Foulkes Conjecture

Foulkes' conjecture asserts that for integers and a finite-dimensional complex vector space , there is a linear, -equivariant injection

Here is the general linear group, and is the th symmetric power of . Equivalently, the difference of the characters, written as the plethysm , is a linear combination of Schur polynomials with nonnegative integer coefficients. Here is the complete homogeneous symmetric polynomial. The assertion requires an injection but does not specify a canonical map.