Charles Eaton: Created page with "Let $G$ be a finite group and $B$ a block of $kG$ or $\mathcal{O}$ . Let $(D,b_D)$ be a maximal $B$ -subpair, a..."

2018-12-19T09:41:12Z

Created page with "Let <math>G</math> be a finite group and <math>B</math> a block of <math>kG</math> or <math>\mathcal{O}</math>. Let <math>(D,b_D)</math> be a maximal <math>B</math>-subpair, a..."

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Let <math>G</math> be a finite group and <math>B</math> a block of <math>kG</math> or <math>\mathcal{O}</math>. Let <math>(D,b_D)</math> be a maximal <math>B</math>-subpair, and write <math>\mathcal{F}_{(D,b_D)}(G,B)</math> for the associated [[Glossary#fusion system|fusion system]]. Then <math>B</math> is ''nilpotent'' if <math>\mathcal{F}_{(D,b_D)}(G,B)=\mathcal{F}_D(D)</math>. Equivalently, <math>B</math> is nilpotent if and only if for each <math>B</math>-subpair <math>(Q,b_Q)</math> we have that <math>N_G(Q,b_Q)/C_G(Q)</math> is a <math>p</math>-group. Nilpotent blocks were introduced by Broué and Puig in [[References#B|[BP80]]], and the main results regarding Morita equivalence proved in [[References#P|[Pu88]]]. See [[References#L|[Li18d, 8.11]]] for a treatment of nilpotent blocks, including proofs of all of the following results.

*A block <math>B</math> of <math>kG</math> is nilpotent if and only if the corresponding block of <math>\mathcal{O}G</math> is.
*The principal block is nilpotent if and only if <math>G</math> is <math>p</math>-nilpotent, that is, if <math>G=O_{p'}(G)P</math> for <math>P</math> a Sylow <math>p</math>-subgroup of <math>G</math>.<ref>By a well-known theorem of Frobenius, for which see [[References#L|[Li18d,8,11,7]]].</ref>
*Suppose that <math>D</math> is abelian. Then <math>B</math> is nilpotent if and only if <math>N_G(D,b_D)/C_G(D)=1</math>.
*A block <math>B</math> of <math>\mathcal{O}G</math> is nilpotent if and only it is Morita equivalent to <math>\mathcal{O}D</math>.<ref>See Theorem 8.2 of [[References#P|[Pu99]]].</ref>
*If <math>B</math> is nilpotent, then it is basic Morita equivalent to <math>kD</math> (or <math>\mathcal{O}D</math> as appropriate).<ref> That nilpotent blocks are Morita equivalent to the unique block of a defect group was originally proved in [[References#P|[Pu88]]].</ref> Consequently, a nilpotent block satisfies <math>l(B)=1</math> and its decomposition and Cartan matrices are determined.
*<math>B</math> has [[Glossary#Source algebra|source algebra]] of the form <math>S \otimes_\mathcal{O} \mathcal{O}D</math> for <math>S={\rm End}_\mathcal{O}(V)</math> for some indecomposable endopermutation <math>\mathcal{O}D</math>-module <math>V</math> with vertex <math>D</math> and determinant one.<ref>By [[References#P|[Pu88]]]. See also [[References#L|[Li18d,8.11.5]]]. Note that this does not in general imply a source algebra equivalence.</ref>
*If the principal block is nilpotent, then it is source algebra equivalent to <math>kP</math>.

== Notes ==

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Nilpotent blocks - Revision history

Charles Eaton: Created page with "Let G be a finite group and B a block of kG or \mathcal{O}. Let (D,b_D) be a maximal B-subpair, a..."

Charles Eaton: Created page with "Let $G$ be a finite group and $B$ a block of $kG$ or $\mathcal{O}$ . Let $(D,b_D)$ be a maximal $B$ -subpair, a..."