Periodicity of Weyl Modules for SL(2,9)

Weyl modules play a central role in the representation theory of the general and special linear groups. There has been great interest recently in the modular case of these groups over fields of prime characteristic p. In particular, Glover [l] has proved some remarkable and comprehensive results about these modules for SL(2,p). We shall extend one of these theorems to the case of SL(2, q) where q =pe. First, let us fix some notation. We set S = ,X(2, k), k a field of q elements and R = k[x, y] the polynomial algebra in two variables considered as a module for kS, the group algebra, in the usual way. Let V, be the kS-module of dimension n consisting of the homogenous polynomials of degree n 1. Hence, V, is the trivial kS-module and the V, are the duals of the Weyl modules for S. Let V, = IV,, + P, be a direct decomposition where P, is a projective kS-module and W,, has no non-zero projective direct summand. Our main result is as follows: