# Nilpotent blocks

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 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 [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 [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].
^{[1]} - 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].
^{[2]} - If [math]B[/math] is nilpotent, then it is basic Morita equivalent to [math]kD[/math] (or [math]\mathcal{O}D[/math] as appropriate).
^{[3]}Consequently, a nilpotent block satisfies [math]l(B)=1[/math] and its decomposition and Cartan matrices are determined. - [math]B[/math] has 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.
^{[4]} - If the principal block is nilpotent, then it is source algebra equivalent to [math]kP[/math].

## Notes

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