Symmetric group action on polynomials in ${2n}$ variables

Question

There is a well developed theory of symmetric functions on $n$ variable including Schur functions as a basis etc. Consider a a $2n$ variable polynomial and define the action of $S_n$ on a polynomial in $2n$ variables by requiring $S_n$ acting on the first $n$ variables and the last $n$ variable simultaneously. For example $x_1y_1$ will be $x_2y_2$ under the permutation (12). Is there a similar theory of Symmetric polynomials on these $2n$ variables when the group acts simultaneously? For example analogue of Schur functions? etc?