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

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Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance Jessica Striker Communicated by Benjamin Braun Dynamical Algebraic Combinatorics: 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 group 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 mathematics 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 partially ordered set. 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.
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