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Core and atomic double cosets

PDF p. 20: Remark 4.22, second paragraph

Definition

Let $(W, S)$ be a Coxeter system with $S = {s_{i} : i \in Γ_{0}}$ . For $J \subseteq Γ_{0}$ , let $W_{J} = ⟨ s_{j} : j \in J ⟩$ be its standard parabolic subgroup. For $I, K \subseteq Γ_{0}$ and $x \in W$ , the $(I, K)$ -double coset of $x$ is the set $W_{I} x W_{K} = {u x v : u \in W_{I}, v \in W_{K}}$ . The set $W_{I} x = {u x : u \in W_{I}}$ is the right coset of $W_{I}$ represented by $x$ , and $x W_{K} = {x v : v \in W_{K}}$ is the left coset of $W_{K}$ represented by $x$ .

The $(I, K)$ -double coset $W_{I} x W_{K}$ is core if $W_{I} x = W_{I} x W_{K} = x W_{K}$ as sets. No finiteness condition is imposed on $W_{I}$ or $W_{K}$ in this definition.

For $L \subseteq Γ_{0}$ with $W_{L}$ finite, write $w_{L}$ for its unique longest element, meaning the element of maximal Coxeter length in $W_{L}$ . For $a, a^{'} \in L$ , write $L - a = L ∖ {a}$ and $L - a^{'} = L ∖ {a^{'}}$ . An atomic double coset, or atom, is a core double coset of the form $W_{L - a} w_{L} W_{L - a^{'}}$ for such $L, a, a^{'}$ . The core condition is part of the definition; it is not asserted for arbitrary choices of $a, a^{'}$ .

Definition references

Standard parabolic subgroups and parabolic roots

Longest parabolic elements and opposition permutation

Left and right cosets gH and Hg of a subgroup

Justification

The standard parabolic construction supplies subgroups $W_{I}$ and $W_{K}$ of $W$ . Their left, right and double cosets are sets of group products, so the displayed equality is a meaningful condition. It depends only on the double coset and its specified pair $(I, K)$ : if it holds for $x$ and $y$ lies in that double coset, then $y \in W_{I} x$ and $y \in x W_{K}$ , whence $W_{I} y = W_{I} x$ and $y W_{K} = x W_{K}$ ; also $W_{I} y W_{K} = W_{I} x W_{K}$ . Thus the same equality holds for $y$ .

When $W_{L}$ is finite, the longest-elements definition supplies the unique $w_{L}$ . The two deleted subsets are subsets of $Γ_{0}$ , so their standard parabolics and the displayed double coset exist. Selecting those double cosets that satisfy the core condition therefore defines the atoms.

Depends on

Used by · 5

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External dependencies

An atomic Coxeter presentation · Definitions 1.1 and 1.6, PDF pp. 1 and 4

A core parabolic double coset is simultaneously the corresponding left and right coset. An atom is a core coset formed from the longest element of a finitary parabolic and two corank-one parabolics.

Hypotheses: A Coxeter system; in the atomic definition the parabolic indexed by the larger subset is finite.

Reading limits: Read Definitions 1.1 and 1.6 and their immediate context. No theorem about atomic compositions or their translation to groupoid products is used here.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 20, Remark 4.22, second paragraph
An atomic Coxeter presentation · Definitions 1.1 and 1.6, PDF pp. 1 and 4