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
?
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 $r+1$.
$r$ in the question was the rank. (Too many letters used in this subject.)
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Commented
Oct 29, 2012 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 $red(w)$.
