标题： 正的组合公式用于对合矩阵流形和轨道调和 显示英文标题

标题： Positive combinatorial formulae for involution matrix loci and orbit harmonics

摘要： 设$\mathcal{M}_{n,a}$为对称群$\mathfrak{S}_n$中恰好具有$a$个不动点的对合组成的集合，并应用轨道调和方法得到一个分次$\mathfrak{S}_n$-模$R(\mathcal{M}_{n,a})$。 Liu、Ma、Rhoades 和 Zhu 推导出了分次 Frobenius 映射$\grFrob(R(\MMM_{n,a});q)$的带符号组合公式，该公式对应于$R(\MMM_{n,a})$。 我们的目标是消除这些符号。 最后，我们找到了两个正的组合公式用于$\mathrm{grFrob}(R(\mathcal{M}_{n,a});q)$，这为寻找$R(\mathcal{M}_{n,a})$的线性基提供了潜在的方法，并找到了一个统计量$\mathrm{stat}:\mathcal{M}_{n,a}\rightarrow\mathbb{Z}_{\ge 0}$来解释 Hilbert 系列$\mathrm{Hilb}(R(\mathcal{M}_{n,a});q)$的$R(\mathcal{M}_{n,a})$。 显示英文摘要

摘要： Let $\mathcal{M}_{n,a}$ be the set consisting of involutions in symmetric group $\mathfrak{S}_n$ with exactly $a$ fixed points and apply the orbit harmonics method to obtain a graded $\mathfrak{S}_n$-module $R(\mathcal{M}_{n,a})$. Liu, Ma, Rhoades, and Zhu figured out a signed combinatorial formula for the graded Frobenius image $\grFrob(R(\MMM_{n,a});q)$ of $R(\MMM_{n,a})$. Our goal is to cancel these signs. Finally, we find two positive combinatorial formulae for $\mathrm{grFrob}(R(\mathcal{M}_{n,a});q)$, yielding potential ways to find a linear basis for $R(\mathcal{M}_{n,a})$ and find a statistic $\mathrm{stat}:\mathcal{M}_{n,a}\rightarrow\mathbb{Z}_{\ge 0}$ to interpret the Hilbert series $\mathrm{Hilb}(R(\mathcal{M}_{n,a});q)$ of $R(\mathcal{M}_{n,a})$.