I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E\_i$ and $F\_i$. There are good geometric reasons the coefficents have to be positive in simply-laced situations, but no such arguments can work for non-simply laced examples. However, this is quite a bit weaker than knowing the result is false.



> 
> Does anyone have a good example or reference for a situation where this positivity fails?
> 
> 
> 



When does Lusztig's canonical basis have non-positive structure coefficients?
Hi, 


The following formulas are examples of non-positive structure coefficients
for non-symmetric cases which are easily verified by the algorithm presented
in Leclerc's paper "Dual Canonical Bases, Quantum Shuffles, and q-characters" 
or quagroup package in GAP4.


Professor Masaki Kashiwara told me that he has known such non-positive 
structure coefficient for $G\_2$ since Shigenori Yamane found it in 1994
as treated in his master thesis at Osaka University (written in Japanese).
You can see similar negative coefficients in at least case $A\_{2n}^{(2)}, D\_{n+1}^{(2)}$.
Anyway, conjecture 52 in Leclerc's paper is false 
(I already told Professor Leclerc about it).


Shunsuke Tsuchioka


Notation: $G(i\_1,\cdots,i\_n)$ stands for the canonical basis element
corresponds to a crystal element 
$b(i\_1,\cdots,i\_n)=\tilde{f}\_{i\_n}b(i\_1,\cdots,i\_{n-1})=\cdots$. 


$G\_2$ (1 is the short root) : 
$f\_2 G(121112211)
= G(1211122211)
 + [2]G(1111222211)
 + G(2111112221)$


$ + [2]G(1211112221)
 + G(1111122221)
 - G(1112211122)
 + [2]G(1122111122)$


$C\_3$ (1,2 are short roots) :
$f\_3 G(23122312)
= [2]G(222333121)
 + [2]G(312222331)
 + [2]G(231222331)$


$ + [2]G(122223331)
 + G(231223312)
 + [2]G(122233312)
 - G(223112233)
 + [2]G(231122233)$


$B\_4$ (1,2,3 are long roots) :
$f\_1G(4342341234)
= [2]G(43344423211)
 + [2]G(43423443211)
 - G(44233443211)$


$ + [2]G(43423344211)
 + [2]G(43423442311)
 + [2]G(34234442311)$


$ + [2]G(43422334411)
 + G(43423412341)$


