# What makes a $$\mathbf{D}_0$$ D 0 graph Schur positive?

Research paper by Jonah Blasiak

Indexed on: 22 Oct '16Published on: 01 Nov '16Published in: Journal of Algebraic Combinatorics

#### Abstract

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.AbstractWe 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. $${D}_0$$ $${D}_0$$graphn $$\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {bca} \cdots$$ $$\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {bca} \cdots$$ $$\cdots \mathsf {acb} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots$$ $$\cdots \mathsf {acb} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots$$Knuth transformations $$\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {acb} \cdots$$ $$\cdots \mathsf {bac} \cdots \leftrightsquigarrow \cdots \mathsf {acb} \cdots$$ $$\cdots \mathsf {bca} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots$$ $$\cdots \mathsf {bca} \cdots \leftrightsquigarrow \cdots \mathsf {cab} \cdots$$rotation transformations $$i-1, i, i+1$$ $$i-1, i, i+1$$generating function $$_0$$ $$_0$$http://www-bcf.usc.edu/~shassaf/degs.pdf2014http://www-bcf.usc.edu/~shassaf/positivity.pdf2014 $$q^t$$ $$q^t$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$ $$_0$$