  We introduce new families of cylindric symmetric functions as subcoalgebras
in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which
have non-negative structure constants. Combinatorially these cylindric
symmetric functions are defined as weighted sums over cylindric reverse plane
partitions or - alternatively - in terms of sets of affine permutations. We
relate their combinatorial definition to an algebraic construction in terms of
the principal Heisenberg subalgebra of the affine Lie algebra
$\mathfrak{\widehat{sl}}_n$ and a specialised cyclotomic Hecke algebra. Using
Schur-Weyl duality we show that the new cylindric symmetric functions arise as
matrix elements of Lie algebra elements in the subspace of symmetric tensors of
a particular level-0 module which can be identified with the small quantum
cohomology ring of the $k$-fold product of projective space. The analogous
construction in the subspace of alternating tensors gives the known set of
cylindric Schur functions which are related to the small quantum cohomology
ring of Grassmannians. We prove that cylindric Schur functions form a
subcoalgebra in $\Lambda$ whose structure constants are the 3-point genus 0
Gromov-Witten invariants. We show that the new families of cylindric functions
obtained from the subspace of symmetric tensors also share the structure
constants of a symmetric Frobenius algebra, which we define in terms of tensor
multiplicities of the generalised symmetric group $G(n,1,k)$.
