Proposition 1

Tag 0ALO, permanent. Cite it as [Bourbaki, Tag 0ALO].

Lie Groups and Lie Algebras, chapter IV, § 1. Coxeter Groups

Let w and w' be in W. We have the formulas:

l(ww') \leq l(w) + l(w'), \tag{1}

l(w^{-1}) = l(w), \tag{2}

|l(w) - l(w')| \leq l(w{w'}^{-1}). \tag{3}

Let (s_1, \ldots, s_p) and (s'_1, \ldots, s'_q) be reduced decompositions of w and w' respectively. Thus l(w) = p and l(w') = q. Since ww' = s_1 \ldots s_p s'_1 \ldots s'_q, we have l(ww') \leq p + q, proving (1). Since S = S^{-1} and w^{-1} = s_p^{-1} \ldots s_1^{-1}, we have l(w^{-1}) \leq p = l(w). Replacing w by w^{-1} gives the opposite inequality l(w) \leq l(w^{-1}), proving (2). Replacing w by ww'^{-1} in (1) and (2) gives the relations

l(w) - l(w') \leq l(w{w'}^{-1}), \tag{4}

l(w{w'}^{-1}) = l(w'w^{-1}). \tag{5}

Exchanging $w$ and $w'$ in (4) gives $l(w') - l(w) \leq l(w{w'}^{-1})$ by (5), proving (3).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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