Christopher Miller: LLT Polynomials, k-Schur Functions, and Positivity

Abstract

Both LLT polynomials and k-Schur functions were derived from the study of Macdonald polynomials, and have proved to be fruitful areas of study. A conjecture due to Haglund and Haiman states that k-bandwidth LLT polynomials expand positively into k-Schur functions. This is trivial in the case k=1 and has been recently proved for k=2. In this talk, I will present a proof for the case k=3. To this end, I will introduce a new computational method for establishing linear relations among LLT polynomials.