The analysis of the characters of the Lie representations of the general linear group

Abstract

1. If a free Lie ring has n generators and A is a nonsingular n Xn matrix of complex elements, then when the generators undergo a linear transformation of matrix A, the module of all forms of degree m in the generators is mapped into itself by a linear transformation of matrix Lm(A) on a set of basis elements. The mapping A+-Lm(A) is a representation of the full linear group known as the mth Lie representation [1]. The character of this representation has been shown [2 ] to be ym = m-1Ed Im u(d)sd 1d, where 4(k) is the Mobius function of the integer k, and sr is the sum of the rth powers of the eigenvalues of A. The decomposition of the mth Lie representation into its irreducible constituents is in exact correspondence with the analysis of the symmetric function 7m into Schur functions. When m is prime there is a simple rule for the coefficient ax of any S-function { X} in (m [2], but there is no such rule when m is composite. Since sd4 Xx Xdmld {X },where XX is the irreducible character of the class (p) of the symmetric group 5m corresponding to the partition (X) of m, it follows that [10],