Comparing skew Schur functions: a quasisymmetric perspective

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.

Keywords

Skew Schur function, quasisymmetric function, $F$-positive, support containment, dominance order

2010 Mathematics Subject Classification

Primary 05E05. Secondary 05E10, 06A07, 20C08, 20C30.

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Published 12 February 2014