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On Terwilliger \({\mathbb {F}}\)-algebras of quasi-thin association schemes

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Abstract

In Hanaki (Graphs Combin 37:1521–1529, 2021), Hanaki defined the Terwilliger algebras of association schemes over a commutative unital ring. In this paper, we call the Terwilliger algebras of association schemes over a field \({\mathbb {F}}\) the Terwilliger \({\mathbb {F}}\)-algebras of association schemes and study the Terwilliger \({\mathbb {F}}\)-algebras of quasi-thin association schemes. As main results, we determine the \({\mathbb {F}}\)-dimensions, the semisimplicity, the Jacobson radicals, and the algebraic structures of the Terwilliger \({\mathbb {F}}\)-algebras of quasi-thin association schemes.

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Acknowledgements

This present work is motivated by [2]. The author gratefully thanks Professor Akihide Hanaki for letting him know some interesting problems on the modular representations of association schemes. He also gratefully thanks his postdoctoral mentor Professor Andrey V. Vasil’ev for his constant encouragement. Finally, he thanks a referee for careful reading and many valuable comments.

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  1. School of Mathematical Sciences, Anhui University, No. 111 Jiulong Road, Hefei, 230601, China

    Yu Jiang

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  1. Yu Jiang
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Correspondence to Yu Jiang.

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Jiang, Y. On Terwilliger \({\mathbb {F}}\)-algebras of quasi-thin association schemes. J Algebr Comb 57, 1219–1251 (2023). https://doi.org/10.1007/s10801-023-01223-9

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  • DOI: https://doi.org/10.1007/s10801-023-01223-9

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