I'm interested in finite Iwahori-Hecke algebras.

If $\mathcal{H}$ is such a Hecke algebra, defined over $\mathbb{Z}[q^{\pm 1/2}]$, and $\Lambda$ an irreductible representation, there is the notion of a *Schur element* $S_\Lambda$. Roughly speaking, to $\Lambda$ you can associate a central element whose matrix in $\Lambda$ is given by $S_\Lambda$ times the identity.

I computed several of these Schur elements in type $A_n$ and I noticed a remarkable positivity property: all the coefficients in $q$ seem to be positive (see below)! I looked up in the literature, but I don't find explicitely this property.

*Is there a concrete reference for this positivity? Does it hold in other types?*

Here a small list of Schur elements:

- For $\mathfrak{sl}_2$: $1+q$, $1+q^{-1}$
- For $\mathfrak{sl}_3$: $q+1+q^{-1}$, $1+2q+2q^2+q^3$, $1+2q^{-1}+2q^{-2}+q^{-3}$
- For $\mathfrak{sl}_4$: $[4]!$ (quantum factorial), $q^2+3q+4+3q^{-1}+q^{-2}$, $q^{-1}+2+2q+2q^2+q^3$, ...

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