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Empty-subset wall crossings are simple reflections

PDF p. 11: Remark 3.16

Statement

Let $(W, S)$ be a Coxeter system with simple generators $s_{i}$ indexed by the vertices $i$ of a Coxeter diagram $Γ$ . For $K \subseteq Γ_{0}$ , write $W_{K} = ⟨ s_{k} : k \in K ⟩$ . An associate of $J \subseteq Γ_{0}$ is a subset $I$ for which $W_{I}$ is conjugate to $W_{J}$ , as in Associates of a parabolic subset. The empty subset is its only associate.

For every $a \in Γ_{0}$ , the connected-support notation of Connected component notation [I,a] gives $[\emptyset, a] = {a}$ , the group $W_{[\emptyset, a]}$ is cyclic of order two, and the simple wall-crossing element of Simple wall-crossing elements and maps is $v_{a, \emptyset} = s_{a}$ .

Proof

Given: the Coxeter system and vertex $a$ in the Statement.

[F1] The generators satisfy $s_{i}^{2} = 1$ , and in the standard geometric representation $s_{i} α_{i} = - α_{i}$ for a nonzero basis vector $α_{i}$ . Word length is the least length of a word in the simple generators. Coxeter-system, roots, length and weak-order conventions

[F2] The standard parabolic $W_{K}$ is generated by the $s_{k}$ for $k \in K$ ; in particular $W_{\emptyset} = {1}$ . Standard parabolic subgroups and parabolic roots

[F3] The set $[I, a]$ is the component containing $a$ in the diagram induced on $I \cup {a}$ . Connected component notation [I,a]

[F4] For finite $W_{L}$ , $w_{L}$ is its maximum-length element; $w_{\emptyset} = 1$ . Longest parabolic elements and opposition permutation

[F5] A pair $(x, I)$ lies in $C h a m (J)$ exactly when $W_{J} x = x W_{I}$ and $x$ has minimum word length in $x W_{I}$ . For such a label, if $a \in Γ_{0} ∖ I$ , $D = [I, a]$ and $W_{D}$ is finite, the wall-crossing definition gives $v_{a, I} = w_{D ∖ {a}} w_{D}$ . Algebraic chamber-label set Simple wall-crossing elements and maps

Proof technique: direct.

Goal: prove the associate, component, group-order and wall-crossing assertions.

1.1F1F2given

Associates. By [F1], $s_{i}$ acts nontrivially on $α_{i}$ , so $s_{i} \neq 1$ for every vertex $i$ . By [F2], a nonempty $I$ has $s_{i} \in W_{I}$ for some $i \in I$ , so $W_{I}$ is nontrivial. Since a conjugate of $W_{\emptyset} = {1}$ is again ${1}$ , no nonempty $I$ is an associate of $\emptyset$ . The empty subset is an associate of itself by conjugation by $1$ .

1.2F1F2F3F4

Rank one. The induced diagram on ${a}$ has only the vertex $a$ , so [F3] gives $D = [\emptyset, a] = {a}$ . By [F1] and [F2], $W_{D} = {1, s_{a}}$ and its two elements are distinct; hence it is cyclic of order two. In particular, $W_{D}$ is finite. The identity has length zero, while the nonidentity generator $s_{a}$ has length one. Thus [F4] gives $w_{D} = s_{a}$ and $w_{D ∖ {a}} = w_{\emptyset} = 1$ .

2.1F1F2F5givenstep 1.1step 1.2∎

Chamber label and wall crossing. Set $J = I = \emptyset$ and $x = 1$ . By [F2], $W_{\emptyset} 1 = 1 W_{\emptyset} = {1}$ . The identity is the only element in this coset and has word length zero by [F1], so it has minimum length there. The chamber-label criterion in [F5] therefore gives $(1, \emptyset) \in C h a m (\emptyset)$ . Also $a \in Γ_{0} ∖ \emptyset$ by the given choice of vertex. Step 1.2 establishes the remaining finiteness hypothesis, so [F5] applies and gives $v_{a, \emptyset} = w_{\emptyset} w_{{a}} = 1 s_{a} = s_{a}$ . Together with steps 1.1 and 1.2 this proves the Statement.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 11, Remark 3.16