Flip classes and elementary expansions of vertical-strip LLT polynomials
Abstract
After the substitution q↦ 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 qasc(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.
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BibTeX
@misc{alexandersson2026flip,
title = {{Flip classes and elementary expansions of vertical-strip LLT polynomials}},
author = {Per Alexandersson and Lilan Dai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26294},
eprint = {2610.26294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26294},
}
RIS
TY - UNPB
TI - Flip classes and elementary expansions of vertical-strip LLT polynomials
AU - Per Alexandersson
AU - Lilan Dai
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26294
UR - https://arxiv.org/abs/2610.26294
AB - After the substitution q↦ 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^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.
ER -