A survey of the Shi arrangement (Q2417459)

The presented survey provides an interesting outline on remarkable properties of Shi arrangements (which come as byproducts of Shi's proof of Lusztig's conjecture that the number of two-sided cells for the affine Weyl group of type $A_{n-1}$ is the number of partitions of $n$). Let us recall the key construction. ``Let $V$ be a finite dimensional real vector space with fixed inner product $\langle \cdot | \cdot \rangle$. We will use $\Delta$ to denote a root system: a finite set of vectors in $V$ which satisfies: (1) $\Delta \cap \mathbb{R} \alpha = \{ \alpha,-\alpha \}$ for all $\alpha \in \Delta$, and (2) $s_{\alpha} \cdot \Delta = \Delta$ for all $\alpha \in \Delta$, where $s_{\alpha}$ is the reflection about the hyperplane with normal $\alpha$. [\dots] Let $W$ be a group with a set of generators $S \subset W$. Let $m_{st}$ be the order of the element $st$, with $s, t \in S$. If there is no relation between $s$ and $t$, we set $m_{st} =\infty$. If $W$ has a presentation such that (1) $m_{ss} = 1$, (2) for $s, t \in S$, $s \neq t$, $1 < m_{st} \leq \infty$, then $W$ is a Coxeter group. [\dots] If $m_{st} \in \{2, 3, 4, 6 \}$ when $s \neq t$, then the Coxeter group is called crystallographic and, if finite, is a Weyl group.'' Let $\Delta$ be a root system. The roots (plus the integers) define a system of affine hyperplanes \[ H_{\alpha,k} = \{v\in V \mid \langle v | \alpha \rangle = k\}. \] Now for any irreducible and crystallographic root system $\Delta$ the Shi arrangement $\text{Shi}_{\Delta}$ is the collection of hyperplanes \[ \{ H_{\alpha ,k} \mid \alpha \in \Delta^{+}, 0 \leq k \leq 1 \}, \] where we use $\Delta^{+}$ to denote a choice of positive roots of $\Delta$. Let us present very briefly the content of the survey. In Chapter 2, one can see how the Shi arrangement arose, Chapter 3 is devoted to enumerative properties of regions determined by Shi arrangements, in Chapter 4 the author presents interesting connections between Shi arrangements and some topics in algebra (for instance decomposition numbers for certain Hecke algebras), or how to use Shi arrangements in order to build an automaton which recognizes reduced expressions. For the entire collection see [Zbl 1410.05001].