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

Foster Tom

doi:10.5070/c61055380

Abstract

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) in some special cases. We associate a weighted graph \(\Pi\) to \(\boldsymbol\lambda\) 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 \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) whenever \(\Pi\) 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

Highlights

LLT polynomials are remarkable symmetric functions with many connections in algebraic combinatorics
Haiman, and Loehr [8] found a combinatorial formula for Macdonald polynomials, which implies a positive expansion in terms of these LLT polynomials Gλ(x; q)
Grojnowski and Haiman [7] proved that LLT polynomials, and Macdonald polynomials, are Schur-positive using Kazhdan–Lusztig theory, but it remains a major open problem to find an explicit combinatorial Schur-positive expansion

Summary

Introduction

LLT polynomials are remarkable symmetric functions with many connections in algebraic combinatorics. Nam, and Yoo [10] found an explicit Schur-positive expansion whenever this graph is a “melting lollipop”, namely They proved that for arbitrary unit interval graphs, this formula gives the correct coefficient of sμ whenever the partition μ is a hook. Alexandersson and Uhlin [3] found a generalization of cocharge to prove an analogous formula when the rows of λ come from a skew shape σ/τ with no column having more than two cells. They formulated it for vertical-strips but we can equivalently state it as. We prove that the largest power of q in the LLT polynomial Gλ(x; q) is the total edge weight of Π

Background

A weighted graph description of horizontal-strip LLT polynomials

A combinatorial formula