2 added 1 characters in body

Here is another notion of a reduced decomposition for the long word, as used for example in a recent preprint of Baumann, Kamnitzer and TigleyTingley.

Recall the theorem from the finite type case that reduced decompositions of w0 are in bijection with convex orderings on the set of positive roots. The latter notion is easier to generalise, and can be considered an appropriate analogue.

Definition: A convex order on the set of all positive roots Φ+ is a preorder ≤ such that

i) For all α, β in Φ+, we have α ≤ β or β ≤ α (OR not XOR).

ii) If α ≤ β and α+β is a root, then α ≤ β+α ≤ β.

iii) If α ≤ β and β ≤ α then α and β are proportional.

Voila.

1

Here is another notion of a reduced decomposition for the long word, as used for example in a recent preprint of Baumann, Kamnitzer and Tigley.

Recall the theorem from the finite type case that reduced decompositions of w0 are in bijection with convex orderings on the set of positive roots. The latter notion is easier to generalise, and can be considered an appropriate analogue.

Definition: A convex order on the set of all positive roots Φ+ is a preorder ≤ such that

i) For all α, β in Φ+, we have α ≤ β or β ≤ α (OR not XOR).

ii) If α ≤ β and α+β is a root, then α ≤ β+α ≤ β.

iii) If α ≤ β and β ≤ α then α and β are proportional.

Voila.