On the blocks of the infinitesimal Schur algebras

For a reductive algebraic group scheme G, much can be learnt about its repr esentations over a field k of characteristic p > 0 by studying the represen tations of a related group scheme, G(r)T, associated to the rth Frobenius k ernel G(r) and a maximal torus T of G. In the case G = GL(n, k) one can als o consider the polynomial representations, and reduce to the study of repre sentations of the Schur algebras. In [8] these two approaches were combined , and gave rise to the construction of a monoid scheme MrD whose representa tions are equivalent to the polynomial representations of G(r)T. Just as in the ordinary case, this leads naturally to the study of certain finite dim ensional algebras, the infinitesimal Schur algebras. In this paper we deter mine 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.