Noncommutative Shifted Symmetric Functions

TL;DR

This paper develops a new algebraic framework for noncommutative shifted symmetric functions, introducing basis elements, formulas, and realizations that extend classical symmetric function theory into a noncommutative, multiparameter setting.

Contribution

It introduces a novel ring of noncommutative shifted symmetric functions with bases, formulas, and realizations, expanding the theory of symmetric functions into a multiparameter noncommutative context.

Findings

Defined shifted ribbon Schur functions forming a basis

Derived analogues of Jacobi-Trudi and Giambelli formulas

Provided a realization as rational functions with shifted symmetry

Abstract

We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi-Trudi and N\"agelsbach-Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these noncommutative shifted symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables and are well-behaved under extension of the list of variables.