I'm trying to decompose the Kazhdan-Lusztig C' basis element associated to the longest word in $S\_n$, $C'\_{w\_0}$ into products and sums of elements $C'\_w$ where $w < w\_0$ in the Bruhat order. For example I know that $C'\_{s\_1s\_2s\_1}=C'\_{s\_1}C'\_{s\_2}C'\_{s\_1}-C'\_{s\_1}$. 


The question I have is the following: Are such decompositions known as a general fact, or would I have to go through the process of changing the basis back to the standard basis and mapping back to the KL basis after the fact? Of course the latter can be done by hand (or computer for higher n) but I was just curious if there was a better way to approach this that I've overlooked as a non-expert. 


Thanks!



Decomposition of C' Kazhdan-Lusztig basis element associated to longest word in S_n
Is the expansion
 $$C'\_w = C'\_{ws}C'\_s - \sum\_{z\leq ws;\, zs < z} \mu(z,ws) C'\_z$$
for $ws < w$ what you're looking for? ($s$ is a simple reflection.) Here, $\mu(z,ws)$ is the coefficient of $q^{(\ell(ws)-\ell(z)-1)/2}$ in the Kazhdan-Lusztig polynomial $P\_{z,ws}(q)$. Perhaps there is a simplification for the specific case of $w=w\_0$, but I don't know it. For general $w$, a simpler formula would be tantamount to having a handle on the combinatorially elusive $\mu$-coefficients.


