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).

The Cartan invariant matrix for the group SL(5, 2) and related results are determined in this note.

Extensions of simple modules for the finite Chevalley group G2(p) with p＞-13 are completely determined in this note.

Let G be a connected simply-connected simple algebraic group of type G2 over an algebraically closed field K of characteristic p＞-13,and G1 the kernel of the Frobenius morphism F on G. In the present paper we show how one can obtain the extensions of any two simple modules for G by using the same method as[7] and[9]. Note that there are 6 positive roots in the root system of G, and 12 alcove types corresponding to the 12 alcoves in the restricted box, hence there are 12 "generic decomposition patterns" which indicate composition factor multiplicities in the Weyl modules of G. In particular, all patterns involve the same number of alcoves and the same distribution of multiplicities, and the total number of composition factors is 119. Moreover, composition factor multiplicities are from 1 to 4. All these facts make the calculation quite complicated. We shall omit almost all details of proofs which are not essentially defferent from those in[7]in order to make the size of this paper not too large. The paper is organized as follows. In Section 1 we introduce our basic notations and preliminaries on the extensions of simple modules for G and G1. In Section 2 and 3 we determine the extensions of any two simple Gmodules with "small" highest weights, and the extensions of any two simple G1-modules. Section 4 is devoted to determining the G-socle of the tensor product of two simple G-modules. Finally in Section 5 we obtain our main results on Ext1 G. The authors wish to thank the referee for his helpful comments, especially for his pointing out both mistakes of (8) and the definition of B(λ) in the original version. (The same mistakes also occur in[7].)

Let G be a simply-connected semisimple algebraic group over an algebraically closed field K of characteristic p>0. Letπbe an automorphism of G coming from that of the Dynkin diagram of G, and Fn the n-th Frobenius morphism of G. We denote by Gπ(n) the finite group consisting of fixed points underπ·Fn in G, which is called a finite group of Lie type. In the present paper, we shall show how to determine the first Cartan invariant C(n)00 of Gπ(n), i.e. the multiplicity of the trivial module in its projective cover. The method used here is due to Jantzen's lectures at ECNU in 1984(cf.[J2]). As an example, we calculate this value for G being of type C2 and p=7, i.e. C(n)00 for Sp(4, 7n). When p>7, C(n)00 for Sp(4, pn) has been determined in[Ye4]. We shall use the notations mentioned in Jantzen's book[J4]unless otherwise indicated.

The purpose of this paper is to obtain a formula for computing the first Caftan invariant C(m)11 of the group Sp(4,pm). The main results are as follows: