Comparing skew Schur functions: a quasisymmetric perspective

TL;DR

This paper explores the relationship between skew Schur functions and quasisymmetric functions, revealing that support containment in the fundamental basis implies certain shape inequalities and proposing conjectures about their equivalence.

Contribution

It introduces new support-based criteria for skew Schur function equality and dominance relations, extending previous shape-based results and proposing conjectures supported by computational evidence.

Findings

Support containment implies row overlap dominance.

Support equality suggests shape equality.

Evidence supports conjectures on support and shape relations.

Abstract

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A and s_B are equal, then the skew shapes A and B must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than skew Schur equality: that s_A and s_B have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of s_A contains that of s_B, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.