Let $Δ$ be a root system with a subset of positive roots, $Δ^+$. We consider edges of the Hasse diagrams of some posets associated with $Δ^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $Δ^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $Δ$. We prove that the number of edges of the corresponding Hasse diagram is $(n+1)2^{n-2}$. For $Δ^+$ and the Abelian ideals, we compute the number of edges of each type and prove that the number of edges of type $α$ depends only on the length of $α$.
We introduce two polynomials (in $q$) associated with a finite poset $P$ that encode some information on the covering relation in $P$. If $P$ is a distributive lattice, and hence $P$ is isomorphic to the poset of dual order ideals in a poset $L$, then these polynomials coincide and the coefficient of $q$ equals the number of $k$-element antichains in $L$. In general, these two covering polynomials are different, and we introduce a deviation polynomial of $P$, which measures the difference between these two. We then compute all these polynomials in the case, where $P$ is one of the posets assoc…
We discuss properties of root posets for finite crystallographic root systems, and show that these properties uniquely determine root posets for the noncrystallographic dihedral types and type $H_3$, while proving that there does not exist a poset satisfying all of the properties in type $H_4$. We do this by exhaustive computer searches for posets having these properties. We further give a realization of the poset of type $H_3$ as restricted roots of type $D_6$, and conjecture a Hilbert polynomial for the $q,t$-Catalan numbers for type $H_4$.
Let $Δ$ be a (connected) Dynkin diagram of rank $n\ge 2$ and $Φ_+ = Φ_+(Δ)$ the corresponding root poset (it consists of all positive roots with respect to a fixed root basis). The width of $Φ_+$ is $n$. We will show that $Φ_+$ is "conical": it is the disjoint union of $n$ solid chains. The rich antichains in $Φ_+$ are the antichains of cardinality $n-1$. It is well known that the number of rich antichains is equal to the cardinality of $Φ_+$. The set $\mathcal R(Δ)$ of rich antichains in $Φ_+$ can itself be considered as a poset which is quite similar, but not always isomorphic, to $Φ_+$. We …
The entries of frieze patterns may be interpreted as coordinates of roots of a finite Weyl groupoid of rank two. We prove the existence of maximal elements in their root posets and classify those frieze patterns which can be used to build an affine simplicial arrangement.
We prove that the posets of connected components of intersections of toric and elliptic arrangements defined by root systems are EL-shellable and we compute their homotopy type. Our method rests on Bibby's description of such posets by means of "labeled partitions": after giving an EL-labeling and counting homology chains for general posets of labeled partitions, we obtain the stated results by considering the appropriate subposets.
Inspired by the infinite families of finite and affine root systems, we consider a "stretching" operation on general crystallographic root systems which, on the level of Coxeter diagrams, replaces a vertex with a path of unlabeled edges. We embed a root system into its stretched versions using a similar operation on individual roots. For a fixed root, we study the growth of two associated structures as we lengthen the stretched path: the downset in the root poset (in the sense of Björner and Brenti [3]) and the arrangement of shards, introduced by Nathan Reading. We show that both eventually a…
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…
For any finite poset P, there is a natural operator $X$ acting on the antichains of P. We discuss conjectural properties of this operator for some graded posets associated with irreducible root systems. In particular, if $Δ^+$ is the set of positive roots and $Π$ is the set of simple roots in $Δ^+$, then we consider the cases $P=Δ^+$ and $Δ^+\setminus Π$. For the root system of type $A_n$, we consider an $X$-invariant integer-valued function on the set of antichains of $Δ^+$ and establish some properties of it.
The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure ("weight poset"). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with $\mathbb Z$-gradings of simple Lie algebras (arXiv: math.CO 1411.7683). If $\mathfrak g$ is a simple Lie algebra, then a $\mathbb Z$-grading of $\mathfrak g$ induces a $\mathbb Z$-grading of the corresponding root system $Δ$. In this article, we elaborate on a general theory of lower ideals (or antich…
Let $g$ be a simple Lie algebra and $Ab$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $g$. In 2003 (IMRN, no.35, 1889--1913), we constructed a partition of $Ab$ into the subposets $Ab_μ$, parameterised by the long positive roots of $g$, and established some properties of these subposets. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation of abelian ideals and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $g$ is a join-semilattice.
The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type $B$ analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen--Macaulay posets. As a result, new families of convex polytopes whose face lattices have real-rooted chain polynomials are presented. An application to the face enumeration of the second barycentri…
The purpose of the present paper is to give a realization of a cylindric diagram as a subset of root systems of type $A_{κ-1}^{(1)}$ and several characterization of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the Weyl group.
Greene's rational function $Ψ_P({\bf x})$ is a sum of certain rational functions in ${\bf x}=(x_1, \ldots, x_n)$ over the linear extensions of the poset $P$ (which has $n$ elements), which he introduced in his study of the Murnaghan-Nakayama formula for the characters of the symmetric group. In recent work Boussicault, Féray, Lascoux and Reiner showed that $Ψ_P({\bf x})$ equals a valuation on a cone and calculated $Ψ_P({\bf x})$ for several posets this way. In this paper we give an expression for $Ψ_P({\bf x})$ for any poset $P$. We obtain such a formula using dissections of root polytopes. Mo…
Baden-Wuerttemberg Cooperative State University (DHBW)（巴登-符腾堡州合作应用大学）；Jagiellonian University（雅盖隆大学）；University of Kassel（卡塞尔大学）；Georgia Institute of Technology（佐治亚理工学院）
Let $\b$ be a Borel subalgebra of a simple Lie algebra $\g$ and let $\Ab$ denote the set of all Abelian ideals of $\b$. We consider $\Ab$ as poset with respect to inclusion, the zero ideal being the unique minimal element of $\Ab$. It was shown in my paper with G.Roehrle (Adv. Math. v.159 (2001)) that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of $\g$. But the very existence of it was demonstrated in a case-by-case fashion. Here a conceptual explanation for that empirical observation is given. The main results are: 1) there is a natural ma…
Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.
It is elementary and well-known that if an element x of a bounded modular lattice L has a complement in L then x has a relative complement in every interval [a,b] containing x. We show that the relatively strong assumption of modularity of L can be replaced by a weaker one formulated in the language of so-called modular triples. We further show that, in general, we need not suppose that x has a complement in L. By introducing the concept of modular triples in posets, we extend our results obtained for lattices to posets. It should be remarked that the notion of a complement can be introduced a…
A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in $A_{n-1}$ with respect to a slightly different order and prove that this poset is graded.
For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of repre…
Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which $\g$ has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of al…
A root ideal arrangement $A_I$ is the set of reflecting hyperplanes corresponding to the roots in an order ideal $I$ of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, $A_I$ is supersolvable if and only if $I$ is chain peelable, meaning that it is possible to reach the empty poset from $I$ by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the minimal ideals that correspond to non-su…
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.
We introduce the notion of a \emph{Whitney dual} of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge and chain-edge labelings which we call \emph{Whitney labelings}. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly construct a Whitney dual using a technique involving quotient posets. As applications of our main theorem, we show that geometric lattices, the lattice of noncrossi…
We introduce a Morse theory for posets of Bestvina-Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.
We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is …
We provide necessary and sufficient conditions on the unimodality of a convolution of two sequences of binomial coefficients preceded by a finite number of ones. These convolution sequences arise as as rank sequences of posets of vertex-induced subtrees for a particular class of trees. The number of such trees whose poset of vertex-induced subgraphs containing the root is not rank unimodal is determined for a fixed number of vertices $i$.
We consider the poset of weighted partitions $Π_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $Π_n^w$ provide a generalization of the lattice $Π_n$ of partitions, which we show possesses many of the well-known properties of $Π_n$. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\{1,2,\dots,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and …
A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. Our pr…
Previously, Erdős, Kierstead and Trotter investigated the dimension of random height~$2$ partially ordered sets. Their research was motivated primarily by two goals: (1)~analyzing the relative tightness of the Füredi-Kahn upper bounds on dimension in terms of maximum degree; and (2)~developing machinery for estimating the expected dimension of a random labeled poset on $n$ points. For these reasons, most of their effort was focused on the case $0<p\le 1/2$. While bounds were given for the range $1/2\le p <1$, the relative accuracy of the results in the original paper deteriorated as $p$ approa…
We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of comparable pairs in a subset of $P$ of cardinality $m$? We provide a framework for obtaining lower bounds on this quantity based on counting comparable pairs relative to a random chain and apply this framework to obtain supersaturation results for three classical posets: the boolean lattice, the collection of subspaces of $\mathbb{F}_q^n$ ordered by set inclusion a…
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}…
Let $Φ$ be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check{Q}$, spanning a Euclidean space $V$. Let $m$ be a positive integer and $\aA^m_Φ$ be the arrangement of hyperplanes in $V$ of the form $(α, x) = k$ for $α\in Φ$ and $k = 0, 1,...,m$. It is known that the number $N^+ (Φ, m)$ of bounded dominant regions of $\aA^m_Φ$ is equal to the number of facets of the positive part $Δ^m_+ (Φ)$ of the generalized cluster complex associated to the pair $(Φ, m)$ by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of $\aA^m_Φ$…
We introduce a natural partial order in structurally natural finite subsets the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers` triangular array are then calculated and the explicit formula for them is provided. Next - in the second construction - we endow the set sums of prefabiants with such an another partial order that their their bell like numbers include fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binom…
We introduce a natural partial order in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling like numbers triangular array are then calculated and the explicit formula for them is provided. Next, in the second construction we endow the set sums of prefabiants with such an another partial order that their Bell like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial New…
The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum_{u,w\in P} ^{}μ(u,w) x^{\rk u}y^{\rk w}$, where $μ(.,.)$ is the Möbius function of $P$. We compute the $M$-triangle of Armstrong's poset of $m$-divisible non-crossing partitions for the root systems of type $E_7$ and $E_8$. For the other types except $D_n$ this had been accomplished in the earlier paper "The $F$-triangle of the generalised cluster complex." Altogether, this almost settles Armstrong's $F=M$ Conjecture predicting a surprising relation between the $M$-triangle of the $m$-divisible partitions po…
