2011/01/31 by Wuxing Cai, Naihuan Jing
Mathematics · Physics and Astronomy · #Action (physics) #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Classical orthogonal polynomials #Differential operator #Eigenvalues and eigenvectors #Elementary symmetric polynomial #Fock space #Laplace–Beltrami operator #Mathematical analysis #Mathematics #Operator (biology) #Orthogonal polynomials #Pure mathematics #Random Matrices and Applications #Symmetric function #hep-th #math.CO #math.QA #msc:05E05 #msc:05E10 #msc:17B69
paper · pdf · doi:10.1016/j.ejc.2011.11.003
published as European J Combin. 33 (2012) 556--571 · 19 pages. Corrected new version with new formulas on Jack functions
arxiv created 2011/05/30 · openalex publication_date 2012/01/07 · arxiv updated 2020/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We use a new method to study the Laplace-Beltrami type operator on the Fock space of symmetric functions, and as an example of our explicit computation we show that the Jack symmetric functions are the only family of eigenvectors of the differential operator. As applications of this explicit method we find a combinatorial formula for Jack symmetric functions and the Littlewood-Richardson coefficients in the Jack case. As further applications, we obtain a new determinantal formula for Jack symmetric functions. We also obtained a generalized raising operator formula for Jack symmetric functions, and a formula for the explicit action of Virasoro operators. Special cases of our formulas imply Mimachi-Yamada's result on Jack symmetric functions of rectangular shapes, as well as the explicit formula for Jack functions of two rows or two columns.