Frobenius Categories versus Brauer Blocks pp 151-177 | Cite as

# The nilcentralized chain *k*^{*}-functor of a block

## Abstract

11.1 Let us come back to our setting in chapter 7. Let *G* be a finite group, *b* a block of *G*, (*P, e*) a maximal Brauer (*b, G*)-pair and F_{(b,G)} the associated Frobenius *P*-category, and let us denote by (F_{(b,G)})^{nc} the full subcategory of F_{(b,G)} over the set of nilcentralized objects. In chapter 7 we have seen that, for any F_{(b,G)}-nilcentralized subgroup *Q* of *P*, the group F_{(b,G)}(*Q*) can be *canonically* lifted to a *k*^{*}-group \(
\hat F_{(b,G)} (Q)\)
. In this chapter, we discuss the possibility of lifting *whole* subcategories of (F_{(b,G)})^{nc} to “*k*^{*}-categories”†. These results allow us to construct a suitable functor \(
\widehat{\mathfrak{a}\mathfrak{u}\mathfrak{t}}_{(F_{(b,G)} )^{nc} } \)
from the *proper category of chains* of (F_{(b,G)})^{nc} (cf. A2.8) to the category of *finite k*^{*}-*groups* *k*^{*}-Gr, which lifts the *automorphism functor* \(
\mathfrak{a}\mathfrak{u}\mathfrak{t}_{(F_{(b,G)} )^{nc} } \)
introduced in Proposition A2.10 below. This is the main tool for the definition of the *Grothendieck group of the pair* \(
(F_{(b,G)} , \widehat{\mathfrak{a}\mathfrak{u}\mathfrak{t}}_{(F_{(b,G)} )^{nc} } )\)
in chapter 14.

## Keywords

Full Subcategory Grothendieck Group Group Isomorphism Primitive Idempotent Dade Group## Preview

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