1,056 research outputs found
A Conjecture about Raising Operators for Macdonald Polynomials
A multivariable hypergeometric-type formula for raising operators of the
Macdonald polynomials is conjectured. It is proved that this agrees with Jing
and Jozefiak's expression for the two-row Macdonald polynomials, and also with
Lassalle and Schlosser's formula for partitions with length three.Comment: 13 page
A Lindenstrauss theorem for some classes of multilinear mappings
Under some natural hypotheses, we show that if a multilinear mapping belongs
to some Banach multlinear ideal, then it can be approximated by multilinear
mappings belonging to the same ideal whose Arens extensions simultaneously
attain their norms. We also consider the class of symmetric multilinear
mappings.Comment: 11 page
Recurrence formulas for Macdonald polynomials of type A
We consider products of two Macdonald polynomials of type A, indexed by
dominant weights which are respectively a multiple of the first fundamental
weight and a weight having zero component on the k-th fundamental weight. We
give the explicit decomposition of any Macdonald polynomial of type A in terms
of this basis.Comment: 18 pages, LaTeX, revised version, to appear in Journal of Algebraic
Combinatoric
Some Properties of the Calogero-Sutherland Model with Reflections
We prove that the Calogero-Sutherland Model with reflections (the BC_N model)
possesses a property of duality relating the eigenfunctions of two Hamiltonians
with different coupling constants. We obtain a generating function for their
polynomial eigenfunctions, the generalized Jacobi polynomials. The symmetry of
the wave-functions for certain particular cases (associated to the root systems
of the classical Lie groups B_N, C_N and D_N) is also discussed.Comment: 16 pages, harvmac.te
Ideal structures in vector-valued polynomial spaces
This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for E and F Banach spaces, whether the class of n-homogeneous polynomials, 'P-w((n) E, F), which are weakly continuous on bounded sets, is an HB-subspace or an M(1, C)-ideal in the space of continuous n-homogeneous polynomials, P((n) E, F). We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from P-w((n) E, F) as an ideal in P((n) E, F) to the range space F as an ideal in its bidual F**
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
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