The Fon-Der-Flaass action partitions the order ideals of a poset into disjoint orbits. For a product of two chains, Propp and Roby observed --- across orbits --- the mean cardinality of the order ideals within an orbit to be invariant. That this phenomenon, which they christened homomesy, extends to all minuscule posets is shown herein.
Given a minuscule poset $P$, there exists a complex simple Lie algebra $\mathfrak{g}$ and a representation $V$ of $\mathfrak{g}$ such that the lattice of order ideals of $P$ coincides with the weight lattice of $V$. For a weight $μ$ with corresponding order ideal $I$, it is demonstrated that the behavior of the Weyl group simple reflections on $μ$ not only uniquely determines $μ$, but also encodes the cardinality of $I$. After recourse to work of Rush and Shi mapping the anatomy of the lattice isomorphism, the upshot is a uniform proof that the cardinality statistic exhibits homomesy.
A further application of these ideas shows that the statistic tracking the number of maximal elements in an order ideal is also homomesic, extending another result of Propp and Roby.
Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistic. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and pro…
Given a permutation $τ$ defined on a set of combinatorial objects $S$, together with some statistic $f:S\rightarrow \mathbb{R}$, we say that the triple $\langle S, τ,f \rangle$ exhibits homomesy if $f$ has the same average along all orbits of $τ$ in $S$. This phenomenon was noticed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). After Propp and Roby's paper, homomesy has received a lot of attention and a number of mathematicians are intrigued by it. While seeming ubiquitous, homomesy is often surprisingly non-trivial to prove. Propp and Roby studied homomesy…
J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion exhibits homomesy. In this paper, we prove an analogous result in the case of the product of three chains where one chain has two elements. In order to prove this result, we generalize from two to $n$ dimensions the recombination technique that D. Einstein and Propp developed to study homomesy. We see that our main homomesy result…
Recent work of Hopkins establishes that the lattice of order ideals of a minuscule poset satisfies the coincidental down-degree expectations property of Reiner, Tenner, and Yong. His approach appeals to the classification of minuscule posets. A uniform proof is presented herein. The blueprint follows that of Rush and Wang in their uniform proof that various cardinality statistics are homomesic on orbits of order ideals of minuscule posets under the Fon-Der-Flaass action. The underpinning remains the original insight of Rush and Shi into the structure of the isomorphism between the weight latti…
An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets, it is shown that the Fon-Der-Flaass action exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White. A uniform proof is given by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements. This bijection is proven to be equivariant with respect to a conjugate of the Fon-Der-Flaass action and an arbitrary C…
Reiner, Tenner, and Yong recently introduced the coincidental down-degree expectations (CDE) property for finite posets and showed that many nice posets are CDE. In this paper we further explore the CDE property, resolving a number of conjectures about CDE posets put forth by Reiner-Tenner-Yong. A consequence of our work is the completion of a case-by-case proof that any minuscule lattice is CDE. We also explain two major applications of the study of CDE posets: formulas for certain classes of set-valued tableaux; and homomesy results for rowmotion and gyration acting on sets of order ideals.
The $Γ$-colored $d$-complete and $Γ$-colored minuscule posets unify and generalize multiple classes of colored posets introduced by R.A. Proctor, J.R. Stembridge, and R.M. Green. In previous work, we showed that $Γ$-colored minuscule posets are necessary and sufficient to build from colored posets certain representations of Kac--Moody algebras that generalize minuscule representations of semisimple Lie algebras. In this paper we classify $Γ$-colored minuscule posets, which also classifies the corresponding representations. We show that $Γ$-colored minuscule posets are precisely disjoint unions…
The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the O(1) dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.
The $Γ$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $Γ$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $Γ$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $Γ$-colored $d$-complete posets are precisely order filters of the connected full hea…
Let $P$ be a graded poset of rank $r$ and let $\mathbf{c}$ be a $c$-element chain. For an order ideal $I$ of $P \times \mathbf{c}$, its rowmotion $ψ(I)$ is the smallest ideal containing the minimal elements of the complementary filter of $I$. The map $ψ$ defines invertible dynamics on the set of ideals. We say that $P$ has NRP ("not relatively prime") rowmotion if no $ψ$-orbit has cardinality relatively prime to $r+c+1$. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product $P = \mathbf{a} \times \mathbf{b}…
R.M. Green described structural properties that ``doubly infinite'' colored posets should possess so that they can be used to construct representations of most affine Kac--Moody algebras. These representations are analogs of the minuscule representations of the semisimple Lie algebras, and his posets (``full heaps'') are analogs of the finite minuscule posets. Here only simply laced Kac--Moody algebras are considered. Working with their derived subalgebras, we provide a converse to Green's theorem. Smaller collections of colored structural properties are also shown to be necessary and sufficie…
This paper gives another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$ equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be interpreted as sub-families of certain root posets. We guess these r…
Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of Hamaker, Patrias, Pechenik and Williams between order ideals of rectangle and trapezoid posets, thereby affirming a conjecture of Hopkins that the rectangle and trapezoid posets have the same rowmotion orbit structures. Our main tools in proving this are $K$-jeu-de-taquin and (weak) $K$-Knuth equivalence of increasing tableaux. We define $almost$ $minimal$ $tablea…
A rooted tree T is a poset whose Hasse diagram is a graph-theoretic tree having a unique minimal element. We study rowmotion on antichains and lower order ideals of T. Recently Elizalde, Roby, Plante and Sagan considered rowmotion on fences which are posets whose Hasse diagram is a path (but permitting any number of minimal elements). They showed that in this case, the orbits could be described in terms of tilings of a cylinder. They also defined a new notion called homometry which means that a statistic takes a constant value on all orbits of the same size. This is a weaker condition than the…
Let $G$ be a group acting on a set $X$ of combinatorial objects, with finite orbits, and consider a statistic $ξ: X \to \mathbb{C}$. Propp and Roby defined the triple $(X, G, ξ)$ to be \emph{homomesic} if for any orbits $\mathcal{O}_1, \mathcal{O}_2$, the average value of the statistic $ξ$ is the same, that is \[\frac{1}{{|\mathcal{O}_1|}}\sum_{x \in \mathcal{O}_1} ξ(x) = \frac{1}{|\mathcal{O}_2|}\sum_{y \in \mathcal{O}_2} ξ(y).\] In 2013 Propp and Roby conjectured the following instance of homomesy. Let $\mathrm{SSYT}_k(m \times n)$ denote the set of semistandard Young tableaux of shape $m \t…
This article illustrates the dynamical concept of $homomesy$ in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley, and then lifted to birational operations on the positive orthant in $\mathbb{R}^{|P|}$ and indeed to a dense subset of $\mathbb{C}^{|P|}$. When the poset $P$ is a product of a chain of length $a$ and a chain of…
We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of $\mathfrak{sl}_n$ having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-…
The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of "unique rectification targets" in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's "$d$-complete posets" to extend the Thomas-Yong rule from minu…
For each minuscule flag variety $X$, there is a corresponding minuscule poset, describing its Schubert decomposition. We study an action on plane partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass (1995). For plane partitions of height at most $2$, D. Rush and X. Shi (2013) proved an instance of the cyclic sieving phenomenon, completely describing the orbit structure of this action. They noted their result does not extend to greater heights in general; however, when $X$ is one of the two minuscule flag varieties of exceptional Lie type $E$, they conjectured explicit in…
The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of (small order) periodicity, cyclic sieving, and homomesy. Many of these nice features still hold when the action is extended to $[0,1]$-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting). In this work, we parallel the birational lifting already done for order-ideal rowmotion to antichain rowmotion. We …
It is known that there exists an order isomorphism between the Weyl group orbit through a minuscule weight of a simply-laced finite-dimensional simple Lie algebra and the set of all order filters in a self-dual connected d-complete poset. In this paper, we try to extend this fact to the case of multiply-laced finite-dimensional simple Lie algebras by using the "folding" technique with respect to a Dynkin diagram automorphism.
