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I have been studying Coxeter Groups and started reading on Bruhat Order in the same context. I came across the following definition: Consider $(W,S)$ a Coxeter system with the set of reflections:

$T = \{(wsw)^{-1} \mid w \in W, s\in S \}$.

for any $u, w\in W$ we write:

  1. $u\rightarrow w$ meaning that $l(u) < l(w)$ and there exists some $t \in T$ such that $u^{-1}w = t$
  2. $u \leq w$ meaning that there exists $u_i \in W$ such that: $u=u_0 \rightarrow u_1 \rightarrow \dots \rightarrow u_k = w$

The Bruhat order is defined to be the relation in 2.

I don't understand how is this order a partial order. It is not even refelexive as it is not true that $l(w) < l(w)$ Can someone please enlighten me or point out what I am missing.

Thanks.

FAF
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1 Answers1

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I'll do 2 easy steps (they don't really have anything to do with Coxeter groups, but should get you over the problem about reflexivity). You still need to check antisymmetry.

It is reflexive because when $u=w$ we can take $k=0$ in item 2 (there will not be any $\to$ in that case), and it is transitive because we can concatenate two such chains in item 2.

Ted
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