It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function, whose
expansion in terms of the fundamental quasisymmetric function is known. For example,
formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two
Schur functions, while the Schur expansions of these expressions are still elusive.
Egge, Loehr and Warrington provided a method to obtain the Schur expansion
from the fundamental expansion by replacing each quasisymmetric function by a Schur function
(not necessarily indexed by a partition) and using straightening rules to obtain the Schur
expansion. Here we provide a new method which only involves the coefficients of the quasisymmetric functions
indexed by partitions and the quasi-Kostka matrix. As an application, we prove the lexicographically largest term
in the Schur expansion of the plethysm of two Schur functions and the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$
using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box.
This is based on joint work with Rosa Orellana, Franco Saliola and Mike Zabrocki.
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