Splendid Morita equivalences for principal 2-blocks with dihedral defect groups

Abstract

Given a dihedral 2-group P of order at least 8, we classify the splendid Morita equivalence classes of principal 2-blocks with defect groups isomorphic to P . To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.

Splendid Morita equivalences for principal blocks with generalised quaternion defect groups

“…Theorem 2.1 ([CEKL11] and [KL17]). Let G be an arbitrary finite group with a dihedral Sylow 2-subgroup P = D 2 n of order 2 n with n ≥ 2.…”

“…In [KL17] the proof of the statements for n ≥ 3 partly relies on the Brauer indecomposability of the Scott module Sc(G × H, ∆P ) inducing the splendid Morita equivalence. Passing to generalised quaternion Sylow 2-subgroup we won't need to use arguments involving Brauer indecomposability, however we note that the other way around a Scott module realising a splendid Morita equivalence is necessarily Brauer indecomposable.…”

Brauer indecomposability of Scott modules with semidihedral vertex

“…These theorems in a sense generalize [11][12][13][14], and there are results on Brauer indecomposability of Scott modules also in [15,21]. Notation 1.3.…”

The Brauer Indecomposability of Scott Modules for the Quadratic Group Qd(p)

“…We claim here that By (13), the Mackey formula and (15), (∆Q) )(k) = Ind C C∩N ∆P (∆Q) (k) = Ind C C ∆P (∆Q) (k). Hence, by noting that C P (Q) = P 0 by the assumption (I), we have (16) Res…”

“…Thus, (14) and (16), so that X | Ind C ∆P 0 (k). So, let us restrict these kC-modules to P 0 × P 0 .…”