Abstract
In this paper, we prove homological stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category OrI(R) and prove a local Noetherianity theorem for the category of OrI(R)-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients.
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Acknowledgements
This paper is the result of PRIMES-USA, a program in which high school students (the second author) engage in research-level mathematics led by a mentor (the first author). This paper is based upon work supported by The National Science Foundation Graduate Research Fellowship Program under Grant No. 1842490 awarded to the first author. The authors would like to thank the organizers of the PRIMES-USA program for providing the opportunity for this research. We are also grateful to Dr. Tanya Khovanova and Dr. Kent Vashaw for their helpful comments and to Prof. Steven V. Sam for suggesting this research topic and providing valuable suggestions.
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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. The authors have no conflicts of interest to disclose.
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Presented by: Henning Krause
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Kannan, A.S., Wang, Z. Representation Stability and Finite Orthogonal Groups. Algebr Represent Theor 26, 3119–3141 (2023). https://doi.org/10.1007/s10468-023-10202-4
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DOI: https://doi.org/10.1007/s10468-023-10202-4