# Positivity of Schur elements in Iwahori-Hecke algebras

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), $$q^2+3q+4+3q^{-1}+q^{-2}$$, $$q^{-1}+2+2q+2q^2+q^3$$, ...
• How does one compute these? Is there a formula, or similar that you can reference? Mar 11, 2021 at 19:00
• The representation $\Lambda$ is given by a left cell (at least in case $A_n$). Let $(C_w)$ denote the Kazhdan-Lusztig basis and let $(C^w)$ denote its dual basis, such that $tr (C_wC^v) = \delta_w^v$ where you use the standard trace. Now, for a given left cell $\Lambda$, the element $Z_\Lambda=\sum_{x\in \Lambda}C_xC^x$ is central in $\mathcal{H}$. Its action on $\Lambda$ gives the Schuer element $S_\Lambda$. There is also a formula, given in Neunhöffers article sciencedirect.com/science/article/pii/S0021869306001955, Definition 5.1. Mar 12, 2021 at 8:42
• Of course you are no doubt aware of this, but there are many positivity phenomena in Kazhdan-Lusztig theory (e.g. arxiv.org/abs/1212.0791). Mar 12, 2021 at 13:22
• In type A, the Schur element is a q-deformation of the product of the hook lengths. I would then naturally expect the Schur element to be the product of the corresponding quantum integers. Mar 13, 2021 at 4:54
• @PeterMcNamara: This is great! It seems indeed that the Schur elements are the produt of the quantum integers of the hook lengths. Is there a similar interpretation in other types? Mar 13, 2021 at 16:54

The Schur elements are positive in all classical types $$A_n, B_n$$ and $$D_n$$ (use the formula of Theorem 4.3. in https://hal.archives-ouvertes.fr/tel-01411063/document, which uses generalized hook lengths). For dihedral groups $$I_2(m)$$, Theorem 8.3.4. in the book of Geck-Pfeiffer on characters of finite Coxeter groups and Hecke algebras shows that the Schur elements have negative coefficients for $$m>4$$. For the exceptional cases, by brute force, there are Schur elements with negative coefficients only in type $$H_4, F_4$$ and $$E_8$$.