Birational rowmotion and coxeter-motion on minuscule posets

Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistic. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.


Introduction
Rowmotion (at the combinatorial level) is a bijection R on the set J (P ) of order ideals of a finite poset P , which assigns to I ∈ J (P ) the order ideal R(I) generated by the minimal elements of the complement P \ I. The map R can be also described in terms of toggles. For each v ∈ P , let t v : J (P ) → J (P ) be the map given by and call it the toggle at v, Then the rowmotion map R is expressed as the composition where (v 1 , v 2 , . . . , v N ) is any linear extension of P , i.e., a list of elements of P such that v i < v j in P implies i < j. This rowmotion has been studied from several perspectives and under various names. See [19] and [20] for the history and references. Rowmotion exhibits nice properties such as periodicity and homomesy on special posets including root posets (see [11,1]) and minuscule posets (see [15,16]). In general, given a set S and a bijection f : S → S, we say that a statistic θ : S → R is homomesic with respect to f if there exists a constant C such that for any f -orbit T 1 #T x∈T θ(x) = C.
We refer the reader to [14] for the homomesy phenomenon. For a minuscule poset P and a simple root α ∈ Π, we put where c : P → Π is the coloring of P with color set Π, the set of simple roots. This subset P α is called the file corresponding to α. (See Section 3 for a definition of minuscule posets and related terminologies.) If P is a minuscule poset, then the associated rowmotion map R has the following properties: Theorem 1.1. Let P be a minuscule poset associated to a minuscule weight λ of a simple Lie algebra g. Then we have (a) (periodicity, Rush-Shi [15,Thoerem 1.4]) The rowmotion map R has a finite order equal to the Coxeter number h of g.
(b) (file homomesy, Rush-Wang [16, Theorem 1.2]) For each simple root α ∈ Π, the refined order ideal cardinality #(I ∩ P α ) is homomesic with respect to R. More precisely, for any I ∈ J (P ), we have where ̟ ∨ is the fundamental coweight corresponding to α.
One motivation of this paper is to lift the results in the above theorem to the birational level.
Einstein-Propp [4] introduced birational rowmotion by lifting the notion of toggles from the combinatorial level to the piecewise-linear level, and then to the birational level. Given a finite poset P , let P = P ⊔ { 1, 0} be the poset obtained from P by adjoining an extra maximum element 1 and an extra minimum element 0. For positive real numbers A and B, we put where R >0 denotes the set of positive real numbers. For v ∈ P , we define the birational toggle τ A,B v : K A,B (P ) → K A,B (P ) at v by where the symbol x ⋗ y means that x covers y, i.e., x > y and there is no element z such that x > z > y. It is clear that τ A,B v is an involution. (See (12) for a definition of piecewise-linear toggles.) Then we define birational rowmotion ρ A,B : K A,B (P ) → K A,B (P ) by where (v 1 , . . . , v N ) is a linear extension of P . It can be shown that the definition of ρ A,B is independent of the choices of linear extensions. Since rowmotion is defined by toggling from top to bottom, we have a recursion formula for the values of the birational rowmotion map: We omit the superscript A,B and simply write K(P ), τ v and ρ when there is no confusion. For birational rowmotion on a product of two chains, periodicity and (multiplicative) file homomesy are obtained by Grinberg-Roby [7] and Einstein-Propp [4] respectively. In this paper we generalize their results from products of two chains (type A minuscule posets) to arbitrary minuscule posets.
For a minuscule poset and a simple root α ∈ Π, we define for F ∈ K A,B (P ). Our main results for birational rowmotion are summarized as follows: Let P be the minuscule poset associated to a minuscule weight λ of a finite dimensional simple Lie algebra g. Let ρ = ρ A,B be the birational rowmotion map. Then we have (a) (periodicity) The map ρ has finite order equal to the Coxeter number h of g.
(b) (reciprocity) For any v ∈ P and F ∈ K A,B (P ), we have where rk : P → {1, 2, . . . , h−1} is the rank function of the graded poset P and ι : P → P is the canonical involutive anti-automorphism of P (see Proposition 3.3).
(c) (file homomesy) For a simple root α, we have for any F ∈ K A,B (P ), where w 0 is the longest element of the Weyl group W of g, and ̟ ∨ is the fundamental coweight corresponding to α.
Part (a) of this theorem is established in [6,7] except for the type E 7 minuscule poset. In this paper we provide a way to settle the E 7 case by using a computer. For a type A minuscule poset, Part (b) is obtained in [7,Theorem 30]. Our proof of Part (b) is based on a case-bycase analysis (with a help of computer in types E 6 and E 7 ). Part (c) in type A follows from Einstein-Propp [4, Theorems 7.3 and 8.5] (see Musiker-Roby [9, Theorem 2.16] for another proof). We will give an almost uniform proof to Part (c). Also we can use tropicalization (or ultradiscretization) to deduce the results for combinatorial rowmotion in Theorem 1.1 (see Section 2).
Another aim of this paper is to introduce and study birational Coxeter-motion on minuscule posets, which is regarded as a generalization of birational promotion on a product of two chains (see [4,Definition 5.3]). For a simple root α ∈ Π, we define σ A,B α : which is independent of the order of composition. Then a Coxeter-motion map is a product of all the σ A,B α 's in any order. Our results for birational Coxeter-motion are stated as follows: Theorem 1.3. Let P be the minuscule poset. Let γ = γ A,B be a birational Coxeter-motion map. Then we have (a) (periodicity) The map γ has finite order equal to the Coxeter number h.
If P is a type A minuscule poset and π is the birational promotion map (a special case of birational Coxeter-motion maps), then there is an explicitly defined "recombination map" R such that Rρ = πR (see [4,Theorem 8.2]), which, together with Theorem 1.2 (a), implies Part (a) of the above theorem. We prove Part (a) for arbitrary minuscule posets by showing that any birational Coxeter-motion map is conjugate to the birational rowmotion map in the birational toggle group (Theorem 4.3 below). By applying tropicalization to Part (a), we obtain the periodicity of piecewise-linear promotion, which is proved in [5, Theorem 1.12] via quiver representation. Part (b) in type A is obtained in [4,Theorem 7.3].
Hopkins [8] obtains another example of homomesy for the birational rowmotion for a wider class of posets including minuscule posets. Theorem 1.4. (Hopkins [8,Theorem 4.43]) Let P be a minuscule poset and ρ = ρ A,B the birational rowmotion map. For F ∈ K A,B (P ), we define .

Then we have
Via tropicalization, this theorem reduces to the homomesy phenomenon of the antichain cardinality statistic, which was proved in [16,Theorem 1.4]. In a forthcoming paper [10], we use explicit formulas for iterations of the birational rowmotion map to give refinements of Theorem 1.4. Our refinement in type A provides a birational lift of the homomesy given in [4,Proof of Theorem 27].
The remaining of this paper is organized as follows. We collect some general facts concerning birational rowmotion in Section 2, and give a definition and properties of minuscule posets in Section 3. In Sections 4 to 6 we give a proof of our main theorems. The periodicity in Theorem 1.2 (a) and Theorem 1.3 (a) is proved in Section 4, and the reciprocity in Theorem 1.2 (b) is verified in Section 5. In Section 6, after investigating local properties around a file, we complete the proof of file homomesy in Theorem 1.2 (c) and Theorem 1.3 (b).

Acknowledgements
This work was partially supported by JSPS Grants-in-Aid for Scientific Research No. 18K03208. The author is grateful to Tom Roby for fruitful discussions.

Generalities on rowmotion
In this section, we explain how combinatorial and birational rowmotion are related, and give some general facts about birational rowmotion.

Combinatorial, piecewise-linear and birational rowmotion
We begin with recalling the definition of piecewise-linear toggles and rowmotion. Given a finite poset P and real numbers a, b, we put where P = P ⊔ { 1, 0}. We define the piecewise-linear toggles t ±,a,b v : P a,b (P ) → P a,b (P ) at v ∈ P by the formulas and t ±,a,b v f (x) = f (x) for x = v. For an order ideal I ∈ J (P ), let χ ± I be the characteristic functions defined by Then it follows from the definition (1) and (12) that the toggle t ±,a,b v is a piecewise-linear lift of the combinatorial toggle t v in the following sense: The piecewise-linear rowmotion map R ±,a,b : P a,b (P ) → P a,b (P ) is defined by where (v 1 , . . . , v N ) is a linear extension of P . A rational function F (X 1 , · · · , X m ) ∈ Q(X 1 , · · · , X m ) is called subtraction-free if F is expressed as a ratio F = G/H of two polynomials G(X 1 , · · · , X m ) and H(X 1 , · · · , X m ) ∈ Z[X 1 , . . . , X m ] with nonnegative integer coefficients. By using lim ε→+0 ε log(e a/ε + e b/ε ) = max{a, b}, lim ε→−0 ε log(e a/ε + e b/ε ) = min{a, b}, we can see that, if F (X 1 , . . . , X m ) is subtraction-free, then for any real numbers x 1 , . . . , x m ∈ R the limits f ± (x 1 , · · · , x m ) = lim ε→±0 ε log F (e x 1 /ε , · · · , e xm/ε ) exist and f + (x 1 , . . . , x m ) (resp. f − (x 1 , . . . , x m )) is the piecewise-linear function in x 1 , . . . , x m obtained from F by replacing the multiplication ·, the division / and the addition + with the addition +, the subtraction − and the maximum max (resp. the minimum min). This procedure from F to f ± are called the tropicalization (or ultradiscretization).
Proposition 2.1. Let P be a finite poset. Let R : J (P ) → J (P ) and ρ = ρ A,B : K A,B (P ) → K A,B (P ) be the combinatorial and birational rowmotion maps respectively. Let m : P ×Z → Z be a map with finite support. If there is a integers p and q such that for any F ∈ K A,B (P ), then where χ[S] = 1 if S is true and 0 if S is false.
Proof. By applying the tropicalization procedure to (14), we obtain for any f ∈ P a,b (P ). Then specializing f = χ ± I and using (13), we obtain (15).
for any F ∈ K A,B (P ) and v ∈ P , then R h (I) = I any I ∈ J (P ).
(b) Let v and w ∈ P and k a positive integer. If ρ k F (v) · F (w) = AB, then v ∈ R k (I) and w ∈ I are equivalent for any I ∈ J (P ).
(c) Let M be a subset of P and h be a positive integer. If h−1 k=0 v∈M ρ k F (v) = A p B q for any F ∈ K A,B (P ), then we have h−1 k=0 # R k (I) ∩ M = q for any I ∈ J (P ).
Similar statements hold for birational Coxter-motion.

Birational rowmotion on graded posets
In this subsection we present some properties of birational rowmotion on graded posets. A poset P is called graded of height n if there exists a rank function rk : P → {1, 2, . . . , n} satisfying the following three conditions: (i) If v is minimal in P , then rk(v) = 1; (ii) If v is maximal in P , then rk(v) = n; (iii) If v covers w, then rk(v) = rk(w) + 1.

Lemma 2.3.
If P is a graded poset of height n and the birational rowmotion map ρ A,B has a finite order N , then N is divisible by n + 1.
Proof. By Corollary 2.2 (a), we have R N (I) = I for all I ∈ J (P ). On the other hand, it is easy to see that the R -orbit of the empty order ideal ∅ has length n + 1. Hence we see that n + 1 divides N .
The following lemma gives a relation between ρ A,B and ρ 1,1 .
Lemma 2.4. Let P be a graded poset of height n. For a map F : P → R >0 and positive real numbers A, B ∈ R >0 , we denote by F A,B ∈ K A,B (P ) the extension of F to P such that Proof. We can use the recursive formula (6) to proceed by the double induction on k and n − rk(v).

Change of variables
Let P be a finite poset. Given an initial state This change of variables is used in [9] to describe a lattice path formula for birational rowmotion on a type A minuscule poset. Then the inverse change of variables is given by where the sum is taken over all saturated chains v 1 ⋗ · · · ⋗ v r in P such that v 1 = v and v r is minimal in P . Note that this change of variables is a birational lift of Stanley's transfer map between the order polytope and the chain polytope of a poset (see [17,Section 3]).

Minuscule posets
In this section we review a definition and properties of minuscule posets.

Definition and properties of minuscule posets
Let g be a finite dimensional simple Lie algebra over the complex number field C of type X n , where X ∈ {A, B, C, D, E, F, G} and n is the rank of g. We fix a Cartan subalgebra h and choose a positive root system ∆ + of the root system ∆ ⊂ h * . Let Π = {α 1 , . . . , α n } be the set of simple roots, where we follow [2, Planche I-IX] for the numbering of simple roots. We denote by ̟ i the fundamental weight corresponding to the ith simple root α i . Let ∆ ∨ + ⊂ h be the positive coroot system. Let W be the Weyl group of g, which acts on h and h * . The simple reflections {s α : α ∈ Π} generate W .
For a dominant integral weight λ, we denote by V Xn,λ the irreducible g-module with highest weight λ and by L Xn,λ the set of weights of V Xn,λ . We say that λ is minuscule if L Xn,λ is a single W -orbit. See [3, VIII, §7, n • 3] for properties of minuscule weights. It is known that minuscule weights are fundamental weights. Table 1 is the list of minuscule weights.
Let λ be a minuscule weight of a simple Lie algebra g of type X n . We equip the set of wegiths L Xn,λ with a poset structure by defining µ ≥ ν if ν − µ is a linear combination of simple roots with nonnegative integer coefficients. We note that λ is the minimum element of the poset L Xn,λ . Definition 3.1. Let g be a simple Lie algebra of type X n and λ a minuscule weight. Then the minuscule poset P Xn,λ is defined by where the partial ordering on P Xn,λ is given by saying that α ∨ ≥ β ∨ if α ∨ − β ∨ is a linear combination of simple coroots with nonnegative integer coefficients. (b) ([12, Theorem 11]) There exists a unique map c : P Xn,λ → Π, called the coloring of P Xn,λ , such that the map gives an isomorphism of posets.
If λ is a minuscule weight, then the stabilizer W λ of λ in W is the maximal parabolic subgroup generated by {s β : β ∈ Π \ {α}}, where α is the simple root corresponding to the fundamental weight λ. Proposition 3.3. Let P Xn,λ be the minuscule poset corresponding to a minuscule weight λ, and w λ the longest element of the stabilizer W λ . Then the map gives an involutive anti-automorphism of the poset P Xn,λ .
The following properties of minuscule posets can be checked easily (e.g., by using a description given in the next subsection). (d) For each α ∈ Π, the subposet P α = {v ∈ P : c(v) = α} is a chain.
(e) If v, w ∈ P α , then the difference rk(v) − rk(w) is even.

Description of minuscule posets
In this subsection we give an explicit description of minuscule posets and their colorings. The minuscule posets can be embedded into the poset Type A n . The positive coroot system ∆ ∨ + of type A n can be described as ∆ ∨ + = {e i − e j : 1 ≤ i < j ≤ n + 1} with e 1 + · · · + e n+1 = 0. Then we have and tha map e i − e j → (r − i, j − r − 1) gives an isomorphism of posets from P An,̟r to the subposet The poset P An,̟r is a product poset [0, r − 1] × [0, n − r] of two chains, where [0, m] = {0, 1, . . . , m} is a chain. We call this poset P An,̟r a rectangle poset. The involution ι is the 180 • rotation of the Hasse diagram. For example, the Hasse diagram and the coloring of P A 7 ,̟ 3 are given in Figure 1 Type B n . If we realize the positive coroot system ∆ ∨ + of type B n as ∆ ∨ and the map e i + e j → (n − j, n − i) gives an poset isomorphism from P Bn,̟n to the subposet We call P Bn,̟n a shifted staircase poset. The involution ι is the horizontal flip of the Hasse diagram. For example the Hasse diagram of P B 4 ,̟ 4 and its coloring are given in Figure 2. Type C n . If we realize the positive coroot system ∆ ∨ + of type C n as ∆ ∨ . . , e 1 − e n , e 1 , e 1 + e n , . . . , e 1 + e 2 }.
For example the Hasse diagram of P C 4 ,̟ 1 and its coloring are given in Figure 3. Note that P Cn,̟ 1 is isomorphic to P A 2n−1 ,̟ 1 , but they have different colorings.
Type D n . We realize the positive coroot system ∆ ∨ + of type D n as ∆ ∨ For the minuscule weight ̟ 1 , we have See Figure 4 for the Hasse diagram of P D 5 ,̟ 1 and its coloring. The poset P Dn,̟ 1 is called a double-tailed diamond poset. The involutive anti-automorphism ι is given by For the minuscule weights ̟ n and ̟ n−1 , we have P Dn,̟n = {e i + e j : 1 ≤ i < j ≤ n} and P Dn,̟ n−1 is obtained from P Dn,̟n by replacing e i + e n with e i − e n for 1 ≤ i ≤ n − 1. Both posets P Dn,̟n and P Dn, For example, the Hasse diagram and the coloring of P D 5 ,̟ 5 are given in Figure 5. Note that P Dn,̟ n−1 ∼ = P Dn,̟n and they are isomorphic to P B n−1 ,̟ n−1 , but they have different colorings.

Periodicity
The goal of this section is to prove the periodicity of birational rowmotion and Coxeter-motion (Theorem 1.2 (a) and Theorem 1.3 (a)).

Periodicity of birational rowmotion
For the birational rowmotion map on minuscule posets, the periodicity has been established in [6,7] except for the type E 7 minuscule poset. Let P be a minuscule poset associated to a Lie algebra g, and ρ A,B : K A,B (P ) → K A,B the birational rowmotion map. Since the periodicity depends only on the poset structure, we may assume that g is simply-laced. And by Proposition 3.4 (a), Lemmas 2.3 and 2.4, it is enough to show that ρ = ρ 1,1 satisfies ρ h = 1, where h is the Coxeter number of g.
• If P = P Dn,̟ 1 is a double-tailed diamond poset, then P is a skeletal poset of height 2n − 3, and it follows from [6, Propositions 61, 74 and 75] that ρ has order 2n − 2 (see [6, Section 10] for a definition of skeletal posets and details).
• If P = P E 6 ,̟ 6 is the minuscule poset of type E 6 , then by using a computer we can verify that ρ has order 12.
• Let P = P E 7 ,̟ 7 be the minuscule poset of type E 7 . Given an initial state X ∈ K 1,1 (P ), we regard {X(v) : v ∈ P } as indeterminates and introduce new indeterminates {Z(v) : v ∈ P } by (17). With the author's laptop, it takes about 20 seconds for Maple19 to compute all the values ρ k X (v) (0 ≤ k ≤ 18, v ∈ P ) as rational functions in This completes the proof of Theorem 1.2 (a).

Periodicity of birational Coxeter-motion
In order to prove the periodicity of birational Coxeter-motion (Theorem 1.3 (a)), we work with the birational toggle group and show that any birational Coxeter-motion maps are conjugate to the birational rowmotion map in this group. Let P be a finite poset and fix positive real numbers A and B. We define the birational toggle group, denote by G(P ), to be the subgroup generated by birational toggles τ v = τ A,B v (v ∈ P ) in the group of all bijections on K A,B (P ).
A key tool here is the non-commutativity graph. Given elements g 1 , . . . , g n of a group G, the non-commutativity graph Γ(g 1 , . . . , g n ) is defined as the graph with vertex set {1, 2, . . . , n}, in which two vertices i and j are joined if and only if g i g j = g j g i . The following lemma is useful. . . , g n be elements of a group G. If the noncommutativity graph Γ(g 1 , . . . , g n ) has no cycle, then g ν(1) . . . g ν(n) is conjugate to g 1 . . . g n in G for any permutation ν ∈ S n .
First we prove that all birational Coxeter-motion maps are conjugate. Proof. Note that birational toggles τ v and τ w are commutative unless v ⋖ w ore v ⋗ w. It follows from Proposition 3.4 (c) that, if simple roots α and β are not adjacent in the Dynkin diagram of g, then the corresponding elements σ α and σ β commute with each other in G(P ). Hence the non-commutativity graph Γ(σ α 1 , . . . , σ αn ), where α 1 , . . . , α n are the simple roots, is a subgraph (of the underlying simple graph) of the Dynkin diagram. Since the Dynkin diagram of g has no cycle, we can use Lemma 4.1 to conclude that any two Coxeter-motion maps are conjugate in G(P ).
The periodicity of birational Coxeter-motion maps (Theorem 1.3 (a)) immediately follows from the following thoerem and the periodicity of the birational rowmotion map (Theorem 1.2 (a)). This theorem is a birational lift of [15,Theorem 1.3]. In order to prove this theorem, we use a notion of rc-poset, which is introduced by Striker-Williams [19,Section 4.2]. We put Λ = {(i, j) ∈ Z 2 : i + j is even}. A poset P is called a rowed-and-columned poset (rc-poset for short) if there is a map π : P → Λ such that, if v covers u in P and π(v) = (i, j), then π(u) = (i + 1, j − 1) or (i − 1, j − 1). Minuscule posets P = P Xn,λ are rc-posets with respect to the composition map π : P → Λ of the embedding P ֒→ Z 2 given in Subsection 3.2 and the map Z 2 ∋ (i, j) → (j − i, j + i) ∈ Λ. A row (resp. column) of an RC-poset P is a subset M of P of the form M = {v ∈ P : the second coordinate of π(v) equals r}, (resp. M = {v ∈ P : the first coordinate of π(v) equals c}) for some r (resp. c). If M is a subset of a row or a column of P , then the composition of toggles τ v (v ∈ M ) is independent of the order of composition, so we denote by τ [M ] the resulting element of the toggle group G(P ). If R 1 , . . . , R n are the non-empty rows of an rc-poset P from bottom to top, then the rowmotion map ρ = ρ A,B is given by The following Lemma is proved by exactly the same argument as in [19]. We prove Theorem 4.3 by using this lemma.

Reciprocity
In this section we prove the reciprocity for birational rowmotion (Theorem 1.2 (b)) and propose a conjectural reciprocity for a particular birational Coxeter-motion map. The proof of the reciprocity for birational rowmotion is based on a case-by-case analysis. Let P be a minuscule poset associated to a simple Lie algebra g and ρ A,B the birational rowmotion map. We may assume that g is simply-laced and that A = B = 1 (see Lemma 2.4). For a type A minuscule poset, the reciprocity was proved by Grinberg-Roby [7,Theorem 30] and Musiker-Roby [9,Corollary 2.13]. Also we can verify the reciprocity for the minuscule posets of types E 6 and E 7 by using a computer. The remaining minuscule posets are the shifted staircase posets P Dn,̟n and the double-tailed diamond posets P Dn,̟ 1 .
Lemma 5.1. For F ∈ K 1,1 (P ), we define F ∈ K 1,1 ( P ) by Then we have By using this lemma and the reciprocity for the rectangle poset P , we have This is the desired identity for a shifted staircase poset.

Double-tailed diamond posets
In this subsection, we prove the reciprocity for double-tailed diamond posets. Let P = P Dn,̟ 1 be the minuscule poset associated to the minuscule weight λ = ̟ 1 of the Lie algebra of type D n . We label elements of P by Note that v 1 is the maximum element and v 2n−3 is the minimum element.
(iii) If n + 2 ≤ i ≤ 2n − 3, then we put Then the original indeterminates X(v) can be expressed in terms of Z(v) as follows: The values X(v) (v ∈ P ) are expressed in terms of C(i; l) and C ± (i; l) as follows: Recall that P is a graded poset with rank function rk given by rk Then it is straightforward to prove the following explicit formulas by using the induction on k and i. (We omit the proof.) Proposition 5.3. Let v ∈ P and k a positive integer. If 1 ≤ k ≤ rk(v), then the value ρ k X (v) of iterations of birational rowmotion is expressed in terms of C(i; l) and C ± (i; l) as follows: .
Since the involution ι : P → P is given by n−1 , we obtain the desired reciprocity by comparing formulas in Lemma 5.2 and Proposition 5.3. This completes the proof of Theorem 1.2 (b) for all minuscule posets.

Reciprocity for birational Coxeter-motion
We have the following conjectural reciprocity for a particular birational Coxeter-motion map.
Conjecture 5.4. Let P be a minuscule poset. We decompose the simple root system Π into a disjoin union of two subsets Π 1 and Π 2 such that any roots in ∆ i are pairwise orthogonal for each i. We define γ 1 and γ 2 by where h is the Coxeter number. Then we conjecture that for any F ∈ K A,B (P ) and v ∈ P .

File homomesy
This section is devoted to the proof of the file homomesy phenomenon (Theorem 1.2 (c) and Theorem 1.3 (b)).

Local properties
First we investigate local properties of birational rowmotion and Coxeter-motion around a given file. Let P be a minuscule poset with coloring c : P → Π. We regard the Hasse diagram of the poset P = P ⊔ { 1, 0} as a directed graph, where a directed edge u → v corresponds to the covering relation u ⋖ v. For α ∈ Π, let N α be the neighborhood of P α = {x ∈ P : c(x) = α} given by N α = {x ∈ P : there is an element y ∈ P α such that x ⋖ y or x ⋗ y}.
We define G α to be the bipartite directed subgraph of the Hasse diagram of P with black vertex set P α and white vertex set N α . It follows from Proposition 3.4 (c) that where β runs over all simple roots adjacent to α in the Dynkin diagram, and α max (resp. α min ) is the color of the maximum (resp. minimum) element of P .
To describe the graph structure of G α , we introduce two series of posets G m and H m . For a positive integer m, let G m be the poset consisting of 3m elements x 1 , · · · , x m , y 1 , · · · , y m−1 , z 1 , · · · , z m−1 , u, v with covering relations Note that G 1 is the three-element chain. And, for an integer m ≥ 2, let H m be the (2m + 1)element chain We regard the Hasse diagrams of G m and H m as bipartite directed graphs with black vertices x 1 , . . . , x m . For example, the Hasse diagrams of G 4 and H 4 are shown in Figures 10 and 11 respectively.
x 3 x 2 x 3 x 2 x 1 u y 3 y 2 y 1 Figure 11: H 4 Lemma 6.1. Each bipartite directed graph G α is decomposed into a disjoint union of graphs of the form G m or H m as follows: • If P = P An,̟r , then where r + s = n + 1 and r ≤ s.
• If P = P Bn,̟n , then • If P = P Cn,̟ 1 , then • If P = P Dn,̟ 1 , then • If P = P Dn,̟n , then where G ⊔m 1 is the disjoint union of m copies of G 1 , and ⌊x⌋ stands for the largest integer not exceeding x.
• If P = P E 6 ,̟ 6 , then • If P = P E 7 ,̟ 7 , then The following relations are a key to the proof of the file homomesy phenomenon.
If H m appears as a connected component of G α , then we have Proof. (a) It follows from (6) that By replacing F with ρ m−1 F (resp. ρ i−1 F ) in the first (resp. second) equation, and then by multiplying the resulting equations together, we obtain (21). (b) can be checked by a similar computation.
If H m appears as a connected component of G α , then we have Proof. (a) By the definition (4), we have Multiplying them together, we obtain (23). (b) can be checked by a similar computation.

File homomesy for birational rowmotion
In this subsection, we prove the file homomesy phenomenon for birational rowmotion (Theorem 1.2 (c)).
The following properties of Coxeter elements will be useful in the proof of Theorem 1.2 (c) and Theorem 1.3(b); the proof of the latter will be given in the next subsection. A Coxeter element in a Weyl group W = s α : α ∈ Π is a product of all simple reflections s α in any order. Then it is known that all Coxeter elements are conjugate. By definition, the Coxeter number is the order of any Coxeter element. (c) Let α ∈ Π be a simple root and ̟ the corresponding fundamental weight. If c = s α 1 · · · s αn is a Coxeter element with Π = {α 1 , . . . , α n } and β = s α 1 · · · s α k−1 α k , where α = α k , then we have Hence we see that In order to prove Theorem 1.2 (c), we consider Here v α 0 is the minimum element of P α . Note that P α is a chain and rk(v) − rk(v α 0 ) is an even integer (see Proposition 3.4 (d) and (e)). Since ρ has a finite order h, we have Remark 6.5. It is worth mentioning that Φ ′ α (ρ k X) are Laurent monomials in the variables Z(v) defined by (17). In a forthcoming paper [10], we will give explicit formulas for Φ ′ α (ρ k X) in classical types. Proposition 6.6. For α ∈ Π and F ∈ K A,B (P ), we have where β runs over all simple roots adjacent to α in the Dynkin diagram and Proof. We explain the proof in the case where g is of type E 7 , λ = ̟ 7 and α = α 5 . (The other cases can be proved in a similar way.) We label elements of P α as v α 0 , v α 1 , v α 2 , . . . from bottom to top. By the definition (28), we have The subgraph G α 5 has three connected components By applying (21) to three connected components of G α 5 , we obtain By replacing F with ρ 2 F (resp. ρ 6 F ) in the second (resp. third) equation, and then by multiplying three resulting equations together, we have Since v α 6 0 < v α 5 0 < v α 4 0 (see Figure 7), we obtain (30) in this case.
Proof of Theorem 1.2 (c). We define an element µ(F ) ∈ h * for F ∈ K A,B (P ) by putting Note that, if ̟ ∨ is the fundamental coweight corresponding to α, then we have log Φ α (F ) = ̟ ∨ , µ(F ) .
This completes the proof of (32) and hence of Theorem 1.2 (c).

File homomesy for birational Coxeter-motion
In this subsection we prove Theorem 1.3 (b). The following proposition is a consequence of Lemma 6.1 and Equations (23), (24).
By using this proposition, we can complete the proof of the file homomesy phenomenon for birational Coxeter-motion.
By the definition of µ(F ), we have Then we can complete the proof by taking the pairing , ̟ ∨ .