A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions

2012/08/31 by Chris Berg, Nantel Bergeron, Franco Saliola +2 · 45 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Analytic and geometric function theory #Basis (linear algebra) #Commutative property #Construct (python library) #Lift (data mining) #Schur algebra #Symmetric function #math.CO #math.QA

paper · pdf · doi:10.4153/cjm-2013-013-0

published in Canadian Journal of Mathematics 66(3), 525-565 (Cambridge University Press) · new version includes edits, references to forthcoming research, and a 'hook-length' formula for the number of standard immaculate tableaux

openalex publication_date 2013/04/01 · arxiv created 2013/06/13 · openalex created_date 2016/06/24 · arxiv updated 2016/11/08 · openalex updated_date 2026/08/05

Abstract

Abstract We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions that expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of theHall–Littlewood symmetric functions with similar properties to their commutative counterparts.

Citations

• Vertex operators, symmetric functions, and the spin group Γn

Cited by

• The Pieri Rule for Dual Immaculate Quasi-Symmetric Functions
• A lift of Schur's Q-functions to the peak algebra

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