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

TL;DR

This paper develops noncommutative versions of classical symmetric functions and identities, providing new tools for noncommutative algebra with positive expansions and a noncommutative scalar product analogous to Hall's scalar product.

Contribution

It introduces noncommutative analogs of monomial and fundamental symmetric functions, and derives a noncommutative Cauchy identity and scalar product.

Findings

Ribbon Schur functions expand nonnegatively in these bases

Derived a noncommutative Cauchy identity

Proposed a noncommutative scalar product analogous to Hall's scalar product

Abstract

This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product seems be the noncommutative analog of Hall scalar product in the commutative theory.