论文标题
Nabla操作员的组合公式
A combinatorial formula for the nabla operator
论文作者
论文摘要
我们为Nabla操作员的一般力量提供了一个LLT型公式，该公式适用于库奇产品，用于修改后的麦克唐纳多项式，并使用它来推断出描述$ \ nabla^k e_n $的广义定理的新证明，以$ \ nabla^k e__n和$ elias-hogancamp for $ nabla^nabla^k-nabla^k-nabla^k p-nabla^k p^nabla^k p-nabla^k p p p。我们通过验证LLT扩展能够满足$ \ nabla^k $的定义属性，例如在优势顺序中的三角形，以及基于对第二作者的$ \ Mathbb {p}^1 $ contiment $ \ Mathbb {p}^1 $的几何证明来提供直接证明该定理的定义证明。这些公式与Goresky，Kottwitz和MacPherson研究的A型A仿型Springer纤维的仿射铺路有关，也与Stanley的彩色对称功能有关。
We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$, and the Elias-Hogancamp formula for $(\nabla^k p_1^n,e_n)$ as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $\nabla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $\mathbb{P}^1$ due to the second author. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and also to Stanley's chromatic symmetric functions.
