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.

Equivalences of LLT polynomials via lattice paths

“…Using these techniques we are also able to construct linear relations between different LLT polynomials. Knowing these relations has been instrumental in proving results about the expansion of LLT polynomials into Schur and k-Schur polynomials [2,11,12,15]. Our new techniques give a systematic way to determine these relations.…”

“…As one possible application of this type of calculation, we can reprove a relation between LLT polynomials indexed by single rows given in [15]. Note that the precise powers of t differ than that of [15] as we are working with coinversion LLT polynomials.…”