Consider the Hecke algebra $H\_n$ of type $A\_{n-1}$ with standard basis $T\_w$, $w \in S\_n$ with the quadratic relations $(T\_s - u) (T\_s + u^{-1}) = 0$ and braid relations. The unsigned canonical basis $C'\_w$, $w \in S\_n$ gives rise to a basis for the irreducible $H\_n$-module $M\_\lambda$ of shape $\lambda$: fix some SYT $T$ of shape $\lambda$; then this basis is {$C'\_w : P(w) = T$} after quotienting by cells lower down in the Kazhdan-Lusztig preorder and $P(w)$ is the insertion tableau of $w$.


Similarly, $M\_\lambda$ has a basis coming from the signed canonical basis $C\_w$, $w \in S\_n$.



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> What is known about the transition matrix between these two bases? Does it become the identity matrix at $u=0$? 
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This seems to be trickier to understand than the transition matrix between all the $C$s and all the $C'$s. I expect I will be able to prove the second question on my own, but I'd rather cite it if it's in the literature somewhere.



Signed and unsigned Hecke algebra canonical basis
Not much seems to be known about this matrix in general, and it does become the identity at $u=0$. I show this in the paper *Quantum Schur-Weyl duality and projected canonical bases* <http://arxiv.org/abs/1102.1453> 
using quantum Schur-Weyl duality and its compatibility with canonical bases. This only proves it in type A. I do not know what happens in other types.


