Iterative properties of birational rowmotion

We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we prove that birational rowmotion has order p+q on the (p, q)-rectangle poset (i.e., on the product of a p-element chain with a q-element chain); we furthermore compute its orders on some triangle-shaped posets and on a class of posets which we call"skeletal"(this class includes all graded forests). In all cases mentioned, birational rowmotion turns out to have a finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the AA case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets. We also make a digression to study classical rowmotion on skeletal posets, since this case has seemingly been overlooked so far.


Introduction
The present paper continues our study of periodicity of the birational rowmotion map on finite ranked posets. While in our first paper [GrRo14a] we consider the case of "skeletal posets", which generalize the class of graded forests, here posets of rectangular and triangular shape are the primary focus. Our main motivation for proving periodicity of rectangles (Theorem 30) came from the work of David Einstein and James Propp's [EiPr13,EiPr14], where they consider the lifting of the combinatorial rowmotion operator on a poset and the "homomesy" phenomenon (well-understood for products of chains, i.e., rectangles) to piecewise-linear and birational settings. For the notion of homomesy used therein, the maps considered need to have finite order, a fact which is no longer obvious when rowmotion operates on infinite sets. Our paper can nevertheless be read independently of their work or our earlier paper.
A shorter exposition of the main points of this work and [GrRo14a] appears in a 12page extended abstract for FPSAC 2014 [GrRo13]. A more detailed exposition is available on the ArXiv [GrRo14b] and is updated more frequently on the first author's website. 1 Let P be a finite poset, and J (P ) the set of the order ideals of P . (See [Stan11, Chapter 3] for poset basics.) Rowmotion is a classical map J (P ) → J (P ) which can be defined in various ways, one of which is as follows: For every v ∈ P , let t v : J (P ) → J (P ) be the map sending every order ideal S ∈ J (P ) to ∈ S and S ∪ {v} ∈ J (P ) ; S \ {v} , if v ∈ S and S \ {v} ∈ J (P ) ; S, otherwise.
These maps t v are called (classical) toggles, since all they do is "toggle" an element into or out of an order ideal. Let (v 1 , v 2 , . . . , v m ) be a linear extension of P . Then, (classical) rowmotion is defined as the composition t v 1 •t v 2 •. . .•t vm (which, as can be seen, does not depend on the choice of the particular linear extension (v 1 , v 2 , . . . , v m )). This rowmotion map has been studied from various perspectives; in particular, it is isomorphic to the map f of Fon-der-Flaass [Flaa93], the map F −1 of Brouwer and Schrijver [BrSchr74], and the map f −1 of Cameron and Fon-der-Flaass [CaFl95]. More recently, it has been studied (and christened "rowmotion") in Striker and Williams [StWi11], where further sources and context are also given. We have also covered the case of P being a "skeletal poset" (such as a graded forest with all leaves on the same rank) in [GrRo14a].
Among the questions that have been posed about rowmotion, the most prevalent was probably that of its order: While it clearly has finite order (being a bijective map from the finite set J (P ) to itself), it turns out to have a much smaller order than one would naively expect when the poset P has certain "special" forms (e.g., a rectangle, a root poset, a product of a rectangle with a 2-chain, or -as shown in [GrRo14a] -a graded forest). Most strikingly, when P is the product of two chains [p] × [q] (denoted Rect (p, q) in Definition 27), then the (p + q)-th power of the rowmotion operator is the identity map. This is proven in [BrSchr74,Theorem 3.6] and [Flaa93, Theorem 2] (and a proof can also be constructed from the ideas given in [PrRo13,§3.3.1]).
In [EiPr13], David Einstein and James Propp (inspired by work of Arkady Berenstein and Anatol Kirillov) have lifted the rowmotion map from the set J (P ) of order ideals to the progressively more general setups of: (b) even more generally, the affine variety of K-labellings of P for K an arbitrary infinite field.
In case (a), order ideals of P are replaced by points in the order polytope O (P ), and the role of the map t v (for a given v ∈ P ) is assumed by the map which reflects the v-coordinate of a point in O (P ) around the midpoint of the interval of all values it could take without the point leaving O (P ) (while all other coordinates are considered fixed). The operation of "piecewise linear" rowmotion is still defined as the composition of these reflection maps in the same way as rowmotion is the composition of the toggles t v . This "piecewise linear" rowmotion extends (interpolates, even) classical rowmotion, as order ideals correspond to the vertices of the order polytope O (P ) (see [Stan86,Corollary 1.3]). We will not study case (a) here, since all of the results we could find in this case can be obtained by tropicalization from similar results for case (b).
In case (b), instead of order ideals of P one considers maps from the poset P := {0} ⊕ P ⊕ {1} (where ⊕ stands for the ordinal sum 2 ) to a given infinite field K (or, more graphically, labellings of the elements of P by elements of K, along with two additional labels "at the very bottom" and "at the very top"). The maps t v are then replaced by certain birational maps which we call birational v-toggles (Definition 5); the resulting composition is called birational rowmotion and denoted by R. By a careful limiting procedure (the tropical limit), we can "degenerate" R to the "piecewise linear" rowmotion of case (a), and thus it can be seen as an even higher generalization of classical rowmotion. We refer to the body of this paper for precise definitions of these maps. Note that birational v-toggles (but not birational rowmotion) in the case of a rectangle poset have also appeared in [OSZ13,(3.5)], but (apparently) have not been composed there in a way that yields birational rowmotion.
As in the case of classical rowmotion on J (P ), the most interesting question is the order of this map R, which in general no longer has an obvious reason to be finite (since the affine variety of K-labellings is not a finite set like J (P )). Indeed, for some posets P this order is infinite. We have shown that R has finite order for a wide class of graded posets P in [GrRo14a]; this class covers (in particular) all forests which are graded as posets (i.e., have their leaves all at the same rank). In this paper we will prove the following: • Birational rowmotion on a p × q-rectangle has order p + q and satisfies a further symmetry property (Theorem 32). These results have originally been conjectured by James Propp and the second author, and can be used as an alternative route to certain properties of (Schützenberger's) promotion map on semistandard Young tableaux.
• Birational rowmotion on certain triangle-shaped posets (this is made precise in Sections 9, 10, 11) also has finite order (computed explicitly below). We show this for three kinds of triangle-shaped posets (obtained by cutting the p × p-square in two along either of its two diagonals) and conjecture it for a fourth (a quarter of a p × p-square obtained by cutting it along both diagonals).
The proof of the most difficult and fundamental case -that of a p × q-rectangleis inspired by Volkov's proof of the "rectangular" (type-AA) Zamolodchikov conjecture [Volk06], which uses a similar idea of parametrizing (generic) K-labellings by matrices (or tuples of points in projective space). There is, of course, a striking similarity between the fact itself and the Zamolodchikov conjecture; yet, we were not able to reduce either result to the other.
Applications of the results of this paper (specifically Theorems 30 and 32) are found in [EiPr13]. Directions for further study include relations to the totally positive Grassmannian and generalizations to further classes of posets.
An extended (12-page) abstract [GrRo13] of this paper and [GrRo14a] was presented at the FPSAC 2014 conference.

Leitfaden
This paper can be read in linear order and independently of [GrRo14a] (provided the reader is willing to trust a few results quoted from [GrRo14a] or supply the rather simple proofs on their own). If the reader is not interested in proofs, it is also sufficient to cherrypick the results from Sections 1, 3, 9, 10, 11, 12 and 13 only.

Acknowledgments
When confronted with the (then open) problem of proving what is Theorem 30 in this paper, Pavlo Pylyavskyy and Gregg Musiker independently suggested reading [Volk06]. This suggestion proved highly useful and built the cornerstone of this paper.
Our conventions and notations for posets and related notions (such as linear extension) closely follow that of [Stan11,Chapter 3]. In particular, we write u v to mean "u is covered by v" (i.e., u < v and there is no w ∈ P such that u < w < v). Our notion of "n-graded" is slightly nonstandard, as explained after Definition 14.
Definition 1. Let P be a finite poset. Then, P will denote the poset defined as follows: As a set, let P be the disjoint union of the set P with the two-element set {0, 1}. The smaller-or-equal relation on P will be given by (a b) ⇐⇒ (either (a ∈ P and b ∈ P and a b in P ) or a = 0 or b = 1) (where "either/or" has a non-exclusive meaning). Here and in the following, we regard the canonical injection of the set P into the disjoint union P as an inclusion; thus, P becomes a subposet of P . In the terminology of Stanley's [Stan11, section 3.2], this poset P is the ordinal sum {0} ⊕ P ⊕ {1}.
Definition 2. Let P be a finite poset and K be a field (henceforth). A K-labelling of P will mean a map f : P → K. Thus, K P is the set of all K-labellings of P . If f is a K-labelling of P and v is an element of P , then f (v) will be called the label of f at v.
Our basic object of study, birational rowmotion, will be defined as a map between K-labelings of P . Unfortunately, it can happen that for certain choices of labels, this map will lead to division by zero and not be well-defined. To handle this we make use of some standard notions in basic algebraic geometry: algebraic varieties, the Zariski topology and dominant rational maps. However, the only algebraic varieties that we consider are products of affine spaces and their open subsets.
Definition 3. We use the punctured arrow to signify rational maps (i.e., a rational map from a variety U to a variety V is called a rational map U V ). A rational map U V is said to be dominant if its image is dense in V (with respect to the Zariski topology).
Whenever we are working with a field K, we will tacitly assume that K is either infinite or at least can be enlarged when necessity arises. This assumption is needed to clarify the notions of rational maps and generic elements of algebraic varieties over K. (We will not require K to be algebraically closed.) The words "generic" and "almost" will always refer to the Zariski topology. For example, if U is a finite set, then an assertion saying that some statement holds "for almost every point p ∈ K U " is supposed to mean that there is a Zariski-dense open subset D of K U such that this statement holds for every point p ∈ D. A "generic" point on an algebraic variety V (for example, this can be a "generic matrix" when V is a space of matrices, or a "generic K-labelling of a poset P " when V is the space of all K-labellings of P ) means a point lying in some fixed Zariski-dense open subset S of V ; the concrete definition of S can usually be inferred from the context (often, it will be the subset of V the electronic journal of combinatorics 22(3) (2015), #P3.40 on which everything we want to do with our point is well-defined), but of course should never depend on the actual point. (Note that one often has to read the whole proof in order to be able to tell what this S is. This is similar to the use of the "for small enough" wording in analysis, where it is often not clear until the end of the proof how small exactly the needs to be.) We are sometimes going to abuse notation and say that an equality holds "for every point" instead of "for almost every point" when it is really clear what the S is. (For example, if we say that "the equality x 3 − y 3 x − y = x 2 + xy + y 2 holds for every x ∈ K and y ∈ K", it is clear that S has to be the set Remark 4. Most statements that we make below work not only for fields, but also more generally for semifields such as the semifield Q + of positive rationals or the tropical semiring. We will not concern ourselves with stating them for semifields; a reader curious about this possibility is referred to [GrRo14a,§2] for details on how identities between subtraction-free rational functions can be transferred from fields to semifields. We are now ready to introduce the concepts of a birational toggle and of birational rowmotion. These concepts originate in [EiPr13] (where they have been studied over R + rather than over fields as we do) and are the focus of our work here and in [GrRo14a].
Definition 5. Let v ∈ P . We define a rational map T v : K P K P , called the v-toggle, by for all w ∈ P and f ∈ K P . Note that this rational map T v is well-defined, because the right-hand side of (1) is well-defined on a Zariski-dense open subset of K P .
The following simple properties of these maps T v are proven in [GrRo14a,§2].
Proposition 6. Let v ∈ P . Then, the rational map T v is an involution, i.e., the map T 2 v is well-defined on a Zariski-dense open subset of K P and satisfies T 2 v = id on this subset. As a consequence, T v is a dominant rational map.
The reader should remember that dominant rational maps (unlike general rational maps) can be composed, and their compositions are still dominant rational maps. Of course, in using the notion of dominant maps, we are relying on our assumption that K is infinite. We can now define birational rowmotion: is a linear extension of P . This rational map is well-defined (in particular, it does not depend on the linear extension (v 1 , v 2 , . . . , v m ) chosen), as has been proven in [GrRo14a,§2]. This rational map will be denoted by R, or by R P when we wish to make its dependence on P explicit.
Then f and the output of toggling f at the element (2, 2) are visualized as follows: The label at (2, 2), which is the only one that changed, was computed via To compute Rf , we need to toggle at each vertex of P along a linear extension. Computing successively T (2,1) T (2,2) f , T (1,2) T (2,1) T (2,2) f , and Rf = T (1,1) T (1,2) T (2,1) T (2,2) f gives (respectively) There are two surprises here. First, it turns out that R 4 f = f . This is not obvious, but generalizes in at least two ways: On the one hand, our poset P is a particular case of what we called a "skeletal poset" in [GrRo14a, §9], a class of posets which all were shown in [GrRo14a, §9] to satisfy R n = id for some sufficiently high positive integer n (which can be explicitly computed depending on P ). On the other hand, our poset P is a particular case of rectangle posets, which turn out (Theorem 30) to satisfy R p+q = id with p and q being the side lengths of the rectangle. Second, on a more subtle level, the rational functions appearing as labels in Rf , R 2 f and R 3 f are not as "wild" as one might expect. The values (Rf ) ((1, 1)), (R 2 f ) ((1, 2)), (R 2 f ) ((2, 1)) and (R 3 f ) ((2, 2)) each have the form ab f (v) for some v ∈ P . This is a "reciprocity" phenomenon which turns out to generalize to arbitrary rectangles (Theorem 32).
A different example for birational rowmotion is given in [GrRo14a,§2]. The next proposition merely describe the situation when one is partway through the toggling process. Here (and elsewhere), we tacitly assume that Rf is well-defined because these assumptions are satisfied when the parameters belong to some Zariski-dense open subset of their domains.
Proposition 10. Let v ∈ P . Let f ∈ K P . Then, . (2) Here (and in later statements such as Proposition 10), we take the liberty of leaving assumptions such as "Assume that Rf is well-defined" unsaid. These assumptions are satisfied when the parameters belong to some Zariski-dense open subset of their domains.
Proposition 13. Let f ∈ K P and g ∈ K P be such that f (0) = g (0) and f (1) = g (1). Assume that for every v ∈ P. (3) (This means, in particular, that we assume that all denominators in (3) are nonzero.) Then, g = Rf . 3

Auxiliary results
In this section, we collect further results from [GrRo14a] (which the interested reader may consult), needed for the proofs (but not the statements) of our results.
Definition 14. Let n ∈ N. We call a finite poset P n-graded if there exists a surjective map deg : P → {1, 2, . . . , n} satisfying: Assertion 1: Any two elements u and v of P such that u v satisfy deg u = deg v + 1. Assertion 2: We have deg u = 1 for every minimal element u of P . Assertion 3: We have deg v = n for every maximal element v of P .
Throughout the rest of this paper, unless otherwise stated, P will denote an n-graded poset (for some n ∈ N).
Definition 16. Let P be an n-graded poset. Then, there exists a surjective map deg : P → {1, 2, . . . , n} that satisfies the Assertions 1, 2 and 3 of Definition 14. A moment of thought reveals that such a map deg is also uniquely determined by P .
Moreover, we extend this map deg to a map P → {0, 1, . . . , n + 1} by letting it map 0 to 0 and 1 to n + 1. This extended map will also be denoted by deg. Notice that this extended map deg still satisfies Assertion 1 of Definition 14 if P is replaced by P in that assertion.
The notion of an "n-graded poset" we just defined is identical with the notion of a "graded finite poset of rank n − 1" as defined in The rationale for setting deg 0 = 0 and deg 1 = n + 1 in Definition 16 was to make the following hold: Proposition 17. Let P be an n-graded poset. Let u ∈ P and v ∈ P . Consider the map deg : P → {0, 1, . . . , n + 1} defined in Definition 16.
(a) If u v in P , then deg u = deg v − 1.
More generally, direct computation easily shows that Corollary 21. In the situation of Proposition 19, we have (a) If α and β are two partial maps from the set S, then we write "α = β" to mean: every s ∈ S for which both α (s) and β (s) are well-defined satisfies α (s) = β (s). This is, per se, not a well-behaved notation (e.g., it is possible that three partial maps α, β and γ satisfy α = β and β = γ but not α = γ). However, we are going to use this notation for rational maps and their quotients (and, of course, total maps) only; in all of these cases, the notation is well-behaved (e.g., if α, β and γ are three rational maps satisfying α = β and β = γ, then α = γ, because the intersection of two Zariski-dense open subsets is Zariski-dense and open). we are disregarding the fact that ϕ is only a partial map; we will be working only with dominant rational maps and their quotients (and total maps), so nothing will go wrong.
We denote the order of a partial map ϕ : S S as ord ϕ.
Definition 23. Let P be a poset. Then, P op will denote the poset defined on the same ground set as P but with the order relation defined by for all a ∈ P and b ∈ P ) (where < P denotes the smaller-than relation of the poset P , and where < P op denotes the smaller-than relation of the poset P op which we are defining). The poset P op is called the opposite poset of P .
Note that P op is called the dual of the poset P in [Stan11].
Remark 24. It is clear that (P op ) op = P for any poset P . Also, if n ∈ N, and if P is an n-graded poset, then P op is an n-graded poset.
Proposition 25. Let P be a finite poset. Let K be a field. Then, ord (R P op ) = ord (R P ).
We notice one further result, which was never explicitly stated in [GrRo14a] but follows from [GrRo14a,Proposition 62]. This lemma will only be used to show that ord R is equal to (rather than only a divisor of) a certain value.
Lemma 26. Let n ∈ N. Let K be a field. Let P be an n-graded poset. Then, n+1 | ord R.
(We understand that m | ∞ for any positive integer m.)

The rectangle: statements of the results
We now are ready to state our main results.
The following periodicity theorem was conjectured by James Propp and the second author: Theorem 30. The order of birational rowmotion on the K-labelings of a p × q-rectangle This is a birational analogue (and, using the reasoning of [EiPr13], generalization) of the classical fact (appearing in [StWi11, Theorem 3.1] and [Flaa93, Theorem 2]) that ord r Rect(p,q) = p + q (where r P denotes the classical rowmotion map on the order ideals of a poset P ). When p 2 and q 2, Theorem 30 follows rather easily from results in [GrRo14a, §9] (because Rect (p, q) is a skeletal poset in this case).
Remark 31. Theorem 30 generalizes a well-known property of promotion on semistandard Young tableaux of rectangular shape, albeit not in an obvious way. Let N be a nonnegative integer, and let λ be a partition. Let SSYT N λ denote the set of all semistandard Young tableaux of shape λ whose entries are all N . One can define a map Pro : SSYT N λ → SSYT N λ called jeu-de-taquin promotion (or Schützenberger promotion, or simply promotion when no ambiguities can arise); see [Russ13, §5.1] for a precise definition. This map has some interesting properties already for arbitrary λ, but the most interesting situation is that of λ being a rectangular partition (i.e., a partition all of whose nonzero parts are equal). In this situation, a folklore theorem states that Pro N = id. (The particular case of this theorem when Pro is applied only to standard Young tableaux is well-known (see, e.g., [Haiman92, Theorem 4.4]), but the only proof of the general theorem that we were able to find in the literature is Rhoades [EiPr13] are equivalent, while the definition in [Rhoa10,§2] defines the inverse of the map defined in the other two sources. Unfortunately, we were unable to find the proofs of these facts in existing literature; they are claimed in [KiBe95, Propositions 2.5 and 2.6], and can be proven using the concept of tableau switching [Leeu01, Definition 2.2.1].
Besides Theorem 30, our other main theorem states a symmetry property of birational rowmotion on the p×q-rectangle (referred to as the "pairing property" in [EiPr13]), which was also conjectured by Propp and the second author. It generalizes the "reciprocity phenomenon" observed on the 2 × 2-rectangle in Example 9. Theorem 32. Let f ∈ K Rect(p,q) . Assume that R Rect(p,q) f is well-defined for every ∈ {0, 1, . . . , i + k − 1}. Then for any (i, k) ∈ Rect (p, q) we have .
Remark 33. While Theorem 30 only makes a statement about R Rect(p,q) , it can be used (in combination with results from [GrRo14a]) to derive upper bounds on the order of R P for some other posets P . For example, let N denote the (eponymously named) . Then for details). It can actually be shown that ord (R N ) = 15 by direct computation.
In the same vein it can be shown that ord R Rect(p,q)\{(1,1),(p,q)} | lcm (p + q − 2, p + q) for any integers p > 1 and q > 1. This doesn't, however, generalize to arbitrary posets obtained by removing some ranks from Rect (p, q) (indeed ord R P is infinite for some posets of this type, cf. Section 12).

Reduced labellings
The proof that we give for Theorem 30 and Theorem 32 is largely inspired by the proof of Zamolodchikov's conjecture in case AA given by Volkov in [Volk06] 4 . This is not very surprising because the orbit of a K-labelling under birational rowmotion appears superficially similar to a solution of a Y -system of type AA. Yet we do not see a way to derive Theorem 30 from Zamolodchikov's conjecture or vice versa. (Here the Y-system has an obvious "reducibility property", consisting of two decoupled subsystems -a property not obviously satisfied in the case of birational rowmotion.) The first step towards our proof of Theorem 30 is to restrict attention to so-called reduced labellings, which are not much less general than arbitrary labellings: Many results can be proven for all labellings by means of proving them for reduced labellings first, and then extending them to general labellings by fairly simple arguments. We will use this tactic in our proof of Theorem 30. A slightly different way to reduce the case of a general labelling to that of a reduced one is taken in [EiPr13,§4].
The set of all reduced labellings f ∈ K Rect(p,q) will be identified with K Rect(p,q) in the obvious way.
Note that fixing the values of f (0) and f (1) like this makes f "less generic", but still the operator R Rect(p,q) restricts to a rational map from the variety of all reduced labellings f ∈ K Rect(p,q) to itself. (This is because the operator R Rect(p,q) does not change the values at 0 and 1, and does not degenerate from setting f (0) = f (1) = 1.) Proposition 35. Assume that almost every (in the Zariski sense) reduced labelling f ∈ Proof. Let g ∈ K Rect(p,q) be any K-labelling of Rect (p, q) which is sufficiently generic for R p+q Rect(p,q) g to be well-defined. We can easily find a (p + q + 1)-tuple (a 0 , a 1 , . . . , a p+q ) ∈ (K × ) p+q+1 such that (a 0 , a 1 , . . . , a p+q ) g is a reduced K-labelling (in fact, set a 0 = 1 g (0) and a p+q = 1 g (1) , and choose all other a i arbitrarily). Corollary 21 then yields We have assumed that almost every (in the Zariski sense) reduced labelling f ∈ can be written as an equality between rational functions in the labels of f , and thus it must hold everywhere if it holds on a Zariski-dense open subset). Applying this to f = (a 0 , a 1 , . . . , a p+q ) g, we obtain that R p+q Rect(p,q) ((a 0 , a 1 , . . . , a p+q ) g) = (a 0 , a 1 , . . . , a p+q ) g. Thus, (4)). We can cancel the "(a 0 , a 1 , . . . , a p+q ) " from both sides of this equality (because all the a i are nonzero), and thus obtain g = R p+q Rect(p,q) g. Now, forget that we fixed g. We thus have proven that g = R p+q Rect(p,q) g holds for every K-labelling g ∈ K Rect(p,q) of Rect (p, q) which is sufficiently generic for R p+q Rect(p,q) g to be well-defined. In other words, R p+q Rect(p,q) = id as partial maps. Hence, ord R Rect(p,q) | p+q. On the other hand, Lemma 26 yields that ord R Rect(p,q) is divisible by (p + q − 1) + 1 = p + q. Combined with ord R Rect(p,q) | p + q, this yields ord R Rect(p,q) = p + q.
Let us also formulate the particular case of Theorem 32 for reduced labellings, which we will use a stepping stone to the more general theorem. .

The Grassmannian parametrization: statements
In this section, we introduce the main actor in our proof of Theorem 30: an assignment of a reduced K-labelling of Rect (p, q), denoted Grasp j A, to any integer j and almost any matrix A ∈ K p×(p+q) (Definition 44). This assignment will give us a family of K-labellings of Rect (p, q) which is large enough to cover almost all reduced K-labellings of Rect (p, q) (Proposition 49), while at the same time the construction of this assignment makes it easy to track the behavior of the K-labellings in this family through multiple iterations of birational rowmotion. Indeed, we will see that birational rowmotion has a very simple effect on the reduced K-labelling Grasp j A (Proposition 48).
Moreover, we extend this definition to all i ∈ Z as follows: For every i ∈ Z, let Consequently, the sequence (A i ) i∈Z is periodic with period dividing 2v, and if u is odd, the period also divides v.) (c) For any two integers a and b satisfying a b, we let A [a : b] be the matrix whose columns (from left to right) are A a , A a+1 , . . . , A b−1 .
(d) For any four integers a, b, c and d satisfying a b and c d, Example 38. If A = 3 5 7 4 1 9 , then Remark 39. Some parts of Definition 37 might look accidental and haphazard; here are some motivations and aide-memoires: The choice of sign in Definition 37 (b) is not only the "right" one for what we are going to do below, but also naturally appears in [Post06,Remark 3.3]. It guarantees, among other things, that if A ∈ R u×v is totally nonnegative, then the matrix having columns A 1+i , A 2+i , . . ., A v+i is totally nonnegative for every i ∈ Z.
The convention to define det (A [a : b | c : d]) as 0 in Definition 37 (e) can be motivated using exterior algebra as follows: If we identify ∧ u (K u ) with K by equating with 1 ∈ K the wedge product e 1 ∧ e 2 ∧ . . . ∧ e u of the standard basis vectors, then det ( has to be 0 in this case.
The following four propositions are all straightforward observations.
Proof. This follows from the fact that permuting the columns of a matrix multiplies its determinant by the sign of the corresponding permutation.
Proof. Straightforward from the definition and basic properties of the determinant. Definition 44. Let p and q be two positive integers. Let A ∈ K p×(p+q) . Let j ∈ Z.
(a) We define a map Grasp j A ∈ K Rect(p,q) by This is well-defined when the matrix A is sufficiently generic (in the sense of Zariski topology), since the matrix A [j : j + i | j + i + k : j + p + k] is obtained by picking p distinct columns out of A, some possibly multiplied with (−1) u−1 . This map Grasp j A will be considered as a reduced K-labelling of Rect (p, q) (since we are identifying the set of all reduced labellings f ∈ K Rect(p,q) with K Rect(p,q) ).
The term "Grasp" is meant to suggest "Grassmannian parametrization", as we will later parametrize (generic) reduced labellings on Rect (p, q) by matrices via this map Grasp 0 . The reason for the word "Grassmannian" is that, while we have defined Grasp j as a rational map from the matrix space K p×(p+q) , it actually is not defined outside of the Zariski-dense open subset K p×(p+q) rk=p of K p×(p+q) formed by all matrices whose rank is Proposition 46. Let p and q be two positive integers. Let A ∈ K p×(p+q) be a matrix. Then, Grasp j A = Grasp p+q+j A for every j ∈ Z (provided that A is sufficiently generic in the sense of Zariski topology for Grasp j A to be well-defined).
Proposition 47. Let A ∈ K p×(p+q) . Let (i, k) ∈ Rect (p, q) and j ∈ Z. Then (provided that A is sufficiently generic in the sense of Zariski topology for Grasp j A ((i, k)) and Grasp j+i+k−1 A ((p + 1 − i, q + 1 − k)) to be well-defined).
Each of the next two propositions has one of the following sections devoted to its proof. These are the key lemmas that will allow us fairly easily to prove our main Theorems 30, 36 and 32 in Section 8.
(provided that A is sufficiently generic in the sense of Zariski topology for the two sides of this equality to be well-defined).
Proposition 49. For almost every (in the Zariski sense) f ∈ K Rect(p,q) , there exists a matrix A ∈ K p×(p+q) satisfying f = Grasp 0 A.
Proof of Theorem 50. Theorem 50 follows from the well-known Plücker relations (see, e.g., [KlLa72,(QR) The extended versions [GrRo14b] of this paper have a self-contained proof, which we briefly outline here. First we reduce Theorem 50 to its special case, Lemma 51, by shifting columns. The latter can now be derived by (a) using row-reduction to transform as many columns as possible into standard basis vectors; (b) permuting columns to bring the matrices in (6) into block triangular form; and (c) using that the determinant of such a matrix is the product of the determinants of its blocks.
We are now ready to prove the key lemma that birational rowmotion acts by a cyclic shifted on Grasp-labelings.
Proof of Proposition 48. Let f = Grasp j+1 A and g = Grasp j A. We want to show that g = R Rect(p,q) (f ). By Proposition 13 this will follow once we can show that for every v ∈ Rect (p, q) .
Let v = (i, j) ∈ Rect (p, q). We are clearly in one of the following four cases: Case 1: We have v = (1, 1) and v = (p, q).
Due to these two equalities, (8) becomes (by Theorem 50, applied to a = j + 2, b = j + i, c = j + i + k and d = j + p + k).
So we can rewrite the terms , g (v) and f (v) in (7) using the equalities (10), (11), (12) and (13), respectively. The resulting equation is a tautology because all determinants cancel out. This proves (7) in Case 1. Proofs of the other three cases follow the same lines of argument, but are simpler. Note, however, that it is only in Cases 3 and 4 that we use the fact that the sequence (A n ) n∈Z is "(p + q)-periodic up to sign" as opposed to an arbitrary sequence of length-p column vectors.

Dominance of the Grassmannian parametrization
In this section we prove Proposition 49 that the space of K-labelings that we can obtain in the form Grasp 0 A is sufficiently diverse to cover everything we need. Before plunging the electronic journal of combinatorics 22(3) (2015), #P3.40 into the details of the general case, we illustrate the approach we take with an example.
Clearly the condition f = Grasp 0 A imposes 4 equations on the eight entries of A; thus, we are trying to solve an underdetermined system. However, we can get rid of the superfluous freedom if we additionally try to ensure that our matrix A has the form The requirement f = Grasp 0 (I p | B) therefore translates into the following system, which is solved by elimination (in order w, y, z, x) as shown: 1)) ; x = −f ((1, 2))f ((2, 2)) [f ((1, 2)) + f ((2, 1))]f ((1, 1)) ; y = −1 f ((1, 1)) ; z = f ((2, 2)) f ((2, 1)) While the denominators in these fractions can vanish, leading to underdetermination or unsolvability, this will not happen for generic f . We apply this same technique to the general proof of Proposition 49. For any fixed f ∈ K Rect(p,q) , solving the equation f = Grasp 0 A for A ∈ K p×(p+q) can be considered as a system of pq equations on p (p + q) unknowns. While this (nonlinear) system is usually underdetermined, we can restrict the entries of A by requiring that the leftmost p columns of A form the p×p identity matrix, leaving us with only pq unknowns only; for f sufficiently generic, the resulting system will be uniquely solvable by "triangular elimination" (i.e., there is an equation containing only one unknown; then, when this unknown is eliminated, the resulting system again contains an equation with only one unknown, and once this one is eliminated, one gets a further system containing an equation with only one unknown, and so forth).
We will sketch the ideas of this proof, leaving all straightforward details to the reader. We word the argument using algebraic properties of families of rational functions instead of using the algorithmic nature of "triangular elimination" (similarly to how most applications of linear algebra use the language of bases of vector spaces rather than talk about the process of solving systems by Gaussian elimination). While this clarity comes at the cost of a slight disconnect from the motivation of the proof, we hope that the reader will still see how the wind blows. We first introduce some notation to capture the essence of "triangular elimination" without having to talk about actually moving around variables in equations.
Definition 53. Let F be a field. Let P be a finite set.
(a) Let x p be a new symbol for every p ∈ P. We will denote by F (x P ) the field of rational functions over F in the indeterminates x p with p ranging over all elements of P (hence altogether |P| indeterminates). We also will denote by F [x P ] the ring of polynomials over F in the indeterminates x p with p ranging over all elements of P. (Thus, F (x P ) = F (x p 1 , x p 2 , . . . , x pn ) and F [x P ] = F [x p 1 , x p 2 , . . . , x pn ] if P is written in the form P = {p 1 , p 2 , . . . , p n }.) The symbols x p are understood to be distinct, and are used as commuting indeterminates. We regard F [x P ] as a subring of F (x P ), and F (x P ) as the field of quotients of F [x P ].
(b) If Q is a subset of P, then F (x Q ) can be canonically embedded into F (x P ), and F [x Q ] can be canonically embedded into F [x P ]. We regard these embeddings as inclusions.
(c) Let K be a field extension of F. Let f be an element of F (x P ). If (a p ) p∈P ∈ K P is a family of elements of K indexed by elements of P, then we let f (a p ) p∈P denote the element of K obtained by substituting a p for x p for each p ∈ P in the rational function f .
This f (a p ) p∈P is defined only if the substitution does not render the denominator equal to 0. If K is infinite, this shows that f (a p ) p∈P is defined for almost all (a p ) p∈P ∈ K P (with respect to the Zariski topology).
(d) Let P now be a finite totally ordered set, and let be the smaller-than relation of P. For every p ∈ P, let p ⇓ denote the subset {v ∈ P | v p} of P. For every p ∈ P, let Q p be an element of F (x P ).
We say that the family (Q p ) p∈P is P-triangular if and only if the following condition holds: Algebraic triangularity condition: For every p ∈ P, there exist elements α p , β p , γ p , δ p of F (x p⇓ ) such that α p δ p − β p γ p = 0 and Q p = α p x p + β p γ p x p + δ p .
We will use P-triangularity via the following fact: Lemma 54. Let F be a field. Let P be a finite totally ordered set. For every p ∈ P, let Q p be an element of F (x P ). Assume that (Q p ) p∈P is a P-triangular family. Then: (a) The family (Q p ) p∈P ∈ (F (x P )) P is algebraically independent (over F).
(b) There exists a P-triangular family (R p ) p∈P ∈ (F (x P )) P such that every q ∈ P Proof. The proof of this lemma -an exercise in elementary algebra and induction -is omitted; it can be found in [GrRo14b,Lemma 15.3].
Armed with this definition, we are ready to tackle the proof of Proposition 49 that K-labelings can be generically parametrized by Grasp 0 A.
Proof of Proposition 49. Let F be the prime field of K. (This means either Q or F p depending on the characteristic of K.) In the following, the word "algebraically independent" will always mean "algebraically independent over F" (rather than over K or over Z).
Let P be a totally ordered set such that P = {1, 2, . . . , p} × {1, 2, . . . , q} as sets, and such that (i, k) (i , k ) for all (i, k) ∈ P and (i , k ) ∈ P satisfying (i i and k k ) , where denotes the smaller-or-equal relation of P. Such a P clearly exists (in fact, there usually exist several such P, and it doesn't matter which of them we choose). We denote the smaller-than relation of P by . We will later see what this total order is good for (intuitively, it is an order in which the variables can be eliminated; in other words, it makes our system behave like a triangular matrix rather than like a triangular matrix with permuted columns), but for now let us notice that it is generally not compatible with Rect (p, q).
Consider the field F (x P ) and the ring F [x P ] defined as in Definition 53. In order to prove Proposition 49, it is enough to show that there exists a matrix D ∈ (F (x P )) p×(p+q) satisfying x p = Grasp 0 D (p) for every p ∈ P. (14) 6 Notice that the fraction α p x p + β p γ p x p + δ p is well-defined for any four elements α p , β p , γ p , δ p of F (x p⇓ ) such that α p δ p − β p γ p = 0. (Indeed, γ p x p + δ p = 0 in this case, as can easily be checked.) For then we can obtain a matrix A ∈ K p×(p+q) satisfying f = Grasp 0 A for almost every f ∈ K Rect(p,q) simply by substituting f (p) for every x p in all entries of the matrix D Now define a matrix C ∈ (F [x P ]) p×q by This is simply a matrix whose entries are all the indeterminates x p of the polynomial ring F [x P ], albeit in a strange order (tailored to make the "triangularity" argument work nicely). This matrix C is not directly related to the D we will construct, but will be used in its construction. For every (i, k) ∈ P, define element Our plan from here is the following: Step 1: We will find alternate expressions for the polynomials N (i,k) and D (i,k) which will give us a better idea of what variables occur in these polynomials.
Step 2: We will show that N (i,k) and D (i,k) are nonzero for all (i, k) ∈ P.
Step 3: We will define a Q p ∈ F (x P ) for every p ∈ P by Q p = N p D p , and we will show that Q p = (Grasp 0 (I p | C)) (p).
Step 4: We will prove that the family (Q p ) p∈P ∈ (F (x P )) P is P-triangular.
Step 5: We will use Lemma 54 (b) and the result of Step 4 to find a matrix D ∈ (F (x P )) p×(p+q) satisfying (14).
We now fill in a few details for each step. Details of Step 1: We introduce two more pieces of notation pertaining to matrices: • If ∈ N, and if A 1 , A 2 , . . ., A k are several matrices with rows each, then (A 1 | A 2 | . . . | A k ) will denote the matrix obtained by starting with an (empty) × 0-matrix, then attaching the matrix A 1 to it on the right, then attaching the matrix A 2 to the result on the right, etc., and finally attaching the matrix A k to the result on the right. For example, I 2 | 1 −2 3 0 = 1 0 1 −2 0 1 3 0 .
• If ∈ N, if B is a matrix with rows, and if i 1 , i 2 , . . ., i k are some elements of {1, 2, . . . , }, then rows i 1 ,i 2 ,...,i k B will denote the matrix whose rows (from top to bottom) are the rows labelled i 1 , i 2 , . . ., i k of the matrix B.
We will use without proof a standard fact about determinants of block matrices: • Given a commutative ring L, two nonnegative integers a and b satisfying a b, and a matrix U ∈ L a×b , we have Using this we can rewrite = det (rows i,i+1,...,p (( Also, Although these alternative formulas (20) and (21) for N (i,k) and D (i,k) are not shorter than the definitions, they involve smaller matrices (unless i = 1) and are more useful in understanding the monomials appearing in N (i,k) and D (i,k) .
Details of Step 2: We claim that N (i,k) and D (i,k) are nonzero for all (i, k) ∈ P.
Proof. Let (i, k) ∈ P. Let us first check that N (i,k) is nonzero. This follows from observing that, if 0's and 1's are substituted for the indeterminates x (i,k) in an appropriate way, then the columns of the matrix (I p | C) [1 : i | i + k − 1 : p + k] become the standard the electronic journal of combinatorics 22(3) (2015), #P3.40 basis vectors of K p (in some order), and so the determinant N (i,k) of this matrix becomes ±1, which is nonzero.

Details of
Step 3: Define Q p ∈ F (x P ) for every p ∈ P by Q p = N p D p . This is welldefined because Step 2 has shown that D p is nonzero. Moreover, it is easy to see that every p = (i, k) ∈ P satisfies k)) , i.e., Q p = (Grasp 0 (I p | C)) (p) .

Details of
Step 4: To prove the family (Q p ) p∈P ∈ (F (x P )) P is P-triangular, we need that for every p ∈ P, there exist elements α p , β p , γ p , δ p of F (x p⇓ ) such that α p δ p − β p γ p = 0 and Q p = α p x p + β p γ p x p + δ p (where p ⇓ is defined as in Definition 53 (d)). So fix We will actually do something slightly better than we need. We will find elements α p , β p , γ p , δ p of F [x p⇓ ] (not just of F (x p⇓ )) such that α p δ p − β p γ p = 0 and N p = α p x p + β p and D p = γ p x p + δ p . (Of course, the conditions N p = α p x p + β p and D p = γ p x p + δ p combined imply Q p = α p x p + β p γ p x p + δ p , hence the yearned-for P-triangularity.) We first handle two "boundary" cases: (a) k = 1, and (b) k = 1 but i = p. The case when k = 1 is very easy: we get that N p = 1 (using (20)) and that D p = (−1) i+p x p (using (21)). Consequently, we can take α p = 0, β p = 1, γ p = (−1) i+p and δ p = 0, and it is clear that all three requirements α p δ p − β p γ p = 0 and N p = α p x p + β p and D p = γ p x p + δ p are satisfied.
The case when k = 1 but i = p is not much harder. In this case, (20) simplifies to N p = x p , and (21) simplifies to D p = x (p,1) . Hence, we can take α p = 1, β p = 0, γ p = 0 and δ p = x (p,1) to achieve α p δ p −β p γ p = 0 and N p = α p x p +β p and D p = γ p x p +δ p . Note that this choice of δ p is legitimate because x (p,1) does lie in F [x p⇓ ] (since (p, 1) ∈ p ⇓).
The remaining case, where neither k = 1 nor i = p takes a bit more work. Consider the matrix rows i,i+1,...,p ((I p | C) [i + k − 1 : p + k]) (this matrix appears on the right hand side of (20)). Each entry of this matrix comes either from the matrix I p or from the matrix C. If it comes from I p , it clearly lies in F [x p⇓ ]. If it comes from C, it has the form x q for some q ∈ P, and this q belongs to p ⇓ unless the entry is the (1, p − i + 1)th entry. Therefore, each entry of the matrix (I p | C) [i + k − 1 : p + k] apart from the (1, p − i + 1)-th entry lies in F [x p⇓ ], whereas the (1, p − i + 1)-th entry is x p . Hence, if we use the Laplace expansion with respect to the first row to compute the determinant of this matrix, we obtain a formula of the form det (rows i,i+1,...,p ((I p | C) [i + k − 1 : p + k])) = x p · (some polynomial in entries lying in F [x p⇓ ]) + (more polynomials in entries lying in In other words, there exist elements α p and β p of F [x p⇓ ] such that det (rows i,i+1,...,p ((I p | C) [i + k − 1 : p + k])) = α p x p + β p . Consider these α p and β p . We have We can similarly deal with the matrix rows i,i+1,...,p (C q | (I p | C) [i + k : p + k]) which appears on the right hand side of (21). Again, each entry of this matrix apart from the (1, p − i + 1)-th entry lies in F [x p⇓ ], whereas the (1, p − i + 1)-th entry is x p . Using the Laplace expansion again, we thus see that Hence, there exist elements γ p and δ p of F [x p⇓ ] such that (−1) p−i det (rows i,i+1,...,p (C q | (I p | C) [i + k : p + k])) = γ p x p + δ p . Consider these γ p and δ p . We have We thus have found elements α p , β p , γ p , δ p of F [x p⇓ ] satisfying N p = α p x p + β p and D p = γ p x p + δ p . In order to finish the proof of P-triangularity, we only need to show that α p δ p − β p γ p = 0.
In order to achieve this goal, we notice that Hence, proving α p δ p − β p γ p = 0 is equivalent to proving α p D p − N p γ p = 0. It is the latter that we are going to do, because α p , D p , N p and γ p are easier to get our hands on than β p and δ p . Recall that our proof that det (rows i,i+1,...,p (( proceeded by applying the Laplace expansion with respect to the first row to the matrix rows i,i+1,...,p ((I p | C) [i + k − 1 : p + k]). The only term involving x p was The second factor above is actually the (1, p − i + 1)-th cofactor of the matrix rows i,i+1,...,p ((I p | C) [i + k − 1 : p + k]). Hence, α p = (the (1, p − i + 1) -th cofactor of rows i,i+1,...,p ((I p | C) [i + k − 1 : p + k])) Similarly, (note that we lost the sign (−1) p−i from (25) since it got cancelled against the (−1) p−(i+1) arising from the definition of a cofactor). Now, since neither k = 1 nor i = p, (i + 1, k − 1) also belongs to P; hence, we can apply (20) to (i + 1, k − 1) in lieu of (i, k), and obtain In light of this, (26) becomes Similarly, applying (21) to (i + 1, k − 1) in lieu of (i, k), rewrites (27) as Hence, Thus, we can shift our goal from proving α p D p − N p γ p = 0 to proving N (i+1,k−1) D p + N p D (i+1,k−1) = 0. But this turns out to be surprisingly simple: Since p = (i, k), we have by definition and Theorem 50. On the other hand, (i, k − 1) and (i + 1, k) also belong to P and satisfy and Hence, (28) becomes Step 2. This finishes our proof that N (i+1,k−1) D p + N p D (i+1,k−1) = 0, thus also that α p D p − N p γ p = 0, hence also that α p δ p − β p γ p = 0, and ultimately of the P-triangularity of the family (Q p ) p∈P .

Details of
Step 5: Recall that our goal is to prove the existence of a matrix D ∈ (F (x P )) p×(p+q) satisfying (14). By Step 4, we know that the family (Q p ) p∈P ∈ (F (x P )) P is P-triangular. Hence, Lemma 54 (b) shows that there exists a P-triangular family (R p ) p∈P ∈ (F (x P )) P such that every q ∈ P satisfies Q q (R p ) p∈P = x q . Applying Lemma 54 (a) to this family (R p ) p∈P , we conclude that (R p ) p∈P is algebraically independent. In Step 3, we have shown that Q p = (Grasp 0 (I p | C)) (p) for every p ∈ P. Renaming p as q, we rewrite this as follows: for every q ∈ P.
Now, let C ∈ (F (x P )) p×(p+q) denote the matrix obtained from C ∈ (F [x P ]) p×(p+q) by substituting (R p ) p∈P for the variables (x p ) p∈P . Since (29) is an identity between rational functions in the variables (x p ) p∈P , we thus can substitute (R p ) p∈P for the variables (x p ) p∈P in (29) 7 , and obtain Q q (R p ) p∈P = Grasp 0 I p | C (q) for every q ∈ P (since this substitution takes the matrix C to C). But since Q q (R p ) p∈P = x q for every q ∈ P, this rewrites as Upon renaming q as p again, this becomes x p = Grasp 0 I p | C (p) for every p ∈ P.
We could also consider the subset (i, k) ∈ {1, 2, . . . , p} 2 | i k , but that would yield a poset isomorphic to Tria (p) and thus would not be of any further interest.
Theorem 58. Let p be a positive integer. Let K be a field. Then, ord R Tria(p) = 2p. In order to prove Theorem 58, we need a way to turn labellings of Tria (p) into labellings of Rect (p, p) in a rowmotion-equivariant way. It turns out that the obvious "unfolding" construction (with some fudge coefficients) works: Lemma 59. Let p be a positive integer. Let K be a field of characteristic = 2.
(b) Define a map dble : K Tria(p) → K Rect(p,p) by setting for all v ∈ Rect (p, p) for all f ∈ K Tria(p) . This is well-defined. We have Also, The coefficients 1 2 and 2 in the definition of dble ensure that the labellings R Rect(p,p) • dble and dble •R Tria(p) in part (c) of the Lemma are equal at every element of the poset, without extraneous factors appearing in certain ranks.
Proof. The proofs of (a) and (b) are easy, following in a few lines from the definitions. The proof of (c) involves a few pages of rewriting formulas and case-checking, but there are no surprises. Full details are available in [GrRo14b].
Proof of Theorem 58. Applying Lemma 26 to 2p − 1 and Tria (p) instead of n and P , we see that ord R Tria(p) is divisible by 2p − 1 + 1 = 2p. Now, if we can prove that ord R Tria(p) | 2p, then we will immediately obtain ord R Tria(p) = 2p, and Theorem 58 will be proven. So it suffices to show that R 2p Tria(p) = id. Since this statement boils down to a collection of polynomial identities in the labels of an arbitrary K-labelling of Tria (p), it is clear that it is enough to prove it in the case when K is a field of rational functions in finitely many variables over Q. So let us WLOG assume we are in this case; then the characteristic of K is 0 = 2, so that we can apply Lemma 59(c) to get From this, it follows (by induction over k) that for every k ∈ N. Applied to k = 2p, this yields But Theorem 30 (applied to q = p) yields ord R Rect(p,p) = p + p = 2p, so that R 2p Rect(p,p) = id. Hence, (34) simplifies to dble = dble •R 2p Tria(p) . We can cancel dble from this equation, because dble is an injective and therefore leftcancellable map. As a consequence, we obtain id = R 2p Tria(p) . In other words, R 2p Tria(p) = id. This proves Theorem 58. 10 The ∆ and ∇ triangles The next kind of triangle-shaped posets is more interesting.
Clearly, Eq (p) is an antichain with p elements. (The name Eq comes from "equator".) The posets ∆ (p) and ∇ (p) are (p − 1)-graded posets. They have the form of a "Deltashaped triangle" and a "Nabla-shaped triangle", respectively (whence the names).
Example 61. Here is the Hasse diagram of the poset Rect (4, 4), where the elements belonging to ∆ (4) have been underlined and the elements belonging to Eq (4) have been boxed: (4, 4) Lemma 67. Let p be a positive integer. Let P be a (2p − 1)-graded finite poset. Let hrefl : P → P be an involution such that hrefl is a poset antiautomorphism of P . We extend hrefl to an involutive poset antiautomorphism of P by setting hrefl (0) = 1 and hrefl (1) = 0. Assume that every v ∈ P satisfies deg (hrefl v) = 2p − deg v. Let N be a positive integer. Assume that, for every v ∈ P satisfying deg v = p − 1, there exist precisely N elements u of P satisfying u v.
Define three subsets ∆, Eq and ∇ of P by Clearly, ∆, Eq and ∇ become subposets of P . The poset Eq is an antichain, while the posets ∆ and ∇ are (p − 1)-graded.
Assume that hrefl | Eq = id. It is easy to see that hrefl (∆) = ∇. Let K be a field such that N is invertible in K.
(a) Define a rational map wing : K ∆ K P by setting for all v ∈ P for all f ∈ K ∆ . This is well-defined.
(b) There exists a rational map wing : K ∆ K P such that the diagram The rational map wing defined in Lemma 67 (b) satisfies (d) Almost every (in the sense of Zariski topology) labelling f ∈ K ∆ satisfying f (0) = N satisfies R P (wing f ) = wing (R ∆ f ) .
The condition f (0) = N in part (d) of this lemma has been made to ensure that we obtain a honest equality between R P (wing f ) and wing (R ∆ f ), without "correction factors" in certain ranks. choosing all other a i arbitrarily). Fix such a (p + 1)-tuple, and set f = (a 0 , a 1 , . . . , a p ) g. Then, f (0) = 2. We are going to prove that R p ∆(p) f = f • vrefl. Until we have done this, we can forget about g; all we need to know is that f is a sufficiently generic K-labelling of ∆ (p) satisfying f (0) = 2.
Since (p + 1 − k, p + 1 − i) = v and Rect(p,p) (wing f ) = wing f, this equality rewrites as By Corollary 12 and the definition of wing this simplifies to .
The proof of Corollary 66 is now a simple exercise (or can be looked up in [GrRo14b]).

The quarter-triangles
We have now studied the order of birational rowmotion on all four triangles (two of which are isomorphic as posets) which are obtained by cutting the rectangle Rect (p, p) along one of its diagonals. But we can also cut Rect (p, p) along both diagonals into four smaller triangles. These are isomorphic in pairs, and we will analyze them now. The following definition is an analogue of Definition 60 but using Tria (p) instead of Rect (p, p): Definition 70. Let p be a positive integer. Define three subsets NEtri (p), Eqtri (p) and SEtri (p) of Tria (p) by NEtri (p) = {(i, k) ∈ Tria (p) | i + k > p + 1} ; Eqtri (p) = {(i, k) ∈ Tria (p) | i + k = p + 1} ; SEtri (p) = {(i, k) ∈ Tria (p) | i + k < p + 1} . These subsets NEtri (p), Eqtri (p) and SEtri (p) inherit a poset structure from Tria (p). In the following, we will consider NEtri (p), Eqtri (p) and SEtri (p) as posets using this structure.
Next we display the Hasse diagrams of the poset NEtri (4) (on the left) and SEtri (4) (on the right): The following conjecture has been verified using Sage for small values of p: Conjecture 73. Let p be an integer > 1. Then, ord R SEtri(p) = p and ord R NEtri(p) = p.
In the case when p is odd, we can prove this conjecture using the same approach that was used to prove Theorem 58 (see [GrRo14b] for details): Theorem 74. Let p be an odd integer > 1. Then, ord R SEtri(p) = p and ord R NEtri(p) = p.
However, this reasoning fails in the even-p case (although the order of classical rowmotion is again known to be p in the even-p case -see [StWi11, Conjecture 3.6]).
This conjecture has been verified using Sage for all p 7. Williams (based on a philosophy from his thesis [Will13]) suspects there could be a birational map between K NEtri (p) and K Rect(s−1,p−s+1) which commutes with the respective birational rowmotion operators for all s > p 2 ; this, if shown, would obviously yield a proof of Conjecture 75. This already is an interesting question for classical rowmotion; a bijection between the antichains (and thus between the order ideals) of NEtri (p) and those of Rect (s − 1, p − s + 1) was found by Stembridge [Stem86,Theorem 5.4], but does not commute with classical rowmotion.

Negative results
Generally, it is not true that if P is an n-graded poset, then ord (R P ) is necessarily finite. When char K = 0, the authors have proven the following 8 : • If P is the poset {x 1 , x 2 , x 3 , x 4 , x 5 } with relations x 1 < x 3 , x 1 < x 4 , x 1 < x 5 , x 2 < x 4 and x 2 < x 5 (this is a 5-element 2-graded poset), then ord (R P ) = ∞.
The situation seems even more hopeless for non-graded posets.

The root system connection
A question naturally suggesting itself is: What is it that makes certain posets P have finite ord (R P ), while others have not? Can we characterize the former posets? It might be too optimistic to expect a full classification, given that our examples are already rather diverse (skeletal posets, rectangles, triangles, posets like that in Remark 33). As a first step (and inspired by the general forms of the Zamolodchikov conjecture), we were tempted to study posets arising from Dynkin diagrams. It appears that, unlike in the Zamolodchikov conjecture, the interesting cases are not those having P be a product of Dynkin diagrams, but those having P be a positive root poset of a root system, or a parabolic quotient thereof. The idea is not new, as it was already conjectured by Panyushev [Pan08, Conjecture 2.1] and proven by Armstrong, Stump and Thomas [AST11, Theorem 1.2] that if W is a finite Weyl group with Coxeter number h, then classical rowmotion on the set J (Φ + (W )) (where Φ + (W ) is the poset of positive roots of W ) has order h or 2h (along with a few more properties, akin to our "reciprocity" statements) 9 . In the case of birational rowmotion, the situation is less simple. Specifically, the following can be said about positive root posets of crystallographic root systems (as considered in [StWi11, §3.2]) 10 : • If P = Φ + (A n ) for n 2, then ord (R P ) = 2 (n + 1). This is just the assertion of Corollary 66. Note that for n = 1, the order ord (R P ) is 2 instead of 2 (1 + 1) = 4.
Nathan Williams has suggested that the behavior of Φ + (A n ) and Φ + (B n ) ∼ = Φ + (C n ) to have finite orders of R P could generalize to the "positive root posets" of the other "coincidental types" H 3 and I 2 (m) (see, for example, Table 2.2 in [Will13]). And indeed, computations in Sage have established that ord (R P ) = 10 for P = Φ + (H 3 ), and we also have ord (R P ) = lcm (2, m) for P = Φ + (I 2 (m)) (this is a very easy consequence of Lemma 26).
It seems that minuscule heaps, as considered e.g. in [RuSh12,§6], also lead to small ord (R P ) values. Namely: • The heap P w J 0 of type E 6 in [RuSh12, Figure 8 (b)] satisfies ord (R P ) = 12. • The heap P w J 0 of type E 7 in [RuSh12, Figure 9 (b)] seems to satisfy ord (R P ) = 18 (this was verified on numerical examples, as the poset is too large for efficient general computations).
(These two posets also appear as posets corresponding to the "Cayley plane" and the "Freudenthal variety" in [ThoYo07, p. 2].) Various other families of posets related to root systems (minuscule posets, d-complete posets, rc-posets, alternating sign matrix posets) remain to be studied.