# Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020 pp. 93–126.

Noncommutative Shifted Symmetric Functions

**Authors**:
Robert Laugwitz (1) and Vladimir Retakh (2)

**Author institution:**(1) University of Nottingham, Nottingham, NG7 2RD United Kingdom

(2) Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019

**Summary: **

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 Nägelsbach–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.

2010 Math. Subj. Class. 05E05.

**Keywords:**Noncommutative symmetric functions, shifted symmetric functions, Schur functions, quasideterminants

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