The freeness and trace conjectures for parabolic Hecke subalgebras

Abstract

The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic Hecke subalgebra $\mathcal{H}(W^{\prime })$ associated with a parabolic subgroup $W^{\prime }$ of $W$ state that $\mathcal{H}(W)$ is a free left and right $\mathcal{H}(W^{\prime })$-module of rank $|W|/|W^{\prime }|$, and that the canonical symmetrising trace of $\mathcal{H}(W^{\prime })$ is the restriction of the canonical symmetrising trace of $\mathcal{H}(W)$ to $\mathcal{H}(W^{\prime })$. Until now, these two conjectures have only be known to be true for real reflection groups. We prove them for all complex reflection groups of rank $2$ for which the BMM symmetrising trace conjecture is known to hold.

arXiv comment24 pages

References in corpus (9)

• Zariski theorems and diagrams for braid groups
• Constructing representations of Hecke algebras for complex reflection groups
• Simple extensions of reflection subgroups of primitive complex reflection groups
• The center of pure complex braid groups
• The BMM symmetrising trace conjecture for groups $G_4,G_5,G_6,G_7,G_8$
• Normalisers of parabolic subgroups in finite unitary reflection groups
• Markov traces on the Birman-Wenzl-Murakami algebras
• New deformations of group algebras of Coxeter groups
• The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G26

Cited by in corpus (3)

• Rational Catalan Numbers for Complex Reflection Groups
• Centers of Hecke Algebras of Complex Reflection Groups
• The principal block of a ${\mathbb{Z}}_{\ell }$-spets and Yokonuma type algebras