Abstract.
For a simple Chevalley group G an explicit version of the Solomon-Tits theorem is proved by describing the generators of the kernel of the map Z[G(K)]→S K , where K is any field and where S K is the Steinberg module of the group G(K). As a corollary it is shown that if is a Euclidean domain whose fraction field is K, then S K is cyclic as a G() module when G is either a classical group or an exceptional group of type E6, or E7.
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Acknowledgement I would like to thank Matt Emerton, Özlem Imamoglu, Paul Gunnells for several helpful comments.
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Tóth, Á. On the Steinberg module of Chevalley groups. manuscripta math. 116, 277–295 (2005). https://doi.org/10.1007/s00229-004-0524-3
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DOI: https://doi.org/10.1007/s00229-004-0524-3