I am reading the book: Anders Björner, Francesco Brenti --- Combinatorics of Coxeter Groups.

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I would like to know whether a variation of Corollary 2.2.8 is true. In other words, does the following holds:

For $s\in S$, $t\in T$, $s\neq t$, $w<sw<stw\implies tw<stw$.

Where $<$ is the Bruhat ordering. Note that $sw<stw\implies w<tw$.

The assertion is true under the assumption $\ell(tw)=\ell(w)+1$ by the proof of Corollary 2.2.8.

A proof of Corollary 2.2.8 can be found on James E. Humphreys---Reflection Groups and Coxeter Groups (Lemma 5.11).


1 Answer 1


No, it is not true. Take $\mathfrak{sl}_3$ with simple reflections $r, s$. Then the longest element $t := rsr=srs$ is also a reflection. Put $w=e$.

Then $w=e < sw=s < stw = rs$, but $tw = srs \nless stw = rs$.


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