From quasi-symmetric to Schur expansions with applications to symmetric chain decompositions and plethysm

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 that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of sw[sₕ](x,y) for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a w× h box. For $w=4$, this is first known combinatorial expression for the coefficient of sλ in sw[sₕ] for two-row partitions λ, and for $w=3$ the combinatorial expression is new.