The freeness and trace conjectures for parabolic Hecke subalgebras

Eirini Chavli (1) , Maria Chlouveraki (2)

1 Universität Stuttgart [Stuttgart]
2 LMV - Laboratoire de Mathématiques de Versailles

The two most fundamental conjectures on the structure of the generic Hecke algebra H(W) associated with a complex reflection group W state that H(W) is a free module of rank |W | over its ring of definition, and that 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 H(W') associated with a parabolic subgroup W' of W state that H(W) is a free left and right H(W')-module of rank |W |/|W'|, and that the canonical symmetrising trace of H(W') is the restriction of the canonical symmetrising trace of H(W) to H(W'). 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.

Domaines: Mathématiques [math]

Eirini Chavli, Maria Chlouveraki. The freeness and trace conjectures for parabolic Hecke subalgebras. Journal of Pure and Applied Algebra, 2022, 226 (10), pp.107050. ⟨10.1016/j.jpaa.2022.107050⟩. ⟨hal-03592527⟩