A Combinatorial Perspective on the Noncommutative Symmetric Functions

The noncommutative symmetric functions NSym were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants ₙ∈ N that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions Λ. Giving noncommutative analogues of generating function relations for other bases of Λ allowed Gelfand et al. to define additional bases of NSym and then determine change-of-basis formulas using quasideterminants. In this paper, we aim for a self-contained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of change-of-basis in Λ: both as a product of a minimal number of matrices, mimicking Macdonald's exposition of Λ in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by E{g}ecio{g}lu and Remmel, 1990.