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$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two permutations $\sigma_1,\sigma_2\in S_n$, $\sigma_1\leq \sigma_2$ in the weak Bruhat order iff the set of "upsets" in the ordering $1<2<\dotsb<n$ induced by the action of $\sigma_1$ on the balls is contained in the set of upsets induced by the action of $\sigma_2$ (more formally, $\sigma_1\leq \sigma_2$ iff $\Inv(\sigma_1)\subset \Inv(\sigma_2)$, where $\Inv(\sigma)=\{(i,j) \mathrel\vert i\leq j, \sigma(i)\geq \sigma(j)\}$). This also gives an interpretation of the rank function for the poset of permutations ordered this way (the number of inversions).

I'm wondering if anyone knows similar interpretations for the weak Bruhat order in more general Coxeter groups. For instance, is there always a set of labelled objects for which the Bruhat order tracks "upsets"?

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    $\begingroup$ The containment of inversions description directly generalizes to any Weyl group (where inversions = positive roots that are sent to negative roots). $\endgroup$ Sep 27 at 22:56
  • $\begingroup$ Welcome to mathoverflow. Just to add to what Sam said: A good source for this is Björner and Brenti's book "Combinatorics of Coxeter groups." They have a whole chapter (Chapter 3) on the weak order, although they develop a lot of the tools in earlier chapters. Your goal would be Proposition 3.1.3. This uses "reflections" for inversions rather than "roots", but it's completely equivalent. If you are happier using roots, it's probably in Humphreys' book "Reflection groups and Coxeter groups", but I don't have that in front of me for an exact reference. $\endgroup$ Sep 28 at 10:45
  • $\begingroup$ I think it’s in Bourbaki as well. $\endgroup$ Sep 28 at 14:25

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