On Brauer indecomposability of Scott modules of Park-type groups

We give a sufficient condition for the kG-Scott module with vertex P to remain indecomposable under taking the Brauer construction for any subgroup Q of P as k[Q C G (Q)]-module, where k is a field of characteristic 2, and P is a wreathed 2subgroup of a finite group G. This generalizes results for the cases where P is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a p-permutation bimodule (p is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method that then can possibly lift to a splendid derived equivalence.

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“…Then N P (Q) = C P (Q), so that N ∆P (∆Q) = C ∆P (∆Q), which implies from (28) that N G (∆Q) = C G (∆Q). Hence (26) automatically yields (27).…”

Section: The Proof Of the Main Theoremmentioning

confidence: 86%

“…Therefore Apparently, (26) holds for the case Q = P 0 . Now, our final aim is to prove that (27) Res…”

Section: The Proof Of the Main Theoremmentioning

confidence: 91%

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

Koshitani

Tuvay

2018

Algebr Represent Theor

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“…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.…”

Section: Introduction and Notationmentioning

confidence: 89%

Brauer indecomposability of Scott modules with semidihedral vertex

Koshitani¹,

Tuvay²

2021

Proceedings of the Edinburgh Mathematical Society

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We present a sufficient condition for the $kG$ -Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$ -module, where $k$ is a field of characteristic $2$ , and $P$ is a semidihedral $2$ -subgroup of a finite group $G$ . This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a $p$ -permutation bimodule (where $p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.

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“…The goal of this paper is to prove that the Scott module whose vertex is isomorphic to a direct product of a generalized quaternion 2-group and a cyclic 2-group is Brauer indecomposable. The Brauer indecomposability of Scott modules is an important notion because it serves a key ingredient for the Scott module to realize a splendid Morita equivalence between certain principal blocks with isomorphic defect groups (see [11][12][13][14][15][16][17]22]).…”

Section: Introductionmentioning

confidence: 99%