rt.representation theory - Signed and unsigned Hecke algebra canonical basis

You probably know all of this already, but here goes...

Write $C'_w = T_w + sum_{x < w} p_{x,w} T_x$ where $p_{x,w} in umathbb{Z}[u]$. Now, the other basis can be defined by applying the involutive automorphism $b: mathcal{H}_n to mathcal{H}_n$, given by $b(T_w)=T_w$ and $b(u)=-u^{-1}$.

Define $C_w := b(C'_w)$.

Since, $b$ commutes with the bar involution, this basis is bar invariant as well.

Explicitly, $C_w = T_w + sum_{x < w} (-1)^{ell(w)+ell(x)} bar p_{x,w} T_x$.

So $C_w = bar{P}^{-1} P C'_w$ which seems hard to compute in general.

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