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


?



elements in the weyl group
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 $r+1$.


