Beyond a question of Markus Linckelmann

TL;DR

This paper proves the existence and uniqueness of a regular central k*-extension for folded Frobenius P-categories, confirming a conjecture related to Brauer pairs and automizer groups in block theory.

Contribution

It establishes the existence and uniqueness of a regular central k*-extension for any folded Frobenius P-category, resolving a conjecture from the 2002 Durham Symposium.

Findings

Confirmed the existence of the k*-extension.

Proved the uniqueness of the k*-extension.

Connected the extension to folder structures in Frobenius categories.

Abstract

In the 2002 Durham Symposium, Markus Linckelmann [1] conjectured the existence of a regular central k*-extension of the full subcategory over the selfcentralizing Brauer pairs of the Frobenius P-category F_{(b,G)} associated with a block b of defect group P of a finite group G, which would include, as k*-automorphism groups of the objects, the k*-groups associated with the automizers of the corresponding selfcentralizing Brauer (b,G)-pairs, introduced in [3, 6.6]; as a matter of fact, in this question the selfcentralizing Brauer pairs can be replaced by the nilcentralized Brauer pairs, still getting a positive answer. But the condition on the k*-automorphism groups of the objects is not precise enough to guarantee the uniqueness of a solution, as showed by Sejong Park in [2, Theorem 1.3]. This uniqueness depends on the folder structure [5,~Section~2] associated with F_{(b,G)} in [4,~Theorem~11.32], and here we prove the existence and the uniqueness of such a regular central k*-extension for any folded Frobenius P-category.