The freeness and trace conjectures for parabolic Hecke subalgebras

Abstract

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 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 ′ ). We prove the last two conjectures for several exceptional irreducible complex reflection groups of rank 2.

The principal block of a $\mathbb{Z}_\ell$-spets and Yokonuma type algebras

“…Note that the assertion that (2) holds is referred to as the parabolic trace conjecture in [11]. Strong symmetry is known to be satisfied in many cases; here we use the Shephard-Todd notation for irreducible complex reflection groups: Proof.…”

Defect in cyclotomic Hecke algebras

“…• the cyclic groups G(l, 1, 1) [ChCh,Proposition 4.2]. Nevertheless, the Schur elements with respect to the canonical symmetrising trace have been determined for all complex reflection groups.…”

Centers of Hecke Algebras of Complex Reflection Groups

“…Apart from the real case, the BMM symmetrising trace conjecture holds for a few exceptional groups and for the infinite family (detailed references can be found in [6,Conjecture 3.3]).…”

“…, |W/W ′ |} is a basis of H as H ′ -module. Since H is a free H ′ -module of dimension |W/W ′ | (see, for example, [6]), we only have to prove that {x i } is a spanning set for H. By construction, we always have x 1 = 1 and, hence, it is enough to prove that for each x i , the elements x i .σ are linear combinations of the form i h i • x i , where h i ∈ H ′ .…”