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A sufficient criterion about group homomorphism of the first row map

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Abstract

In [3, Examples 4.16, 4.13], van der Kallen gave an example that the first row map

$$\begin{aligned}{} & {} \textrm{GL}_{d+1}(R)\longrightarrow \textrm{Um}_{d+1}(R)/E_{d+1}(R)\\{} & {} \sigma \longmapsto [e_{1}\sigma ] \end{aligned}$$

is not a group homomorphism. In this article, we give a sufficient condition on the ring R so that the first row map becomes a group homomorphism.

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Acknowledgements

The author is thankful to Inspire Faculty Fellowship (DST/INSPIRE/04/2021/002849) and Startup Research Grant (SRG/2022/000056) for their support.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi, 175 005, India

    Sampat Sharma

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  1. Sampat Sharma
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Correspondence to Sampat Sharma.

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Communicated by B Sury.

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Sharma, S. A sufficient criterion about group homomorphism of the first row map. Proc Math Sci 133, 20 (2023). https://doi.org/10.1007/s12044-023-00740-8

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  • DOI: https://doi.org/10.1007/s12044-023-00740-8

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