Stable representations of symmetric groups on polynomials, and actions on eventually symmetric functions

Haglund, Rhoades, and Shimozono generalized the co-invariant quotient $R_n$ of Borel, as a representation of the symmetric group, to quotients $R_{n,k}$. Gillespie and Rhoades constructed higher Specht bases for these quotients, using the higher Specht polynomials of Ariki, Terasoma, and Yamada.

We show how to decompose these quotients into ones that sit inside natural stable representations, which have explicit limits as representations of infinite symmetric groups on eventually symmetric functions.
 

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