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

Rosa Orellana¹ ; Franco Saliola² ; Anne Schilling³ ; Mike Zabrocki⁴

¹Dartmouth College
²UQAM
³University of California at Davis
⁴York University

The electronic journal of combinatorics, Tome 31 (2024) no. 4

Résumé

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. Based on work of Egge, Loehr and Warrington, Garsia and Remmel 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 $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\times h$ box. For $w=4$, this is the first known combinatorial expression for the coefficient of $s_{\lambda}$ in $s_{w}[s_{h}]$ for two-row partitions $\lambda$, and for $w=3$ the combinatorial expression is new.

Détail

DOI : 10.37236/12950

Classification : 05E05, 05E10, 05E18, 14M15
Mots-clés : Schur expansion of the plethysm of two Schur functions, quasi-Kostka matrix