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LLT polynomials are $q$-analogues of product of Schur functions that areknown to be Schur-positive by Grojnowski and Haiman. However, there is no knowncombinatorial formula for the coefficients in the Schur expansion. Finding sucha formula also provides Schur positivity of Macdonald polynomials. On the otherhand, Haiman and Hugland conjectured that LLT polynomials for skew partitionslying on $k$ adjacent diagonals are $k$-Schur positive, which is much strongerthan Schur positivity. In this paper, we prove the conjecture for $k=2$ byanalyzing unicellular LLT polynomials. We first present a linearity theorem forunicellular LLT polynomials for $k=2$. By analyzing linear relations betweenLLT polynomials with known results on LLT polynomials for rectangles, weprovide the $2$-Schur positivity of the unicellular LLT polynomials as well asLLT polynomials appearing in Haiman-Hugland conjecture for $k=2$.

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