论文标题
阳性，结和$ Q，T $ -CATALAN数字
Positroids, knots, and $q,t$-Catalan numbers
论文作者
论文摘要
我们将混合的Hodge结构与开放式阳性品种的共同体学（尤其是它们的betti数字超过$ \ Mathbb {C} $，并且点数与$ \ Mathbb {f} _Q $）与Khovanov- khovanov--rozansky链接链接的rozansky同源性。我们推断出，顶级开放式阳性品种的混合霍奇多项式由理性$ q，t $ -catalan数字提供。通过群集品种的奇怪lefschetz属性，这意味着有理$ Q，t $ Q，t $ -catalan数字的$ q，t $ - 符号对称性和非模式性属性。我们表明，$ q，t $ -smmetry现象是类别$ \ Mathcal {o} $的Koszul二元性的一种体现，并与开放的Richardson品种和Verma模块的扩展组讨论了关系。
We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $\mathbb{C}$ and point counts over $\mathbb{F}_q$) to Khovanov--Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $\mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.
