Schur functions in noncommuting variables

In 2004 Rosas and Sagan asked whether there was a way to define basis in the algebra of symmetric functions noncommuting variables, NCSym, having properties analogous classical Schur functions. This because they had constructed partial such set that not basis. We answer their question by defining variables using noncommutative analogue Jacobi-Trudi determinant. Our NCSym map under commutation, subset them indexed partitions forms for NCSym. Amongst other properties, also satisfy product rule terms skew show how are related Specht modules, naturally refine Rosas-Sagan Moreover, generalizing natural way, we prove analogues Littlewood-Richardson coproduct them. Finally, relate our proving extensions ribbon functions, immaculate integer partitions.