Cyclic Sieving for Plane Partitions and Symmetry

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades’s result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades’s result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

1 Introduction and statement of results

Plane partitions
An a × b plane partition of height m is an a × b array π = (π i,j ) 1≤i≤a, 1≤j≤b of nonnegative integers π i,j ∈ N which is weakly decreasing in rows and columns (i.e., π i,j ≥ π i+1,j and π i,j ≥ π i,j+1 for all i, j) and for which the largest entry is less than or equal to m (i.e., π 1,1 ≤ m). We denote the set of such plane partitions by PP m (a × b). For a plane partition π ∈ PP m (a × b), we define its size to be |π| := 1≤i≤a, 1≤j≤b π i,j .
MacMahon's celebrated product formula [53,Section 495] for the size generating function for a × b plane partitions of height m is: See [82,Theorem 7.21.7] for a modern presentation of this result. Note Mac(a, b, m; q) is (essentially) a principal specialization of a Schur polynomial s λ (x 1 , x 2 , . . . , x a+b ): Mac(a, b, m; q) = q −κ(m a ) s m a 1, q, q 2 , . . . , q a+b−1 , where m a is the a × m rectangle shape, and κ(λ) := 0λ 1 + 1λ 2 + 2λ 3 + · · · . The Schur polynomials s λ occur in many contexts, but of particular relevance is the fact that they are characters of general linear group representations.
Then for −a + 1 ≤ k ≤ b − 1 we define F k := 1≤i≤a, 1≤j≤b, j−i=k τ i,j to be the composition of all the toggles along the "kth diagonal" of our array (note that all these toggles commute). Finally, we define promotion Pro : PP m (a × b) → PP m (a × b) as the composition of these diagonal toggles F k from left to right: Example 1.1. Suppose a := 2, b := 2, and m := 4. We can compute an application of promotion on a plane partition π ∈ PP 4 (2 × 2) as follows: This description of promotion in terms of piecewise-linear involutions goes back to Berenstein and Kirillov [42] and Berenstein and Zelevinsky [8], building on work of Bender and Knuth [7] and Gansner [27]. More recently, interest in these piecewise-linear toggles has been rekindled in connection with another related operator called rowmotion (see, e.g., [16] or the survey [67]). We will discuss rowmotion later.
It is more common, following the seminal work of Schützenberger [73,74,75] (see also Haiman [34]), to consider promotion as an operator on semistandard Young tableaux defined in terms of "jeu de taquin" sliding moves. But in fact, via a simple change of coordinates using Gelfand-Tsetlin patterns, promotion of plane partitions in PP m (a × b) as we have just defined it exactly corresponds to the usual promotion of semistandard tableaux of shape a × m with entries in the set {1, . . . , a + b} (see Appendix A for the details of this correspondence).

The sieving phenomenon
The sieving phenomenon of Reiner, Stanton, and White [61] is, loosely speaking, the philosophy that we can often count fixed points for a nice group action on a set of combinatorial objects by plugging roots of unity into a polynomial related to this set. Initially the philosophy was considered only for cyclic group actions, and in this context it is usually called the cyclic sieving phenomenon; but there has also been interest in broadening the philosophy to include other groups as well [6,60]. In fact, as we explain below, the sieving phenomenon grew out of Stembridge's "q = −1" phenomenon, which is basically the case where the group has order two.
Sieving phenomena involving polynomials which have simple product formulas in terms of ratios of q-numbers (like the MacMahon formula) are especially valuable, because these imply that every symmetry class has a product formula.
Theorem 1.2 says that promotion acting on plane partitions has a very regular orbit structure. For instance, note that one consequence of Theorem 1.2 is that Pro acting on PP m (a × b) has order dividing a + b. But, as mentioned, Theorem 1.2 implies much more than this: it also means that every symmetry class has a product formula. Theorem 1.2 was first proved by Rhoades [63]. To prove this theorem he used Kazhdan-Lusztig theory and the related theory of quantum groups. In particular, he employed the dual canonical basis for representations of the general linear group [41,49]. More recently, Shen and Weng [79] gave a different proof of Theorem 1.2. Their approach was from the perspective of cluster algebras and the "cluster duality" conjecture of Fock and Goncharov [18]. Specifically, their proof employed the Gross-Hacking-Keel-Kontsevich [33] canonical basis (or "theta basis") for cluster algebras, in the particular case of the coordinate ring of the Grassmannian.
In either case, the proof of Theorem 1.2 followed the "linear algebraic" paradigm. This means that the desired equality is established by computing the trace of a linear operator on a vector space in two different bases. The first basis should be indexed by the combinatorial set in question, and the linear operator should permute this basis according to the cyclic action, so that its trace computes the number of fixed points of the action. The second basis is an eigenbasis where we can compute trace by considering eigenvalues. See, e.g., [62] or [70,Section 4].
In both the Rhoades [63] and Shen-Weng [79] proofs the vector space in question actually carries more structure: it is a GL(a + b) representation. And in both proofs the linear operator corresponding to promotion is the action of a particular lift to the general linear group of the long cycle (a.k.a. standard Coxeter element) c ∈ S a+b in the symmetric group. Geometrically, this map is the twisted cyclic shift χ ∈ GL(a + b) acting on the Grassmannian Gr(a, a + b).

Symmetries of plane partitions
In the present paper, we extend Theorem 1.2 by considering symmetries of plane partitions. The study of plane partitions with symmetry goes back to MacMahon [52], but really took off in the 1970s and 80s: see for instance the seminal paper of Stanley [81] which identified 10 symmetry classes of plane partitions, and see [44] for a modern update to Stanley's paper. Here we will be concerned exclusively with the involutive symmetries of plane partitions.
The first symmetry we consider is complementation Co : PP m (a × b) → PP m (a × b), which is defined by Co(π) i,j := m − π a+1−i,b+1−j . A plane partition π ∈ PP m (a × b) can be viewed as the 3-dimensional stack of cubes inside of an a × b × m rectangular box which has π i,j cubes stacked at position (i, j), in which case complementation is set-theoretic complementation inside this box.
Plane partitions π ∈ PP m (a × b) with Co(π) = π are called self-complementary. Stanley [81] was the first to enumerate self-complementary plane partitions. The enumeration of selfcomplementary plane partitions is one of the prototypical examples of Stembridge's "q = −1" phenomenon [85,87], the precursor to the cyclic sieving phenomenon. Namely: Stembridge [85] (and, independently, Kuperberg [45]) gave a "linear algebraic" proof of Theorem 1.3, computing the trace of a linear operator in two ways. The linear operator correspond-ing to complementation is the action of a particular lift w 0 ∈ GL(a + b) of the longest element w 0 ∈ S a+b in the symmetric group. Geometrically, this map is the twisted reflection on the Grassmannian.
Complementation and promotion together generate a dihedral group: Co · Pro = Pro −1 · Co. (In the context of tableaux, complementation is more often referred to as the "Schützenberger involution" or "evacuation"; see for instance Stanley's survey paper [83].) This means that Co · Pro k is conjugate to Co · Pro j whenever k and j have the same parity. So if we want to count the number of fixed points of Co·Pro k , there are only two cases to consider: k even (which is addressed by Theorem 1.3 above), and k odd.
Recently there has been interest in sieving for dihedral group actions [60,88], where fixed points of both the rotations and reflections are counted by plugging roots of unity into a polynomial. And in fact a "dihedral sieving" result for the action of Pro, Co on plane partitions has already been considered by Abuzzahab-Korson-Li-Meyer [1] and obtained by Rhoades [63] [63,Theorem 7.6]). For any even k ∈ Z, we have # π ∈ PP m (a × b) : Co · Pro k (π) = π = Mac(a, b, m; q → −1). For any odd k ∈ Z, we have We note that there are product formulas for the Schur function evaluation appearing in Theorem 1.4: see [1,Lemma 8.2]. However, Theorem 1.4 is certainly not quite as clean a result as Theorem 1.2. By considering additional plane partition symmetries, we will actually discover fixed point enumerations which are as pleasant as Theorem 1.2.
The next symmetry we consider is transposition, or in other words, reflection across the main diagonal. In order for transposition to act on a fixed set of plane partitions, we need a = b. So let n := a = b. Then Tp : PP m (n × n) → PP m (n × n) is defined by Tp(π) i,j := π j,i . Plane partitions π ∈ PP m (n × n) with Tp(π) = π are usually just called symmetric plane partitions. In 1899 MacMahon conjectured [52], and in 1978 Andrews [4] proved, the following product formula for the size generating function for symmetric plane partitions: [45] gave a linear algebraic proof of Theorem 1.5 where transposition corresponds to the outer automorphism of GL(2n) induced by the symmetry of the Dynkin diagram of type A 2n−1 . Geometrically, the outer automorphism is the symplectic orthogonal complement on the Grassmannian.
Stembridge [85] gave a linear algebraic proof of Theorem 1.6, but this time using representations of the special orthogonal group SO(2n + 1) (or at least its Lie algebra) instead of the general linear group.

New sieving results
To summarize the above, the interaction of promotion and complementation of plane partitions, and also the interaction of transposition and complementation, are understood. The main undertaking of this paper is to understand how promotion and transposition, and promotion and transpose-complementation, interact.
Transposition and promotion together generate a dihedral group: Tp · Pro = Pro −1 · Tp. Our first main result is the following "dihedral sieving"-style result concerning fixed points for elements of Pro, Tp : To prove Theorem 1.7, we note that the evaluation SymMac(n, m; q → −1) is nonzero if and only if m = 2M is even, and in this case is equal to Using algebraic techniques, Proctor [59] demonstrated that a couple of different combinatorial sets of plane partition flavor are enumerated by 1≤i≤j≤n−1 i+j+2M i+j . We will establish that the set π ∈ PP 2M (n × n) : Tp · Pro(π) = π is in bijection with one of the sets Proctor showed is counted by 1≤i≤j≤n−1 i+j+2M i+j . Transpose-complementation and promotion commute. Hence the group they generate is a product of two cyclic groups: Pro, Tp · Co Z/2nZ × Z/2Z. Enumerating fixed points for elements of a cyclic group times Z/2Z is a bit more interesting than for elements of a dihedral group because there are more conjugacy classes. Barcelo,Reiner,and Stanton [6] considered an extension of cyclic sieving to products of two cyclic groups, which they called "bicyclic sieving". Our second main result is the following "bicyclic sieving"-style result concerning fixed points for elements of Pro, Tp · Co : where ζ := e πi/n is a primitive (2n)th root of unity.
The proof of Theorem 1.8 is more involved than the proof of Theorem 1.7. We use a linear algebraic approach, extending the work of Rhoades [63]. Basically, we show that w 0 ∈ GL(2n) (corresponding to complementation) and the outer automorphism of GL(2n) (corresponding to transposition) behave in the appropriate way on the dual canonical basis of the relevant GL(2n) representation. In fact, this has essentially already been done: Berenstein-Zelevinsky [8] and Stembridge [87] showed that w 0 behaves as evacuation on the dual canonical basis of any irreducible general linear group representation; and Berenstein-Zelevinsky [8] also described the effect of the outer automorphism on the dual canonical basis. (General results of Lusztig [50] imply that these automorphisms permute the canonical basis in some way.) We just have to put all these results together and compute the trace of the appropriate composition of these operators. Theorems 1.2, 1.4, 1.7 and 1.8 together imply that for any element g ∈ Pro, Co, Tp , the number of plane partitions in PP m (n × n) fixed by g is given by some kind of evaluation at a root of unity of a polynomial which has a nice product formula representation as a rational expression. However, it is unclear how to package all of these results together into one theorem. Example 1.9. In this example we consider the case m = 1.
For instance, Theorem 1.2 says that where ζ := e 2πi/(a+b) is a primitive (a + b)th root of unity, and Mac(a, b, 1; q) is the usual q-binomial coefficient: This is one of the most prototypical cyclic sieving results, going back to the original Reiner-Stanton-White paper [61, Theorem 1.1(b)]. Theorem 1.4 offers a dihedral extension of this prototypical cyclic sieving result. This dihedral extension, which combines rotation of subsets with reversal of subsets, is less well known, but is discussed for instance in [60,Proposition 4.1] (at least for a + b odd). Now let us assume a = b = n so that we can also consider what Theorems 1.7 and 1.8 say about the case m = 1. Actually, Theorem 1.7 is not so interesting in this case because, as mentioned, SymMac(n, m; q → −1) = 0 when m is odd, and moreover it is clear that there are no I with −w 0 (I) = c(I) since 1 ∈ c(I) ⇐⇒ n ∈ I while 1 ∈ −w 0 (I) ⇐⇒ n / ∈ I. But Theorem 1.8 is quite interesting in this case. It says that # I ⊆ {1, . . . , 2n} of size n : c n+k (I) = −I = SymMac n, 1; q → ζ k , where ζ := e πi/n is a primitive (2n)th root of unity, and SymMac(n, 1; q) has the simple form SymMac(n, 1; q) = I⊆{1,3,5,...,2n−1} For example, taking n = 2, we have SymMac(2, 1; q) = 1 + q + q 3 + q 4 .

The relevant evaluations are
In agreement with these evaluations: the I ⊆  where ζ := e πi/n is a primitive (2n)th root of unity, and SymMac (n, 1; q) has the simple form: This result also appears in the original Reiner-Stanton-White paper [61,Corollary 8.5], and is further discussed in [2, Section 6.1]. Work of Rush and Shi [69] implies that subsets under twisted rotation are in equivariant bijection with PP 1 ( n ) under Row (these notions are defined in Section 5); hence, this result can be seen as the case m = 1 of Conjecture 5.1 as well. While evidently quite similar, we know of no direct connection between counting subsets fixed by powers of twisted rotation, and counting subsets whose rotations are equal to their complements. Remark 1.10. Some of the root of unity evaluations of polynomials appearing in Rhoades's paper [63] had prior combinatorial interpretations, for instance in terms of border-strip tableaux (see [47]). However, we are not aware of any prior combinatorial interpretations of the evaluations appearing in Theorems 1.7 and 1.8.

Rowmotion
In the last sections of the paper we consider another invertible operator on plane partitions called rowmotion, Row : PP m (a × b) → PP m (a × b). Rowmotion and promotion are closely related. Rowmotion is, like promotion, a composition of all of the piecewise-linear toggles acting on PP m (a×b). However, whereas promotion is a composition of these toggles "from left to right", rowmotion is a composition "from top to bottom". Striker and Williams [89] explained that the actions of promotion and rowmotion are conjugate; in particular, there is some composition D of toggles so that D · Row · D −1 = Pro. We show that this conjugating map D behaves nicely with respect to complementation and transposition. We conclude that versions of Theorems 1.2, 1.4, 1.7 and 1.8 hold for Row (but with slight differences since, e.g., Row commutes with Tp, while (Tp · Co) · Row = Row −1 · (Tp · Co), et cetera). In particular, for any g ∈ Row, Co, Tp , we can again count the number of plane partitions in PP m (n × n) fixed by g by some kind of sieving phenomenon evaluation.
One reason to consider rowmotion instead of promotion is because rowmotion makes sense acting on any partially ordered set (not all posets have a notion of left and right, but they all have a notion of top and bottom). Our original motivation for studying the way symmetries interact with rowmotion was a series of cyclic sieving conjectures we made in [35] concerning rowmotion acting on the P -partitions of other posets P besides the rectangle poset. Many of the posets with conjectured cyclic sieving for rowmotion are "triangular" posets which can be obtained from the rectangle by enforcing certain symmetries. More precisely, in the final section we show, following Grinberg and Roby [30], that the P -partitions for these various triangular posets P are in rowmotion-equivariant bijection with the set of plane partitions in PP m (n × n) fixed by various subgroups of Row, Tp . While our results concerning plane partitions fixed by elements of Row, Tp do not directly imply anything about rowmotion for these triangular posets, they do lend credence to the idea that there are nice sieving phenomenon formulas counting plane partitions fixed by many subgroups of Row, Tp .
Denote the set of such triangular arrays π by CY(n, 2M ). Proctor [ (and see also the addendum of that paper for other references for this formula).
With this result in hand we are now ready to prove Theorem 1.7.

Promotion and transpose-complementation
In this section we prove Theorem 1.8. We do this by extending Rhoades's [63] approach to cyclic sieving for tableaux using the dual canonical basis of GL(a + b) representations. But actually, rather than hew closely to Rhoades's presentation, we instead follow the presentation of Lam [46]. Lam explained how the relevant GL(a + b) representation is the coordinate ring of the Grassmannian Gr(a, a + b). We find this geometric perspective useful. Also, as hinted at in Section 1, we owe a great debt to the papers of Stembridge [85,87] and especially Kuperberg [45] for explaining how involutive symmetries of plane partitions can be realized as algebra automorphisms on these coordinate rings.
In this section we work with semistandard tableaux rather than plane partitions. We recall that the correspondence between plane partitions and tableaux of rectangular shape is explained in Appendix A. For a partition λ we use SSYT(λ, k) to denote the set of semistandard Young tableaux of shape λ with entries less than or equal to k. We will freely use the bijection Ψ : PP m (a × b) ∼ − → SSYT(m a , a + b) defined in the appendix. Via this bijection promotion Pro and complementation Co are viewed as operators on SSYT(m a , a + b) and transposition Tp is viewed as an operator on SSYT(m n , 2n). The behavior of these operators on tableaux is explained in Proposition A.9.

Background on Grassmannian coordinate rings
Before we can prove Theorem 1.8 we have to review a bit about Grassmannians and the representation theory arising from their study. We start with the Grassmannian.
The Grassmannian Gr(a, a + b) is the space of a-dimensional subspaces of the complex vector space V = C a+b . There is a very well-known system of coordinates on the Grassmannian called the Plücker coordinates. Let U ∈ Gr(a, a + b) and choose an ordered basis v 1 , . . . , v a ∈ C a+b of U ; let I = {i 1 , i 2 , . . . , i a } ⊆ {1, 2, . . . , a + b} be a subset of size a; then the Plücker coordinate ∆ I (U ) is equal to the maximal minor of the (a + b) × a matrix with column vectors v 1 , . . . , v a given by selecting rows i 1 , . . . , i a . The Grassmannian is a projective variety and the map is an embedding of Gr(a, a + b) into P ( a+b a )−1 known as the Plücker embedding. We use the notation Gr(a, a+b) ⊆ C ( a+b a ) to denote the affine cone over Gr(a, a+b) in its Plücker embedding. And we use R(a, a + b) to denote the coordinate ring of Gr(a, a + b). In other words, R(a, a + b) is the commutative ring where the Plücker relations are the well-known relations cutting out Gr(a, a + b) as a subset of P ( a+b a )−1 . Equivalently we may think of R(a, a + b) as the homogeneous coordinate ring of Gr(a, a + b).
See for instance [77,Chapter 1] for the basics concerning the coordinate ring of the Grassmannian. We use R(a, a + b) m to denote the functions in R(a, a + b) of homogeneous degree m. Now we review representations of the general linear group and canonical bases. We will find the following notation for matrices useful: diag( is the anti-diagonal k × k matrix with anti-diagonal entries x 1 to x k from upper-right to lowerleft; and of course Id k := diag( k 1, 1, . . . , 1) is the k × k identity matrix. We also often write matrices in block form.
The general linear group GL(a + b) is the group of invertible linear transformations acting on V = C a+b . We usually think of the elements of the general linear group as (a + b) × (a + b) C-matrices with nonzero determinant, having implicitly chosen an ordered basis e 1 , e 2 , . . . , e a+b of V . The special linear group SL(a + b) ⊆ GL(a + b) is the subgroup of matrices in GL(a + b) of determinant one. The Lie algebra corresponding to the Lie group SL(a+b) is the Lie algebra sl a+b of (a + b) × (a + b) C-matrices with trace zero, with Lie bracket given by the commutator. The Lie algebra sl a+b is simple.
The symmetric group S a+b on a + b letters is the quotient of the normalizer in GL(a + b) of T by T . In other words, S a+b is the Weyl group of GL(a + b). The symmetric group is also the Weyl group of SL(a+b). Thus elements of the symmetric group can be lifted to the general linear group in various ways; but conjugation by elements of the symmetric group gives a well-defined action on the torus of GL(a + b), and on the torus of SL(a + b).
See for instance [25,Chapter 15] for the basics concerning the representation theory of Lusztig [49] and Kashiwara [41] constructed a canonical (or global ) basis of the irreducible U q (sl a+b )-module V q (λ) (and their two constructions are known to give the same basis [32]). Here U q (sl a+b ) is the quantized universal enveloping algebra of sl a+b , a deformation of the universal enveloping algebra U (sl a+b ). By setting q → 1 in their work, and by fixing a particular highest We can view the Grassmannian as a quotient Gr(a, a is the space of (a + b) × a C-matrices of rank a, and GL(a) acts on Mat • (a + b, a) on the right in the obvious way. This space of matrices Mat • (a + b, a) caries an obvious left action of GL(a + b) which commutes with the right GL(a) action, and in this way we obtain an action of GL(a + b) on Gr(a, a + b). Similarly, we can view Gr(a, a + b) \ {0} as the quotient Gr(a, a + b) \ {0} = Mat • (a + b, a)/SL(a), and in this way we obtain an action of GL(a + b) on Gr(a, a + b) \ {0} which is compatible with the action of GL(a + b) on Gr(a, a + b). We extend this action to an action of GL(a + b) on all of Gr(a, a + b) by declaring g · 0 = 0 for all g ∈ GL(a + b). We then get an action of GL(a + b) on R(a, a + b) via algebra automorphisms by inverting and pulling back: for g ∈ GL(a + b) and f ∈ R(a, a + b) we set (g · f )(U ) := f g −1 U for all U ∈ Gr(a, a + b). It is well known, for instance via the classical Borel-Weil theorem, is a representation of a group G, then the dual representation ρ * : G → GL(V * ) is the representation where g ∈ G acts on the dual space V * by ρ(g −1 ) T , with the T superscript denoting transpose.) This means we can consider the dual canonical basis {H(T ) : T ∈ SSYT(m a , a + b)} as a basis of R(a, a + b) m .

Grassmannian coordinate ring automophisms
We now define several algebra automorphisms of R(a, a + b). These automorphisms are at the heart of our proof of Theorem 1.8: they will correspond to promotion, complementation, and transposition of plane partitions.
Some of these automorphisms are (the actions of) elements of GL(a + b). First we define the twisted cyclic shift χ ∈ GL(a + b): The matrix χ is a lift of the long cycle c := 1 2 ... a+b−1 a+b 2 3 ... a+b 1 ∈ S a+b . Note that χ has order a + b acting on Gr(a, a + b) because χ a+b multiplies each vector in C a+b by (−1) (a−1) and hence each Plücker coordinate by (−1) a(a−1) = 1. We next define the twisted reflection w 0 ∈ GL(a + b): We denote this element of GL(a + b) by w 0 because it is a particular lift of the longest element w 0 := 1 2 ... a+b a+b a+b−1 ... 1 ∈ S a+b in the symmetric group. Note that w 0 is an involution acting on Gr(a, a + b) because w 2 0 multiplies each vector in C a+b by (−1) (a−1) and hence each Plücker coordinate by (−1) a(a−1) = 1.
For the next several paragraphs we suppose that a = b = n. Let B be the following skewsymmetric 2n × 2n matrix: Then B defines a symplectic form ·, · B on V = C 2n by x, y B := x T ·B·y for all x, y ∈ C 2n . This symplectic form defines an outer automorphism φ B of GL(2n) where a linear transformation is sent by φ B to the inverse of its transpose with respect to the identification of V and V * induced by ·, · B . At the level of matrices we have where the superscript T denotes usual matrix transposition. Clearly φ B is an involution. And φ B restricts to an involutive outer automorphism φ B : SL(2n) → SL(2n) of the special linear group.
The symplectic group Sp(2n) ⊆ GL(2n) is the subgroup of GL(2n) consisting of those matrices A ∈ GL(2n) with φ B (A) = A. In fact Sp(2n) ⊆ SL(2n) (this can be seen by consideration of Pfaffians). Inside of Sp(2n) we have the algebraic torus consisting of those diagonal matrices The analog of the symmetric group here is the hyperoctahedral group. The hyperoctahedral group (Z/2Z) S n is the subset of the symmetric group S 2n consisting of those permutations σ ∈ S 2n for which σ(i) = (2n + 1) − σ((2n + 1) − i) for all 1 ≤ i ≤ 2n. The hyperoctahedral group is the Weyl group of Sp(2n). Thus, elements of (Z/2Z) S n act on the torus of Sp(2n) by conjugation. The symplectic form ·, · B also gives rise to the following automorphism (of projective algebraic varieties) on the middle-dimensional Grassmannian Gr(n, 2n): where U ⊥ is the orthogonal complement of the subspace U with respect to ·, · B , that is, U ⊥ := x ∈ C 2n : x, y B = 0 for all y ∈ U . Clearly φ B is an involution. We extend φ B to an automorphism φ B : Gr(n, 2n) → Gr(n, 2n) (of affine algebraic varieties) in a unique way by requiring that ∆ {1,2,...,n} (U ) = ∆ {1,2,...,n} φ B (U ) for all U ∈ Gr(n, 2n). By abuse of notation we also use φ B to denote the induced algebra automorphism on the coordinate ring R(n, 2n) of Gr(n, 2n) given by pulling back φ B . All of these φ B remain involutions.
The essential property connecting φ B and φ B is that for any A ∈ GL(2n), we have for . This is easy to see from the fact that Ax, φ B (A)y B = x, y B for all x, y ∈ C 2n . Moreover, the amount that the action of a matrix A ∈ GL(2n) scales ∆ {1,2,...,n} (U ) for U ∈ Gr(n, 2n) is given by the principal n × n minor of A; a simple computation with block matrices shows that for A ∈ SL(2n) this principal minor is the same for A and for φ B (A). Hence for any A ∈ SL(2n), we in fact have that 2n). That is to say, for any A ∈ SL(2n) we have the following equality of automorphisms of the coordinate ring R(n, 2n):

Behavior of coordinate ring automorphisms on bases
Now we study how these automorphisms behave on the various bases of R(a, a + b).
Lemma 3.1. The actions of the above automorphisms on the Plücker coordinates generating the coordinate ring R(a, a + b) are: Proof . The first two bulleted items are stating simple facts about how matrix minors behave when rows are permuted. The factor i (a−1) in the definition of w 0 is there because i (a−1) Id a+b multiplies each Plücker coordinate by i −a(a−1) = (−1) a(a−1) , which is exactly the right number of minus signs to cancel the number of row transpositions we need to vertically flip an a × a submatrix. Similarly, the entry of (−1) a−1 in the definition of χ cancels out the row transpositions needed to bring the last row of an a × a submatrix to the front.
The statement about φ B is explained, in the somewhat different but equivalent language of alternating forms and the Hodge star, in [45, proof of Theorem 4.1]. It is also not hard to see directly. Let U ∈ Gr(a, a + b), and suppose U lies in the dense open subset of the Grassmannian where ∆ {1,2,...,n} (U ) = 0. Let's represent U by a matrix in reduced column echelon form whose column span is U (and note that ∆ {1,2,...,n} (U ) = 0 implies the upper n × n square submatrix of this matrix is Id n ). Then the effect of φ B is to "transpose" the lower n×n square submatrix of this matrix across its main anti-diagonal, while also multiplying the entries in this square submatrix by ±1 in a checkerboard pattern, as the following diagrams depict in the cases n = 2, 3, 4: This matrix representation makes it easy to check that ∆ I φ B (U ) = ∆ −w 0 (I) (U ). This is because the maximal minors of the 2n × n matrix correspond, up to sign, to all of the minors of its lower n × n square submatrix (see [56,Lemma 3.9]); and the checkerboard pattern of signs exactly addresses the sign issue. Then we observe that Proof . These follow immediately from Lemma 3.1 if we recall the effects of Co and Tp on tableaux as described in Proposition A.9. For a tableau T ∈ SSYT(m a , a + b), the columns of the complementary tableau Co(T ) are w 0 (I m ), w 0 (I m−1 ), . . . , w 0 (I 1 ), where I 1 , I 2 , . . . , I m are the columns of T . Similarly, for a tableau T ∈ SSYT(m n , 2n), the columns of the transposed tableau Tp(T ) are −w 0 (I 1 ), −w 0 (I 2 ), . . . , −w 0 (I m ), where I 1 , I 2 , . . . , I m are the columns of T .
Essentially via Corollary 3.2, Stembridge [85] and Kuperberg [45] were able to deduce the q = −1 results discussed in the Section 1: Theorems 1.3 and 1.5. Note crucially, however, that we do not have χ(M (T )) = M (Pro(T )). Indeed, the whole point of using sophisticated bases like the dual canonical basis is that the naive bases like the standard monomial basis fail to behave well under the action of the long cycle.
This brings us to the main algebraic result we need to prove Theorem 1.8: The actions of the above automorphisms on the dual canonical basis of the ring R(a, a + b) m are: Proof . The first bulleted item is [46,Theorem 2]. But in fact this result is essentially due to Rhoades [63]. To obtain this result Rhoades used a particular realization of the canonical basis in terms of Kazhdan-Lusztig immanants [64,65] which was first introduced by Du [14] and further developed by Skandera [80]. (In [15] Du showed that his canonical basis is the same as that of Lusztig [49].) Very recently, Rush [68] gave a new proof of Rhoades's result which is more conceptual/abstract. For the second bulleted item: Berenstein-Zelevinsky [8,Proposition 8.8], and Stembridge [87,Theorem 1.2] showed that for any irreducible GL(a + b) representation V (λ), multiplication by a lift of w 0 corresponds (up to an overall ±1 sign) to evacuation of tableaux in the dual canonical basis. As explained in the appendix, in the case of rectangular tableaux evacuation is the same as complementation.
For the third bulleted item: Berenstein-Zelevinsky [8, Section 7] again explained the effect of twisting by the GL(2n) automorphism φ B on the dual canonical basis of any irreducible representation V (λ). (This automorphism is denoted by ψ in [8,Section 7].) This effect is described in terms of the "multisegment duality" [43]. It should be possible to show that this involution defined in terms of the multisegment duality reduces to transposition in the case of rectangular partitions, just like evacuation also radically simplifies in the rectangular case.
However, there are some annoying technicalities we would have to deal with in order to directly apply the work of Berenstein-Zelevinsky [8]. For instance, we would have to show that their indexing of the dual canonical basis is compatible with that of Du [14]. Let us instead explain a different way to conclude the second and third bulleted items. The idea is still to observe that these involutions correspond to automorphisms of the quantized universal enveloping algebra. But we can exploit the fact that we are working in the particularly nice "rectangular" setting where the standard monomial basis also behaves well with respect to these involutions (which is not always the case). It turns out that we can piggyback off of the result for the standard monomial basis to obtain the result we want for the dual canonical basis.
In discussing automorphisms of the quantized universal enveloping algebra we follow the presentation of Berenstein-Zelevinsky [8, Section 7]. If φ : U q (sl a+b ) → U q (sl a+b ) is an algebra automorphism, then from a U q (sl a+b )-module V we get another module φ V by twisting by φ: φ V = V as a vector space but we have g ∈ U q (sl a+b ) act on φ V by φ(g). If V = V q (λ) is irreducible then so is φ V : say φ V V q (φ(λ)) for the corresponding highest weight φ(λ). Thus, abusing notation, we get an isomorphism of vector spaces φ : V q (λ) → V q (φ(λ)) also denoted φ which satisfies φ(gv) = φ(g)φ(v) for all g ∈ U q (sl a+b ) and v ∈ V q (λ). Because we have a choice of highest weight vector, the map φ : V q (λ) → V q (φ(λ)) is uniquely defined only up to an overall scalar.
The quantized universal enveloping algebra U q (sl a+b ) is generated by elements E i , F i , K i , K −1 i for 1 ≤ i < a + b subject to certain relations involving the parameter q. There are two involutive automorphisms of U q (sl a+b ) we want to consider. Following [8, Section 7] we call these automorphisms η and ψ. They are given as follows At the level of weights we have η(λ) = λ for all λ, and ψ(mω n ) = mω n for all m in the case a = b = n. Thus, we get involutions η : V q (mω a ) → V q (mω a ), and ψ : V q (mω n ) → V q (mω n ) in the case a = b = n. General results of Lusztig [50, Proposition 21.1.2] (see also Berenstein-Zelevinsky [8, Proposition 7.1]) imply that both η and ψ permute the canonical bases of these U q (sl a+b )-modules; indeed, η is the so-called "Lusztig involution", while ψ is more-or-less the involutive automorphism induced by the Dynkin diagram symmetry in type A 2n−1 . These descend to involutions of the GL(a+b) representations η : V (mω a ) → V (mω a ) and ψ : V (mω n ) → V (mω n ) which permute the corresponding canonical bases. In this way we also get involutions η : V (mω a ) * → V (mω a ) * and ψ : V (mω n ) * → V (mω n ) * on the dual spaces which permute the dual canonical bases. (We are glossing over the fact that the maps η and ψ between irreducible modules are only defined up to an overall scalar; to really get a permutation of the canonical or dual canonical bases we have to normalize properly.) When we take the limit q → 1, the automorphisms of U q (sl a+b ) above reduce to the following automorphisms of U (sl a+b ), the ordinary universal enveloping algebra of sl a+b : So φ B agrees with ψ up to a scalar, and hence permutes the dual canonical basis up to a sign.
To summarize the preceding, we have argued that w 0 and φ B permute the dual canonical basis in some way (at least up to overall signs). But we want to conclude that these permutations correspond to complementation and transposition of tableaux. To do this, we note that the transition matrix between the dual canonical and standard monomial basis is upper unitriangular with respect to a certain order on tableaux. This is proved in a paper of Brundan [11,Theorem 26], where he in fact gives an explicit formula for this transition matrix in terms of Kazhdan-Lusztig polynomials (and he notes [11,Remark 10] that his indexing of the canonical basis is consistent with that of Du [14]). We know from Corollary 3.2 that w 0 and φ B permute the standard monomial basis in the appropriate way. So finally we observe that if a permutation matrix is conjugated by an upper unitriangular matrix to another permutation matrix, then the two permutation matrices have to be the same (and this remains true if one of the permutation matrices is a priori only a permutation matrix up to an overall sign). Thus, the fact that w 0 and φ B permute the standard monomial basis in the appropriate way in fact implies that they permute the dual canonical basis in the appropriate way.

Proof of bicyclic sieving for Grassmannian coordinate ring
We are now ready to prove Theorem 1.8. Thus, It is easy to check that X, Y ∈ SL(2n) and that φ B (Y ) = X −1 . So we have the following equality of algebra automorphisms R(n, 2n) → R(n, 2n): Next we observe that the matrix XBX has the block form from which it is easy to see that Then we can compute that In other words, X −1 · i −1 χ · X is a symplectic matrix. By looking at the characteristic polynomial, we see that the eigenvalues of i −1 χ are all distinct; in fact they are precisely −ζ 1/2 , −ζ 1/2+1 , −ζ 1/2+2 , . . . , −ζ 1/2+(2n−1) , where ζ := e πi/n is a primitive (2n)th root of unity. Set D q := q 1/2−n diag 1, q, q 2 , . . . , q 2n−1 . With this notation, i −1 χ is conjugate to D ζ . Moreover, since X · i −1 χ · X −1 is symplectic, and it is diagonalizable, a basic result in symplectic linear algebra says this matrix is symplectically diagonalizable; that is, we can find a symplectic matrix S ∈ Sp(2n) and a diagonal matrix D for which we have X · i −1 χ · X −1 = S · D · S −1 . In fact, by conjugating by an element of the hyperoctahedral group, we can assume that this diagonal matrix is D = D ζ . (Note that D q is in fact symplectic since its ith and (2n + 1 − i)th entries along the diagonal are inverses.) We then also clearly have that To complete the proof, we compute tr R(n,2n)m φ B · w 0 · i −k χ (n+k) in two ways. On the one hand, = i knm · # T ∈ SSYT(m n , 2n) : (Tp · Co) · Pro n+k (T ) = T = i knm · # π ∈ PP m (n × n) : (Tp · Co) · Pro n+k (π) = π .
Here the factor of i knm comes from the fact that i −k Id 2n ∈ GL(2n) scales each Plücker coordinate by i kn and hence scales elements of R(n, 2n) m by i knm . Meanwhile, the interpretation of the term tr R(n,2n)m φ B · w 0 · χ (n+k) in terms of tableaux fixed by (Tp · Co) · Pro n+k follows from working in the dual canonical basis and recalling Theorem 3.3. Finally the interpretation in terms of plane partitions fixed by (Tp · Co) · Pro n+k comes from the bijection Ψ between plane partitions and tableaux described in Appendix A.
On the other hand, from our work above we have In general, the trace tr V (λ) (φ · D), where φ is the twist by an automorphism of a simple Lie group induced from a Dynkin diagram automorphism and D is a torus element, is what is called a twining character. The twining character formula, originally due to Jantzen [36] and later rediscovered for instance in [23,24], expresses such a twining character as an ordinary character of the so-called "orbit Lie group". In our case, that orbit Lie group would be the special orthogonal group SO(2n + 1). But in fact, it is easy to compute tr R(n,2n)m φ B · D ζ k directly. Indeed, as Kuperberg explained in [45,Section 4], just by considering the action on the standard monomial basis we can see tr R(n,2n)m φ B ·D ζ k = ζ −k −n 2 m/2 SymMac n, m; q → ζ −k . (Here we have ζ −k instead of ζ k because R(n, 2n) m V (mω n ) * is isomorphic to the dual representation.) But ζ −k −n 2 m/2 = i knm , so we conclude # π ∈ PP m (n × n) : (Tp · Co) · Pro n+k (π) = π = SymMac n, m; q → ζ −k .
Remark 3.4. By now the significance (in combinatorics, algebra, geometry, et cetera) of the twisted cyclic shift acting on the Grassmannian is well appreciated. See the recent paper of Karp [37] for a nice survey of many places in which the cyclic shift arises. Another related paper which studied the cyclic shift as well as involutive symmetries of the Grassmannian is the recent paper of Frieden [22]. In that paper, Frieden constructed an affine geometric crystal on the Grassmannian and in doing so showed that (a deformation of) the twisted cyclic shift tropicalizes to promotion of rectangular semistandard Young tableaux.
Remark 3.5. As mentioned in Section 1, there is another proof of cyclic sieving for plane partitions under promotion (Theorem 1.2) due to Shen and Weng [79]. The main difference from Rhoades's proof is that, rather than use the Lusztig/Kashiwara dual canonical basis, Shen-Weng used a basis of the coordinate ring of the Grassmannian coming from its cluster algebra structure. Let us quickly review their setting. A "cluster ensemble" is a pair of an "X -cluster variety" and an "A-cluster variety" associated to a quiver, which are "dual" to one another. The "Fock-Goncharov conjecture" [18] predicts that the tropical points of an X -cluster variety parameterize a canonical basis of the coordinate ring of its dual A-cluster variety, and vice-versa. Breakthrough work of Gross-Hacking-Keel-Kontsevich [33] establishes that the Fock-Goncharov conjecture holds as long as certain combinatorial conditions on the quiver are met. Under these conditions we have a canonical basis for the cluster algebra, the so-called "theta basis". Very roughly speaking, the Grassmannian carries the structure of both an X -and an A-cluster variety, and it is its own dual in a cluster ensemble. Shen-Weng verified the Gross-Hacking-Keel-Kontsevich combinatorial conditions for the quiver associated to the Grassmannian and so showed that the Fock-Goncharov conjecture holds in this case. (Their work is closely related to work of Rietsch and Williams [66], which also studied cluster duality for the Grassmannian.) Moreover, Shen-Weng showed that the twisted cyclic shift corresponds to an element of the "cluster modular group", a certain group of automorphisms of the cluster structure. The Fock-Goncharov conjecture says that the parametrization of the canonical cluster basis of one variety by the tropical points of its dual variety should be equivariant under the action of the cluster modular group. Shen-Weng showed that the action of the twisted cyclic shift on tropical points corresponds to promotion of plane partitions. They deduced that the theta basis of the coordinate ring of the Grassmannian is permuted by the twisted cyclic shift according to promotion of plane partitions (just like the dual canonical basis). It is reasonable to ask how the involutive symmetries w 0 and φ B behave on the theta basis of the coordinate ring of the Grassmannian (this could, for instance, yield a different proof of Theorem 1.8.) Understanding the behavior of the Dynkin diagram automorphism φ B seems tractable. Indeed, in the case a = b = n, the quiver Γ a,a+b (see [79,Section 2.4]) defining the Grassmannian cluster ensemble has a transposition symmetry A i,j → A j,i (ignoring arrows between frozen vertices, which are largely irrelevant). Moreover, this A i,j → A j,i symmetry exactly corresponds to the Plücker coordinate map ∆ I → ∆ −w 0 (I) which Lemma 3.1 says defines the action of φ B . The behavior of w 0 , however, is less clear to us. As Fraser explained in [21,Section 5], the twisted reflection is, together with the twisted cyclic shift, one of the most well-known and significant cluster automorphisms of the Grassmannian. (This follows from the combinatorics of weakly separated collections as elucidated in the seminal work of Scott [76] and Postnikov [56].) However, the twisted reflection is, unlike the twisted cyclic shift, an "orientationreversing" cluster automorphism, which means it cannot be an element of the cluster modular group. Hence, the Fock-Goncharov conjecture says nothing about the behavior of the twisted reflection on the theta basis. At the moment we have no good way of understanding the behavior of w 0 on the theta basis.

Promotion and rowmotion
In this section we introduce another piecewise-linear operator on plane partitions called rowmotion, which is different from but closely related to promotion. Let us very briefly review the history of rowmotion. Combinatorial rowmotion is a certain invertible operator acting on the set of order ideals of any finite 2 poset P which has been studied by many authors over a long period [5,12,10,19,55,69,89]. Piecewise-linear rowmotion is a generalization of combinatorial rowmotion which was introduced by Einstein and Propp [16,17] about 5 years ago. Piecewise-linear rowmotion, as well as its further generalization birational rowmotion, have already been the subject of a good deal of research [30,31,54], with interesting connections to topics ranging from integrable systems to quiver representations [26,28]. In the next section we will define piecewise-linear rowmotion acting on the P -partitions of an arbitrary poset P ; in this section we focus exclusively on plane partitions (which corresponds to P being the rectangle poset, i.e., the product of two chains). Since our focus throughout will be on piecewise-linear rowmotion (as opposed to combinatorial or birational rowmotion), from now on we will drop the "piecewise-linear" adjective and speak simply of "rowmotion".
Our goal in this section is to show that versions of Theorems 1.2, 1.4, 1.7 and 1.8 hold for rowmotion. So now let us formally define rowmotion and explain its relationship to promotion. As in the preceding sections, we fix the parameters a, b, and m defining our set of plane partitions Figure 1. The various toggle compositions F k , R k , P k and N k . PP m (a × b); and sometimes (e.g., when we want to consider the transposition symmetry) we also assume that a = b = n.
We define rowmotion acting on PP m (a × b) as a composition of the piecewise-linear toggles τ i,j : PP m (a×b) → PP m (a×b) introduced in Section 1. Recall that τ i,j and τ i ,j commute unless (i, j) and (i , j ) are directly adjacent. For 1 ≤ k ≤ a + b − 1, we define R k := 1≤i≤a, 1≤j≤b, i+j−1=k τ i,j to be the composition of all the toggles along the "kth antidiagonal" of our array (this composition is well-defined because these toggles commute).
Promotion was defined similarly but in terms of the diagonal toggles F k := 1≤i≤a, 1≤j≤b, j−i=k τ i,j . Observe that promotion is a composition of the piecewise-linear toggles "from left to right", whereas rowmotion is a composition of the toggles "from top to bottom". In a moment we will explain the precise relationship between promotion and rowmotion. First let us review a few other ways to express both promotion and rowmotion. In order to do this we must introduce some other compositions of toggles in addition to the F k and R k . For 1 ≤ k ≤ a we define P k := τ k,b · τ k,b−1 · · · τ k,2 · τ k,1 and for 1 ≤ k ≤ b we define N k := τ a,k · τ a−1,k · · · τ 2,k · τ 1,k . Note that while the toggles constituting F k or R k all commute, this is not true for P k or N k , so we have to be careful to specify the order of composition like we have just done. Relatedly, Fig. 1 is a diagram depicting all of these various compositions of toggles. Let us explain our notation for these compositions, which perhaps appears quite strange at first. This notation derives from terminology due to Einstein and Propp [16, Section 2]: the F k are toggles along a file of the rectangle; the R k are toggles along a rank of the rectangle; the P k are toggles along a positive fiber of the rectangle; and the N k are toggles along a negative fiber of the rectangle. (When we view the rectangle as a poset, like we will do in the next section, the terms "row" and "column" do not cohere with the usual way of depicting a poset via its Hasse diagram, which is why Einstein and Propp avoided those terms.) Both promotion and rowmotion can also be written in terms of P k (or N k ): and Row = P a · P a−1 · · · P 2 · P 1 = N b · N b−1 · · · N 2 · N 1 .
Proof . This is explained, for instance, in [16,Section 2]. It follows easily from the commutativity properties of the toggles. As we will see in the next section, we can in fact define rowmotion to be the composition of all the toggles in the order of any linear extension of the underlying rectangle poset (and something similar is true for promotion).
We now explain the relationship between promotion and rowmotion. This relationship was first discovered and explored by Striker and Williams [89]. It turns out that promotion and rowmotion are conjugate to one another. Moreover, there is a simple, explicit composition of toggles which conjugates Row to Pro. Namely, define Then we have: Proof . This was essentially proved by Striker-Williams [89,Theorem 5.4]. They were working at the level of combinatorial rowmotion (i.e., the case m = 1) but they only used the facts that the toggles are involutions and that non-adjacent toggles commute (in other words, they were really working in the corresponding "right angled Coxeter group"). At any rate, this lemma follows from the description of promotion and rowmotion in terms of the P k given in Proposition 4.1, together with the observation that P i and P j commute unless |i − j| = 1. Proof . This follows from combining Theorem 1.2 and Lemma 4.2.
Observe that Corollary 4.3 implies that Row has order dividing a + b.
Next we want to understand how rowmotion interacts with the involutive symmetries of complementation and transposition. First of all, observe that Co · Row = Row −1 · Co, while Tp · Row = Row · Tp (which is slightly different than for promotion).
In order to count fixed points of elements of the group Row, Co, Tp acting on the plane partitions in PP m (a × b), we need to understand how the conjugating map D interacts with complementation and transposition. First, we give a simple proposition about how Co and Tp interact with the individual toggles and with the P k : and hence for 1 ≤ k ≤ a we have Similarly, if a = b = n, then for 1 ≤ i, j ≤ n we have Tp · τ i,j = τ j,i · Tp and hence for 1 ≤ k ≤ n we have Proof . The statements about how to commute the τ i,j past Co or Tp are immediate from the definitions of complementation, transposition, and the toggles. The statements about the P k then follow easily. Now we can explain how D and Co interact.
Corollary 4.6. For any k ∈ Z, we have with an explicit formula for this number given by Theorem 1.4.

Proof . From Lemmas 4.2 and 4.5 we have
In other words, Co · Row k and Co · Pro k+(a+1) are conjugate, and hence in particular have the same number of fixed points. But by looking at the explicit formula in Theorem 1.4, we see that whether a is even or odd, the number of fixed points of Co · Pro k+(a+1) and of Co · Pro k are the same, and hence the corollary follows.
In order to explain how D and Tp interact, we a need a few more preparatory results. First we need to explain how Co actually can be written as a composition of toggles.
Lemma 4.7. We have Proof . This follows from various results in Appendix A. Namely, in Definition A.3, dual evacuation ε * is defined as the corresponding composition of Bender-Knuth involutions; in Proposition A.7 it is shown that these Bender-Knuth involutions correspond to the diagonal toggles F k ; and in Proposition A.9 it is shown that complementation of plane partitions corresponds to dual evacuation of semistandard tableaux under the bijection Ψ : PP m (a × b) → SSYT(m a , a + b) studied in the appendix. Remark 4.8. In addition to the description of Co in Lemma 4.7, we also have that Co = (R 1 ) · (R 2 · R 1 ) · · · (R a+b−2 · · · R 2 · R 1 ) · (R a+b−1 · · · R 2 · R 1 ). This follows from a "reciprocity" property of rowmotion that was established by Grinberg and Roby [30,Theorem 32]. (We discuss this reciprocity property more in the next section; see Theorem 5.8.) Furthermore, there are other similar ways of writing Co as a composition of toggles which one can obtain by considering evacuation ε or by considering the reciprocity property for Row −1 . The fact that complementation is an involution gives even a few more ways of writing it as a composition of toggles. Proposition 4.9. We have Proof . We claim that By Lemma 4.7, establishing this claim proves the proposition. To show that (4.1) and (4.2) are equal, first we observe that when we expand (4.2) as a product of toggles, a toggle τ i,j appears i − j + b times; this is the same number of times as when we expand (4.1) as a product of toggles. Moreover, if we read the toggles in (4.2) from right to left, then whenever we see a toggle τ i,j for the kth time, we have already seen the toggles τ i+1,j (assuming i = a) and τ i,j−1 (assuming j = 1) exactly k times. This implies that we can indeed commute the toggles that make up (4.2) to fit the form of (4.1), as claimed.

Proof . From Lemmas 4.2 and 4.5 we have
In other words, Tp · Row k and Tp · Co · Pro k−n are conjugate, and hence in particular have the same number of fixed points. By Theorem 1.8, this number is the claimed evaluation of SymMac(n, m, q) (where we note that ζ k−2n = ζ k since ζ 2n = 1).
Finally, we can explain how D and Tp · Co interact.
Lemma 4.12. If a = b = n, then we have Proof . By Lemmas 4.5 and 4.10 we have where we used that Co is an involution and Row has order dividing 2n.
Proof . From Lemmas 4.2 and 4.12 we have In other words, Tp · Co · Row k and Tp · Pro k+1 are conjugate, and hence in particular have the same number of fixed points. By Theorem 1.7, this number is the claimed evaluation of SymMac(n, m, q).
In direct analogy with what we showed for promotion, Corollaries 4.3, 4.6, 4.11 and 4.13 together imply that for any element g ∈ Row, Co, Tp , the number of plane partitions in PP m (n × n) fixed by g is given by some kind of sieving phenomenon evaluation of a nice polynomial at a root of unity.

Rowmotion for triangular posets
Since orbit structures are our main interest in this paper, and since, as we explained in the last section, rowmotion is conjugate to promotion, it might not be clear why we care about rowmotion at all. The reason we do care about rowmotion is that rowmotion can be defined as an action on the set of P -partitions of any poset P . Rowmotion acting on plane partitions corresponds to taking P to be the rectangle poset. Moreover, rowmotion still behaves remarkably well on the P -partitions of other posets P besides the rectangle poset, especially certain nice posets coming from Lie theory (namely, minuscule posets and root posets of coincidental type). In [35] we made a number of cyclic sieving conjectures regarding the action of rowmotion on the P -partitions of these other nice posets P . The major examples of the nice posets P , beyond the rectangle itself, are certain "triangular" posets. As we will explain in this section (following Grinberg-Roby [30]), these triangular posets can be obtained from the rectangle by enforcing certain symmetries. Understanding the behavior of rowmotion on these triangular posets was our original motivation for studying how rowmotion interacts with the symmetries of complementation and transposition. As we will see, while the results we obtained above concerning the interaction of rowmotion with these symmetries do not directly imply anything about cyclic sieving for the triangular posets, morally they are very closely related to our conjectures from [35].
So now we define rowmotion for arbitrary posets. We assume the reader is familiar with the basics of posets as laid out for instance in [84,Chapter 3]. Let P be a finite partially ordered set. We use ≤ for the partial order of P and for its cover relation. A P -partition of height m is a weakly order-preserving map π : P → {0, 1, . . . , m}, i.e., one for which p ≤ q ∈ P implies that π(p) ≤ π(q). We use PP m (P ) to denote the set of P -partitions of height m. For any element p ∈ P , the piecewise-linear toggle at p is the involution τ p : PP m (P ) → PP m (P ) defined by (τ p π)(q) := π(q) if p = q, min({π(r) : p r}) + max({π(r) : r p}) − π(p) if p = q, with the conventions that min(∅) = m and max(∅) = 0. Note that toggles τ p and τ q commute unless there is a cover relation between p and q.
We then define rowmotion Row : PP m (P ) → PP m (P ) by where p 1 , p 2 , . . . , p n is any linear extension of P . The commutativity properties of the toggles imply that this definition does not depend on the choice of linear extension. The rectangle poset, denoted a × b, is the Cartesian product of an a-element chain and a b-element chain. Rowmotion on the rectangle poset is the same as rowmotion of a × b plane partitions. However, note that we are working with order-preserving maps in order to match the conventions of [16] and [35]; and on the other hand in order to match the traditional conventions for plane partitions we put the maximal entry of a plane partition in its upper-left corner. Thus to satisfy all of our conventions we need to view the rectangle poset P = a × b as the set {(i, j) : 1 ≤ i ≤ a, 1 ≤ j ≤ b} with the "backwards" partial order (i, j) ≤ (i , j ) if and only if i ≤ i and j ≤ j. Fig. 2 depicts the Hasse diagram of the rectangle poset: observe how we have simply rotated the usual matrix coordinates 45 • clockwise. With this convention for the naming of elements of the rectangle poset, our notation PP m (a × b) is consistent whether we think of this as a set of plane partitions (π i,j ) or of P -partitions π(i, j). And of course, as mentioned, the two definitions of rowmotion acting on PP m (a × b) agree as well.
The other posets we care about in this section are three families of triangular posets which we denote n , n , and n . The Hasse diagrams of these triangular posets are depicted in Fig. 3. These three families of triangular posets are, in addition to the rectangle poset, the major examples of the nice posets for which we conjectured cyclic sieving under rowmotion in [35]. Namely: Then for all k ∈ Z we have # π ∈ PP m ( n ) : Row k (π) = π = F q → ζ k , where ζ := e πi/n is a primitive (2n)th root of unity.
Remark 5.4. The F (q) appearing in Conjectures 5.1, 5.2 and 5.3 are all actually polynomials with nonnegative integer coefficients F (q) ∈ N[q]. For example, the F (q) in Conjecture 5.1 is the size generating function for P -partitions in PP m ( n ): that is, F (q) = π∈PP m (P ) q |π| , where |π| := p∈P π(p). In fact, this is the same as SymMac (n, m; q) from Section 1. Meanwhile, the F (q) in Conjecture 5.2 is q m·( n+1 2 ) times the quantity denoted "(CGI)" by Proctor in [59]. Proctor explained how this expression is the generating function for P -partitions in PP m ( n ) with respect to a certain statistic (the statistic in question is slightly more complicated than size-it involves an alternating sum of entries). Finally, the F (q) in Conjecture 5.3 is the result of applying the substitution q → q 2 to Mac(n, n, m; q). For all these conjectures, the case k = 0 is known; that is, F (1) is known to be equal to the total number of height m plane partitions of the poset. Also, for all these conjectures, the case m = 1 is known [5,69]. Remark 5.6. In [20], Fontaine and Kamnitzer use some ideas from geometric representation theory to obtain a kind of refinement of Rhoades's Theorem 1.2 in which, rather than considering the action of promotion on the whole set SSYT(m a , a + b), they consider only those tableaux with a fixed (cyclically symmetric) content. The relevant sieving polynomial turns out to be the corresponding Kostka-Foulkes polynomial (in fact, in [63] Rhoades obtained a less precise version of this result in which only the absolute value of the root of unity evaluation was considered). It is possible that Conjectures 5.1, 5.2 and 5.3 have similar "content" refinements as well. Actually, this possibility is discussed in [35,Remark 4.25] where it is suggested that the appropriate Lusztig's q-analog of weight multiplicity could be the sieving polynomial. We will not discuss "content" refinements further here. Now we explain, following Grinberg-Roby [30], how the P -partitions in PP m (P ) for the triangular posets P are in Row-equivariant bijection with the plane partitions in PP m (n × n) which satisfy certain symmetry properties. This allows us to reformulate Conjectures 5.1, 5.2 and 5.3 as assertions about the number of plane partitions in PP m (n × n) fixed by various subgroups of Row, Tp .
First let us explain how to relate n to the rectangle, which is very easy.
Lemma 5.7. There is a Row-equivariant bijection between PP m ( n ) and the subset of those π ∈ PP m (n × n) for which Tp(π) = π.
Proof . This is basically proved by Section 9]. They were working at the birational level; the result we want, at the piecewise-linear level, is even simpler than what they did.
From such a π we obtain a π ∈ PP m (n × n) by setting This map π → π gives the desired bijection. In particular, it is easily seen to be rowmotion equivariant.
In order to relate rowmotion of n to rowmotion of the rectangle, we have to review a remarkable "reciprocity" property of rowmotion acting on the rectangle that was established by Grinberg-Roby [30]. Theorem 5.8 ([30,Theorem 32]). For any π ∈ PP m (a + b) we have Actually, Grinberg-Roby proved the birational version of Theorem 5.8; but the birational version implies the piecewise-linear version we have stated via tropicalization. Theorem 5.8 allows us to relate n to the rectangle, as follows.
Lemma 5.9. There is a Row-equivariant bijection between PP M ( n−1 ) and the subset of those π ∈ PP 2M (n × n) for which Tp · Row n (π) = π.
Proof . This is basically proved by Grinberg-Roby in [30, Section 10] (they worked at the birational level, but via tropicalization their results imply the corresponding piecewise-linear statements). They explained how to embed PP M ( n−1 ) into PP 2M (n×n) in a Row-equivariant way. We now review their embedding. Let us view a P -partition π ∈ PP M ( n−1 ) as triangular array π = (π i,j ) 1≤i,j≤n−1 i+j≤n of nonnegative integers π i,j ∈ N such that: π is weakly decreasing in rows and columns (i.e., π i,j ≥ π i+1,j , π i,j ≥ π i,j+1 for all i, j), the maximal entry of π is less than or equal to m (i.e., π 1,1 ≤ m).
From such a π we obtain a π ∈ PP 2M (n × n) by setting An example of the map π → π in the case n = 3, M = 2 is In [30,Lemma 67] it is shown that the map π → π is indeed an embedding of PP M ( n−1 ) into PP 2M (n × n) which is equivariant under rowmotion. Moreover, [30,Theorem 65] implies that if π ∈ PP 2M (n×n) is in the image of this embedding of PP M ( n−1 ) into PP 2M (n × n) then we will have Row n (π) = Tp(π). Indeed, that essentially follows from the reciprocity result, Theorem 5.8. It also can be shown using Theorem 5.8 that the only π ∈ PP 2M (n × n) with Row n (π) = Tp(π) are in the image of this embedding. But in fact, we can also establish that there are no other such π via a counting argument. Namely, Corollary 4.11 implies that # π ∈ PP 2M (n × n) : Tp · Row n (π) = π = SymMac(n, 2M ; q → −1).
Recall from Section 1 that And it is known that #PP M ( n−1 ) = 1≤i≤j≤n−1 i+j+2M i+j ; see, for instance, [59, Case "(CG)" of Theorem 1]. This completes the proof of the lemma.
Remark 5.10. If m is odd, there are no π ∈ PP m (n×n) for which Tp·Row n (π) = π. This can be seen either from the Grinberg-Roby reciprocity result, Theorem 5.8, or from our Corollary 4.11.
Remark 5.11. Recall the set CY(n, 2M ) defined in Section 2: this can be thought of as the subset of PP 2M ( n−1 ) with even entries along the leftmost "file". In Section 2 we explained how CY(n, 2M ) is naturally in bijection with the set of plane partitions π ∈ PP 2M (n × n) for which Tp · Pro(π) = π. Meanwhile, we just explained in the proof of Lemma 5.9 that PP M ( n−1 ) is naturally in bijection with the set of plane partitions π ∈ PP 2M (n × n) for which Tp · Row n (π) = π. Then observe that #CY(n, 2M ) = as was proved for instance in the paper of Proctor [59]. However, it is not at all a priori clear that CY(n, 2M ) and PP M ( n−1 ) have the same size, and constructing a bijection between these two sets is rather difficult (a bijection appears for instance in [78]). Thus, by studying the way these two operators interact with transposition, we uncovered another interesting "duality" between promotion and rowmotion.
We can relate n to the rectangle by combining the previous two ideas.
Proof . This follows from combining the ideas in the proofs of Lemmas 5.7 and 5.9. There is an obvious "transposition" symmetry of 2n−1 which reflects the poset across the vertical line of symmetry. Let Tp : PP M ( 2n−1 ) → PP M ( 2n−1 ) denote the induced symmetry of P -partitions. Then, by the same argument as in the proof of Lemma 5.7, PP M ( n ) is in Rowequivariant bijection with those π ∈ PP M ( 2n−1 ) with Tp(π) = π. Via the proof of Lemma 5.9 we can further embed PP M ( n ) into PP 2M (2n × 2n) in desired way.
The preceding lemmas about how to embed the triangular posets into the rectangle allow us to reformulate Conjectures 5.1, 5.2 and 5.3, as follows: Conjecture 5.13 (reformulation of Conjecture 5.1 in light of Lemma 5.7). For any k ∈ Z we have that # π ∈ PP m (n × n) : Tp(π) = π, Row k (π) = π = F q → ζ k , where ζ := e πi/n is a primitive (2n)th root of unity and Conjecture 5.14 (reformulation of Conjecture 5.2 in light of Lemma 5.9). For any k ∈ Z we have that # π ∈ PP 2M (n × n) : Tp · Row n (π) = π, Row k (π) = π = F q → ζ k , where ζ := e πi/n is a primitive (2n)th root of unity and Conjecture 5.15 (reformulation of Conjecture 5.3 in light of Lemma 5.12). For any k ∈ Z we have that # π ∈ PP 2M (2n × 2n) : Tp(π) = π, Row 2n (π) = π, Row k (π) = π = F q → ζ k , where ζ := e πi/2n is a primitive (4n)th root of unity and Corollaries 4.3 and 4.11 from the previous section say that for any g ∈ Row, Tp , the number of plane partitions in PP m (n × n) fixed by g is given by some kind of sieving phenomenon evaluation of a polynomial with a simple product formula as a rational expression. In other words, for any cyclic subgroup H ⊆ Row, Tp , the number of plane partitions in PP m (n × n) fixed by H is given by such an evaluation. Meanwhile, Conjectures 5.13, 5.14 and 5.15 assert that for various noncyclic subgroups H ⊆ Row, Tp , the number of plane partitions in PP m (n × n) fixed by H is given by such an evaluation. This leads us to wonder the following: Question 5.17. Is the number of plane partitions in PP m (n × n) fixed by H, where H is any subgroup of Row, Tp , given by a sieving phenomenon-type evaluation at a root of unity of a polynomial with a simple product formula as a rational expression?
If Question 5.17 had a positive answer, it would be very similar to what happens with the "classical" symmetries of plane partitions, where the 10 symmetry classes all have product formulas for their enumeration (again see [44,81]).
Remark 5.18. The polynomial F (q) appearing in Conjecture 5.1 is the principal specialization of the character of the irreducible SO(2n + 1) representation V (mω n ), where ω n is the minuscule weight of type B n . (Technically one might have to work with the simply connected double cover SO(2n + 1), i.e., the so-called "spin group".) Geometrically, this representation is the dual of the mth homogeneous component of the coordinate ring of the maximal orthogonal Grassmannian OG(n, 2n + 1). Similarly, the polynomial F (q) appearing in Conjecture 5.2 is (essentially) the principal specialization of the character of the irreducible Sp(2n) representation V (mω n ), where ω n is the cominuscule weight of type C n . Geometrically, this representation is the dual of the mth homogeneous component of the coordinate ring of the Lagrangian Grassmannian LG(n, 2n). See Proctor [58,59] or Stembridge [85] for more information about these polynomials. At any rate, the fact that these polynomials are more-or-less Lie group characters naturally suggests an approach for resolving Conjectures 5.1 and 5.2: find a basis of the corresponding representation indexed by the set of P -partitions in question and such that an appropriate group element (e.g., the lift of a Coxeter element) permutes the basis according to rowmotion (or, more likely, according to a conjugate "promotion"-like action). In other words, extend Rhoades's approach [63] to other types. The problem with this approach is that the naive bases like the standard monomial basis fail to behave in the appropriate way, while the sophisticated bases like the dual canonical basis or the theta basis are extremely hard to concretely get one's hands on, and doubly so outside of type A. (We do not mean to suggest that it is totally hopeless to work in other types. For instance, the theory of plabic graphs [56,76] is ultimately the combinatorial underpinning of the Shen-Weng [79] proof of cyclic sieving, and the work of Karpman [38,39,40] extends much of the theory of plabic graphs to the Lagrangian Grassmannian.) Our results in this paper point the way to an alternative but ultimately complementary approach to Conjectures 5.1 and 5.2: stay in the type A world but impose symmetries.

A Plane partitions and semistandard tableaux
In this appendix we explain the correspondence between plane partitions and semistandard tableaux of rectangular shape, and how the operators we are interested in (e.g., promotion) behave under this correspondence.
A partition λ = (λ 1 ≥ λ 2 ≥ · · · ) is an infinite nonincreasing sequence of integers for which λ i = 0 for all i 0. The nonzero λ i are called the parts of λ. If we write a partition as λ = (λ 1 , . . . , λ k ) that means that λ i = 0 for i > k. A particularly important family of partitions for us will be the rectangle partitions m a := ( a m, m, . . . , m). We represent a partition λ via its Young diagram, which is the collection of boxes in rows with λ i boxes left-justified in row i. For example, the Young diagram of (4, 3, 1, 1) is A semistandard Young tableau of shape λ is a filling of the boxes of the Young diagram of λ with positive integers that is weakly increasing in rows and strictly increasing in columns. For example, the following is a semistandard Young tableau of shape (4, 3, 1, 1): We use SSYT(λ, k) to denote the set of semistandard Young tableaux of shape λ whose entries belong to {1, 2, . . . , k}. Note that this set is empty if k is less than the number of parts of λ. Now we define the promotion operator ρ acting on SSYT(λ, k). 3 Roughly speaking, promotion behaves as follows: first we delete all entries of k in our tableau, leaving holes in their places; then we slide the remaining entries into the holes so that the holes occupy the upper-left of the Young diagram; then we increment by one all entries in the tableau; and finally we fill the holes in the upper-left with 1's. For example, with k = 6, an application of promotion might look like the following: To formalize this definition would require more explanation of the sliding procedure. A precise description is given in [63,Section 2]. In fact, these slides are the "jeu de taquin" moves of Schützenberger [73,74,75] (see also the presentation of Haiman [34]). Another closely related operator acting on SSYT(λ, k) is evacuation ε. It can also be defined in terms of jeu de taquin slides. Evacuation roughly behaves as follows: first we rotate the tableau 180 • ; then we replace every entry by k + 1 minus that entry; finally, we slide the entries into the upper-left so that we get back to a Young diagram shape. For example, with k = 6, an application of evacuation might look like the following: Again, to formalize this definition of evacuation we would need to explain the jeu de taquin slides in more detail; a precise description is given in [63, Section 2]. However, rather than use jeu de taquin, we will instead work with different but equivalent definitions of promotion and evacuation in terms of the so-called "Bender-Knuth involutions" [7].
Definition A.1. The ith Bender-Knuth involution, denoted BK i : SSYT(λ, k) → SSYT(λ, k), for 1 ≤ i < k, is the operator which acts on a tableau T ∈ SSYT(λ, k) as follows: first we "freeze" in place all i's directly above (i + 1)'s, and all (i + 1)'s directly below i's; and then, in each row, we change unfrozen i's into (i + 1)'s and unfrozen (i + 1)'s into i's in the unique way which preserves the semistandardness condition and so that the number of unfrozen i's in that row in the resulting tableau is the number of unfrozen (i + 1)'s in that row in the original tableau, and vice-versa. In other words, considering just the unfrozen i's and (i + 1)'s in a row, we perform the transformation i x (i + 1) y → i y (i + 1) x on these entries. Then we "freeze" in place 4's directly above 5's and 5's directly below 4's. In the picture below these frozen boxes have been shaded: It is clear that the BK i are indeed involutions; however, unlike the reflection operators s i which act on SSYT(λ, k) thanks to its crystal structure, note that the BK i do not satisfy the braid relations, and hence do not give an action of the symmetric group on these tableaux.
We can define promotion and evacuation as a composition of the Bender-Knuth involutions.
Dual evacuation ε * : SSYT(λ, k) → SSYT(λ, k) is We did not discuss dual evacuation earlier but it turns out to be useful in understanding the behavior of promotion and evacuation. It is a theorem of Gansner [27] (see also [9,13]) that the definitions of promotion and evacuation in terms of the Bender-Knuth involutions are the same as those in terms of Schützenberger's jeu de taquin moves.
As we will see in the next proposition, evacuation has a very simple behavior on tableaux of rectangular shape. In order to record that behavior we need a little notation. So for a tableau T ∈ SSYT(m a , k) we define T + ∈ SSYT(m a , k) to be the tableau obtained by rotating T 180 • and replacing every entry by k + 1 minus that entry.
The following proposition records some basic properties of promotion and evacuation which are well known but are "folklore". The best reference we have for these results is the paper of Bloom-Pechenik-Saracino [9], who adapt the arguments presented by Stanley [83] in the case of standard Young tableaux, and also use the fundamental connection of promotion and evacuation to the Robinson-Schensted-Knuth correspondence.
Proposition A.4 (see [9,Theorem 2.9], building off of [83, Theorem 2.1]). For any shape λ and any k we have the following relationship among the operators ρ, ε, ε * : SSYT(λ, k) → SSYT(λ, k): Furthermore, if λ = m a is a rectangle then ε(T ) = ε * (T ) = T + for all tableaux T ∈ SSYT(m a , k). Consequently, ρ k = id if λ = m a is a rectangle. Now we explain an equivalent, but very useful, way to think about semistandard Young tableaux: namely, as Gelfand-Tsetlin patterns. In particular, Gelfand-Tsetlin patterns will serve as the bridge between semistandard tableaux and plane partitions. The usefulness of Gelfand-Tsetlin patterns for understanding operations on semistandard tableaux in terms of piecewiselinear expressions was especially emphasized in the papers of Berenstein and Kirillov [42] and Berenstein and Zelevinsky [8].
Definition A.5. Let λ be a partition and k an integer greater than or equal to the number of parts of λ. A Gelfand-Tsetlin pattern of shape λ and length k is a triangular array π = (π i,j ) 1≤i≤j≤k of nonnegative integers π i,j ∈ N such that: π is weakly decreasing in rows and columns (i.e., π i,j ≥ π i+1,j , π i,j ≥ π i,j+1 for all i, j), the main diagonal (π 1,1 , π 2,2 , . . . , π k,k ) of π is equal to the partition λ.
We denote the set of such Gelfand-Tsetlin patterns by GT(λ, k).
There is a well-known bijection Φ : GT(λ, k) → SSYT(λ, k): for π ∈ GT(λ, k), the tableau T = Φ(π) is the unique semistandard tableau such that for all 1 ≤ i ≤ k the diagonal (π i,1 , π i+1,2 , . . . , π k,k+1−i ) of π is the shape of the restriction of T to the entries {1, 2, . . . , k+1−i}. To see that this is really a bijection, observe that π k,1 is the number of 1's in T , and these must all go in the first row; similarly π k−1,1 − π k,1 is the number of 2's in the first row and π k−1,2 is the number of 2's in the second row; and so on. In this way we can clearly reconstruct a unique tableau T from π, and the inequalities imposed on the π i,j exactly correspond to the semistandardness condition.
Example A.6. Let λ = (3, 2, 1, 1, 0) and let π ∈ GT(λ, 5) be the following Gelfand-Tsetlin pattern: Naturally we want to understand how promotion behaves in terms of Gelfand-Tsetlin patterns. This is where the piecewise-linear toggles come in. We define the piecewise-linear toggle τ i,j : GT(λ, k) → GT(λ, k) for 1 ≤ i < j ≤ k by (τ i,j π) p,q := π p,q if (p, q) = (i, j), min(π i,j−1 , π i−1,j ) + max(π i+1,j , π i,j+1 ) − π i,j if (p, q) = (i, j), where we ignore π i,j with i, j outside of the bounds of the triangle (at least one term in each max and min will exist). Observe that these are exactly the same as the τ i,j defined in Section 1. We again define F l := 1≤i≤j≤k, j−i=l τ i,j for 1 ≤ l ≤ k − 1 to be the composition of all the toggles along the "lth diagonal" of our array. As Berenstein-Kirillov [42] explained, these diagonal toggles are the same as the Bender-Knuth involutions: 4 Proposition A.7 ([42, Proposition 2.2]). Viewing the Bender-Knuth involutions as operators on GT(λ, k) via the bijection Φ, we have BK i = F k−i for 1 ≤ i ≤ k − 1.
So ρ and ε can be described in terms of piecewise-linear dynamics on GT(λ, k). Finally, let us concentrate on the rectangular case and the correspondence with plane partitions. So suppose that λ = m a , and let us take k = a + b to match our indexing of plane partitions. Then consider what a Gelfand-Tsetlin pattern π ∈ GT(m a , a + b) looks like. In the upper-left, π has a length a triangle of entries which must all be m's; in the lower-right, π has a length b triangle of entries which must all be 0's; and the other entries in π, whose values are not forced, form an a × b rectangle. For example, with a = 3, b = 2, and m = 5, we have π = 5 5 5 * * 5 5 * * 5 * * 0 0 0 where the asterisks denote the entries whose values are not forced. What condition is placed on these asterisk entries? Well, they certainly must be weakly decreasing in rows and columns, and they must all be integers between 0 and m. In other words, they exactly form a plane partition in PP m (a × b). And clearly any such plane partition can be placed in the asterisk entries.
In this way we obtain a bijection Ψ : PP m (a × b) → SSYT(m a , a + b): we extend a plane partition π ∈ PP m (a × b) to a Gelfand-Tsetlin pattern in GT(m a , a + b) by appending a length a triangle of m's to its left and a length b triangle of 0's below it; and then we map that Gelfand-Tsetlin pattern to a semistandard tableau in SSYT(m a , a + b) via Φ. This is depicted in the following example for a plane partition π ∈ PP 4 (2 × 2): This construction is discussed, very briefly, in [17, pp. 516-517].
Remark A.8. There is a very naive way to obtain a semistandard Young tableau of rectangular shape from a plane partition π ∈ PP m (a × b): rotate π 180 • , and then add i to all entries in the ith row. The bijection Ψ is not this naive procedure. Indeed, this naive procedure produces a tableau in SSYT(b a , m + a), which is not the same set of tableaux that Ψ maps into.
Let us describe another way to view the bijection Ψ, which is also useful.
We start with the case m = 1. Note that a single column tableau T ∈ SSYT(1 a , a + b) is exactly the same as a subset I ⊆ {1, 2, . . . , a + b} of size a. So in this case Ψ is some bijection Ψ : PP 1 (a × b) ∼ − → {I ⊆ {1, . . . , a + b} of size a}. In fact, this bijection is a correspondence between Young diagrams that fit in an a × b rectangle and size a subsets of {1, 2, . . . , a + b} which is ubiquitous in algebraic combinatorics, as we now explain. Let π ∈ PP 1 (a × b). The boundary between the entries of 1 and 0 in π determines a lattice path of down and left steps from the upper-right corner of the a × b grid to the lower-left corner. Writing this lattice path as a word in the alphabet {D, L} with a D's and b L's (where D's correspond to down steps and L's to left steps), the subset Ψ(π) ⊆ {1, 2, . . . , a + b} is the set of positions of D's in this word. This is depicted in the following example with a = 4 and b = 5: Now we extend the construction from the previous paragraph to greater values of m. For π, π ∈ PP m (a × b) we write π ≥ π to mean that π is entrywise greater than or equal to π ; and we define π + π ∈ PP m+m (a × b) for π ∈ PP m (a × b), π ∈ PP m (a × b) to be their entrywise sum. Let π ∈ PP m (a × b). There are unique plane partitions π 1 , π 2 , . . . , π m ∈ PP 1 (a × b) for which π = π 1 + π 2 + · · · + π m and π 1 ≥ π 2 ≥ · · · ≥ π m ; explicitly, we have (π k ) i,j = 1 if π i,j ≥ k, 0 otherwise, for all 1 ≤ k ≤ m. Then Ψ(π) ∈ SSYT(m a , a + b) is the tableau whose columns are the subsets Ψ(π 1 ), Ψ(π 2 ), . . . , Ψ(π m ) in order. This is easily proven inductively: the condition π m−1 ≥ π m means that placing the column Ψ(π m ) to the right of the column Ψ(π m−1 ) will preserve the tableau's semistandardness; conversely, appending the column Ψ(π m ) changes the entries in the Gelfand-Tsetlin pattern of the tableau in exactly the way that corresponds to adding π m . This description of Ψ is depicted in the following example for a plane partition π ∈ PP 4 (2 × 2): We end this section by describing how the operators Pro, Co, and Tp on plane partitions defined in Section 1 behave when viewed as operators on semistandard tableaux via the bijection Ψ. Unsurprisingly, Pro behaves as ρ (thus justifying the name promotion), while Co behaves as evacuation. In order to record the behavior of transposition we need a little notation. So for T ∈ SSYT(m n , 2n) we define T † ∈ SSYT(m n , 2n) to be the tableau obtained from T by first replacing each entry by 2n + 1 minus that entry, and then replacing each column I by its set-theoretic complement {1, 2, . . . , 2n} \ I. Proof . The first bulleted item is immediate from the original description of Ψ in terms of Gelfand-Tsetlin patterns, together with the description of the Bender-Knuth involutions as compositions of toggles which appears in Proposition A.7 above.
The second and third bulleted items are easier to see from the alternate description of Ψ. (Of course, with the second bulleted item we are implicitly applying the folklore Proposition A.4.) It is easily checked that the behaviors of Co and Tp are as claimed for single column tableaux. Furthermore, for T ∈ SSYT(m a , a + b), if the columns of T are I 1 , . . . , I m then the columns of T + will be I + m , . . . , I + 1 ; and for T ∈ SSYT(m n , 2n), if the columns of T are I 1 , . . . , I m then the columns of T † will be I † 1 , . . . , I † m . Finally, we have π ≥ π ⇐⇒ Co(π ) ≥ Co(π) for any pair π, π ∈ PP 1 (a × b); and similarly we have π ≥ π ⇐⇒ Tp(π) ≥ Tp(π ) for any pair π, π ∈ PP 1 (n × n). These observations, together with the alternate description of Ψ, imply the second and third bulleted items.