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Antipodal Sets and Designs on Unitary Groups

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Abstract

The unitary group U(n) is a symmetric space and has the point-symmetry for every point \(x\in U(n)\). A great antipodal set on U(n) is a “good” finite subset of U(n) related to the point-symmetries. On the other hand, a great antipodal set on U(n) is an analogue of a pair of antipodal points on spheres. It is known that a finite subset of a sphere is a tight spherical 1-design if and only if it is a pair of antipodal points. In this paper, we investigate a relation between great antipodal sets on U(n) and design theory on U(n). Moreover, we give a relation between a great antipodal set on U(n) and a Hamming cube graph \({\mathcal {Q}}_n\).

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Acknowledgements

The author would like to thank Takayuki Okuda and Takahiro Hashinaga for the fruitful discussion. The author also wishes to thank the anonymous referees for careful reading and valuable suggestions that improved the presentation. Eiichi Bannai is a supervisor of the author. The author sincerely congratulate Bannai and Enomoto in their 75th birth-year.

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  1. National Institute of Technology, Kitakyushu College, 5-20-1 Shii, Kokuraminamiku, Kitakyushu, Fukuoka, 802-0985, Japan

    Hirotake Kurihara

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Correspondence to Hirotake Kurihara.

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This work was supported by JSPS KAKENHI Grant Number JP16K17604.

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Kurihara, H. Antipodal Sets and Designs on Unitary Groups. Graphs and Combinatorics 37, 1559–1583 (2021). https://doi.org/10.1007/s00373-021-02287-9

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