Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations

An ideal of polynomials is symmetric if it closed under permutations variables. We relate general ideals to the so called Specht generated by all a given shape. show connection between leading monomials in and contained ideal. This provides applications several contexts. Most notably, this gives information about solutions corresponding set equations. From another perspective, restricts isotypic decomposition viewed as representation group.