Zonal polynomials for wreath products. (English) Zbl 1112.05100
Summary: The pair of groups, symmetric group \(S_{2n}\) and hyperoctohedral group \(H_{n}\), form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of \((S_{2n},H_{n})\) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials. A wreath product generalization of the Gelfand pair \((S_{2n},H_{n})\) is discussed in this paper. Then a multipartition versions of the theory is constructed. The multipartition versions of zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the graded Hecke algebra into a multipartition version of the ring of symmetric functions.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |
| 05E10 | Combinatorial aspects of representation theory |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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