Poset Structure Research Articles

The facial weak order in finite Coxeter groups

We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.

Signed ring families and signed posets

ABSTRACT The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978, especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner's result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections.

Extended symmetric spaces and θ-twisted involution graphs

For a Weyl group G and an automorphism θ of order 2, the set of involutions and θ-twisted involutions can be generated by considering actions by basis elements, creating a poset structure on the elements. Haas and Helminck showed that there is a relationship between these sets and their Bruhat posets. We extend that result by considering other bases and automorphisms. We show for G = Sn, θ an involution, and any basis consisting of transpositions, the extended symmetric space is generated by a similar algorithm. Moreover, there is an isomorphism of the poset graphs for certain bases and θ.

Periodic Pólya urns, the density method and asymptotics of Young tableaux

Polya urns are urns where at each unit of time a ball is drawn and replaced with some other balls according to its colour. We introduce a more general model: the replacement rule depends on the colour of the drawn ball and the value of the time ($\operatorname{mod}p$). We extend the work of Flajolet et al. on Polya urns: the generating function encoding the evolution of the urn is studied by methods of analytic combinatorics. We show that the initial partial differential equations lead to ordinary linear differential equations which are related to hypergeometric functions (giving the exact state of the urns at time $n$). When the time goes to infinity, we prove that these periodic Polya urns have asymptotic fluctuations which are described by a product of generalized gamma distributions. With the additional help of what we call the density method (a method which offers access to enumeration and random generation of poset structures), we prove that the law of the southeast corner of a triangular Young tableau follows asymptotically a product of generalized gamma distributions. This allows us to tackle some questions related to the continuous limit of random Young tableaux and links with random surfaces.

Polynomial bases: Positivity and Schur multiplication

We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters, and Demazure atoms; the quasi-key, fundamental, and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and that an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason, and van Willigenburg’s (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux.

Dn Dynkin quiver moduli spaces

We study 3d quiver gauge theories with gauge nodes forming a Dn Dynkin diagram and their relation to nilpotent varieties in . The class of good Dn Dynkin quivers is completely characterised and the moduli space singularity structure fully determined for all such theories. The class of good Dn Dynkin quivers is denoted where is an integer, and are integer partitions and denotes membership of one of two broad subclasses. Small subclasses of these quivers are known to realise some nilpotent varieties with their moduli space branches. We fully determine which nilpotent varieties are realisable as Dn Dynkin quiver moduli spaces and which are not. Quiver addition is introduced and is used to give large subclasses of Dn Dynkin quivers poset structure. The partial ordering is determined by inclusion relations for the moduli space branches. The resulting Hasse diagrams are used to both classify Dn Dynkin quivers and determine the moduli space singularity structure for an arbitrary good theory. The poset constructions and local moduli space analyses are complemented throughout by explicit checks utilising moduli space dimension matching.

A poset structure on the alternating group generated by 3-cycles

We investigate the poset structure on the alternating group that arises when the latter is generated by 3-cycles. We study intervals in this poset and give several enumerative results, as well as a complete description of the orbits of the Hurwitz action on maximal chains. Our motivating example is the well-studied absolute order arising when the symmetric group is generated by transpositions, i.e. 2-cycles, and we compare our results to this case along the way. In particular, noncrossing partitions arise naturally in both settings.

Models for configuration space in a simplicial complex

We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve $y^2 z = x^3 + x^2 z$, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its two- and three-strand local braid groups near the singular point.

Hibi Algebras and Representation Theory

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory (Bruns and Herzog 1993; De Concini et al. 1982; Eisenbud 1980; Gonciulea and Lakshmibai 2001; Seshadri 2007). In particular, a special type of such algebras introduced by Hibi (1987) provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.

Spherical subcategories in representation theory

We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the cardinality and the height of the poset. We explicitly describe spherical subcategories and their poset structure for derived categories of certain finite-dimensional algebras.

The primitive spectrum for $\mathfrak{gl}(m|n)$

We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical simple Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, a description of the poset structure on the set primitive ideals is at the moment not known, apart from some low dimensional specific cases. We derive an alternative definition of the left Kazhdan-Lusztig quasi-order which extends to classical Lie superalgebras. We denote this quasi-order by $\unlhd$ and show that a relation in $\unlhd$ implies an inclusion between primitive ideals. For $\mathfrak{gl}(m|n)$ the new quasi-order $\unlhd$ is defined explicitly in terms of Brundan's Kazhdan-Lusztig theory. We prove that $\unlhd$ induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of $\mathfrak{gl}(m|n)$ and in general for $\mathfrak{gl}(2|2)$. An important tool is a new translation principle for primitive ideals, based on the crystal structure underlying Brundan's categorification on category ${\mathcal O}$. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for $\mathfrak{gl}(m|1)$.

The facial weak order and its lattice quotients

We investigate a poset structure that extends the weak order on a finite Coxeter group $W$ to the set of all faces of the permutahedron of $W$. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjorner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of their classes. As application, we describe the facial boolean lattice on the faces of the cube and the facial Cambrian lattice on the faces of the corresponding generalized associahedron.

On Prime Numbers and The Riemann Zeros

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: {\em adelic duality} and the {\em POSet of prime numbers}. The article presents computational evidence of the structure of the imaginary parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$, called in this article the {\em Riemann Spectrum}, using the study of their distribution. The novelty represents in considering the associated characters $p^{it}$, towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied, is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and $\omega$-eigen-distributions, are explored following.

Perfect codes in poset spaces and poset block spaces

The paper begins by giving a counter example to show that the algorithm for construction of new perfect poset codes from a given perfect poset code by removal of a coordinate as given by Lee (2004) [11] does not hold. The algorithm has been improved and generalized to obtain new perfect poset block codes from a given perfect poset block code. The modified necessary and sufficient conditions for the construction of new perfect poset codes have been derived as a particular case. A bound has been obtained on the height of poset Ps that turns a given π-code into r-perfect (Ps,π)-code. We show that there does not exist a poset which admits the binary Simplex code of order 3 to be a 2-perfect poset code. Also, all the poset structures which admit the extended ternary Golay code to be a 3-perfect poset code have been classified.

Face Rings of Cycles, Associahedra, and Standard Young Tableaux

We show that $J_n$, the Stanley-Reisner ideal of the $n$-cycle, has a free resolution supported on the $(n-3)$-dimensional simplicial associahedron $A_n$. This resolution is not minimal for $n \geq 6$; in this case the Betti numbers of $J_n$ are strictly smaller than the $f$-vector of $A_n$. We show that in fact the Betti numbers $\beta_{d}$ of $J_n$ are in bijection with the number of standard Young tableaux of shape $(d+1, 2, 1^{n-d-3})$. This complements the fact that the number of $(d-1)$-dimensional faces of $A_n$ are given by the number of standard Young tableaux of (super)shape $(d+1, d+1, 1^{n-d-3})$; a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of $J_n$ that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.

Characterization of solutions of non-symmetric algebraic Riccati equations

We give a new characterization of the solution set of non-symmetric algebraic Riccati equations involving real matrices. Our characterization involves the use of invariant subspaces of the coefficient matrices. We also give a poset structure on the solutions of ARE and explore some properties of this poset (partially ordered set).

Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

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 $\Delta$. In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets $\Delta(1)$. In particular, we provide a bijection between the lower ideals in $\Delta(1)$ and certain elements of the Weyl group of $\mathfrak g$. An inspiring observation is that, to a great extent, the theory of lower ideals in $\Delta(1)$ is similar to the theory of upper (= ad-nilpotent) ideals in the whole poset of positive roots $\Delta^+$.

The Minset-Poset Approach to Representations of Graph Connectivity

Various instances of the minimal-set poset (minset-poset for short) have been proposed in the literature, e.g., the representation of Picard and Queyranne for all st -minimum cuts of a flow network. We begin with an explanation of why this poset structure is common. We show any family of sets F that can be defined by a “labelling algorithm” (e.g., the Ford-Fulkerson labelling algorithm for maximum network flow) has an algorithm that constructs the minset poset for F . We implement this algorithm to efficiently find the nodes of the poset when F is the family of minimum edge cuts of an unweighted graph; we also give related algorithms to construct the entire poset for weighted graphs. The rest of the article discusses applications to edge- and vertex connectivity, both combinatorial and algorithmic, that we now describe. For digraphs, a natural interpretation of the minset poset represents all minimum edge cuts. In the special case of undirected graphs, the minset poset is proved to be a variant of the well-known cactus representation of all mincuts. We use the poset algorithms to construct the cactus representation for unweighted graphs in time O ( m +λ 2 n log (n/λ)) (λ is the edge connectivity) improving the previous bound O (λ n 2 ) for all but the densest graphs. We also construct the cactus representation for weighted graphs in time O ( nm log( n 2 / m )), the same bound as a previously known algorithm but in linear space O ( m ). The latter bound also holds for constructing the minset poset for any weighted digraph; the former bound also holds for constructing the nodes of that poset for any unweighted digraph. The poset is used in algorithms to increase the edge connectivity of a graph by adding the fewest edges possible. For directed and undirected graphs, weighted and unweighted, we achieve the time of the preceding two bounds, i.e., essentially the best-known bounds to compute the edge connectivity itself. Some constructions of minset posets for graph rigidity are also sketched. For vertex connectivity, the minset poset is proved to be a slight variant of the dominator tree. This leads to an algorithm to construct the dominator tree in time O ( m ) on a RAM. (The algorithm is included in the appendix, since other linear-time algorithms of similar simplicity have recently been presented.)

On structural decompositions of finite frames

A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be scalable if there exist nonnegative scalars $\{c_i\}_{i\in I}$ such that $\{c_if_i\}_{i\in I}$ is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame $\{f_i\}_{i\in I}$ to be a collection of subsets of $I$ ordered by inclusion so that nonempty $J\subseteq I$ is in the factor poset if and only if $\{f_j\}_{j\in J}$ is a tight frame for $H_n$. A similar definition is given for the scalability poset of a frame. We prove conditions which factor posets satisfy and use these to study the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset $P$. We determine a necessary condition for solving the inverse factor poset problem in $H_n$ which is also sufficient for $H_2$. We describe how factor poset structure of frames is preserved under orthogonal projections. We also consider the enumeration of the number of possible factor posets and bounds on the size of factors posets. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset.

Special matchings and parabolic Kazhdan–Lusztig polynomials

We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in [Advances in Math. {202} (2006), 555-601]. As a consequence, the parabolic Kazhdan-Lusztig polynomial indexed by $u$ and $v$ depends only on the poset structure of the Bruhat interval from the identity element to $v$ and on which elements of that interval are minimal coset representatives.