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On the blocks of the infinitesimal Schur algebras

Cox, A. (2000). On the blocks of the infinitesimal Schur algebras. The Quarterly Journal of Mathematics, 50(1), pp. 39-56. doi: 10.1093/qmathj/50.1.39


For a reductive algebraic group scheme G, much can be learnt about its representations over a field k of characteristic p > 0 by studying the representations of a related group scheme, GrT, associated to the rth Frobenius kernel Gr and a maximal torus T of G. In the case G =GL(n, k) one can also consider the polynomial representations, and reduce to the study of representations of the Schur algebras. In [8] these two approaches were combined, and gave rise to the construction of a monoid scheme Mr D whose representations are equivalent to the polynomial representations of GrT. Just as in the ordinary case, this leads naturally to the study of certain finite dimensional algebras, the infinitesimal Schur algebras. In this paper we determine the blocks of these algebras when n = 2, which extends a result in [9] where the blocks were determined in the case n = 2 and r = 1. We conclude by defining a quantum version of the infinitesimal Schur algebras, and show that the corresponding result also holds in this case.

Publication Type: Article
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
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