Flip classes and elementary expansions of vertical-strip LLT polynomials

Source abstract

After the substitution $q\mapsto q+1$, vertical-strip LLT polynomials are weighted sums of strict $P$-partition generating functions indexed by orientations of decorated unit-interval graphs, with weight $q^{\mathrm{asc}(θ)}$. A flip moves an ascending edge across a triangle while keeping its long edge. We show that the sum over every flip class is elementary-positive: it equals the sum of the elementary symmetric functions given by the highest reachable vertices of its orientations. This answers Hwang's question on the Schur positivity of LLT flip-class sums, and it holds for circular unit-interval graphs as well. Summing over classes gives a new proof of the elementary expansion of Alexandersson and Sulzgruber, and its circular analog, which was open. We also obtain a flip-class refinement of the elementary expansion of LLT cumulants.