Linear relations on LLT polynomials and their k-Schur positivity for k=2

LLT polynomials are $q$-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a formula also provides Schur positivity of Macdonald polynomials. On the other hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions lying on $k$ adjacent diagonals are $k$-Schur positive, which is much stronger than Schur positivity. In this paper, we prove the conjecture for $k=2$ by analyzing unicellular LLT polynomials. We first present a linearity theorem for unicellular LLT polynomials for $k=2$. By analyzing linear relations between LLT polynomials with known results on LLT polynomials for rectangles, we provide the $2$-Schur positivity of the unicellular LLT polynomials as well as LLT polynomials appearing in Haiman-Hugland conjecture for $k=2$.