Plethysm of S -Functions

Abstract

The problem of expressing an invariant matrix of an invariant matrix as a direct sum of irreducible invariant matrices is that of expressing the plethysm { d } O { u } of two S -functions as a sum of S -functions. Several attacks (Littlewood 1944, 1951; Murnaghan 1951 a , b , c ; Robinson 1949, 1950; Thrall 1942; Todd 1949) have been made on this problem, varying considerably in their generality and the degree to which the results obtained have been applicable to numerical cases. As the weights of { d } and { /U> } increase it becomes more and more apparent that there is need to develop a technique of finding the coefficient of any given S -function in a plethysm, rather than a method, inevitably very laborious, of determining the full expansion. Furthermore, there is need to consider the structure of any given plethysm to see whether the terms can be grouped into sets in any systematic way, or whether any relations exist between the coefficients that arise. Something of this sort is known to be the case for { m }O S _r and {l ^m }® Sr (Foulkes 1951), in which every S -function with k parts appearing in the latter result can be characterized by a partition of m together with a set of k non-negative integers.