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Minimal coset representatives remove chamber-label ambiguity

PDF p. 10: Remark 3.11 and following paragraph

Statement

Let $(W, S)$ be a finite-rank Coxeter system, with simple roots $(e_{s})_{s \in S}$ in its canonical real reflection representation and contragredient action on the dual. For $I \subseteq S$ , let $W_{I} = ⟨ I ⟩$ and let $C_{I}$ be the face consisting of dual functionals that vanish on $e_{i}$ for $i \in I$ and are positive on $e_{s}$ for $s \notin I$ . If $x^{'} \in x W_{I}$ , then $x^{'} C_{I} = x C_{I}$ ; consequently distinct group elements can give the same face translate.

Fix $J \subseteq S$ . A label in $C h a m (J)$ is a pair $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ of minimum word length in $x W_{I}$ . These labels identify the chambers of the relative arrangement in the interior of the $J$ -cone bijectively with their face translates $x C_{I}$ . Thus the minimum-length condition removes the ambiguity within each right coset, and distinct labels do not describe the same chamber. Accordingly $C h a m (J)$ may denote either this label set or the corresponding set of geometric chambers, with the bijection understood.

Statement references

Real Coxeter representation and contragredient notation

Algebraic chamber-label set

Tits cone intersections, relative roots and arrangements

Chambers, walls and adjacency

Parabolic root subsystems and unique coset minima

Labelling theorem for relative chambers

Proof

Given: The finite-rank Coxeter system and subsets $I, J \subseteq S$ in the Statement, with arbitrary Coxeter type.

[F1] The simple reflection $s$ acts on the real root space by $v \mapsto v - 2 B (v, e_{s}) e_{s}$ , and the dual action is $(w f) (v) = f (w^{- 1} v)$ . The face $C_{I}$ is defined by the coordinate conditions in the Statement. Real Coxeter representation and contragredient notation

[F2] Every right coset $x W_{I}$ has a unique minimum-length element. Parabolic root subsystems and unique coset minima

[F3] The label set consists of the pairs specified in the Statement. The $J$ -cone is the intersection of the Tits cone with the subspace annihilating the simple roots indexed by $J$ ; the relative arrangement uses the proper traces of root hyperplanes on that subspace, and its chambers are connected components of the complement in the cone's relative interior. Algebraic chamber-label set Tits cone intersections, relative roots and arrangements Chambers, walls and adjacency

[F4] For these finite-rank conventions and arbitrary $J$ , the map $(x, I) \mapsto x C_{I}$ is a bijection from the label set to the chambers described in [F3]. Labelling theorem for relative chambers

Proof technique: direct.

Goal: Prove the equality of translates and the stated uniqueness and notation assertions.

1.1F1given

Let $f \in C_{I}$ and $i \in I$ . Since $f (e_{i}) = 0$ , the reflection formula and dual action in [F1] give $(i f) (v) = f (v - 2 B (v, e_{i}) e_{i}) = f (v)$ for every root-space vector $v$ . Thus each generator in $I$ fixes every point of $C_{I}$ , and so does every $u \in W_{I}$ . If $x^{'} = x u$ , the group action gives $x^{'} C_{I} = x (u C_{I}) = x C_{I}$ . When $I$ is nonempty, an element $i \in I$ is distinct from the identity, so $x$ and $x i$ provide distinct group elements with the same translate.

2.1F2F3F4step 1.1∎

By [F2], each right coset has exactly one minimum-length element, so at most one choice of $x$ from that coset satisfies the length condition for a fixed $I$ . The bijection in [F4], with the label and chamber conventions in [F3], also rules out equality of chamber translates from any two distinct labels, including labels with different subsets. It supplies every geometric chamber and its unique label; hence the two uses of $C h a m (J)$ are identified by that bijection. Together with step 1.1 this proves the assertions.

Source notes

The article citation attributes the remark and its notation. The argument above uses the declared mathematical suppliers, rather than the article remark as a proof of itself. The geometric statement uses the finite-rank convention of the completed relative-cone and labelling suppliers.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 10, Remark 3.11 and the following paragraph