Length function

Length function

Let G be a group. A length function on G is a function Lcolon G o mathbb{R}^+ satisfying:

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: L(e)

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: =

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: 0,

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: L(g)

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: =

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: L(g^{-1}), quadforall g in G,

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: L(g_1 g_2)

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: leq

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: L(g_1) + L(g_2), quadforall 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.

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