STABLE PROPERTIES OF PLETHYSM - ON 2 CONJECTURES OF FOULKES

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).