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This paper introduces noncommutative analogs of monomial symmetric functionsand fundamental noncommutative symmetric functions. The expansion of ribbonSchur functions in both of these basis is nonnegative. With these functions athand, one can derive a noncommutative Cauchy identity as well as study anoncommutative scalar product implied by Cauchy identity. This scalar productseems be the noncommutative analog of Hall scalar product in the commutativetheory.

Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product