Simple Combinatorial Description Research Articles

Combinatorial properties of nonarchimedean convex sets

We study combinatorial properties of convex sets over arbitrary valued fields. We demonstrate analogs of some classical results for convex sets over the reals (e.g. the fractional Helly theorem and B\'ar\'any's theorem on points in many simplices), along with some additional properties not satisfied by convex sets over the reals, including finite breadth and VC-dimension. These results are deduced from a simple combinatorial description of modules over the valuation ring in a spherically complete valued field.

Sublinearly Morse boundaries from the viewpoint of combinatorics

Abstract We prove that the sublinearly Morse boundary of CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} cubulated groups with factor systems continuously injects in the Gromov boundary of a certain hyperbolic graph Γ. We also show that for all CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continuously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

Feynman polytopes and the tropical geometry of UV and IR divergences

We introduce a class of polytopes that concisely capture the structure of UV and IR divergences of general Feynman integrals in Schwinger parameter space, treating them in a unified way as worldline segments shrinking and expanding at different relative rates. While these polytopes conventionally arise as convex hulls---via Newton polytopes of Symanzik polynomials---we show that they also have a remarkably simple dual description as cut out by linear inequalities defining the facets. It is this dual definition that makes it possible to transparently understand and efficiently compute leading UV and IR divergences for any Feynman integral. In the case of the UV, this provides a transparent geometric understanding of the familiar nested and overlapping divergences. In the IR, the polytope exposes a new perspective on soft/collinear singularities and their intricate generalizations. Tropical geometry furnishes a simple framework for calculating the leading UV/IR divergences of any Feynman integral, associating them with the volumes of certain dual cones. As concrete applications, we generalize Weinberg's theorem to include a characterization of IR divergences, and classify space-time dimensions in which general IR divergences (logarithmic as well as power-law) can occur. We also compute the leading IR divergence of rectangular fishnet diagrams at all loop orders, which turn out to have a surprisingly simple combinatorial description.

Geometric conditions for □-irreducibility of certain representations of the general linear group over a non-archimedean local field

Let π be an irreducible, complex, smooth representation of GLn over a local non-archimedean (skew) field. Assuming π has regular Zelevinsky parameters, we give a geometric necessary and sufficient criterion for the irreducibility of the parabolic induction of π⊗π to GL2n. The latter irreducibility property is the p-adic analogue of a special case of the notion of “real representations” introduced by Leclerc and studied recently by Kang–Kashiwara–Kim–Oh (in the context of KLR or quantum affine algebras). Our criterion is in terms of singularities of Schubert varieties of type A and admits a simple combinatorial description. It is also equivalent to a condition studied by Geiss–Leclerc–Schröer.

Alexander invariants of ribbon tangles and planar algebras

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, bounding $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, throught welded diagrams. We give a simple combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. When restricted to diagrams without virtual crossings, $\mathsf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author.

Cohomology of rotational tiling spaces

A spectral sequence is defined which converges to the \v{C}ech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and ePE cohomology groups of the tiling, and the only potentially non-trivial boundary map has a simple combinatorial description in terms of its local patches. Using this spectral sequence, we compute the \v{C}ech cohomology of the Euclidean hull of the Penrose tilings.

Local Pauli stabilizers of symmetric hypergraph states

Hypergraph states of many quantum bits share the rich interplay between simple combinatorial description and nontrivial entanglement properties enjoyed by the graph states that they generalize. In this paper, we consider hypergraph states that are also permutationally invariant. We characterize the states in this class that have nontrivial local Pauli stabilizers and give applications to nonlocality and error correction.

Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra

In this article we provide a simple combinatorial description of morphisms between indecomposable complexes in the bounded derived category of a gentle algebra.

Multi-cluster complexes

We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.

Motors and Impossible Firing Patterns in the Parallel Chip-Firing Game

The parallel chip-firing game is an automaton on graphs in which vertices “fire” chips to their neighbors. This simple model contains much emergent complexity and has many connections to different areas of mathematics. In this work, we study firing sequences, which describe each vertex's interaction with its neighbors in this game. First, we introduce the concepts of motors and motorized games. Motors both generalize the game and allow us to isolate local behavior of the (ordinary) game. We study the effects of motors connected to a tree and show that motorized games can be transformed into ordinary games if the motor's firing sequence occurs in some ordinary game. Then, we completely characterize the periodic firing sequences that can occur in an ordinary game, which have a surprisingly simple combinatorial description.

Lusztig’s partition and sheets (with an Appendix by M. Bulois)

We show that, for a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic, the parts, called strata, in the partition of G recently introduced by Lusztig are unions of sheets of conjugacy classes. For G simple and adjoint we refine the parametrization of such sheets obtained in previous work with F. Esposito. We give a simple combinatorial description of strata containing spherical conjugacy classes, showing that Lusztig's correspondence induces a bijection between unions of spherical conjugacy classes and unions of classes of involutions in the Weyl group. Using ideas from the Appendix by M. Bulois, we show that the closure of a stratum is not necessarily a union of strata.

Firing Patterns in the Parallel Chip-Firing Game

The $\textit{parallel chip-firing game}$ is an automaton on graphs in which vertices "fire'' chips to their neighbors. This simple model, analogous to sandpiles forming and collapsing, contains much emergent complexity and has connections to different areas of mathematics including self-organized criticality and the study of the sandpile group. In this work, we study $\textit{firing sequences}$, which describe each vertex's interaction with its neighbors in this game. Our main contribution is a complete characterization of the periodic firing sequences that can occur in a game, which have a surprisingly simple combinatorial description. We also obtain other results about local behavior of the game after introducing the concept of $\textit{motors}$. Le $\textit{parallel chip-firing game}$, c’est une automate sur les graphiques, dans lequel les sommets “tirent” des jetons à leurs voisins. Ce modèle simple, semblable aux tas de sable qui forment et s’affaissent, contient beaucoup de complexité émergente et a des connections avec différents domaines de mathématiques, incluant le $\textit{self-organized criticality}$ et l’étude du $\textit{sandpile group}$. Dans ce projet, on étudie les $\textit{firing sequences}$, qui décrivent les interactions de chaque sommet avec ses voisins dans le jeu. Notre contribution principale est une caractérisation complète des séquences de tir qui peuvent arriver dans une jeu, qui ont une description combinatoire assez simple. Nous obtenonsaussi d'autres résultats sur le conduite locale du jeu après l’introduction du concept des $\textit{motors}$.

Теория инварианта Бухштабера симплициальных комплексов и выпуклых многогранников

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The problem stated by Victor M. Buchstaber is to find a simple combinatorial description of an s-number. We describe the main properties of s(P) and study the properties of simple n-polytopes with n+3 facets. In particular, we find the value of an s-number for such polytopes, a simple formula for their h-polynomials and the bigraded cohomology rings of the corresponding moment-angle complexes

Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time O(n+m)?(n+m), where ? is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear time split decomposition algorithm (Dahlhaus in J. Algorithms 36(2):205---240, 2000), which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper (Gioan et al. in arXiv:1104.3284 , 2011) uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.

Posetted Trees and Baker-Campbell-Hausdorff Product

In this paper, we use some typical tools of algebraic operads and homotopy transfer theory, in order to give a simple and elementary combinatorial description of the Baker-Campbell-Hausdorff formula. More precisely, we exploit the usual operadic notion of planar rooted trees, enriched with the notion of subroots, and we define a posetted tree as a planar rooted tree endowed with a monotone labelling of leaves, with elements in a partially ordered set. The main result of this paper is an explicit expression of the Baker-Campbell-Hausdorff product as a sum of iterated brackets over an indexing set of posetted binary trees.

Balanced Fiber Bundles and GKM Theory

Let T be a torus and B a compact T-manifold. Goresky et al. show in [3] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H-T*(B) as a subring of H-T*(B-T). In this paper, we prove an analog of this result for T-equivariant fiber bundles: we show that if M is a T-manifold and pi:M -> B a fiber bundle for which pi intertwines the two T-actions, there is a simple combinatorial description of H-T*(M) as a subring of H-T*(pi(-1)(B-T)). Using this result, we obtain fiber bundle analogs of results of Guillemin et al. [4] on GKM theory for homogeneous spaces.

Refined Gelfand models for wreath products

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370–393] there is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be naturally decomposed into the direct sum of submodules indexed by symmetric conjugacy classes, and in this paper we present a simple combinatorial description of the irreducible decomposition of these submodules if the group is the wreath product of a cyclic group with a symmetric group. This is attained by showing that such decomposition is compatible with the generalized Robinson–Schensted correspondence for these groups.

Cluster Multiplication in Regular Components via Generalized Chebyshev Polynomials

We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $ \mathbb{A} $ and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type $ \mathbb{A} $ . We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver.

Free knots and groups

Free knots are a simplification of virtual knots obtained by forgetting arrow/sign information at classical crossings. First non-trivial examples of free knots were constructed recently by the second named author. By using parity considerations, we construct invariants of free knots valued in certain groups. These groups have a simple combinatorial description, the first one being the infinite dihedral group.

Some combinatorially defined subsets of [formula omitted] and their relation to the idempotents

Members of idempotents in ( β N , + ) , especially those in the smallest ideal, have strong combinatorial properties. And the closure Γ of the set of idempotents has a simple combinatorial description. We investigate here the relationships among several subsets of β N that have simple combinatorial descriptions, as well as the semigroups they generate and their closures.