  In a recent paper, the authors introduced a new basis of the ring of
symmetric functions which evaluate to the irreducible characters of the
symmetric group at roots of unity. The structure coefficients for this new
basis are the stable Kronecker coefficients. In this paper we give
combinatorial descriptions for several products that have as consequences
several versions of the Pieri rule for this new basis of symmetric functions.
In addition, we give several applications of the products studied in this
paper.
