Open Collections

Esipova, Maria

University of British Columbia

The dual immaculate functions are an example of a Schur-like basis in the algebra of quasisymmetric functions (QSym). We study the dual immaculate functions and their images under involutions ρ,ψ,ω on QSym, termed variants of dual immaculate functions. Variants of dual immaculate functions form a family of Schur-like bases that share similar combinatorial properties, in particular, are generating functions of immaculate tableaux. A key contribution of this work is the classification of when a variant of a dual immaculate function is equal to a dual immaculate function. To achieve this, the leading term of the expansion of a variant of a dual immaculate function in the dual immaculate basis is identified. This is a first step in developing a full description of the transition matrix between variants of dual immaculate functions and dual immaculate functions. As a consequence, new maps and canonical tableaux associated with compositions are identified. Furthermore, sufficient and necessary conditions for the existence of immaculate tableaux are derived. Subsequently, this gives sufficient and necessary conditions for certain terms to have a nonzero coefficient in the fundamental expansion of variants of dual immaculate functions. These results provide a deeper understanding of the interplay between composition tableaux and transition matrices between bases in QSym related by automorphisms.

Text

2025-04-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/90617

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/