these ideas in the context of P = [3] × [3]. Here we see two orders in which one can toggle all the elements of P to obtain ρ. The first is just the standard order for rowmotion (from top to bottom rank by rank, from left to right within each rank). The second toggles the elements in the topmost positive fiber from top to bottom, then the elements in the middle positive fiber from tom to bottom, and then the elements in the bottommost positive fiber from top to bottom. Since the element of P marked 3 in the left frame neither covers nor is covered by the element of P marked 4, the associated toggles commute, and the same goes for the two toggles associated with the elements of P marked 6 and 7. Hence the composite operation on the left ("rowmotion by ranks") coincides with the composite operation on the right ("rowmotion by fibers"). There is a similar picture for promotion. Here we see two other orders in which one can toggle all the elements of P . The first is just the standard order for promotion (from left to right file by file, from top to bottom within each file). The second toggles the elements in the topmost positive fiber from left to right, then the elements in the middle positive fiber from left to right, and then the elements in the bottommost positive fiber from left to right. As before, the 3 and the 4 can be swapped, as can the 6 and the 7. Hence the composite operation on the left ("promotion by files") coincides with the composite operation on the right ("promotion by fibers"). Note that in both cases we divide the poset into positive fibers, and toggle them from top to bottom; the only difference is whether we toggle the elements within each fiber from left to right (promotion) or right to left (rowmotion). Promotion and rowmotion are even more intimately linked than the preceding discussion might suggest; this link manifests itself if we look at the negative fibers.
Here is a partial orbit of ρ B for P = [3] × [3]: Now consider the array g = ∆f formed by recombining the bottom negative fiber of f (consisting of a 1 , a 2 , a 3 ), the middle negative fiber of ρ B f (consisting of b 4 , b 5 , b 6 ), and the top negative fiber of ρ 2 B f (consisting of c 7 , c 8 , c 9 ), as shown in the left frame of the figure below: We claim that π B g coincides with the array formed by recombining the bottom negative fiber of ρ B f (consisting of b 1 , b 2 , b 3 ), the middle negative fiber of ρ 2 B f (consisting of c 4 , c 5 , c 6 ), and the top negative fiber of ρ 3 B f (consisting of d 7 , d 8 , d 9 ), as shown in the right frame of the figure above. For example, in both cases, when one computes c 5 in terms of previously-computed entries, the governing relation is b 5 c 5 = (b 2 + b 6 )(c 4 c 8 ).
More generally, given P = [a] × [b] and given an array f : P → (R + ) n , let ∆f : P → (R + ) n be the array whose b negative fibers, read from bottom to top, have the same values as the corresponding negative fibers in f , That is, ∆ is an equivariant bijection between the action of rowmotion on (R + ) P and the action of promotion on (R + ) P .
Proof. We will show that for all (i, j) ∈ [a] × [b] by left-to-right induction (starting with (a, 1) and ending with (1, b)). The reader may find it handy to consult the figure which shows the vicinity of a typical element x = (i, j) in the poset [a] × [b] (away from the boundary). In the boundary cases where one or more of the pairs (i + 1, j), (i, j + 1), (i, j − 1), (i − 1, j) does not belong to [a] × [b], the pair(s) may be ignored. Note that (i, j − 1) and (i + 1, j) are to the left of (i, j). If we now assume that (17) holds for (i, j − 1) (which we take to be vacuously true if ) and for (i + 1, j), then and (∆ρ B f )(i + 1, j) = (π B ∆f )(i + 1, j).
Note that ρ B can be described directly via the recurrence where in lieu of including0 and1 we interpret and + of the empty set as 1. We rewrite (20) as where terms f (·, ·) and (ρ B f )(·, ·) are to be ignored if the arguments do not belong to [a] × [b] (and + and of the empty set are taken to be 1). Likewise, promotion can be described by the recurrence From the definition of ∆, we have Hence On the other hand, (18) and (19)) Comparing right-hand sides of the equation blocks, we conclude that (∆ρ B f )(i, j) = (π B ∆f )(i, j), which was to be proved.
For the next theorem, it is helpful to have as a companion to (23). 12 Theorem 9. For any functional F on (R + ) P of the form F f = x∈P f (x) ax with a x ∈ Z for all x ∈ P , In particular, if F is homomesic with respect to promotion, F is homomesic with respect to rowmotion.
Proof. First consider F x f = f (x) a for some particular x ∈ P , say x = (i, j).
Oops; I just discovered that the proof is wrong! The original sketch said: "The recombination picture also tells us that if we look at any particular poset element x ∈ P , the values for f (x) that we see as f ranges over the rowmotion orbit of some f 0 will be the same as the values for g(x) that we see as g ranges over the promotion orbit of g 0 = ∆f 0 . In particular, if the geometric mean of f (x) as f ranges over the rowmotion orbit of f 0 is the same value c for every f 0 . then the geometric mean of g(x) as g ranges over the rowmotion orbit of g 0 is the same value c for every g 0 . This applies not just to the elementary functionals that send f to f (x) for some x ∈ P but also to more general functionals obtained as products of elementary functionals or even more generally, ratios of such products. In this way, we can see that the file-products q i discussed in the proof of Theorem 4 are homomesic under promotion as well as rowmotion." But trying to fill in the details, I learned that I was missing a key detail. So, ignore that end-of-proof symbol: This recombination picture works equally well in the piecewise-linear setting and the combinatorial setting, and in the context of rowmotion and promotion on J(P ) our ∆ is essentially the same equivariant bijection as the one given in Theorem 5.4 of [SW12], which Striker and Williams use to show that rowmotion and promotion, viewed as permutations of J(P ), have the same cycle-structure.

Tropicalization
Theorem 4 is nothing more than a complicated identity involving the operations of multiplication, division, addition, and parallel addition. As such, Theorem 4 can be tropicalized to yield an identity involving the operations of addition, subtraction, min, and max. The resulting identity is nothing other than Theorem 3.
At the same time, we should mention that in a certain sense, Theorem 3 can be proved without relying on Theorem 4. Specifically, one can tropicalize each of the steps in the proof of Theorem 4, using (for instance) the identity min(x, y) + max(x, y) = x + y in place of the identity (5). At the end of section ??, we proved a refined version of Theorem 4, and this tropicalizes, yielding a refined version of Theorem 3: Using our usual conflation of f and v, let us redefine |v| i to be the sum of f (x) over all x belonging to the ith file of P . Then the n − 1 functions v → |v| i have the property that for every v in R P , These n − 1 functions are also homomesic for π P , although now each file has its own (non-zero) value c i with the property that the average of |v| i over every orbit is ai/n if i ≤ b and b(n − i)n if i > b. Thus we obtain a refined version of Theorem 2.
Lastly, if we specialize to the vertices of O(P ), we obtain a refined version of Theorem 1 (also found in [PR13]

More homomesies
This section needs to be written. 12 Directions for future work

Other invariants
This subsection needs to be junked, although parts of it might belong elsewhere in the article.
We have mentioned that for piecewise-linear rowmotion and promotion, the quantity min{f (y) − f (x) : x, y ∈P , x ⋖ y} is invariant, and that for birational rowmotion and promotion, the quantity + {f (x)/f (y) : x, y ∈P , x ⋖ y} is invariant. There is a sense in which the space of invariant functionals and the space of homomesic functionals are dual to one another (see section 2.2 of [PR13]), so it is natural to complement our program of finding all the homomesic functionals for π B and ρ B by the program of finding all the invariants. In the article [KB95], Kirillov and Berenstein show that when P is the triangle poset mentioned in section 3, there is more than one invariant functional; see for instance Lemma 3.1 of their paper and the paragraph that follows its proof. Berenstein has mentioned (in private communication) that the motivation for the Kirillov-Berenstein piecewise-linear invariants comes from the birational setting, which admits birational counterparts of so that φ 12 P sends v k to v k−2 . Therefore there are rational points in O(P ) whose orbits are arbitrarily large.
On the other hand, it appears that some form of homomesy applies even to maps of infinite order. Empirical evidence suggests that for the action of φ P on O(P ) ∩ Q P (which consist of finite orbits of unbounded size), the function f : v → v 1 − v 2 − v 3 + v 4 is homesic with average 0 on all orbits. Indeed, we conjecture that even if v has infinite φ P -orbit, lim n→∞ 1 n n−1 k=0 f (φ k P (v)) = 0.

Other homomesies
A different sort of homomesy, already studied in the combinatorial literature, relates to cardinality of antichains rather than cardinality of order ideals. Instead of α 3 • α 2 • α 1 : J(P ) → J(P ) we can consider α 2 • α 1 • α 3 : A(P ) → A(P ). Various conjectures of Panyushev [Pan08], some of which were proved by Armstrong et al. [AST11], assert that the function A → |A| : A(P ) → Z is homomesic under the action of α 2 •α 1 •α 3 , for P a root poset. Experiments suggest that for P = [a] × [b], this homomesy lifts to the piecewise-linear setting. Putting it differently, we use the transfer map to move the dynamics on O to C, where there are different local evaluation operations (which cannot be expressed as linear combinations of the local evaluation operations for O), and different homomesies. It is natural to suspect that what works for piecewise-linear setting also works in the birational setting, if one detropicalizes the maps in the natural way. We have checked that this works for some small values of a and b.

Other posets
Experiments also suggest that much of this theory carries over from the poset [a] × [b] (a minuscule poset of type A) to other minuscule posets, and root posets as well. For instance, Shahrzad Haddadan (in work to be described elsewhere) has shown that if P is a root poset of type A, then the functional I → |I + | − |I − | is homomesic for the action of rowmotion and promotion on J(P ), where I + and I − denote the elements of I of even and odd rank, respectively.