LLT polynomials are q-analogs of products 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 Haglund 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–Haglund conjecture for \(k=2\).

LLT多项式是Schur函数乘积的q模拟，Grojnowski和Haiman知道它们是Schur正数。但是，对于舒尔展开中的系数，没有已知的组合公式。找到这样的公式还可以提供麦克唐纳多项式的Schur正性。另一方面，Haiman和Haglund推测，位于k个相邻对角线上的倾斜分区的LLT多项式为k -Schur正，比Schur正性强得多。在本文中，我们通过分析单细胞LLT多项式来证明\（k = 2 \）的猜想。我们首先针对\（k = 2 \）的单细胞LLT多项式给出线性定理。通过分析矩形的LLT多项式上具有已知结果的LLT多项式之间的线性关系，我们提供了单细胞LLT多项式的2-Schur正性以及出现在Haiman–Haglund猜想中的\（k = 2 \）的LLT多项式。