Two conjectures made by H.O. Foulkes in 1950 can be stated as follows.
1) Denote by V a finite-dimensional complex vector space, and by S(m)
V its m-th symmetric power. Then the GL(V)-module S(n)(S(m)V) contains
the GL(V)-module S(m)(S(n)V) for n > m. 2) For any (decreasing) parti
tion lambda = (lambda1, lambda2, lambda3, ...), denote by S(lambda)V t
he associated simple, polynomial GL(V)-module. Then the multiplicity o
f S(lambda1+np, lambda2, lambda3,...)V in the GL(V)-module S(n)(S(m+p)
V) is an increasing function of p. We show that Foulkes' first conject
ure holds for n large enough with respect to m (Corollary 1.3). Moreov
er, we state and prove two broad generalizations of Foulkes' second co
njecture. They hold in the framework of representations of connected r
eductive groups, and they lead e.g. to a general analog of Hermite's r
eciprocity law (Corollary 1 in 3.3).