A generalized Macdonald operator

We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.Comment: 11 pages. To appear in Int. Math. Res. Not. IMR

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Jack vertex operators and realization of Jack functions

We give an iterative method to realize general Jack functions from Jack functions of rectangular shapes. We first show some cases of Stanley's conjecture on positivity of the Littlewood-Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that vectors of products of Jack vertex operators form a basis of symmetric functions. In particular this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Thirdly a generalized Frobenius formula for Jack functions was given and was used to give new evaluation of Dyson integrals and even powers of Vandermonde determinant.Comment: Expanded versio

Ancestral genome estimation reveals the history of ecological diversification in Agrobacterium

Horizontal gene transfer (HGT) is considered as a major source of innovation in bacteria, and as such is expected to drive adaptation to new ecological niches. However, among the many genes acquired through HGT along the diversification history of genomes, only a fraction may have actively contributed to sustained ecological adaptation. We used a phylogenetic approach accounting for the transfer of genes (or groups of genes) to estimate the history of genomes in Agrobacterium biovar 1, a diverse group of soil and plant-dwelling bacterial species. We identified clade-specific blocks of cotransferred genes encoding coherent biochemical pathways that may have contributed to the evolutionary success of key Agrobacterium clades. This pattern of gene coevolution rejects a neutral model of transfer, in which neighboring genes would be transferred independently of their function and rather suggests purifying selection on collectively coded acquired pathways. The acquisition of these synapomorphic blocks of cofunctioning genes probably drove the ecological diversification of Agrobacterium and defined features of ancestral ecological niches, which consistently hint at a strong selective role of host plant rhizospheres

Co-evolution of sites under immune selection shapes Epstein-Barr Virus population structure

Epstein-Barr virus (EBV) is one of the most common viral infections in humans and persists within its host for life. EBV therefore represents an extremely successful virus that has evolved complex strategies to evade the host's innate and adaptive immune response during both initial and persistent stages of infection. Here, we conducted a comparative genomics analysis on 223 whole genome sequences of world-wide EBV strains. We recover extensive genome-wide linkage disequilibrium (LD) despite pervasive genetic recombination. This pattern is explained by the global EBV population being subdivided into three main sub-populations, one primarily found in East Asia, one in Southeast Asia and Oceania, and the third including most of the other globally distributed genomes we analyzed. Additionally, sites in LD were overrepresented in immunogenic genes. Taken together, our results suggest that host immune selection and local adaptation to different human host populations has shaped the genome-wide patterns of genetic diversity in EBV

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version of hep-th/0305038 which has been splitted in two articles. In this revised version, the introduction has been rewritten and a new appendix has been added. To appear in JP

TPMS: a set of utilities for querying collections of gene trees

Background: The information in large collections of phylogenetic trees is useful for many comparative genomic studies. Therefore, there is a need for flexible tools that allow exploration of such collections in order to retrieve relevant data as quickly as possible. Results: In this paper, we present TPMS (Tree Pattern-Matching Suite), a set of programs for handling and retrieving gene trees according to different criteria. The programs from the suite include utilities for tree collection building, specific tree-pattern search strategies and tree rooting. Use of TPMS is illustrated through three examples: systematic search for incongruencies in a large tree collection, a short study on the Coelomata/Ecdysozoa controversy and an evaluation of the level of support for a recently published Mammal phylogeny. Conclusion: TPMS is a powerful suite allowing to quickly retrieve sets of trees matching complex patterns in large collection or to root trees using more rigorous approaches than the classical midpoint method. As it is made of a set of command-line programs, it can be easily integrated in any sequence analysis pipeline for an automated use

Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure

Phylogenomics reveals the basis of adaptation of Pseudorhizobium species to extreme environments and supports a taxonomic revision of the genus

The family Rhizobiaceae includes many genera of soil bacteria, often isolated for their association with plants. Herein, we investigate the genomic diversity of a group of Rhizobium species and unclassified strains isolated from atypical environments, including seawater, rock matrix or polluted soil. Based on whole-genome similarity and core genome phylogeny, we show that this group corresponds to the genus Pseudorhizobium. We thus reclassify Rhizobium halotolerans, R. marinum, R. flavum and R. endolithicum as P. halotolerans sp. nov., P. marinum comb. nov., P. flavum comb. nov. and P. endolithicum comb. nov., respectively, and show that P. pelagicum is a synonym of P. marinum. We also delineate a new chemolithoautotroph species, P. banfieldiae sp. nov., whose type strain is NT-26T (=DSM 106348T=CFBP 8663T). This genome-based classification was supported by a chemotaxonomic comparison, with increasing taxonomic resolution provided by fatty acid, protein and metabolic profiles. In addition, we used a phylogenetic approach to infer scenarios of duplication, horizontal transfer and loss for all genes in the Pseudorhizobium pangenome. We thus identify the key functions associated with the diversification of each species and higher clades, shedding light on the mechanisms of adaptation to their respective ecological niches. Respiratory proteins acquired at the origin of Pseudorhizobium were combined with clade-specific genes to enable different strategies for detoxification and nutrition in harsh, nutrient-poor environments