The free Lie ring and Lie representations of the full linear group

Abstract

Introduction. The present paper is a continuation and amplification of R. M. Thrall's paper On symmetrized Kronecker powers and the structure of the free Lie ring [7 ]J(1), which we shall denote as FR. The notation used in FR shall be adopted in this paper. The author wishes to express her appreciation to Professor R. M. Thrall of the University of Michigan, who suggested the probl'em of the present paper and gave valuable guidance during the preparation of it. We propose to study the structure of the free Lie ring, its characteristic ideals, and certain related right ideals in the group ring of the symmetric group. We operate over a field K of characteristic zero and thus insure that the named group ring be semi-simple (which is not the case for fields of characteristic p). In FR a recursion formula [7, p. 386] was developed from which the irreducible constituents of the mth Lie representation for m ? 10 were obtained. The main result of the present investigation is Theorem III (?2) which gives a direct formula for the character of the mth Lie representation. 1. The free Lie ring and Lie representations of the full linear group. The free non-associative K ring Zn, the free Lie ring L =Ln, and the mth Lie representation are defined in FR [7, pp. 372-373]. It is convenient to make the definition of ideal more explicit than in FR. We begin with the concept of a homomorphism of a ring Zn onto a ring Z *, and then call a subset J of Zn an ideal if it is the kernel of some homomorphism. A homogeneous ideal is defined in FR [7, p. 372 ]. The following result was stated without proof in FR [7, p. 372].