Abstract
This paper studies the “reduction mod p” method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group Uζ at a pr-th root of unity. The G-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between ExtGn(V,W) and ExtUζn(V′,W′), when V,W are obtained from V′,W′ by reduction mod p. Since the dimensions of Extn-spaces for Uζ-modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p-th root of unity case and a general pr-th root of unity case are essentially different.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.