论文标题
基本对称函数的垂直条纹LLT多项式的组合扩展
A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
论文作者
论文摘要
我们给出了垂直 - 斯特llt多项式$ \ mathrm {llt} _p（x; q）$的新表征，是满足某些组合关系的唯一对称函数家族。然后，该表征被用来证明垂直条纹LLT多项式的显式组合扩展。这种公式是由A. Garsia等人独立猜想的。和第一个命名的作者，并由单位间隔图的方向组合。如果$ Q $被$ Q+1 $取代，则获得的扩展显然是积极的，从而恢复了M. d'Adderio的最新结果。我们的结果基于D'Adderio的工作中出现的LLT多项式之间的线性关系，以及Carlsson和A. Mellit。在某种程度上，使用单位间隔图的颜色，为这些关系提供了新的两种徒证明。作为奖励，我们获得了单位间隔图色素对称函数的新表征。
We give a new characterization of the vertical-strip LLT polynomials $\mathrm{LLT}_P(x;q)$ as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if $q$ is replaced by $q+1$, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.
