Dynamical Algebraic : Promotion, Rowmotion, and Resonance Jessica Striker Communicated by Benjamin Braun

is the cyclic sieving phenomenon [ReStWh04], which occurs when the evaluation of a generating function for 푋 at the primitive 푛th root of unity 휁푑 (where 휁 = 푒2휋푖/푛) counts the number of elements of 푋 fixed under 푔푑. For example, let 푋 be the set of binary words composed of two zeros and two ones. Let 푔 be the cyclic shift that acts by moving the first digit to the end of the word. Each word has four digits, so 푔 is of order four. Consider the inversion number statistic inv(푤) of 푤 ∈ 푋, which equals Dynamical Algebraic Combinatorics the number of pairs (푖, 푗) with 푖 < 푗 such that the 푖th digit Mathematical inquiry often begins with the study of ob- of 푤 is 1 and the 푗th digit is 0. See Figure 1 for the orbits jects (numbers, shapes, variables, matrices, ideals, metric of 푋 under 푔 and the inversion numbers of each binary spaces, …) and the question, “What are the objects like?” word. It then moves to the study of actions (functions, rotations, g reflections, multiplication, derivatives, shifts, …) and the 0 2 question, “How do the objects behave?” The study of 0011 0110 g actions in various mathematical contexts has been ex- tremely fruitful; consider the study of metric spaces g g 1 0101 1010 3 through the lens of dynamical systems or the study of g symmetries arising from actions. For objects and 1001 1100 actions arising from algebraic combinatorics, we call this 2 4 study dynamical algebraic combinatorics. g Let 푔 be a bijective action on a finite set 푋. Such an Figure 1. The orbits of binary words of length four action breaks the space 푋 into orbits. Often, the study with two ones under a cyclic shift; the inversion of the orbits of 푔 provides insight into the structure numbers corresponding to the binary words are of the objects in 푋, revealing hidden symmetries and shown in red. connections. One typically first seeks to understand the order 푛 of the action and then finds interesting properties The inversion number statistic determines the gener- the action exhibits. One surprisingly ubiquitous property ating function

Jessica Striker is assistant professor of at North (1) 푋(푞) = ∑ 푞inv(푤) = 1 + 푞 + 2푞2 + 푞3 + 푞4. Dakota State University. Her e-mail address is jessica.striker 푤∈푋 @ndsu.edu. 2휋푖/4 Her work is partially supported by National Security Agency Since 푔 is of order four, 휁 = 푒 = 푖. One may check 푑 Grant H98230-15-1-0041. using Figure 1 and (1) that 푋(푖 ) equals the number 푑 For permission to reprint this article, please contact: of elements of 푋 fixed by 푔 , thus this is an instance [email protected]. of the cyclic sieving phenomenon. For example, 푋(푖1) = DOI: http://dx.doi.org/10.1090/noti1539 1+푖+2푖2 +푖3 +푖4 = 1+푖+2(−1)+(−푖)+1 = 0, and zero

June/July 2017 Notices of the AMS 543 Rowmotion /

O O

  1 0 0 1 1 0 1 0 1 1 0 1 1 0

Figure 2. An order ideal in the product of chains poset 3 × 4 and its image under rowmotion. The red paths are boundary paths that divide the order ideal from the rest of the poset; the binary words corresponding to these paths are shown beneath. elements of 푋 are fixed under 푔1. 푋(푖2) = 푋(−1) = 2, and two elements are fixed under 푔2. Another interesting property actions in dynamical algebraic combinatorics often exhibit is the homomesy → → phenomenon [PrRo15], in which the average value of a statistic on every 푔-orbit equals the global average value of that statistic. Each orbit in Figure 1 has an average inversion number of 2. Thus, the inversion number statistic on 푋 is homomesic with respect to 푔. Note that both cyclic sieving and homomesy still hold in the → → → more general case in which 푋 is the set of binary words of length 푛 with 푘 ones [ReStWh04], [PrRo15]. In Figure 1, our action was visibly cyclic, so its order was clear from the definition. In the coming sections, we discuss more complicated combinatorial actions whose orders are difficult to predict. We will see examples in which an action 푔 on a large combinatorial set 푋 has a → → relatively small order; this indicates the elements of 푋 have hidden cyclic symmetry such that 푔 is a rotation in disguise. We will also see examples in which 푔 is not of small order, but rather exhibits resonance, meaning that 푔 maps to an underlying cyclic action with small order. We discuss examples of such actions on order ideals and tableaux and then give a surprising relation between them, → → which we found via this study of dynamics.

Rowmotion on Order Ideals Figure 3. Rowmotion from Figure 2 computed by As our first example, let our set 푋 be the order ideals of toggling rows from top to bottom. The elements that a poset. are being toggled at each step are shown in red, and Definition 1. A poset is a . The set at any step in which the toggles act nontrivially, the of order ideals 퐽(푃) of a poset 푃 is the set of all subsets order ideal resulting from those toggles is shown. 퐼 ⊆ 푃 such that if 푦 ∈ 퐼 and 푧 ≤ 푦, then 푧 ∈ 퐼. Specifically, we will take 푋 to be the set of order ideals 퐽(푃) for 푃 the product of chains poset a × b. That is, counting argument shows the number of order ideals in 푎+푏 for a natural number 푎, a = {1, 2, … , 푎} and the product 퐽(a × b) is given by the binomial coefficient ( 푎 ), since partial order on a × b is (푥, 푦) ≤ (푥′, 푦′) if and only if these order ideals are in bijection with binary words with 푥 ≤ 푥′ and 푦 ≤ 푦′. See Figure 2 for an example. An easy 푎 zeros and 푏 ones via the boundary path that separates

544 Notices of the AMS Volume 64, Number 6 Promotion /

O O

  Cyclic shift / 1 0 0 1 1 0 1 0 0 1 1 0 1 1

Figure 4. An order ideal in 퐽(3 × 4) and its image under promotion. Promotion acts as a cyclic shift on the binary word corresponding to the boundary path. the order ideal from the rest of the poset (where a one showed rowmotion and promotion are conjugate actions indicates an up-step and a zero indicates a down-step); in the toggle group of any ranked poset; this implies they see Figure 2. are equivariant, or have the same orbit structure. Our action 푔 will be the following; see Figure 2. Theorem 6 ([StWi12]). In any ranked poset, there is an Definition 2. Let 푃 be a poset, and let 퐼 ∈ 퐽(푃). Then row- equivariant bijection between the order ideals under row- motion on 퐼 is the order ideal generated by the minimal motion (toggle top to bottom by rows) and promotion (tog- elements of 푃\퐼. gle left to right by columns). The order of rowmotion on general posets is neither By the bijection to binary words, promotion on 퐽(a×b) well behaved nor predictable. But on 퐽(a × b) its order is, is a cyclic shift of a binary word of length 푎 + 푏; each 푎+푏 surprisingly, much smaller than ( 푎 ). toggle swaps adjacent letters, resulting in the first letter Theorem 3 ([BrSc74]). The order of rowmotion on 퐽(a×b) swapping all the way through the word to the end; see is 푎 + 푏. Figure 4. Therefore, by the above theorem, the cyclic nature of promotion on 퐽(a × b) gives a satisfying explanation To see why the order of a global action such as for the order of rowmotion on 퐽(a × b). Furthermore, rowmotion is surprisingly small, it often helps to interpret since binary words under a cyclic shift exhibit the cyclic the action as a composition of local involutions and work sieving phenomenon (as discussed in Section 1), so does within the group generated by those involutions. In the rowmotion on 퐽(a × b). case of rowmotion, we call this group the toggle group. Corollary 7 ([StWi12]). There is an equivariant bijection Definition 4. For each element 푒 ∈ 푃 define its toggle between the order ideals of a × b under rowmotion and 푡푒 ∶ 퐽(푃) → 퐽(푃) as follows. binary words of length 푎 + 푏 with 푏 ones under rotation. ⎪⎧퐼 ∪ {푒} if 푒 ∉ 퐼 and 퐼 ∪ {푒} ∈ 퐽(푃) The cyclic sieving phenomenon follows. 푡푒(퐼) = 퐼\{푒} if 푒 ∈ 퐼 and 퐼\{푒} ∈ 퐽(푃) Rowmotion on these order ideals also exhibits the ⎪⎨ ⎩퐼 otherwise homomesy phenomenon. The toggle group 푇(퐽(푃)) is the subgroup of the symmet- Theorem 8 ([PrRo15]). The order ideal cardinality statis- ric group 푆퐽(푃) generated by {푡푒}푒∈푃. tic in 퐽(a × b) exhibits homomesy (orbit-average = global- Theorem 5 ([CaFo95]). Given any poset 푃, rowmotion is average) with respect to rowmotion or promotion. the toggle group element that toggles the elements of 푃 It is natural to ask whether similar results hold on from top to bottom (in the reverse order of any linear ex- posets constructed as products of more than two chains. tension). Theorem 9 ([CaFo95]). The order of rowmotion on 퐽(a × For example, in Figure 3, we recompute rowmotion on b × 2) is 푎 + 푏 + 1. the order ideal from Figure 2 by toggling from top to bottom by rows. Again, as a corollary of Theorem 6, we used the toggle In joint work with Nathan Williams, we found a toggle group to explain this result by showing promotion on conjugate to rowmotion, which we called 퐽(a × b × 2) is a cyclic action on combinatorial objects in promotion, defined for 퐽(a × b) as toggling the elements bijection with these order ideals, which also exhibit the from left to right (we will soon make this more precise). We cyclic sieving phenomenon; see Figure 5.

June/July 2017 Notices of the AMS 545 Promotion /

1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1

1 1 8 2 8 2

Rotation / 7 3 7 3

6 4 6 4 5 5 Figure 5. An order ideal in 퐽(4 × 3 × 2) and its image under promotion, along with the boundary paths for each layer and the corresponding boundary path matrices, which transform via the parenthesizations shown to the given noncrossing partitions. Promotion on 퐽(a × b × 2) rotates the corresponding noncrossing partition.

Theorem 10 ([StWi12]). There is an equivariant bijection Promotion on Standard Young Tableaux between 퐽(a × b × 2) under rowmotion and noncrossing As our second example, let our set 푋 be composed of the partitions of 푎+푏+1 into 푏+1 blocks under rotation. The following well-loved objects in algebraic combinatorics cyclic sieving phenomenon follows. and .

Homomesy also holds in this case. Definition 12. A standard of partition shape 휆 is a bijective filling of 휆 with the numbers Theorem 11 ([Vo17]). The order ideal cardinality statistic {1, 2, … , 푛}, where 푛 is the number of boxes in 휆, such in 퐽(a × b × 2) exhibits homomesy with respect to rowmo- that labels strictly increase from left to right across tion or promotion. rows and from top to bottom down columns. Let 푆푌푇(휆) denote the set of standard Young tableaux of shape 휆. The above theorems may lead one to guess that rowmotion on 퐽(a × b × c) is of order 푎 + 푏 + 푐 − 1 and See Figures 6 and 7 for examples of tableaux in exhibits the cyclic sieving and homomesy phenomena. 푆푌푇 ( ) = 푆푌푇(2×5). 푆푌푇(휆) is enumerated by the 푛! This is not true in general. For example, when 푎 = 푏 = famous Frame–Robinson–Thrall hook formula: ∏푥∈휆 ℎ(푥) , 푐 = 3, the order of rowmotion is 8, but the generating where 푥 ranges over all boxes in the diagram of 휆. ℎ(푥) is function no longer exhibits the cyclic sieving phenomenon. the hook number of 푥, that is, the number of boxes in 휆 For 푎 = 푏 = 3, 푐 = 4, the cardinality statistic is no in the same row as 푥 and to its right, plus the number of longer homomesic. When 푎 = 푏 = 푐 = 4, the order of boxes in the same column as 푥 and below, plus one (for 푥 rowmotion is not 푎+푏+푐−1 = 11, but rather 33. Similar itself). computations for many other values of 푎, 푏, 푐 show that Our action 푔 will be M.-P. Schützenberger’s promotion the orbits of rowmotion on 퐽(a × b × c) are each of operator (see Figure 6). In general, this is a different cardinality a multiple of a divisor of 푎 + 푏 + 푐 − 1. This action from the toggle-promotion defined in the previous section. A relation between these actions will be made is an admittedly fuzzy notion, which we will make more clear later in this section and the next. precise in the last section of this article. But first, we move to our second example of an action on standard Young Definition 13. Given a partition 휆, promotion on tableaux. 푇 ∈ 푆푌푇(휆) is the product 휌푛−1 ⋯ 휌2휌1(푇) of the

546 Notices of the AMS Volume 64, Number 6 1 2 3 4 7 / 1 2 3 4 7 휌1 / 1 2 3 4 7 휌2 / 1 2 3 4 7 휌3 / 5 6 8 9 10 5 6 8 9 10 5 6 8 9 10 5 6 8 9 10

1 2 3 4 7 휌4 / 1 2 3 5 7 / 1 2 3 5 7 휌5 / 1 2 3 6 7 / 5 6 8 9 10 4 6 8 9 10 4 6 8 9 10 4 5 8 9 10

1 2 3 6 7 휌6 / 1 2 3 6 7 휌7 / 1 2 3 6 8 / 1 2 3 6 8 휌8 / 4 5 8 9 10 4 5 8 9 10 4 5 7 9 10 4 5 7 9 10

1 2 3 6 9 / 1 2 3 6 9 휌9 / 4 5 7 8 10 4 5 7 8 10

Figure 6. Promotion computed as the product 휌9 ⋯ 휌2휌1(푇) of Bender–Knuth involutions. The entries that are being acted on at each step are shown in red, and at any step in which the involution acts nontrivially, the resulting tableau is shown.

Promotion /

O O

  4 3 4 3 5 2 5 2 Rotation / 6 1 6 1

7 10 7 10 8 9 8 9

Figure 7. Promotion on 푆푌푇(2 × 푏) is in equivariant bijection with rotation on noncrossing matchings of 2푏 points. Each top row tableau entry is the smaller number in its matching pair.

Bender–Knuth involutions 휌푖, where each 휌푖 swaps 푖 Theorem 15 ([Rh10]). Promotion on 푆푌푇(푎 × 푏) exhibits and 푖 + 1 if possible. the cyclic sieving phenomenon. Promotion on standard Young tableaux in an 푎 × 푏 Proofs of this result illuminated connections to rep- rectangle has the unexpected property that all the orbit resentation theory and . In the cases 푎 = 2, 3, there are proofs that proceed by giving a bijection to other sizes are divisors of the maximum value 푎푏. combinatorial objects (noncrossing matchings and webs, Theorem 14 ([Hai92]). Promotion on 푆푌푇(푎 × 푏) is of respectively) that sends promotion to rotation [PePyRh09]; order 푎푏. see Figure 7. There are also homomesy results in this and more general settings; see [BlPeSa16]. For example, by the hook formula, |푆푌푇(5 × 7)| = Promotion on two-row rectangular tableaux 푆푌푇(2×푏) 35! 11223344556575849310211 = 278,607,172,289,160, but promo- is equivalent to toggle-promotion on the type A positive tion is of order 7 ⋅ 5 = 35. Rectangles are among a few root poset [StWi12], which is why we gave the name special shapes for which promotion is of a nice order. promotion to the action of toggling from left to right. Promotion on tableaux of general shape does not have A few years later, this name was further validated by a nice order; for example, promotion on the partition the result that hyperplane-toggle promotion on 퐽(a × b × has order 7,554,844,752. c) is equivalent to 퐾-theoretic promotion on increasing Even more surprisingly, 푆푌푇(푎 × 푏) under promotion tableaux [DiPeSt17]; we explain this in the next section. exhibits the cyclic sieving phenomenon, which means the evaluation of the 푞-analogue of the hook length Resonance on Orbits of Increasing Tableaux and formula for 푆푌푇(푎 × 푏) at the (푎푏)th root of unity Plane Partitions (푒2휋푖/푎푏)푑 counts the number of 푇 ∈ 푆푌푇(푎 × 푏) such For our final example, we take our set 푋 to be increasing that Promotion푑(푇) = 푇. tableaux, objects that have appeared in various contexts

June/July 2017 Notices of the AMS 547 within algebraic combinatorics, in particular, in relation 휔 if, for all 푥 ∈ 푋, 푐 ⋅ 푓(푥) = 푓(푔 ⋅ 푥), that is, the following to 퐾-theoretic Schubert calculus of Grassmannians. diagram commutes: 푋 푋 푔⋅ Definition 16. An increasing tableau of partition shape 휆 is a filling of 휆 with positive such that labels strictly increase from left to right across rows and from 푓 푓 top to bottom down columns. Let Inc푞(휆) denote the set of all increasing tableaux of shape 휆 with entries at most 푌 푌 푞. 푐⋅

Resonance is a first step in understanding dynamics 1 4 5 8 of more complicated combinatorial objects, since it is 2 5 7 9 an analogue of an action having a nice order. Resonance 6 7 9 10 is a pseudo-periodicity property, in that the resonant 8 10 frequency 휔 is generally less than the order of 푔. (If 휔 equals the order of 푔, this is an instance of trivial resonance.) For non-trivial resonance, we also need a K-BK3 K-BK8 surjective map 푓 to a simpler underlying set 푌 on which 푔 corresponds to a cyclic action 푐 with smaller order 휔. In 1 3 5 8 1 4 5 8 the second theorem below, this map is the binary content 푞 2 5 7 9 2 5 7 9 vector, defined on an increasing tableau 푇 ∈ Inc (휆) as 6 7 9 10 6 7 8 10 the sequence Con(푇) = (푎1, 푎2, … , 푎푞), where 푎푖 = 1 if 푖 8 10 9 10 is an entry of 푇 and 푎푖 = 0 if it is not. See Figure 9. Theorem 19 ([DiPeSt17]). For a certain map 푓, (퐽(a × b × Figure 8. The action of some 퐾-Bender–Knuth c), ⟨Rowmotion⟩, 푓) exhibits resonance with frequency 푎 + involutions on an increasing tableau in Inc10(4, 4, 4, 2). 푏 + 푐 − 1. See Figures 8 and 9 for examples. Note that Inc푛(휆) Theorem 20 ([DiPeSt17]). (Inc푞(휆), ⟨퐾-promotion⟩, Con) (where 푛 is the number of boxes in 휆) contains 푆푌푇(휆) as exhibits resonance with frequency 푞. a subset; increasing tableaux differ from standard Young The uncanny similarity of the objects and actions in tableaux in that there may be repeated and/or missing these theorems when 휆 = 푎 × 푏 and 푞 = 푎 + 푏 + 푐 − 1 led numbers in the filling. to a nonobvious connection between them. Our action 푔 will be 퐾-promotion (see Figure 8), which Theorem 21 ([DiPeSt17]). Inc푎+푏+푐−1(푎 × 푏) under 퐾- was originally defined globally using 퐾-jeu-de-taquin; promotion is in equivariant bijection with 퐽(a × b × c) we give here our equivalent definition from [DiPeSt17]. under rowmotion. Note that if an increasing tableau is a standard Young This proof relies on the following extension of The- tableau, 퐾-Bender–Knuth involutions are equivalent to orem 6 from the 2- to 푛-dimensional . For this Bender–Knuth involutions. generalization, we introduced and developed the machin- ery of affine hyperplane toggles and 푛-dimensional lattice Definition 17. Given a partition 휆 and a natural num- projections. We obtained a large family of toggle group ber 푞, K-promotion on 푇 ∈ Inc푞(휆) is the product actions {Pro휋,푣}, whose orbit structures are equivalent to that of rowmotion. That is, given a poset 푃, we project the K-BK ⋯ K-BK K-BK (푇) of the 퐾-Bender–Knuth in- 푞−1 2 1 elements in a way consistent with the covering relations volutions K-BK푖, where each K-BK푖 increments 푖 and/or into the 푛-dimensional lattice by the map 휋. Then Pro휋,푣 is decrements 푖 + 1 wherever possible. the toggle group action that toggles elements as it sweeps through 휋(푃) by an affine hyperplane in the direction For Inc푞(푎 × 푏) under 퐾-promotion, it is no longer true determined by the vector 푣. that orbit sizes are divisors of the maximum value 푞, but Theorem 22 ([DiPeSt17]). Let 푃 be a finite poset with an rather, they are multiples of divisors of 푞. For example, 푛-dimensional lattice projection 휋. Let 푣 and 푤 be length 11 Inc (4 × 4) has orbits of size 11 and 33. With Kevin Dilks 푛 vectors with entries in {±1}. Then there is an equivari- and Oliver Pechenik, we made this observation more ant bijection between 퐽(푃) under Pro휋,푣 and 퐽(푃) under precise and gave this phenomenon the name resonance. Pro휋,푤.

It is not hard to show that Pro휋,(1,1,…,1) is rowmotion, so Definition 18 ([DiPeSt17]). Suppose 푔 is a cyclic group this theorem says rowmotion and 2푛 −1 other promotions action on a set 푋, 푐 a cyclic group action of order 휔 acting have the same orbit structure. For 휋 the natural three- nontrivially on a set 푌, and 푓 ∶ 푋 → 푌 a surjection. We say dimensional embedding, Pro휋,(1,1,−1) on 퐽(a × b × c) is the triple (푋, ⟨푔⟩, 푓) exhibits resonance with frequency equivalent to 퐾-promotion on Inc푎+푏+푐−1(푎 × 푏), which

548 Notices of the AMS Volume 64, Number 6 1 2 4 7 1 3 5 6 3 5 6 8 퐾-Promotion / 2 4 7 9 5 7 8 10 4 6 9 11 7 9 10 12 6 8 11 12

Con Con   Cyclic shift / (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1) (1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1)

Figure 9. An increasing tableau in Inc12(4 × 4) and its image under 퐾-promotion, along with the map to the binary content of each. 퐾-promotion cyclically shifts the binary content vector.

when combined with the above theorem gives the proof of [Rh10] B. Rhoades, Cyclic sieving, promotion, and representa- Theorem 21, showing that toggle-promotion and tableaux- tion theory, J. Combin. Theory Ser. A 117 (2010), no. 1, promotion coincide on a much larger set than we had 38–76. previously known. [StWi12] J. Striker and N. Williams, Promotion and rowmotion, Eur. J. Combin. 33 (2012), no. 8, 1919–1942. [Vo17] C. Vorland, Homomesy in products of three chains Conclusion and multidimensional recombination, preprint available at: We have given a flavor of dynamical algebraic combina- https://arxiv.org/abs/1705.02665. torics through examples of combinatorial objects with actions that have small order and other nice properties, Credits such as cyclic sieving and homomesy. Most of the objects Figure 7 is courtesy of Nathan Williams. and actions that behave nicely in these three respects are, Figure 8 is courtesy of Oliver Pechenik. in some sense, planar. These phenomena often still occur Photo of Jessica Striker is courtesy of Ryan Striker. in objects that are only slightly three-dimensional, since such objects may often be mapped bijectively to planar objects, as in Theorem 10. But once such maps are no longer possible, these properties tend to no longer hold. We have seen examples of higher-dimensional combina- torial objects with natural actions that no longer have a nice order, but rather exhibit resonance. We hope that further study of resonance in such combinatorial dynam- ical systems will uncover more interesting properties and surprising connections.

References [BlPeSa16] J. Bloom, O. Pechenik, and D. Saracino, Proofs and generalizations of a homomesy conjecture of Propp and Roby, Discrete Math. 339 (2016), 194–206. [BrSc74] A. Brouwer and A. Schrijver, On the period of an oper- ator, defined on antichains, Math Centrum report ZW 24/74 (1974). [CaFo95] P. Cameron and D. Fon-der-Flaass, Orbits of antichains revisited, Eur. J. Combin. 16 (1995), 545–554. [DiPeSt17] K. Dilks, O. Pechenik, and J. Striker, Resonance in orbits of plane partitions and increasing tableaux, J. Combin. Th. Ser. A 148 (2017), 244–274. [Hai92] M. Haiman, Dual equivalence with applications, includ- ing a conjecture of Proctor, Discrete Math. 99 (1992), no. 1-3, 79–113. [PePyRh09] T. Petersen, P. Pylyavskyy, and B. Rhoades, Pro- motion and cyclic sieving via webs, J. Algebraic Combin. 30 (2009), no. 1, 19–41. [PrRo15] J. Propp and T. Roby, Homomesy in products of two chains, Electron. J. Combin. 22 (2015), no. 3. [ReStWh04] V. Reiner, D. Stanton, and D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004), no. 1, 17–50.

June/July 2017 Notices of the AMS 549