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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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    $\begingroup$ How does one compute these? Is there a formula, or similar that you can reference? $\endgroup$
    – Per Alexandersson
    Commented Mar 11, 2021 at 19:00
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    $\begingroup$ 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. $\endgroup$
    – AThomas
    Commented Mar 12, 2021 at 8:42
  • $\begingroup$ 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). $\endgroup$
    – Sam Hopkins
    Commented Mar 12, 2021 at 13:22
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    $\begingroup$ 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. $\endgroup$
    – Peter McNamara
    Commented Mar 13, 2021 at 4:54
  • $\begingroup$ @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? $\endgroup$
    – AThomas
    Commented Mar 13, 2021 at 16:54

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Together with Maria Chlouveraki, we determined the answer:

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 $E_8$.

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  • $\begingroup$ Did there end up being any connection to fake degrees? $\endgroup$
    – Sam Hopkins
    Commented Mar 16, 2021 at 13:55

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