Vector-Valued Jack Polynomials from Scratch

Symmetry Integrability and Geometry-Methods and Applications

article

Charles F. Dunkl¹  Jean-Gabriel Luque² 

[1] University of Virginia;Université de Rouen

关键词: Jack polynomials;    Yang–Baxter graph;    Hecke algebra;   

DOI  :  10.3842/SIGMA.2011.026

来源: National Academy of Science of Ukraine

【 摘 要 】

Vector-valued Jack polynomials associated to the symmetric group S N are polynomials with multiplicities in an irreducible module of S N 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.

【 授权许可】

Unknown