We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $GL_n$ modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.

35 pages; this is the more complete version of arXiv:1510.00438; v5 differs from previous versions with minor edits and the addition of an appendix about using Sagemath to do computations with these bases (this appendix does not appear in the published journal version)