Abstract
Let \(S_k \times S_{n-k}\) be a maximal Young subgroup of the symmetric group \(S_n\). We introduce a basis \({{\mathcal {B}}}_{n,k}\) for the coset space \(S_n/S_k \times S_{n-k}\) that is naturally parametrized by the set of standard Young tableaux with n boxes, at most two rows, and at most k boxes in the second row. The basis \({{\mathcal {B}}}_{n,k}\) has positivity properties that resemble those of a root system, and there is a composition series of the coset space in which each term is spanned by the basis elements that it contains. We prove that the spherical functions of the associated Gelfand pair are nonnegative linear combinations of the \({{\mathcal {B}}}_{n,k}\).
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Acknowledgements
I am grateful to Nathan Lindzey and Nat Thiem for some helpful conversations, and to Tianyuan Xu for making many helpful comments and suggestions on an earlier version of this paper. I also thank the referees for their corrections and feedback.
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Communicated by Alexander Yong.
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Green, R.M. Positivity Properties for Spherical Functions of Maximal Young Subgroups. Ann. Comb. (2023). https://doi.org/10.1007/s00026-023-00666-y
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DOI: https://doi.org/10.1007/s00026-023-00666-y