Séminaire Lotharingien de Combinatoire, 85B.18 (2021), 12 pp.

Foster Tom

A Combinatorial Schur Expansion of Triangle-Free Horizontal-Strip LLT Polynomials

Abstract. In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial G_λ(x;q) in some special cases. We associate a weighted graph Π to λ 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_λ(x;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.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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