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Let $B_n$ be the group of signed permutations, which is a Coxeter group acting on $\mathbb{R}^n$ with Coxeter generators $\sigma_i=(i\; i+1)\in S_n$ and the change of sign $\tau(x_1,x_2,\dots,x_n)=(-x_1,x_2,\dots,x_n)$.

So the elements can be represented by the action on the vector $(1,2,\dots,n)$ as words $w$ in the signed alphabet $\{\pm 1,\dots,\pm n\}$ where the $|w_i|$ form a permutation. Moreover, $$ \operatorname{inv}(w)=|\{iw(j)\}|+|\{i$\operatorname{inv}(w)$ is just the minimum length of an expression for $w$ written as a product of the Coxeter generators.

How does that characterization follow from this definition? Thank you.

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    Dear moderators, I registered an account. Can it be merged with the old unregistered one: http://math.stackexchange.com/users/6742/hobbie? Thanks.2012-02-06
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    Have you looked in Combinatorics of Coxeter Groups by Bjorner/Brenti? I'm fairly certain it is in there.2012-02-06
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    @MichaelJoyce I know Proposition 1.5.2 proves something similar, but there they use a different definition for the inversion number of elements of $S_n$.2012-02-06
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    @Hobbie: I've merged your accounts.2012-02-07
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    You can take a look at book *The Geometry of the Classical Groups* by *D.E. Taylor*. I don't have a copy at hand, but if I remember correctly (it's a while ago that I read the book), then the book contains a nice proof for the fact you mentioned.2012-02-07
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    @ego Thanks for the reference. I picked up a copy of the book, but don't see it mentioned. Do you recall where exactly it was?2012-02-11
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    @Hobbie: I do not have access to a copy of the book currently, so I cannot check it. As I read the book some years ago, maybe my memory foils me, sorry (also for the late response).2012-02-17
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    This is quite straightforward, if you are familiar with the language of root systems. All you need to do is to count how many positive roots get mapped to negative roots, and that's precisely what the claimed formula does. The dirty work (=that this number is equal to the minimum length of the group element as a product of generators) has then been done for you in full generality!2012-07-06

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