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Orthogonality of super-Jack polynomials and a Hilbert space interpretation of deformed Calogero-Moser-Sutherland operators
- Source :
- Bull. Lond. Math. Soc. 51 (2019), no. 2, 353-370
- Publication Year :
- 2018
-
Abstract
- We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $SP_\lambda((z_1,\ldots,z_n),(w_1,\ldots,w_m);\theta)$ with respect to a natural positive semi-definite, but degenerate, Hermitian product $\langle\cdot,\cdot\rangle_{n,m}^\prime$. In case $m=0$ (or $n=0$), our product reduces to Macdonald's well-known inner product $\langle\cdot,\cdot\rangle_n^\prime$, and we recover his corresponding orthogonality results for the Jack polynomials $P_\lambda((z_1,\ldots,z_n);\theta)$. From our main results, we readily infer that the kernel of $\langle\cdot,\cdot\rangle_{n,m}^\prime$ is spanned by the super-Jack polynomials indexed by a partition $\lambda$ not containing the $m\times n$ rectangle $(m^n)$. As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type $A(n-1,m-1)$.<br />Comment: 19 pages, 1 figure
- Subjects :
- Mathematics - Quantum Algebra
Mathematical Physics
Subjects
Details
- Database :
- arXiv
- Journal :
- Bull. Lond. Math. Soc. 51 (2019), no. 2, 353-370
- Publication Type :
- Report
- Accession number :
- edsarx.1802.02016
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1112/blms.12234