What makes a $\mathbf{D}_0$ graph Schur positive?

Abstract

We define a ${D}_0$ graph to be a graph whose vertex set is a subset of permutations of n, with edges of the form $\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {bca} \cdots$ or $\cdots \mathsf {acb} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots$ (Knuth transformations), or $\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {acb} \cdots$ or $\cdots \mathsf {bca} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots$ (rotation transformations), such that whenever the Knuth and rotation transformations at positions $i-1, i, i+1$ are available at a vertex, exactly one of these is an edge. The generating function of such a graph is the sum of the quasisymmetric functions associated to the descent sets of its vertices. Assaf studied D$_0$ graphs in (Dual equivalence and Schur positivity, http://www-bcf.usc.edu/~shassaf/degs.pdf, 2014) and showed that they provide a rich source of examples of the D graphs of (Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity, http://www-bcf.usc.edu/~shassaf/positivity.pdf, 2014). A key construction of Assaf expresses the coefficient of $q^t$ in an LLT polynomial as the generating function of a certain D$_0$ graph. LLT polynomials are known to be Schur positive by work of Grojnowski-Haiman, and experimentation shows that many D$_0$ graphs have Schur positive generating functions, which suggests a vast generalization of LLT positivity in this setting. As part of a series of papers, we study D$_0$ graphs using the Fomin-Greene theory of noncommutative Schur functions. We construct a D$_0$ graph whose generating function is not Schur positive by solving a linear program related to a certain noncommutative Schur function. We go on to construct a D graph on the same vertex set as this D$_0$ graph.