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Let $W$ be a Weyl group of a semisimple Lie algebra $\mathfrak{g}$. Let $\beta$ be a positive root, $\alpha$ a simple root such that $\beta=w(\alpha)$.

It is a straightforward fact that we can express the reflection through $\beta$ as $$s_\beta=ws_\alpha w^{-1}$$

Question: Does this give a reduced expression for $s_\beta$? If not, is there a nice way to construct reduced expressions for arbitrary reflections in $W$?

SamJeralds
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  • Maybe it helps to remind us (and yourself) what exactly a "reduced expression" is defined as, and in particular, whether that depends on a certain choice (e.g. of simple roots/reflections). – Torsten Schoeneberg Jun 04 '19 at 15:58
  • Not every $w,\alpha$ will work, of course, as there are many. But you can always find some which do. – Christoph Mark Jun 19 '19 at 17:36
  • There is a construction, often called the "root sequence", which carries a lot of useful information. For an expression $s_1 s_2 \ldots s_n$, it is often defined as $\alpha_n$, $s_n (\alpha_{n-1})$, $s_n s_{n-1}(\alpha_{n-2})$, etc. It is the order in which $w$ sends roots negative. Variations in order and whether to use roots or reflections exist in the literature. Regardless, the root sequence identifies pairs of generators you can apply the Deletion Condition to as pairs of roots $\alpha$,$-\alpha$ in the sequence. This is fully general. – Hugh Denoncourt Jul 19 '19 at 13:09
  • Since it is fully general, it doesn't take advantage of an element being a reflection. It's just one of many general ways to keep applying the Deletion Condition until you get a reduced expression. Does something like that interest you, or are you looking for something to take advantage of the fact that your expression is specifically a reflection? – Hugh Denoncourt Jul 19 '19 at 13:11

1 Answers1

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This doesn't always give a reduced expression: for example, in Type $A_2$ writing $s_1$ and $s_2$ for the simple reflections, we have $s_1 = (s_1s_2s_1)s_2(s_1s_2s_1)$, which is not a reduced expression.

But given $\beta$, one can make a choice of $\alpha$ and $w$ such that $s_\beta = ws_\alpha w^{-1}$ is a reduced expression. This is explained in Lemma 2.7 of "Reflection Subgroups of Coxeter Systems" by Matthew Dyer. It is also known that every reduced word for any reflection arises in this way; this is shown in Proposition 2.4 of "Quasiminiscule Quotients and Reduced Words for Reflections" by John Stembridge.

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