CONSTRUCTION OF p−1 IRREDUCIBLE MODULES WITH FUNDAMENTAL HIGHEST WEIGHT FOR THE SYMPLECTIC GROUP IN CHARACTERISTIC p

Abstract

Let K be a field and let V be a vector space of dimension 2m over K. Let ∧V denote the exterior algebra of V and ∧^kV its kth exterior power for 0[les ]k[les ]2m. Let f be a non-degenerate alternating bilinear form defined on V×V. The symplectic group Sp_2m(K) is the group of all isometries of f and it acts as a group of vector space automorphisms on ∧^kV. In the case that K is algebraically closed and 1[les ]k[les ]m, it is known that ∧^kV contains a composition factor corresponding to the fundamental weight ω_k of a root system of type C_m. We shall refer to the irreducible module for Sp_2m(K) given by this composition factor as a fundamental module.