Stable Higher Specht Polynomials and Representations of Infinite Symmetric Groups

We define eventually symmetric functions to be those power series of bounded degree in infinitely many variables that are invariant under interchanging all the variables with large enough indices. We show how this ring $\tilde{\Lambda}$ is the natural place to define the stable versions of the higher Specht polynomials of Ariki, Terasoma, and Yamada and their generalized versions from the prequels to this paper, and investigate its various properties as a representation of the infinite symmetric groups. This requires defining infinite versions of Ferrers diagrams, standard Young tableau, semi-standard ones, and the appropriate representations inside $\tilde{\Lambda}$, which are irreducibe as limits of irreducible representations of finite symmetric groups. The homogeneous parts of $\tilde{\Lambda}$ and of its subring of polynomials in infinitely many variables are no longer completely reducible, and we determine the form of the maximal completely reducible sub-representations there (in several normalizations). After posing a conjecture about the decompositions of polynomials in $n$ variables using the representations of $S_{n+1}$, we obtain explicit filtrations on $\tilde{\Lambda}$ and its subring, whose graded pieces are the maximal completely reducible sub-representations at each step.