Stable functorial equivalence of blocks

Let k be an algebraically closed ﬁeld of characteristic p > 0, let R be a commutative ring and let F be an algebraically closed ﬁeld of characteristic 0. We introduce the category F ∆ Rpp k of stable diagonal p -permutation functors over R . We prove that the category F ∆ F pp k is semisimple and give a parametrization of its simple objects in terms of the simple diagonal p -permutation functors. We also introduce the notion of a stable functorial equivalence over R between blocks of ﬁnite groups. We prove that if G is a ﬁnite group and if b is a block idempotent of kG with an abelian defect group D and Frobenius inertial quotient E , then there exists a stable functorial equivalence over F between the pairs ( G, b ) and ( D ⋊ E, 1).

In [Bouc and Yılmaz 2022] we introduced another equivalence of blocks, namely functorial equivalence, using the notion of diagonal p-permutation functors: Let k be an algebraically closed field of characteristic p > 0, let ‫ކ‬ be an algebraically closed field of characteristic 0 and let R be a commutative ring.We denote by Rpp k the category whose objects are finite groups and for finite groups G and H whose morphisms from H to G are the Grothendieck group RT (G, H ) of diagonal p-permutation (kG, k H )-bimodules.An R-linear functor from Rpp k to R Mod is called a diagonal p-permutation functor.To each pair (G, b) of a finite group G and a block idempotent b of kG, we associate a canonical diagonal p-permutation functor over R, denoted by RT G,b .If (H, c) is another such pair, we say that (G, b) and (H, c) are functorially equivalent over R if the functors RT G,b and RT H,c are isomorphic.
In [Bouc and Yılmaz 2022] we proved that the category of diagonal p-permutation functors over ‫ކ‬ is semisimple, parametrized simple functors and provided three equivalent descriptions of the decomposition of the functor ‫ކ‬T G,b in terms of the simple functors [Bouc and Yılmaz 2022, Corollary 6.15 and Theorem 8.22].We proved that the number of isomorphism classes of simple modules, the number of ordinary irreducible characters, and the defect groups are preserved under functorial equivalences over ‫ކ‬ [Bouc and Yılmaz 2022, Theorem 10.5].Moreover we proved that for a given finite p-group D, there are only finitely many pairs (G, b), where G is a finite group and b is a block idempotent of kG, up to functorial equivalence over ‫ކ‬ [Bouc and Yılmaz 2022, Theorem 10.6] and we provided a sufficient condition for two blocks to be functorially equivalent over ‫ކ‬ in the situation of Broué's abelian defect group conjecture [Bouc and Yılmaz 2022, Theorem 11.1].
In this paper, we introduce the notion of stable diagonal p-permutation functors and stable functorial equivalences.We denote by Rpp k the quotient category of Rpp k by the morphisms that factor through the trivial group.A stable diagonal p-permutation functor over R is an R-linear functor from Rpp k to R Mod, or equivalently, a diagonal p-permutation functor which vanishes at the trivial group.In particular, the simple diagonal p-permutation functors S L ,u,V with L ̸ = 1 are (simple) stable diagonal p-permutation functors.Our first main result is the following.We also consider the blocks with abelian defect groups and Frobenius inertial quotient.(ii) Suppose that E is abelian.Then there exists a functorial equivalence over ‫ކ‬ between (G, b) and (D ⋊ E, 1) if and only if (G, b) and (D ⋊ E, 1) are p-permutation equivalent.
In Section 2 we recall diagonal p-permutation functors and functorial equivalences between blocks.In Section 3 we introduce the category of stable diagonal p-permutation functors and prove Theorem 1.1.In Section 4 we introduce the notion of stable functorial equivalences between blocks and prove Theorem 1.2.Finally, in Section 5 we prove Theorem 1.3.

Preliminaries
(a) Let (P, s) be a pair, where P is a p-group and s is a generator of a p ′ -group acting on P. We write P⟨s⟩ := P ⋊ ⟨s⟩ for the corresponding semidirect product.We say that two pairs (P, s) and (Q, t) are isomorphic and write (P, s) ∼ = (Q, t), if there is a group isomorphism f : P⟨s⟩ → Q⟨t⟩ that sends s to a conjugate of t.We set Aut(P, s) to be the group of the automorphisms of the pair (P, s) and Out(P, s) = Aut(P, s)/Inn(P⟨s⟩).Recall from [Bouc and Yılmaz 2020] that a pair (b) Let G and H be finite groups.We denote by T (G) the Grothendieck group of p-permutation kG-modules and by T (G, H ) the Grothendieck group of ppermutation (kG, k H )-bimodules whose indecomposable direct summands have twisted diagonal vertices.Let Rpp k denote the following category: • objects: finite groups.
• composition is induced from the tensor product of bimodules.
An R-linear functor from Rpp k to R Mod is called a diagonal p-permutation functor over R. Together with natural transformations, diagonal p-permutation functors form an abelian category F Rpp k .
(c) Recall from [Bouc and Yılmaz 2022] that the category F ‫ކ‬ pp k is semisimple.Moreover, the simple diagonal p-permutation functors, up to isomorphism, are parametrized by the isomorphism classes of triples (L , u, V ), where (L , u) is a D -pair, and V is a simple ‫ކ‬Out(L , u)-module (see [loc.cit., Sections 6 and 7] for more details on simple functors).
(d) Let G be a finite group and b a block idempotent of kG.Recall from [loc.cit.] that the block diagonal p-permutation functor RT G,b is defined as

Stable diagonal p-permutation functors
In this section we introduce the category of stable diagonal p-permutation functors.
For a finite group G, let P(G) denote the subgroup of T (G) generated by the indecomposable projective kG-modules.Let also T (G) denote the quotient group T (G)/P(G).For X ∈ T (G), we denote by X the image of X in T (G).If H is another finite group, we define P(G, H ) and T (G, H ) similarly.
Lemma 3.1.For finite groups G and H one has P(G, H Proof.This follows from the fact that the projective indecomposable k(G × H )modules are of the form P ⊗ k Q, where P and Q are projective indecomposable kG and k H -modules, respectively.□ Definition 3.2.Let Rpp k denote the following category: • objects: finite groups.
• composition is induced from the tensor product of bimodules.
Then F is a diagonal p-permutation functor, equal to the quotient of F by the subfunctor generated by F(1).Obviously, F vanishes at the trivial group, so it is a stable diagonal p-permutation functor.The functor : F → F is a left adjoint to the above functor .In particular, Moreover S L ,u,V (1) = 0 for L ̸ = 1, and hence ‫ކ‬T (1, G) = I(1, G).Therefore, Since the ‫-ކ‬dimensions of ‫ކ‬Proj(H, G) and I(H, G) are the same for any finite group H , it follows that ‫ކ‬Proj(−, G) = I(−, G). □ Proof of Theorem 1.1.For a finite group G, the representable diagonal p-permutation functor ‫ކ‬T (−, G) decomposes as a direct sum of simple functors S L ,u,V , and hence we have for some nonnegative integers m L ,u,V , where (L , u, V ) runs over a set of isomorphism classes of D -pairs (L , u) with L ̸ = 1, and simple ‫ކ‬Out(L , u)-modules V .By Lemma 3.5, the representable stable diagonal p-permutation functor ‫ކ‬T (−, G) is isomorphic to the direct sum of simple diagonal p-permutation functors, and each of these simple functors is a simple stable diagonal p-permutation functor.Since the functor category F ‫ކ‬ pp k is generated by the representable functors the result follows.□

Stable functorial equivalences
Let G and H be finite groups.Suppose that E acts freely on D\{1}.This condition is equivalent to requiring D⋊ E be a Frobenius group.Let F b be the fusion system of b with respect to (D, e D ).
Then F b is equal to the fusion system F D⋊E (D) on D determined by D ⋊ E. Under these assumptions, we know from [Linckelmann 2018, Theorem 10.5.1] that there is a stable equivalence of Morita type between (G, b) and (D ⋊ E, 1), induced by a bimodule with an endopermutation source.Since D is abelian, by [Rickard 1996, Theorem 7.2] this bimodule admits an endosplit p-permutation resolution, which yields a stable p-permutation equivalence between (G, b) and (D ⋊ E, 1); see, for instance, [Linckelmann 2018, Remark 9.11.7].In particular (G, b) and (D ⋊ E, 1) are stably functorially equivalent.
Hereafter we give an alternative proof of the existence of this stable functorial equivalence, which relies only on Theorem 1.1.Together with Theorem 1.2, this now implies Part (i) of Theorem 1.3.Part (ii) of Theorem 1.3 follows from Part (i) and [Linckelmann 2018, Theorem 10.5.10].
(b) Let S L ,u,V be a simple diagonal p-permutation functor such that L is nontrivial and isomorphic to a subgroup of D. Recall that by [Bouc and Yılmaz 2022, Theorem 8.22] the multiplicity of S L ,u,V in ‫ކ‬T G,b is equal to the ‫-ކ‬dimension of (P,e P )∈[F b ] π∈[N G (P,e P )\P (P,e P ) (L ,u)/Aut(L ,u)] ‫ކ‬Proj(ke P C G (P), u) ⊗ Aut(L ,u) (P,e P ,π) V, where [F b ] denotes a set of isomorphism classes of objects in F b , P (P,e P ) is the set of group isomorphisms π : L → P with πi u π −1 ∈ Aut F b (P, e P ), and Aut(L , u) (P,e P ,π ) is the stabilizer in Aut(L , u) of the G-orbit of (P, e P , π ).Since b is a block with Frobenius inertial quotient, the block kC G (P)e P is nilpotent for every nontrivial subgroup P of D; see, for instance, [Linckelmann 2018, Theorem 10.5.2].Therefore, we have l(ke P C G (P)) = 1, and hence the multiplicity formula reduces to (P,e P )∈[F b ] π∈[N G (P,e P )\P (P,e P ) (L ,u)/Aut(L ,u)]
Let Q D⋊E, p denote the set of pairs (P, s) of p-subgroups P of D ⋊ E and p ′ -elements s of N D⋊E (P).Let also [Q D⋊E, p ] denote a set of representatives of D ⋊ E-orbits on Q D⋊E, p under the conjugation map.Recall from [Bouc and Yılmaz 2022, Corollary 7.4] that the multiplicity of S L ,u,V in ‫ކ‬T D⋊E is equal to the ‫-ކ‬dimension of (P,s)∈[Q D⋊E, p ] ( P,s) ∼ =(L ,u) V N D⋊E (P,s) , where for a pair (P, s) ∈ Q D⋊E, p with ( P, s) ∼ = (L , u), we fix an isomorphism φ P,s : L → P with φ P,s ( u l) = s φ P,s (l) for all l ∈ L and we view V as an ‫ކ‬N D⋊E (P, s)-module via the group homomorphism (1) that sends g ∈ N G (P, s) to the image of φ −1 P,s • i g • φ P,s in Out(L , u).(c) Let P b (G, L , u) denote the set of triples (P, e, π ) where (P, e) ∈ F b and π ∈ P (P,e P ) (L , u).Let also Q D⋊E, p (L , u) denote the set of pairs (P, s) in Q D⋊E, p with the property that ( P, s) ∼ = (L , u).
If (P, e, π) ∈ P b (G, L , u), then πi u π −1 ∈ Aut F b (P, e P ) by definition and since F b is equal to F D⋊E (D), it follows that there exists a p ′ -element s of N D⋊E (P) with πi u π −1 = i s .This implies by [Bouc and Yılmaz 2022, Lemma 3.3] that ( P, s) ∼ = (L , u) and therefore we have a map L , u), (P, e, π ) → (P, s).
Again, there exists g ∈ G such that i g = i h : P → Q. Define Proof.We have πi u π −1 = i s and hence the image of N D⋊E (P, s) is given by Note that since h s = s, we have i h i s = i s i h , i.e., i h πi u π −1 = πi u π −1 i h .Therefore the image is {π −1 i h π | h ∈ D ⋊ E, i h : P → P, h s = s} = {π −1 i g π | g ∈ N G (P, e), i g πi u π −1 = πi u π −1 i g } = {π −1 i g π ∈ Out(L , u) | π −1 i g π = i g , g ∈ N G (P, e)} = Out(L , u) (P,e,π ) as was to be shown.□ Proof of Theorem 1.3.We need to show that for any L ̸ = 1, the multiplicities of a simple diagonal p-permutation functor S L ,u,V in ‫ކ‬T G,b and in ‫ކ‬T D⋊E are equal.But this follows from Lemmas 5.1 and 5.2.Now Part (i) follows from Theorem 1.2(i), and Part (ii) follows from [Linckelmann 2018, Theorem 10.5.10].□

Theorem 1. 1 .
The category F ‫ކ‬ pp k of stable diagonal p-permutation functors over ‫ކ‬ is semisimple.The simple stable diagonal p-permutation functors are precisely the simple diagonal p-permutation functors S L ,u,V with L ̸ = 1.Given a finite group G and a block idempotent b of kG, we define a stable diagonal p-permutation functor RT G,b similar to RT G,b ; see Definition 4.1.Note that RT G,b is the zero functor if and only if b has defect 0. We say that two pairs (G, b) and (H, c) are stably functorially equivalent over R if the functors RT G,b and RT H,c are isomorphic.For a block algebra kGb, let k(kGb) and l(kGb) denote the number of irreducible ordinary characters and the number of irreducible Brauer characters of b, respectively.Theorem 1.2.Let b be a block idempotent of kG and let c be a block idempotent of k H . (i) The pairs (G, b) and (H, c) are stably functorially equivalent over ‫ކ‬ if and only if the multiplicities of S L ,u,V in ‫ކ‬T G,b and ‫ކ‬T H,c are the same for any simple diagonal p-permutation functor S L ,u,V with L ̸ = 1.In this case, (G, b) and (H, c) are functorially equivalent over ‫ކ‬ if and only if l(kGb) = l(k H c). (ii) If the pairs (G, b) and (H, c) are stably functorially equivalent over ‫,ކ‬ then b and c have isomorphic defect groups and one has k(kGb) − l(kGb) = k(k H c) − l(k H c).
Theorem 1.3.Let G be a finite group, b a block idempotent of kG with a nontrivial abelian defect group D. Let E = N G (D, e D )/C G (D) denote the inertial quotient of b.Suppose that E acts freely on D \ {1}.Then: (i) There exists a functorial equivalence over ‫ކ‬ between (G, b) and (D ⋊ E, 1) if and only if l(kGb) = l(k(D ⋊ E)).
See [loc.cit., Section 8] for the decomposition of ‫ކ‬T G,b in terms of the simple functors S L ,u,V .(e) Let b be a block idempotent of kG and let c be a block idempotent of k H .We say that the pairs (G, b) and (H, c) are functorially equivalent over R, if the corresponding diagonal p-permutation functors RT G,b and RT H,c are isomorphic in F Rpp k [loc.cit., Definition 10.1].By [loc.cit., Lemma 10.2] the pairs (G, b) and (H, c) are functorially equivalent over R if and only if there exists ω ∈ b RT (G, H )c and σ ∈ c RT (H, G)b such that Rpp k is a reflective subcategory of F Rpp k .Let G be a finite group.Recall that by[Bouc and Yılmaz 2022, Corollary 8.23(i)], the multiplicity of the simple diagonal p-permutation functor S ‫ކ,1,1‬ in the representable functor ‫ކ‬T (−, G) is equal to the number l(kG) of the isomorphism classes of simple kG-modules.Let I(−, G) denote the sum of simple subfunctors of ‫ކ‬T (−, G) isomorphic to S ‫ކ,1,1‬ .Let also ‫ކ‬Proj(−, G) denote the subfunctor of ‫ކ‬T (−, G) sending a finite group H to ‫ކ‬Proj(H, G).Lemma 3.5.The subfunctors I(−, G) and ‫ކ‬Proj(−, G) of the representable functor ‫ކ‬T (−, G) are equal.Proof.For finite groups G and H , the number of isomorphism classes of projective indecomposable k(G × H )-modules, or equivalently, the number of isomorphism classes of simple k(G × H )-modules is equal to the number of conjugacy classes of p ′ -elements of G × H . Hence the ‫-ކ‬dimension of the evaluation ‫ކ‬Proj(H, G) is equal to l(k(G × H )) = l(kG)l(k H ) which is equal to the ‫-ކ‬dimension of l(kG)S ‫ކ,1,1‬ (H ) by [Bouc and Yılmaz 2022, Corollary 8.23(i)], and hence to the ‫-ކ‬dimension of I(H, G).Note that ‫ކ‬Proj(−, G) is equal to the functor ‫ކ‬T (−, 1) • ‫ކ‬T (1, G).

Definition 4. 1 .
Let b a block idempotent of kG.The stable diagonal p-permutation functor RT G,b is defined as RT G,b : Rpp k → R Mod, H → RT (H, G) ⊗ kG kGb.See Section 2(d) for the definition of RT G,b and note that RT G,b = (RT G,b ).Definition 4.2.Let b be a block idempotent of kG and let c be a block idempotent of k H .We say that the pairs (G, b) and (H, c) are stably functorially equivalent over R, if their corresponding stable diagonal p-permutation functors RT G,b and RT H,c are isomorphic in F Rpp k .Lemma 4.3.Let b be a block idempotent of kG and let c be a block idempotent of k H .(a) The pairs (G, b) and (H, c) are stably functorially equivalent over R if and only if there exists ω ∈ b RT (G, H )c and σ ∈ c RT (H, G)b such that ω• H σ =[kGb]+[P] in bRT (G, G)b and σ • G ω=[kH c]+[Q] in c RT (H, H )c for some P ∈ RProj(kGb, kGb) and Q ∈ RProj(k H c, k H c). (b) If the pairs (G, b) and (H, c) are functorially equivalent over R, then they are also stably functorially equivalent over R. Proof.By the Yoneda lemma, the pairs (G, b) and (H, c) are stably functorially equivalent over R if and only if there exists ω ∈ b RT (G, H )c and σ ∈ c RT (H, G)b such that ω • H σ = [kGb] in b RT (G, G)b and σ • G ω = [k H c] in c RT (H, H )c. Hence (a) follows and (b) is clear.□ Proof of Theorem 1.2.(i) The first statement follows from Theorem 1.1 and the second statement follows since by [Bouc and Yılmaz 2022, Corollary 8.23(i)] the multiplicity of the simple functor S ‫ކ,1,1‬ in ‫ކ‬T G,b is equal to l(kGb).(ii) The first statement follows from the proof of [loc.cit., Theorem 10.5(iii)] and the second statement follows from the proof of [loc.cit., Theorem 10.5(ii)].□ 5. Blocks with Frobenius inertial quotient (a) Recall the assumptions of Theorem 1.3: Let G be a finite group, b a block idempotent of kG with a nontrivial abelian defect group D. Let (D, e D ) be a maximal b-Brauer pair and let E = N G (D, e D )/C G (D) denote the inertial quotient of b.
Definition 3.3.An R-linear functor Rpp k → R Mod is called a stable diagonal p-permutation functor over R. Together with natural transformations, stable diagonal p-permutation functors form an abelian category F Rpp k .Rpp k → F Rpp k obtained by composition with the projection Rpp k → Rpp k gives a description of F Rpp k as a full subcategory of F Rpp k .Moreover, has a left adjoint , constructed as follows: If F is a diagonal p-permutation functor over R and G is a finite group, set • Id G = [kG].