1,497 research outputs found

    Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space

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    Associated to any finite simple graph Γ\Gamma is the chromatic polynomial PΓ(q)P_\Gamma(q) whose complex zeroes are called the chromatic zeros of Γ\Gamma. A hierarchical lattice is a sequence of finite simple graphs {Γn}n=0∞\{\Gamma_n\}_{n=0}^\infty built recursively using a substitution rule expressed in terms of a generating graph. For each nn, let μn\mu_n denote the probability measure that assigns a Dirac measure to each chromatic zero of Γn\Gamma_n. Under a mild hypothesis on the generating graph, we prove that the sequence μn\mu_n converges to some measure μ\mu as nn tends to infinity. We call μ\mu the limiting measure of chromatic zeros associated to {Γn}n=0∞\{\Gamma_n\}_{n=0}^\infty. In the case of the Diamond Hierarchical Lattice we prove that the support of μ\mu has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added considerably more background on activity currents and especially on the Dujardin-Favre classification of the passive locus. Exposition in the proof of the main theorem was improved. Comments welcome

    Phylogenetic toric varieties on graphs

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    We define phylogenetic projective toric model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. Generators of the pro- jective coordinate ring of the models of graphs with one cycle are explicitly described. The phylogenetic models of graphs with the same topological invariants are deforma- tion equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function.Comment: 36 pages, improved expositio

    Singular structure of Toda lattices and cohomology of certain compact Lie groups

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    We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra g\mathfrak g. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group K(Fq)K({\mathbb F}_q) related to the maximal compact subgroup KK of the group Gˇ\check G with gˇ=Lie(Gˇ)\check{\mathfrak g}={\rm Lie}(\check G) over the finite field Fq{\mathbb F}_q. Here gˇ\check{\mathfrak g} is the Langlands dual of g{\mathfrak g}. The blow-ups of the Toda lattice are given by the zero set of the τ\tau-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the τ\tau-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics

    Toda lattice, cohomology of compact Lie groups and finite Chevalley groups

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    In this paper, we describe a connection that exists among (a) the number of singular points along the trajectory of Toda flow, (b) the cohomology of a compact subgroup KK, and (c) the number of points of a Chevalley group K(Fq)K({\mathbb F}_q) related to KK over a finite field Fq{\mathbb F}_q. The Toda lattice is defined for a real split semisimple Lie algebra g\mathfrak g, and KK is a maximal compact Lie subgroup of GG associated to g\mathfrak g. Relations are also obtained between the singularities of the Toda flow and the integral cohomology of the real flag manifold G/BG/B with BB the Borel subgroup of GG (here we have G/B=K/TG/B=K/T with a finite group TT). We also compute the maximal number of singularities of the Toda flow for any real split semisimple algebra, and find that this number gives the multiplicity of the singularity at the intersection of the varieties defined by the zero set of Schur polynomials.Comment: 28 pages, 5 figure
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