Abstract
We investigate the zonal polynomials, a family of symmetric polynomials that appear in many mathematical contexts, such as multivariate statistics, differential geometry, representation theory, and combinatorics. We present two computer algebra packages, in SageMath and in Mathematica, for their computation. With the help of these software packages, we carry out an experimental mathematics study of some properties of zonal polynomials. Moreover, we derive and prove closed forms for several infinite families of zonal polynomial coefficients.
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Acknowledgements
We are grateful to Akimichi Takemura for helpful comments and to Yi Zhang for stimulating discussions. The first author also would like to thank Raymond Kan, who, after the submission of the original draft, contributed to the SageMath program and now becomes a co-contributor. Both authors were supported by the Austrian Science Fund (FWF): P29467-N32, and the second author was also supported by F5011-N15.
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Jiu, L., Koutschan, C. Calculation and Properties of Zonal Polynomials. Math.Comput.Sci. 14, 623–640 (2020). https://doi.org/10.1007/s11786-020-00458-0
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DOI: https://doi.org/10.1007/s11786-020-00458-0
Keywords
- Zonal polynomial
- Symmetric function
- Integer partition
- Laplace–Beltrami operator
- Wishart matrix
- Hypergeometric function of a matrix argument