Combinatorial Description Research Articles

Geometric model for weighted projective lines of type (p,q)

We give a geometric model for the category of coherent sheaves over the weighted projective line of type (p,q) in terms of an annulus with marked points on its boundary. We establish a bijection between indecomposable sheaves over the weighted projective line and certain homotopy classes of oriented curves in the annulus, and prove that the dimension of extension group between indecomposable sheaves equals to the positive intersection number between the corresponding curves.By using the geometric model, we provide a combinatorial description for the titling graph of tilting bundles, which is a line or composed of quadrilaterals. Moreover, we obtain that the automorphism group of the coherent sheaf category is isomorphic to the mapping class group of the marked annulus, and show the compatibility of their actions on the tilting graph of coherent sheaves and on the triangulation of the geometric model respectively. A geometric description of the perpendicular category with respect to an exceptional sheaf is presented at the end of the paper.

The Nash problem for torus actions of complexity one

Abstract We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration.

Lascoux Polynomials and Subdivisions of Gelfand–Zetlin Polytopes

Abstract We give a new combinatorial description for Grassmannian Grothendieck polynomials in terms of subdivisions of Gelfand–Zetlin polytopes. Moreover, these subdivisions also provide a description of Lascoux polynomials. This generalizes a similar result on key polynomials by Kiritchenko, Smirnov, and Timorin.

A Unified Treatment of Families of Partition Functions

We present a unified framework of combinatorial descriptions, and the analogous asymptotic growth of the coefficients of two general families of functions related to integer partitions. In particular, we resolve several conjectures and verify several claims that are posted on the On-Line Encyclopedia of Integer Sequences. We perform the asymptotic analysis by systematically applying the Mellin transform, residue analysis, and the saddle point method. The combinatorial descriptions of these families of generalized partition functions involve colorings of Young tableaux, along with their “divisor diagrams”, denoted with sets of colors whose sizes are controlled by divisor functions.

Classical Dynamical r-matrices for the Chern–Simons Formulation of Generalized 3d Gravity

AbstractClassical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for $$\mathfrak {su}(2)$$ su ( 2 ) and $$\mathfrak {sl}(2,{\mathbb {R}})$$ sl ( 2 , R ) . We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

Boij-Söderberg conjectures for differential modules

Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring S=k[x1,…,xn]. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential S-modules, which are natural generalizations of chain complexes. We prove several results that lend evidence in support of this conjecture, including a categorical pairing between the derived categories of graded differential S-modules and coherent sheaves on Pn−1 and a proof of the conjecture in the case where S=k[t].

The $s$-Weak Order and $s$-Permutahedra II: The Combinatorial Complex of Pure Intervals

This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the $s$-permutahedron as the complex of pure intervals of the $s$-weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the $s$-associahedron as the complex of pure $s$-Tamari intervals of the $s$-Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen $\nu$-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.

Combinatorial generators for the cohomology of toric arrangements

We give a new combinatorial description of the cohomology ring structure of H⁎(M(A);Z) of the complement M(A) of a real complexified toric arrangement A in (C⁎)d. In particular, we correct an error in the paper [4].

Moduli spaces of ℤ/kℤ-constellations over 𝔸2

Let [Formula: see text] be a representation of a finite abelian group and let [Formula: see text] be the space of generic stability conditions on the set of [Formula: see text]-constellations. We provide a combinatorial description of all the chambers [Formula: see text] and prove that there are [Formula: see text] of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric [Formula: see text]-constellations, it is enough to build all simple chambers. We also prove that there are [Formula: see text] simple chambers. Finally, we provide an explicit formula for the tautological bundles [Formula: see text] over the moduli spaces [Formula: see text] for all chambers [Formula: see text] which only depends upon the chamber stair which is a combinatorial object attached to the chamber [Formula: see text].

Sandpile groups of Cayley graphs of 𝔽r 2

The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of F 2 r , focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for “generic” Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries.

Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [F:Q]=n≥3, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R+n, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

Modular representations in type A with a two-row nilpotent central character

We study the category of representations of slm+2n over a field of characteristic p with p≫0, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n=1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger.

2-Step nilpotent L∞-algebras and hypergraphs

We describe a procedure to attach a nilpotent strong homotopy Lie algebra to every simple hypergraph and prove that two hypergraphs are isomorphic if and only if the corresponding strong homotopy Lie algebras are isomorphic. As an application, we characterize hypergraphs admitting a system of distinct representatives in terms of symplectic forms on the corresponding strong homotopy Lie algebra. We conclude with a combinatorial description of the cohomology of these strong homotopy Lie algebras in low degree.

Polyocollection ideals and primary decomposition of polyomino ideals

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.

Graded local cohomology of modules over semigroup rings

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen–Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen–Macaulay affine semigroup rings.

Norm tori of étale algebras and unramified Brauer groups

Let k be a field, and let L be an étale k-algebra of finite rank. If a∈k×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\in {k^ \ imes }$$\\end{document}, let Xa be the affine variety defined by NL/k(x)=a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${N_{L/k}}(x) = a$$\\end{document}. Assuming that L has at least one factor that is a cyclic field extension of k, we give a combinatorial description of the unramified Brauer group of Xa.

Total Positivity is a Quantum Phenomenon: The Grassmannian Case

The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter q q is transcendental, we show that “quantum positroids” are completely prime ideals in the quantum grassmannian O q ( G m n ( F ) ) {\mathcal O}_q(G_{mn}(\mathbb {F})) . As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-invariant prime ideals in O q ( G m n ( F ) ) {\mathcal O}_q(G_{mn}(\mathbb {F})) , and prove a version of the orbit method for torus-invariant objects. Finally, we construct separating Ore sets for all torus-invariant primes in O q ( G m n ( F ) ) {\mathcal O}_q(G_{mn}(\mathbb {F})) . The latter is the first step in the Brown-Goodearl strategy to establish the orbit method for (quantum) grassmannians.

The partial Temperley–Lieb algebra and its representations

In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra \mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k}) , where {V = V(0) \oplus V(1)} is the direct sum of the trivial and natural module for the quantized enveloping algebra \mathbf{U}_q(\mathfrak{gl}_2) . It is a proper subalgebra of the Motzkin algebra (the \mathbf{U}_q(\mathfrak{sl}_2) -centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.

The bipartite Laplacian matrix of a nonsingular tree

Abstract For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the bipartite Laplacian matrix, is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.

Combinatorics of Centers of 0-Hecke Algebras in Type A

A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis corresponds to certain equivalence classes of the Coxeter group. We consider case of the symmetric group $S_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. We develop an explicit combinatorial description for the equivalence classes that are parametrized by the maximal compositions whose odd parts form a hook.