2005 Fiscal Year Final Research Report Summary
Forrester's conjecture and a generalization of Selberg intengral
| Project/Area Number |
16540198
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | The University of the Ryukyus |
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Principal Investigator |
KANEKO Jyoichi University of the Ryukyus, Faculty of Science, Professor, 理学部, 教授 (10194911)
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| Co-Investigator(Kenkyū-buntansha) |
KATO Mitsuo University of the Ryukyus, Faculty of Education, Professor, 教育学部, 教授 (50045043)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Forrester's conjecture / Jack polynomial / Macdonald polynomial / Koornwinder polynomial / double affine Hecke algebra / Norm formula |
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| Research Abstract |
It was our first intension to verify the N_1=7 case of Forrester's conjecture. We tried to show that, using Macdonald differential operators, when one expands a symmetric polynomilal that comes from the conjecture in terms of Jack polynomials, certain part of terms does not appear. This certainly holds in the cases N_1【less than or equal】6, but we could not find the general procedure to get necessary number of independent linear relations of expansion coefficients to show vanishing of certain terms.
In this year of 2005, we studied mainly (nonsymmetric) Koornwinder polynomials and the double affine Hecke algebra of type BC associated to these polynomials. Especially we have investigated properties of generalized symmetrization operators and generalized alternating operators in the algebra. The reason is this : It is conjectured and verified in some cases that if one applies the generalized alternating operator to a Koornwinder polynomial, then one obtains a Koornwinder polynomial multiplied by a difference product with respect to part of variables. This difference product is exactly the same polynomial appearing in the conjecture of Baker-Forrester (=q-analogue of Forrester's conjecture). Hence, as the norm formula of Koornwinder polynomials is already established, we may expect that the conjecture follows from the norm formula.
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