Length function

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Let G be a group. A length function on G is a function L\colon G \to \mathbb{R}^+ satisfying:

L(e)
=
0,
L(g)
=
L(g^{-1}), \quad\forall g \in G,
L(g1g2)
\leq
L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G.

Coxeter groups (including the symmetric group) have combinatorial important length functions, for which each simple reflection has length 1.

This article incorporates material from Length function on PlanetMath, which is licensed under the GFDL.