The algebra generated by the down and up operators on a differential or Ž . uniform partially ordered set poset encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down᎐up algebras. We show that down᎐up algebras exhibit many Ž . of the important features of the universal enveloping algebra U ᒐ l of the Lie 2 algebra ᒐ ᒉ including a Poincare᎐BirkhoffᎏWitt type basis and a well-behaved 2 representation theory. We investigate the structure and representations of down᎐up algebras and focus especially on Verma modules, highest weight representations, and category O O modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets. ᮊ

version 6.0 (February 25, 2018). This is the arXiv version, not the published version. The published version has been abridged in several places and split into two papers. AbstractWe study a birational map associated to any finite poset P . This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P . Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we prove that birational rowmotion has order p + q on the (p, q)-rectangle poset (i.e., on the product of a p-element chain with a q-element chain); we furthermore compute its orders on some triangle-shaped posets and on a class of posets which we call "skeletal" (this class includes all graded forests). In all cases mentioned, birational rowmotion turns out to have a finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the AA case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets. We also make a digression to study classical rowmotion on skeletal posets, since this case has seemingly been overlooked so far.

Many invertible actions τ on a set S of combinatorial objects, along with a natural statistic f on S, exhibit the following property which we dub homomesy: the average of f over each τ -orbit in S is the same as the average of f over the whole set S. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter's action on certain subposets of Young's Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.In this situation we say that the function f : S → K is homomesic under the action of τ on S, or more specifically c-mesic.When S is a finite set, homomesy can be restated equivalently as all orbit-averages being equal to the global average: 1. reversal of permutations with the statistic that counts inversions [ § 2.1]; 2. cyclic rotation of words on {−1, +1} with the {0, 1}-function that indicates whether a word satisfies the ballot condition [ § 2.2]; 3. cyclic rotation of words on {−1, +1} with the statistic that counts the number of (multiset) inversions in the word [ § 2.3]; 4. linear maps which satisfy T n = 1 acting in a vector space V with statistic the identity function [ § 2.4]; 5. the phase-shift action on simple harmonic motion with statistics given by certain polynomial combinations of position and velocity [ § 2.5]; 6. the Lyness 5-cycle acting on (most of) R 2 with f ((x, y)) = log |x −1 + x −2 | as the statistic [ § 2.6]; 7. the action on recurrent sandpile configurations given by adding 1 grain to the source vertex and then allowing the system to stabilize, with statistic the firing vector [ § 2.7]; 8. Suter's action on Young diagrams with a weighted cardinality statistic [ § 2.8]; 9. promotion in the sense of Schützenberger acting on semistandard Young tableaux of rectangular shape with statistic given by summing the entries in any centrallysymmetric subset of cells of the tableaux [ § 2.9], as studied by Bloom, Pechenik, and Saracino [BPS13]; 10. promotion (in the sense of [SW12]) acting on the set of order ideals of [a] × [b] with the cardinality statistic [ § 3.2]; 11. rowmotion acting on the set of order ideals of [a] × [b] with the cardinality statistic [ § 3.3.1]; and 12. rowmotion acting on the set of antichains of [a] × [b] with the cardinality statistic [ § 3.3.2].The authors are grateful to

International audience Many cyclic actions $τ$ on a finite set $\mathcal{S}$ ; of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit ``homomesy'': the average of $f$ over each $τ$-orbit in $\mathcal{S} $ is the same as the average of $f$ over the whole set $\mathcal{S} $. This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind and discuss old and new results for the actions of promotion and rowmotion on the poset that is the product of two chains.

We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.