Selecta Mathematica最新文献

Abstract

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley–Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron (mathcal {P}_lambda ) , and we give a formula for the dominant weight (lambda ) . Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of ((3+1)) -free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the (zeta ) map from diagonal harmonics.

Abstract

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing (t=0) Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of (U'_q({widehat{mathfrak {sl}}}_2)) -modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials.

We give a Chevalley formula for an arbitrary weight for the torus-equivariant K-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K-theory (QK_{T}(G/B)) of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of (QK_{T}(G/B)). In type (A_{n-1}), we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory (QK(SL_{n}/B)); we also obtain very explicit information about the coefficients in the respective Chevalley formula.

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.

Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. This paper solves the long standing open problem of giving a non-iterative formula for straightening a filling. We apply our formula to give a complete generalization of a theorem of Gonciulea and Lakshmibai.

We prove structure theorems for o-minimal definable subsets (Ssubset G) of definable groups containing large multiplicative structures, and show definable groups do not have bounded torsion arbitrarily close to the identity. As an application, for certain models of n-step random walks X in G we show upper bounds (mathbb {P}(Xin S)le n^{-C}) and a structure theorem for the steps of X when (mathbb {P}(Xin S)ge n^{-C'}).

We investigate the question: for which functions (f(x_1,ldots ,x_n),~g(x_1,ldots ,x_n)) the asymptotic expansion of the integral (int g(x_1,ldots ,x_n) e^{frac{f(x_1,ldots ,x_n)+x_1y_1+dots +x_ny_n}{hbar }}dx_1ldots dx_n) consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form ({(1:x_1:ldots :x_n:f)}). We also construct various examples, in particular we prove that Kummer surface in ({mathbb {P}}^3) gives a solution to our problem.

Telescopers for a function are linear differential (resp. difference) operators annihilating the definite integral (resp. definite sum) of this function. They play a key role in Wilf–Zeilberger theory and algorithms for computing them have been extensively studied in the past 30 years. In this paper, we introduce the notion of telescopers for differential forms with D-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given.

We identify Whittaker vectors for (mathcal {W}^{textsf{k}}(mathfrak {g}))-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over (mathbb {P}^2) for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure (mathcal {N} = 2) four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in ({mathbb {R}}^3). We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of ({mathbb {R}}^3) that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of ({mathbb {S}}^3) which arise naturally when considering toroidal sets.