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

TL;DR

This work addresses the problem of converting a known fundamental quasisymmetric expansion of a symmetric function into its Schur expansion by introducing a new right inverse of the Schur–fundamental transition that uses only partition-indexed coefficients and the quasi-Kostka matrix. The authors develop a combinatorial framework via the inverse quasi-Kostka matrix $\mathsf{Q}^{-1}$ and chains of quasi-Yamanouchi tableaux to express Schur coefficients $b_{\lambda}$ directly from quasisymmetric data, plus a submatrix technique for length-bounded cases. They apply this machinery to extract leading terms in plethysm and to compute Schur expansions of $s_{w}[s_{h}]$ for $w=2,3,4$ in two variables, supported by novel symmetric chain decompositions of Young’s lattice in $w\times h$ boxes with restriction, extension, and pattern properties. The results yield new combinatorial expressions for plethysm coefficients and, in particular, explicit formulas for two-variable plethysms, advancing understanding of Newton polytopes and the structure of plethysm coefficients.

Abstract

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 $s_w[s_h](x,y)$ 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 first known combinatorial expression for the coefficient of $s_λ$ in $s_{w}[s_{h}]$ for two-row partitions $λ$, and for $w=3$ the combinatorial expression is new.

Figures (5)

• Figure 1: The symmetric chain decomposition for $L(3,10)$ satisfying the restriction and pattern conditions. The blue cells are the partitions representing the highest weights. The number entries indicate in which order boxes are added for the given chain.
• Figure 2: Symmetric chain decomposition for $L(3, 10)$ matching that of Figure \ref{['fig:SCD-w3-C0']}. The blue edges follow rule 1, red edges follow rule 2, magenta edges follow rules 3,4 and 5, orange edges follow rules 6,7 and 8.
• Figure 3: The symmetric chain decomposition for $L(4, 7)$ satisfying the restriction, extension and pattern condition. The chains are organized so that the elements of $L"(4,h)$ are all listed first, followed by the elements of $L'(4,h) \backslash L"(4,h)$, followed by the elements of $L(4,h) \backslash L'(4,h)$.
• Figure 4: Lindstrom's Lindstrom1980, Wen's Wen2004, and our SCD of $L(3, 3)$
• Figure 5: Our SCD of $L(4, 3)$

Theorems & Definitions (43)

• Definition 2.1: AS.2017Wang.2019
• Example 2.2
• Definition 3.1
• Example 3.2
• Example 3.3
• Proposition 3.4
• proof
• Corollary 3.5
• Corollary 3.6
• Theorem 3.7