# Nilpotent blocks

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

• A block $B$ of $kG$ is nilpotent if and only if the corresponding block of $\mathcal{O}G$ is.
• The principal block is nilpotent if and only if $G$ is $p$-nilpotent, that is, if $G=O_{p'}(G)P$ for $P$ a Sylow $p$-subgroup of $G$.[1]
• Suppose that $D$ is abelian. Then $B$ is nilpotent if and only if $N_G(D,b_D)/C_G(D)=1$.
• A block $B$ of $\mathcal{O}G$ is nilpotent if and only it is Morita equivalent to $\mathcal{O}D$.[2]
• If $B$ is nilpotent, then it is basic Morita equivalent to $kD$ (or $\mathcal{O}D$ as appropriate).[3] Consequently, a nilpotent block satisfies $l(B)=1$ and its decomposition and Cartan matrices are determined.
• $B$ has source algebra of the form $S \otimes_\mathcal{O} \mathcal{O}D$ for $S={\rm End}_\mathcal{O}(V)$ for some indecomposable endopermutation $\mathcal{O}D$-module $V$ with vertex $D$ and determinant one.[4]
• If the principal block is nilpotent, then it is source algebra equivalent to $kP$.

## Notes

1. By a well-known theorem of Frobenius, for which see [Li18d,8,11,7].
2. See Theorem 8.2 of [Pu99].
3. That nilpotent blocks are Morita equivalent to the unique block of a defect group was originally proved in [Pu88].
4. By [Pu88]. See also [Li18d,8.11.5]. Note that this does not in general imply a source algebra equivalence.