Planar Harmonic Polynomials of Type B

The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to partial derivative operators. This paper finds all the polynomials in N variables which are annihilated by the sum of the squares (T_1)^2+(T_2)^2 and by all T_i for i>2 (harmonic). They are given explicitly in terms of a novel basis of polynomials, defined by generating functions. The harmonic polynomials can be used to find wave functions for the quantum many-body spin Calogero model.Comment: 17 pages, LaTe

Vector valued Macdonald polynomials

This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type $A_{N-1}$. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several objects and properties are analyzed, such as the canonical bilinear form which pairs polynomials with those arising from reciprocals of the original parameters, and the symmetrization of the Macdonald polynomials. The main tool of the study is the Yang-Baxter graph. We show that these Macdonald polynomials can be easily computed following this graph. We give also an interpretation of the symmetrization and the bilinear forms applied to the Macdonald polynomials in terms of the Yang-Baxter graph.Comment: 85 pages, 5 figure

Path Integral Solution of Linear Second Order Partial Differential Equations I. The General Construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schr\"{o}dinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.Comment: 40 page

Generalized Calogero-Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential.Comment: 36 pages; the main change is an improvement of section 7 so that it now deals with an arbitrary complex reflection group; Selecta Math, 201

Vector-Valued Jack Polynomials from Scratch

Vector-valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N) and studied by one of the authors (C. Dunkl) in the specialization r=p=1 (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials

Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model

We study the hole propagator of the Calogero-Sutherland model with SU(2) internal symmetry. We obtain the exact expression for arbitrary non-negative integer coupling parameter $\beta$ and prove the conjecture proposed by one of the authors. Our method is based on the theory of the Jack polynomials with a prescribed symmetry.Comment: 12 pages, REVTEX, 1 eps figur

Algebraic Linearization of Dynamics of Calogero Type for any Coxeter Group

Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are discussed.Comment: LaTeX2e, 13 pages, no figure

Survival of ancient landforms in a collisional setting as revealed by combined fission track and (U-Th)/He thermochronometry: A case study from Corsica (France)

The age of high-elevation planation surfaces in Corsica is constrained using new apatite (U-Th)/He data, field observations, and published work (zircon fission track, apatite fission track [AFT] data and landform/stratigraphical analysis). Thermal modeling results based on AFT and (U-Th)/He data, and the Eocene sediments uncomformably overlapping the Variscan crystalline basement indicate that present-day elevated planation surfaces in Corsica are the remnants of an erosion surface formed on the basement between ∼120 and ∼60 Ma. During the Alpine collision in the Paleocene-Eocene, the Variscan crystalline basement was buried beneath a westward-thinning wedge of flysch, and the eastern portion was overridden by the Alpine nappes. Resetting of the apatite fission track thermochronometer suggests an overburden thickness of >4 km covering Variscan Corsica. Protected by soft sediment, the planation surface was preserved. In the latest Oligocene to Miocene times, the surface was re-exposed and offset by reactivated faults, with individual basement blocks differentially uplifted in several phases to elevations of, in some cases, >2 km.Currently the planation surface remnants occur at different altitudes and with variable tilt. This Corsican example demonstrates that under favorable conditions, paleolandforms typical of tectonically inactive areas can survive in tectonically active settings such as at collisional plate margins. The results of some samples also reveal some discrepancies in thermal histories modeled from combined AFT and (U-Th)/He data. In some cases, models could not find a cooling path that fit both data sets, while in other instances, the modeled cooling paths suggest isothermal holding at temperature levels just below the apatite partial annealing zone followed by final late Neogene cooling. This result appears to be an artifact of the modeling algorithm as it is in conflict with independent geological constraints. Caution should be used when cross-validating the AFT and (U-Th)/He systems both in the case extremely old terrains and in the case of rocks with a relatively simple, young cooling history

The R-matrix structure of the Euler-Calogero-Moser model

We construct the $r$-matrix for the generalization of the Calogero-Moser system introduced by Gibbons and Hermsen. By reduction procedures we obtain the $r$-matrix for the $O(N)$ Euler-Calogero-Moser model and for the standard $A_N$ Calogero-Moser model.Comment: 7 page