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A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

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https://doi.org/10.5070/C61055380Creative Commons 'BY' version 4.0 license
Abstract

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial G\boldsymbolλ(\boldsymbolx;q) in some special cases. We associate a weighted graph Π to \boldsymbolλ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of G\boldsymbolλ(\boldsymbolx;q) whenever Π is triangle-free. We also prove that the largest power of q in the LLT polynomial is the total edge weight of our graph.

Keywords: Charge, chromatic symmetric function, cocharge, Hall--Littlewood polynomial, jeu de taquin, LLT polynomial, interval graph, Schur function, Schur-positive, symmetric function.

Mathematics Subject Classifications: 05E05, 05E10, 05C15

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