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Let W be the Weyl a group of a semisimple simply connected group over C.

Let I={1,...,r} the set of simple roots.

For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple reflexions that appear in a reduced decomposition of w.

Let w an element such that supp(w)=I and length(w)>r+1, is it true that there exist an element w' such that

1/ $w'\leq w$

2/length (w')=r+1

3/supp(w')=I

?

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I guess the answers make it clear that your question is really about arbitrary Coxeter groups (with a finite generating set)? In any case, I've added a tag. But on the other hand, the question itself doesn't concern representation theory or algebraic geometry. –  Jim Humphreys Oct 29 '12 at 15:02

2 Answers 2

up vote 5 down vote accepted

I agree with the answer below, but to turn it into a rigorous proof one could argue in a slightly different fashion:

Let $l$ be the length of $w$ and let $Red(w)$ be the set of $all$ reduced expressions for $w$. Given ${\bf r}=(i_1,\ldots, i_l)\in Red(w)$ denote by $k({\bf r})$ the smallest $k\le l$ such that $i_k$ appeas in $\bf r$ more than once. Let ${\bf t}=(j_1,\ldots, j_l)\in Red(w)$ be such that $k({\bf t})\ge k({\bf r})$ for all ${\bf r}\in Red(w)$. Write $w=s_{j_1}\cdots s_{j_l}$ and let $w'$ be the element of $W$ obtained from $w$ by deleting $s_{j_k}$ from the above expression where $k=k({\bf t})$. By the choice of $k$ any reduced expression for $w'$ will have full support. Repeat this step until the length decreases to $r+1$.

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@Sasha: This looks efficient, though $r$ in the question was the rank. (Too many letters used in this subject.) –  Jim Humphreys Oct 29 '12 at 14:59

Yes. Choose a reduced expression $red(w)$ for $w$. Now delete from $red(w)$ the rightmost letter $s_i$ that appears more than once in $red(w)$. This will yield a reduced expression for an element $v\in W$ with the same support as $ w$ and length one less than $w$, with $v$ less than $w$ in Bruhat order. Repeat until you have decreased the length to $r+1$.

Added later: To prove that my claim holds for an arbitrary reduced expression, rather than making a special choice, one can use the exchange axiom for Coxeter groups as follows. Suppose the new expression obtained by deleting $s_i$ is not reduced, then consider the leftmost letter $s_{i_t}$ to the right of $s_i$ such that $s_{i_1} \cdots \hat{s_i} \cdots s_{i_t}$ is nonreduced. Now consider the rightmost $k$ so that $s_{i_k}\cdots \hat{s_i} \cdots s_{i_t}$ is nonreduced. Then $s_{i_k}\cdots \hat{s_i} \cdots s_{i_{t-1}}$ and $s_{i_{k+1}}\cdots \hat{s_i} \cdots s_{i_t}$ will be reduced expressions for the same Coxeter group element. This implies that the former expression must contain the letter $s_{i_t}$, a contradiction to $s_{i_t}$ only appearing once in $red(w)$.

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I also seem to have double-used the variable $r$, so just corrected this silly notational error. –  Patricia Hersh Oct 29 '12 at 16:34
    
@Patricia: A former student of Mike Artin told me that he liked the term "local notation" when re-assigning previously defined notation during a lecture. (P.S I hope you and Sasha haven't done some graduate student's homework problem.) –  Jim Humphreys Oct 30 '12 at 12:25
    
@Jim: that's a good point. It was easy for me to imagine someone might need this in a lemma in a paper, but it would have been better if I'd asked the OP first in a comment what the context of the question was rather than answering straight away. I've heard of one grad student getting in a lot of trouble for asking homework questions here at MO -- so many people look at this site regularly that I would think there'd be a serious danger for such a student of getting caught. –  Patricia Hersh Nov 1 '12 at 13:20

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