Further Pieri-type formulas for the nonsymmetric Macdonald polynomial
W. Baratta
Department of Mathematics, University of Melbourne, Melbourne, Australia
DOI: 10.1007/s10801-011-0323-3
Abstract
The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial P κ ( z) are known explicitly. These formulas generalise the known r=1 case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials E η ( z). In this paper, we extend beyond the case r=1 for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients \tbinom h n q, t \tbinom{η}{ν}_{q,t} associated with the nonsymmetric Macdonald polynomials.
Pages: 45–66
Keywords: keywords Macdonald polynomial; Pieri formulas; nonsymmetric; q-binomial coefficients
Full Text: PDF
References
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2. Baratta, W.: Pieri-type formulas for nonsymmetric Macdonald polynomials. Int. Math. Res. Not. 15, 2829-2854 (2009)
3. Cherednik, I.: Nonsymmetric Macdonald polynomials. Int. Math. Res. Not. 10, 483-515 (1995)
4. Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Symmetric polynomials vanishing on shifted diagonals and Macdonald polynomials. Int. Math. Res. Not. 18, 1015-1034 (2003)
5. Forrester, P.J., McAnally, D.S.: Pieri-type formulas for the nonsymmetric Jack polynomials. Comment. Math. Helv. 79(1), 1-24 (2004)
6. Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131(1), 1-61 (2005)
7. Knop, F.: Symmetric and nonsymmetric quantum Capelli polynomials. Comment. Math. Helv. 72(1), 84-100 (1997)
8. Knop, F., Sahi, S.: Difference equations and symmetric polynomials defined by their zeros. Int. Math. Res. Not. 10, 473-486 (1996)
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16. Mimachi, K., Noumi, M.: A reproducing kernel for nonsymmetric Macdonald polynomials. Duke Math. J. 91(3), 621-634 (1998)
17. Okounkov, A.: Binomial formula for Macdonald polynomials and applications. Math. Res. Lett. 4(4), 533-553 (1997)
18. Sahi, S.: A new scalar product for nonsymmetric Jack polynomials. Int. Math. Res. Not. 20, 997-1004 (1996)
19. Sahi, S.: Interpolation, integrality, and a generalization of Macdonald's polynomials. Int. Math. Res.
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