UC Berkeley

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 well-known 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 work, we present a proof for the case $k=3$. In doing so, we introduce a new method for establishing linear relations among LLT polynomials.