The weight set of an irreducible module for the algebraic group G of type A over an algebraically closed field of characteristic p>0 is described in the present note by using a quite different method from[3]and[4]. We show that the weight set of an irreducible G-module L(λ) is the same as that of the Weyl module V (λ) when 2 X1(T) is a restricted weight.

A monomial basis and a filtration of subalgebras for the universal enveloping algebra υ(g) of a complex simple Lie algebra gl of type Al is given in this note. In particular, a new multiplicity formula for the Weyl module V(λ) of υ(gl) is obtained in this note

In this note, we determine the irreducible characters for the simple algebraic groups of type A4 and D4 over an algebraically closed field K of characteristic 3, by using a theorem of Xi Nanhua[1] and the MATLAB software.

In this note, we determine the irreducible characters for the special linear group SL(6,K) and SL(7,K) over an algebraically closed field K of characteristic 2, by using the theorem of Xi Nanhua[7] and the MATLAB software.

In this note, we determine the irreducible characters for the special orthogonal group SO(7,K) and the symplectic group Sp(6,K) over an algebraically closed field K of characteristic 2. By applying these results, we get the Cartan invariant matrix of the finite groups SO(7, 2) and Sp(6, 2).