关于Fomin-Kirillov代数的交换商
On the commutative quotient of Fomin-Kirillov algebras
摘要：
Fomin-Kirillov代数$\mathcal E_n$是一个非交换代数，在$n$顶点上的完全图的每条边都有一个生成元。对于$n$顶点上的图$G$，设$\mathcal E_G$是由$G$中的边生成的$\mathcal E_n$的子代数。证明了$\mathcal E_G$的交换商同构于$G$的Orlik-Terao代数.因此，这个商的希尔伯特级数由$(-t)^n χ_G(-t^{-1})$给出，其中$χ_G$是$G$的色多项式。我们还给出了一个约化算法的分次组件的$\mathcal E_G$，不消失的交换商，并表明其结构所描述的组合的非交叉森林。
英文摘要：
The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative algebra with a generator for each edge in the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, let $\mathcal E_G$ be the subalgebra of $\mathcal E_n$ generated by the edges in $G$. We show that the commutative quotient of $\mathcal E_G$ is isomorphic to the Orlik-Terao algebra of $G$. As a consequence, the Hilbert series of this quotient is given by $(-t)^n χ_G(-t^{-1})$, where $χ_G$ is the chromatic polynomial of $G$. We also give a reduction algorithm for the graded components of $\mathcal E_G$ that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
