A New Plethysm Formula for Symmetric Functions

This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for b, where the sum is over semistandard tableaux T of weight a^b, is a root of unity, and maj(T) is a major index like statistic on semistandard tableaux.An Sb-representation, denoted S,b, is defined. In the special case when b, S,b is the Specht module corresponding to . It is shown that the character of S,b on elements of cycle type is where the sum is over semistandard tableaux T of shape and weight ab. Moreover, the eigenvalues of the action of an element of cycle type acting on S,b are {}. This generalizes J. Stembridge's result [11] on the eigenvalues of elements of the symmetric group acting on the Specht modules.