Jack polynomials and affine Yangian. (English) Zbl 1514.81290
Summary: In this paper, we define one kind of new Jack polynomials denoted by \(Jk_\lambda \), which equal the standard Jack polynomials \(P_\lambda^\alpha\) multiplied by a coefficient. We show that the structure constant \(C_{\lambda\mu}^\nu\) in \(Jk_\lambda\cdot Jk_\mu = \sum_\nu C_{\lambda\mu}^\nu Jk_\nu\) can be given from affine Yangian of \(\mathfrak{gl}(1)\). Then we give the Boson-Fermion correspondence for Jack polynomials \(Jk_\lambda\) and show that 3-Schur functions \(S_\pi\) equal Jack polynomials \(Jk_\lambda\) in the special case of \(h_1 = h\), \(h_2 = -h^{-1}\), \(h_3 = -h + h^{-1}\).
MSC:
| 81V72 | Particle exchange symmetries in quantum theory (general) |
| 18D60 | Profunctors (= correspondences, distributors, modules) |
| 12E10 | Special polynomials in general fields |
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