Orthogonal Polynomials of Types A and B and Related Calogero Models

Abstract

Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author in TAMS 1989 (311), 167-183. There is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,...,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. A complete basis of eigenfunctions of Yamamoto's B_N spin Calogero model is obtained by multiplying these polynomials by the ground state.

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• The Polychronakos-Frahm spin chain of BC_N type and Berry-Tabor's conjecture
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• From $s{l}_q(2)$ to a Parabosonic Hopf Algebra
• Evaluation and normalization of Jack superpolynomials
• The spin Sutherland model of D_N type and its associated spin chain
• Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model
• Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models
• The exactly solvable spin Sutherland model of B_N type and its related spin chain
• Exact Charge Dynamics in the Supersymmetric t-J Model with Inverse-Square Interaction
• Exact spin dynamics of the 1/r^2 supersymmetric t-J model in a magnetic field
• Exchange operator formalism for N-body spin models with near-neighbors interactions
• Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
• Dynamical correlation functions of the spin Calogero-Sutherland model
• On the evaluation formula for Jack polynomials with prescribed symmetry
• Algebraic study on the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles
• Planar Harmonic Polynomials of Type B
• Rational quantum integrable systems of D_N type with polarized spin reversal operators
• Supersymmetric analogue of BC_N type rational integrable models with polarized spin reversal operators
• A novel class of translationally invariant spin chains with long-range interactions
• Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions
• Particle Propagator of Spin Calogero-Sutherland Model
• Derivation of Green's Function of Spin Calogero-Sutherland Model by Uglov's Method