Isomorphisms of subcategories of fusion systems of blocks and Clifford theory

Abstract

Abstract Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that ( k ⁢ N ) ⁢ c {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a ( G , N , c ) {(G,N,c)} -Brauer pair ( R , f R ) {(R,f_{R})} consists of a p-subgroup R of G and a block f R {f_{R}} of ( k ⁢ C N ⁢ ( R ) ) {(kC_{N}(R))} . If Q is a defect group of c and f Q ∈ 𝐵 ⁢ ℓ ⁡ ( k ⁢ C N ⁢ ( Q ) ) {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))} , then ( Q , f Q ) {(Q,f_{Q})} is a ( G , N , c ) {(G,N,c)} -Brauer pair. The ( G , N , c ) {(G,N,c)} -Brauer pairs form a (finite) poset. Set H = N G ⁢ ( Q , f Q ) {H=N_{G}(Q,f_{Q})} so that ( Q , f Q ) {(Q,f_{Q})} is an ( H , C N ⁢ ( Q ) , f Q ) {(H,C_{N}(Q),f_{Q})} -Brauer pair. We extend Lemma 8.6.4 of the above book to show that if ( U , f U ) {(U,f_{U})} is a maximal ( G , N , c ) {(G,N,c)} -Brauer pair containing ( Q , f Q ) {(Q,f_{Q})} , then ( U , f U ) {(U,f_{U})} is a maximal ( H , C N ⁢ ( c ) , f Q ) {(H,C_{N}(c),f_{Q})} -Brauer pair containing ( Q , f Q ) {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of ℱ ( U , f U ) ⁢ ( G , N , c ) {\mathcal{F}_{(U,f_{U})}(G,N,c)} and ℱ ( U , f U ) ⁢ ( H , C N ⁢ ( Q ) , f Q ) {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including ( Q , f Q ) {(Q,f_{Q})} and ( U , f U ) {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.