Homomorphisms between neighbouring Weyl modules
Let G be a simply connected semi-simple algebraic group over an algebraically closed field k of positive characteristic, defined and split over the prime field of k. In this paper, the author deals with homomorphisms between the Weyl modules of G. He introduces for the sets of weights the concept of small dominant intervals and shows that if the dominant weights \(\lambda\) and \(\chi\) belong to the same small dominant interval, then there is a non-zero homomorphism \(f: V(\lambda)\to V(\chi)\) between corresponding Weyl modules. Furthermore, he shows that for a special class of small dominant intervals, f is unique up to a scalar multiple and the composite map of two such homomorphisms is non- zero. To deal with the Weyl modules the author also introduces in the set of weights a new partial order called the regular linkage relation. On dominant weights it implies the relation used by \textit{J. C. Jantzen} [Moduln mit einem höchsten Gewicht (Lect. Notes Math. 750, 1979; Zbl 0426.17001)], while for dominant weights in alcoves the relations are equivalent. In various results, including the strong linkage principle [cf. \textit{H. H. Andersen}, J. Reine Angew. Math. 315, 53-59 (1980; Zbl 0439.20026)], the author shows that the new relation can substitute for the above without any significant changes in the proofs and leads to slight improvements of some older theorems.
simply connected semi-simple algebraic group
scientific article; zbMATH DE number 3973164
scientific article