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K-theoretic Gromov-Witten invariants of a compact Kahler manifold $X$ aredefined as super-dimensions of sheaf cohomology of interesting bundles overmoduli spaces of n-pointed holomorphic curves in X. With this article, we begina series of publications on K-theoretic Gromov-Witten invariants, cognizant ofthe $S_n$-module structure on the sheaf cohomology, induced by renumbering ofthe marked points. In the opening paper, we introduce such invariants, explainhow the representation-theoretic information with varying $S_n$ is incorporatedinto generating functions of Gromov-Witten theory, and compute one of them, thesmall J-function for $X=pt$, by using Kapranov's description of Deligne-Mumfordspaces $\overline{\mathcal M}_{0,n}$.

Permutation-equivariant quantum K-theory I. Definitions. Elementary K-theory of $\overline{\mathcal M}_{0,n}/S_n$