Combinatorial Description Research Articles

Colliding holes in Riemann surfaces and quantum cluster algebras

In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesic arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston–Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.

Sturm 3-ball global attractors 1: Thom–Smale complexes and meanders

This is the first of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x), \end{aligned}$$ on the unit interval $$0< x<1$$ with Neumann boundary conditions. Equilibria are assumed to be hyperbolic. Geometrically, we study the resulting Thom–Smale dynamic complex with cells defined by the unstable manifolds of the equilibria. The Thom–Smale complex turns out to be a regular cell complex. Our geometric description involves a bipolar orientation of the 1-skeleton, a hemisphere decomposition of the boundary 2-sphere by two polar meridians, and a meridian overlap of certain 2-cell faces in opposite hemispheres. The combinatorial description is in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $$x=0$$ and $$x=1$$ , respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we show the implications $$\begin{aligned} \text {Sturm attractor}\quad \Longrightarrow \quad \text {Thom--Smale complex} \quad \Longrightarrow \quad \text {meander}. \end{aligned}$$ The sequel, part 2, closes the cycle of equivalences by the implication $$\begin{aligned} \text {meander} \quad \Longrightarrow \quad \text {Sturm attractor}. \end{aligned}$$ Each implication, or mapping, involves certain constructions which are tuned such that the final 3-ball Sturm global attractor defined by the meander combinatorics coincides with the originally given Sturm 3-ball. Many explicit examples and illustrations will be discussed in part 3. The present 3-ball trilogy extends our previous trilogy on planar Sturm global attractors towards the still elusive goal of geometric and combinational characterizations of all Sturm global attractors of arbitrary dimension.

Dimers on rail yard graphs

We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson–fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions. Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick’s theorem. Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimermodel. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.

Quantum K-theoretic geometric Satake: the case

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group$G$and the spherical perverse sheaves on the affine Grassmannian$Gr$of its Langlands dual group. Bezrukavnikov and Finkelberg developed a derived version of this equivalence which relates the derived category of$G^{\vee }$-equivariant constructible sheaves on$Gr$with the category of$G$-equivariant${\mathcal{O}}(\mathfrak{g})$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group$U_{q}\mathfrak{g}$. We define a convolution category$K\operatorname{Conv}(Gr)$whose morphism spaces are given by the$G^{\vee }\times \mathbb{C}^{\times }$-equivariant algebraic K-theory of certain fibre products. We conjecture that$K\operatorname{Conv}(Gr)$is equivalent to a full subcategory of the category of$U_{q}\mathfrak{g}$-equivariant${\mathcal{O}}_{q}(G)$-modules. We prove this conjecture when$G=\operatorname{SL}_{n}$. A key tool in our proof is the$\operatorname{SL}_{n}$spider, which is a combinatorial description of the category of$U_{q}\mathfrak{sl}_{n}$representations. By applying horizontal trace, we show that the annular$\operatorname{SL}_{n}$spider describes the category of$U_{q}\mathfrak{sl}_{n}$-equivariant${\mathcal{O}}_{q}(\operatorname{SL}_{n})$-modules. Then we use quantum loop algebras to relate the annular$\operatorname{SL}_{n}$spider to$K\operatorname{Conv}(Gr)$. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.

Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences

Let L w be the Levi part of the stabilizer Q w in G L N (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,N . For the natural action of L w on $\mathbb {C}[X(w)]$ , the homogeneous coordinate ring of X(w) (for the Plucker embedding), we give a combinatorial description of the decomposition of $\mathbb {C}[X(w)]$ into irreducible L w -modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic subgroup Q that is contained in Q w . This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in G2,N are spherical L w -varieties.

ON THE IRREDUCIBLE COMPONENTS OF MODULI SCHEMES FOR AFFINE SPHERICAL VARIETIES

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid Г. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion’s moduli scheme MГ for such varieties. Moreover, we find several sufficient conditions for MГ to be irreducible and exhibit several examples where MГ is reducible. Finally, we provide examples of non-reduced MГ.

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.

Stuffle Product Formulas of Multiple Zeta Values

Using the combinatorial descriptions of stuffle product, we obtain recursive formulas for the stuffle product of multiple zeta values and of multiple zeta-star values. Then we apply the formulas to prove several stuffle product formulas with one or two strings of $z_p$'s. We also describe how to use our formulas in general cases.

Representation stability for the cohomology of arrangements associated to root systems

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI$_W$-module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.

Crystal Analysis of type C Stanley Symmetric Functions

Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.

On the determinant of the Laplacian matrix of a complex unit gain graph

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ(vivj) to each oriented edge vivj such that φ(vivj)φ(vjvi)=1 for all edges. The Laplacian matrix of G is defined as L(G)=D(G)−A(G), where D(G) is the degree diagonal matrix of Γ and A(G)=(aij) has aij=φ(vivj) if vi is adjacent to vj and aij=0 otherwise. In this paper, we provide a combinatorial description of det(L(G)) that generalizes that for the determinant of the Laplacian matrix of a signed graph.

Combinatorics and topology of Kawai-Lewellen-Tye relations

We revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the un-derlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are the α′-corrected bi-adjoint scalar amplitudes recently defined by the author [1]. We show that they naturally arise as intersection numbers of twisted cycles. In this work we focus on the combinatorial and topological description of twisted cycles relevant for string theory amplitudes. In this setting, each twisted cycle is a polytope, known in combinatorics as the associahedron, together with an additional structure encoding monodromy properties of string integrals. In fact, this additional structure is given by higher-dimensional generalizations of the Pochhammer contour. An open string amplitude is then computed as an integral of a logarithmic form over an associahedron. We show that the inverse of the KLT kernel can be calculated from the knowledge of how pairs of associahedra intersect one another in the moduli space. In the field theory limit, contributions from these intersections localize to vertices of the associahedra, giving rise to the bi-adjoint scalar partial amplitudes.

Completion of tree metrics and rank 2 matrices

Motivated by applications to low-rank matrix completion, we give a combinatorial characterization of the independent sets in the algebraic matroid associated to the collection of m×n rank-2 matrices and n×n skew-symmetric rank-2 matrices. Our approach is to use tropical geometry to reduce this to a problem about phylogenetic trees which we then solve. In particular, we give a combinatorial description of the collections of pairwise distances between several taxa that may be arbitrarily prescribed while still allowing the resulting dissimilarity map to be completed to a tree metric.

Smooth varieties with torus actions

In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the étale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic in the étale topology to the combinatorial data describing affine space endowed with a linear torus action. Finally, we provide an effective method to check the smoothness of a Gm-threefold in terms of the combinatorial data.

Metric uniformization of morphisms of Berkovich curves

We show that the metric structure of morphisms f:Y→X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Γ=(ΓY,ΓX) of f, the sets Nf,≥n of points of Y of multiplicity at least n in the fiber are radial around ΓY with the radius changing piecewise monomially along ΓY. In this case, for any interval l=[z,y]⊂Y connecting a point z of type 1 to the skeleton, the restriction f|l gives rise to a profile piecewise monomial function φy:[0,1]→[0,1] that depends only on the type 2 point y∈ΓY. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φy} with y∈ΓY(2). We prove that this family is piecewise monomial in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φy coincides with the Herbrand function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.

Primary Components of Codimension Two Lattice Basis Ideals

We provide explicit combinatorial descriptions of the primary components of codimension two lattice basis ideals. As an application, we compute the set of parameters for which a bivariate Horn system of hypergeometric differential equations is holonomic.

Geometric bijections between spanning trees and break divisors

The Jacobian group Jac(G) of a finite graph G is a group whose cardinality is the number of spanning trees of G. G also has a tropical Jacobian which has the structure of a real torus; using the notion of break divisors, An et al. obtained a polyhedral decomposition of the tropical Jacobian where vertices and cells correspond to elements of Jac(G) and spanning trees of G, respectively. We give a combinatorial description of bijections coming from this geometric setting. This provides a new geometric method for constructing bijections in combinatorics. We introduce a special class of geometric bijections that we call edge ordering maps, which have good algorithmic properties. Finally, we study the connection between our geometric bijections and the class of bijections introduced by Bernardi; in particular we prove a conjecture of Baker that planar Bernardi bijections are “geometric”. We also give sharpened versions of results by Baker and Wang on Bernardi torsors.

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.

On intersection cohomology with torus actions of complexity one

The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus \({{\mathbb {T}}}\), one of our result determines the intersection cohomology Betti numbers of any normal projective \(\mathbb {T}\)-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.

A combinatorial construction of an M12-invariant code

A ternary [66, 10, 36]3-code admitting the Mathieu group M12 as a group of automorphisms has recently been constructed by N. Pace, see [N. Pace: New ternary linear codes from projectivity groups, Discrete Mathematics 331 (2014), 22-26]. We give a construction of the Pace code in terms of M12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5, 6, 12). We also present a proof that the Pace code does indeed have minimum distance 36.