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数学代写|超平面置换理论代写Hyperplane Arrangements代考|MATH4550 Exponential sequences of arrangements

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数学代写|超平面置换理论代写Hyperplane Arrangements代考| Exponential sequences of arrangements

The braid arrangement (in fact, any Coxeter arrangement) is highly symmetrical; indeed, the group of linear transformations that preserves the arrangement acts transitively on the regions. Thus all regions “look the same.” The Shi arrangement lacks this symmetry, but it still possesses a kind of “combinatorial symmetry” that allows us to express the characteristic polynomials $\chi_{s_n}(t)$, for all $n \geq 1$, in terms of the number $r\left(\mathcal{S}_n\right)$ of regions.

Definition 5.14. A sequence $\mathfrak{A}=\left(\mathcal{A}_1, \mathcal{A}_2, \ldots\right)$ of arrangements is called an exponential sequence of arrangements (ESA) if it satisfies the following three conditions.
(1) $\mathcal{A}_n$ is in $K^n$ for some field $K$ (independent of $n$ ).
(2) Every $H \in \mathcal{A}_n$ is parallel to some hyperplane $H^{\prime}$ in the braid arrangement $\mathcal{B}_n($ over $K)$
(3) Let $S$ be a $k$-element subset of $[n]$, and define $\mathcal{A}_n^S=\left{H \in \mathcal{A}_n: H\right.$ is parallel to $x_i-x_j=0$ for some $\left.i, j \in S\right}$.
Then $L\left(\mathcal{A}_n^S\right) \cong L\left(\mathcal{A}_k\right)$.
Examples of ESA’s are given by $\mathcal{A}_n=\mathcal{B}_n$ or $\mathcal{A}_n=\mathcal{S}_n$. In fact, in these cases we have $\mathcal{A}_n^S \cong \mathcal{A}_k \times K^{n-k}$.

The combinatorial properties of ESA’s are related to the exponential formula in the theory of exponential generating functions $[\mathbf{3 2}, \S 5.1]$, which we now review. Informally, we are dealing with “structures” that can be put on a vertex set $V$ such that each structure is a disjoint union of its “connected components.” We obtain a structure on $V$ by partitioning $V$ and placing a connected structure on each block (independently). Examples of such structures are graphs, forests, and posets, but not trees or groups. Let $h(n)$ be the total number of structures on an $n$-set $V$ (with $h(0)=1$ ), and let $f(n)$ be the number that are connected. The exponential formula states that
$$\sum_{n \geq 0} h(n) \frac{x^n}{n !}=\exp \sum_{n \geq 1} f(n) \frac{x^n}{n !}$$



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