A New Prehomogeneous Vector Space of Characteristic p

Let K be an algebraically closed field of characteristic 3. Let Lambda_1,Lambda_2,Laambda_3 denote all: the fundamental dominant weights of GL(4). Then the K-dimension of the irreducible GL(4)- module V with the highest weight Lambda_1 +Lambda_2 is equal to 16, and it is denoted , by V(16). In this paper, the following results are proved: (1) (GL(4),Lambda_1 +Lambda_2 ,V(16)) is a regular irreducible prehomogeneous vector space.The degree of its irreducible relative invariant is 8, the associated character isX(g) = (det g)^6.(2) There exist only one 6-dimensional GL (4) -orbit and one 9-dimensional GL(4)- orbit in V(16). When m=7, 8 or lleq mleq 5, there are no m-dimensional GL(4)-orbits.