论文标题
Buryak-Feigin-Nakajima的组合证明
A combinatorial proof of Buryak-Feigin-Nakajima
论文作者
论文摘要
Buryak，Feigin和Nakajima通过使用$ C^2 $的Hilbert Spoint中的$ z/cz $固定点集计算了分区统计家庭的生成功能。洛尔（Loehr）和沃灵顿（Warrington）已经展示了海曼（Haiman）在$ c^2 $上使用希尔伯特（Hilbert）方案的几何形状的类似观察结果，可以纯粹是组合。我们扩展了Loehr和Warrington的技术，以说明核心和商。 In particular, we construct a multigraph $M_{r,s,c}$ that is a direct refinement of Loehr and Warrington's multigraphs $M_{r,s}$, retains the relevant partition data, and is preserved by an involution $I_{r,s,c}$ which we use to prove the equidistribution of a family of partition statistics. 结果，我们获得了Buryak，Feigin和Nakajima结果的纯粹组合证明。更准确地说，我们定义一个分区统计家庭$ \ {h_ {x，c}^+，x \ in [0，\ infty）\} $，并给出一个组合证明，证明所有$ x $ and ash oble x $ and ash oble x $ \ begin {equation*} \ sum q^{|λ|} t^{h_ {x，c}^+（λ）} = q^{|μ|} \ prod_ {i \ geq 1} \ frac {1} {1} {（1-q^{ic}}} {c-1}） 1} \ frac {1} {1-q^{jc} t}， \ end {equation*} 总和在所有分区上的范围$λ$带有$ c $ - core $μ$。 第2节回顾了有关分区，核心和商的背景，并用新的主题写作。
Buryak, Feigin and Nakajima computed a generating function for a family of partition statistics by using the geometry of the $Z/cZ$ fixed point sets in the Hilbert scheme of points on $C^2$. Loehr and Warrington had already shown how a similar observation by Haiman using the geometry of the Hilbert scheme of points on $C^2$ could be made purely combinatorial. We extend the techniques of Loehr and Warrington to also account for cores and quotients. In particular, we construct a multigraph $M_{r,s,c}$ that is a direct refinement of Loehr and Warrington's multigraphs $M_{r,s}$, retains the relevant partition data, and is preserved by an involution $I_{r,s,c}$ which we use to prove the equidistribution of a family of partition statistics. As a consequence, we obtain a purely combinatorial proof of a result of Buryak, Feigin, and Nakajima. More precisely, we define a family of partition statistics $\{h_{x,c}^+, x\in [0,\infty)\}$ and give a combinatorial proof that for all $x$ and all positive integers $c$, \begin{equation*} \sum q^{|λ|}t^{h_{x,c}^+(λ)}=q^{|μ|}\prod_{i\geq 1}\frac{1}{(1-q^{ic})^{c-1}}\prod_{j\geq 1}\frac{1}{1-q^{jc}t}, \end{equation*} where the sum ranges over all partitions $λ$ with $c$-core $μ$. Section 2 recalls background on partitions, cores and quotients and is written with those new to the subject in mind.
