Noncommutative Shifted Symmetric Functions

Laugwitz, Robert; Retakh, Vladimir

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 de_ned and form a basis for the ring. Further, we produce analogues of Jacobi{Trudi and Nagelsbach{Kostka formulas, a duality antialgebra 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.

Citation

Laugwitz, R., & Retakh, V. (2020). Noncommutative Shifted Symmetric Functions. Moscow Mathematical Journal, 20(1), 93-126

• Journal Article Type: Article
• Acceptance Date: Feb 21, 2019
• Online Publication Date: Mar 1, 2020
• Publication Date: Mar 1, 2020
• Deposit Date: Jan 21, 2020
• Journal: Moscow Mathematical Journal
• Print ISSN: 1609-3321
• Electronic ISSN: 1609-4514
• Publisher: Nezavisimyi Moskovskii Universitet
• Peer Reviewed: Peer Reviewed
• Volume: 20
• Issue: 1
• Pages: 93-126
• Keywords: Noncommutative Symmetric Functions, Shifted Symmetric Functions, Schur Functions, Quasideterminants
• Public URL: https://nottingham-repository.worktribe.com/output/3774150
• Publisher URL: http://www.mathjournals.org/mmj/2020-020-001/2020-020-001-005.html