Adams operations and symmetries of representation categories
AAdams operations and symmetriesof representation categories
Ehud Meir and Markus SzymikOctober 2018
Abstract: Adams operations are the natural transformations of the representation ring func-tor on the category of finite groups, and they are one way to describe the usual λ –ringstructure on these rings. From the representation-theoretical point of view, they codifysome of the symmetric monoidal structure of the representation category. We show thatthe monoidal structure on the category alone, regardless of the particular symmetry, deter-mines all the odd Adams operations. On the other hand, we give examples to show thatmonoidal equivalences do not have to preserve the second Adams operations and to showthat monoidal equivalences that preserve the second Adams operations do not have to besymmetric. Along the way, we classify all possible symmetries and all monoidal auto-equivalences of representation categories of finite groups.MSC: 18D10, 19A22, 20C15Keywords: Representation rings, Adams operations, λ –rings, symmetric monoidal cate-gories Every finite group G can be reconstructed from the category Rep ( G ) of its finite-dimensionalrepresentations if one considers this category as a symmetric monoidal category. This followsfrom more general results of Deligne [DM82, Prop. 2.8], [Del90]. If one considers the repre-sentation category Rep ( G ) as a monoidal category alone, without its canonical symmetry, thenit does not determine the group G . See Davydov [Dav01] and Etingof–Gelaki [EG01] for such isocategorical groups. Examples go back to Fischer [Fis88].The representation ring R ( G ) of a finite group G is a λ –ring. This structure derives from thesymmetric monoidal structure of the category Rep ( G ) . For rings which are torsion-free asabelian groups, such as R ( G ) , the structure of a λ –ring is equivalent to the lesser-known struc-tures of a τ –ring or Ψ –ring (see [Hof79] and [Wil82]). One way or another, it is determined bya family of commuting Frobenius lifts Ψ p , the Adams operations . The λ –ring structure of R ( G ) gives us more information about the group G than just the ring R ( G ) .For example, the representations rings of the dihedral group D and the quaternion group Q of order 8 are isomorphic as rings, but not as λ –rings: the second Adams operation Ψ canbe used to tell them apart. However, there are also examples (due to Dade [Dad64]) of non-isomorphic finite groups (of order 5 = a r X i v : . [ m a t h . R T ] O c t s λ –rings. (These examples are particularly simple for the Adams operations because thesegroups are p –groups of exponent p , so that the p –th Adams operation Ψ p gives the dimensionfunction, and the other Adams operations Ψ (cid:96) for (cid:96) prime to p are given by Galois actions,regardless of the group. Dade distinguishes these groups using their Lie rings.) The main result of this paper is the following.
Theorem 1.1. If Rep ( G ) → Rep ( G (cid:48) ) is a monoidal equivalence between representation cate-gories of finite groups G and G (cid:48) , then the induced isomorphism R ( G ) ∼ = R ( G (cid:48) ) between theirrepresentation rings preserves the Adams operations Ψ p for all odd p. Of course, when the monoidal equivalence is symmetric , the consequence holds trivially for all
Adams operations, since these are defined using the symmetry of the category. The point hereis that we do not assume this. The result shows that the odd Adams operations do not carryadditional information about the symmetry on the monoidal representation category.In Example 6.1, we show that this result cannot be improved: there exist groups G and G (cid:48) together with a monoidal equivalence Rep ( G ) → Rep ( G (cid:48) ) that does not preserve the secondAdams operations Ψ . Of course, such a monoidal equivalence cannot be symmetric.One might then wonder if the additional preservation of the second Adams operation Ψ isenough to ensure that we have a symmetric monoidal equivalence. We show in Example 6.2that this is also not the case, completing the story. Consider two finite groups G and G (cid:48) and a functor F : Rep ( G ) → Rep ( G (cid:48) ) which is an equiv-alence between the underlying K –linear categories. We say that F preserves an Adams oper-ation Ψ n if the induced map R ( G ) → R ( G (cid:48) ) preserves this operator. Consider the followingstatements about a functor F .(F1) The functor F is symmetric monoidal.(F2) The functor F is monoidal and preserves all Adams operations.(F3) The functor F is monoidal and preserves the odd Adams operations.(F4) The functor F is monoidal.Then, obviously, we have (F1) (cid:43) (cid:51) (F2) (cid:43) (cid:51) (F3) (cid:43) (cid:51) (F4) . Now, as a consequence of our results, we can decide the validity of all other implications: (F3)and (F4) are actually equivalent by Theorem 1.1, and the other implications cannot be reversed,by Examples 6.1 and 6.2. (F1) (cid:107) (cid:115) (cid:54) (F2) (cid:107) (cid:115) (cid:54) (F3) (cid:107) (cid:115) (F4)2he examples that we have in Section 6 lead to more general questions, about the mere exis-tence of functors. Consider the following statements about two finite groups G and G (cid:48) .(G1) The groups G and G (cid:48) are isomorphic.(G2) There exists a symmetric monoidal equivalence Rep ( G ) (cid:39) Rep ( G (cid:48) ) .(G3) There exists a monoidal equivalence Rep ( G ) (cid:39) Rep ( G (cid:48) ) which preserves allAdams operations.(G4) There exists a monoidal equivalence Rep ( G ) (cid:39) Rep ( G (cid:48) ) which preserves the oddAdams operations.(G5) There exists a monoidal equivalence Rep ( G ) (cid:39) Rep ( G (cid:48) ) .Then, obviously, we have (G1) ⇒ (G2) ⇒ (G3) ⇒ (G4) ⇒ (G5). The equivalence of (G1)and (G2) follows from Deligne’s work. The equivalence of (G4) and (G5) follows from ourTheorem 1.1. The examples of Etingof and Gelaki show that (G5) (cid:59) (G1), and we now knowthat this entails (G4) (cid:59) (G2). On the other hand, we do not know if the condition (G4) isstrictly stronger than (G3), or if (G3) is strictly stronger than (G2). Of course, it is not possiblethat (G3) will be equivalent to both (G2) and (G4), because this would contradict (G4) (cid:59) (G2).(G1) (cid:107) (cid:115) (cid:43) (cid:51) (G2) (cid:27) (cid:35) (cid:107) (cid:115) (cid:54) (G4) (cid:107) (cid:115) (cid:43) (cid:51) (G5)(G3) (cid:59) (cid:67) We review the Adams operations in Section 2 and give End’s characterization of these opera-tions as the natural transformations of the representation ring functor, and offer a representa-tion theoretic description in Proposition 2.5. This result nicely complements the usual formulaon the level of characters. In Section 3, we review the Etingof–Gelaki classification of iso-categorical groups. Building on that, we give a new description of all possible symmetrieson the monoidal representation categories
Rep ( G ) in Theorem 4.1 and the monoidal auto-equivalences of Rep ( G ) in Theorem 4.2. Section 5 contains a proof of Theorem 1.1, and wegive the promised examples in the final Section 6. For any finite group G , we will denote by Rep ( K G ) the category of finite-dimensional G –representations over an algebraically closed field K of characteristic 0. This is more precisethan the notation Rep ( G ) that we have used for the purposes of the introduction only. Moregenerally, we will use the notation Rep ( H ) for the category of finite-dimensional represen-tations of a finite-dimensional Hopf algebra H over K . These representation categories havea canonical structure of a rigid monoidal category. (Rigidity means that each object admits aleft and right dual object. See [Kas95], [CE08], [ENO05] or [EGNO15] for expositions.) Inaddition, these representation categories admit a canonical symmetric monoidal structure: forevery two objects V , W in Rep ( K G ) we have a natural isomorphism σ V , W : V ⊗ W → W ⊗ V given by v ⊗ w (cid:55)→ w ⊗ v . This isomorphism satisfies σ V , W σ W , V = id. Our analysis of the Adamsoperations later will lead us to study other symmetric monoidal structures on Rep ( K G ) as well.Notice that in the work [EG01] all possible pairs of isocategorical groups were classified, butthe corresponding monoidal equivalences between the corresponding monoidal categories were3ot described. They are described here in Section 3 and in the preprint [GM], where it is alsoshown that the representation categories of isocategorical groups have isomorphic Witt groups.Throughout the paper, we will denote by R ( G ) the Grothendieck ring K Rep ( K G ) of themonoidal category Rep ( K G ) . This is the representation ring of the group G , and (by ourassumptions on the field K ) independent of our particular choice of K , as the notation suggests.The representation ring has a canonical Z –basis given by the isomorphism classes [ V ] , . . . , [ V r ] of a representative set of irreducible representations V , . . . , V r of G . It can be identified witha subring of the ring of class functions on G by means of the character map that sends a rep-resentation V to its character χ V : g (cid:55)→ tr ( g | V ) . In the following, we will identify a class [ V i ] with its character χ i whenever it will facilitate the text. In this section we review some background material. We describe the Adams operations asthe natural transformations of the representation ring functor on the category of finite groups.They are an equivalent way of encoding the well-known λ –ring structure. We also give aninterpretation in terms of the representation category, and explain how to determine the orderand the exponent of any finite group from these operations. λ –rings The notion of a λ –ring first arose in Grothendieck’s work [Gro58, 4.2] on vector bundlesand K-theory. Nowadays, it is recognized that λ –operations are useful additional structurethat is present in many other contexts. For representation theory, see [AT69], [Ker76], [Kra80],and [Ben84]. There are many ways to present the algebraic theory of λ –rings. See Berth-elot’s chapter [Ber71] in SGA 6 as well as [Knu73], [Pat03], and the recent book [Yau10] byYau. For the purposes of the present text, it is only important to know about Wilkerson’s crite-rion [Wil82]: if a ring is torsion-free as an abelian group, then a λ –ring structure on it is equiv-alent to a family ( Ψ n | n ∈ Z ) of commuting ring endomorphisms that satisfy Ψ mn = Ψ m Ψ n andsuch that Ψ p is a Frobenius lift for each prime number p . In particular, since representationrings are torsion-free, their λ –ring structure is determined by the Adams operations and viceversa.For a natural number k and a representation V of a finite group G , the λ –operations on therepresentation ring R ( G ) are defined using the exterior powers: we have λ k [ V ] = [ Λ k V ] . Noticethat this definition relies on the usual symmetric monoidal structure on the representationcategory Rep ( K G ) . The corresponding Adams operations are defined on the level of classfunctions by the formula Ψ kG ( χ )( g ) = χ ( g k ) . (Of course, formulas like this go back to Frobe-nius, see [Fro07, § G is clear from the context, we willwrite Ψ kG = Ψ k . Notice that for every k and every character χ it holds that Ψ k ( χ ) is a Z –linear combination of the irreducible characters. This is not immediately clear from the givendefinition of the Adams operations. It follows from the fact that they correspond to the afore-mentioned λ –operations. See [Ser67, Ex. 9.3.b] and Remark 2.6. Remark 2.1.
Adams operations in K-theory have been introduced in [Ada62, Sec. 4]. Seealso the exposition [Ati66] by Atiyah. The latter approach was axiomatized and applied to4epresentation rings of finite groups G by Hoffman [Hof79], a former student of Adams. Theidea is, rather than working with the exterior powers Λ k V of a G –representation V , to use thetensor powers V ⊗ k as representations of the wreath products G (cid:111) Σ k . The resulting theory of τ –rings turns out to be equivalent to the theory of λ –rings [Hof79, Thm. 6.6]. We will not makeuse of these facts in the following. The k –th Adams operation Ψ kG depends only the residue class of the integer k modulo theexponent of G , and therefore can be defined for all elements k in the pro-finite completion (cid:98) Z ofthe group Z of integers.The family ( Ψ kG | G ) defines a natural endomorphism Ψ k : R → R of the representation ringfunctor R, thought of as a contravariant functor from the category of finite groups and (all)homomorphisms to the category of (commutative) rings (with unit): restriction along a grouphomomorphism G → G (cid:48) defines a ring homomorphism R ( G (cid:48) ) → R ( G ) . In fact, it is easy to seethat this leads to a characterization of the Adams operations. This was proved by End [End72]after the K-theory case had been proved by tom Dieck [tD67]: Proposition 2.2. (End [End72, Satz (8.4)])
The monoid of endomorphisms of the representationring functor is isomorphic to the multiplicative monoid of pro-finite integers (cid:98) Z . The pro-finiteinteger k corresponds to the k–th Adams operation Ψ k . The idea for a proof is very simple: the group Z of integers under addition generates thecategory of groups (in the sense that it (co-)represents the forgetful functor to sets) and wecould detect every natural transformation on this single example. But the group Z of integers isnot finite, and we have to replace it by the family of its finite quotients, the finite cyclic groups. Proof.
Let us first see that that every natural transformation Φ has the form Ψ k for some pro-finite integer k . Since the restrictions to cyclic subgroups induce an injectionR ( G ) −→ (cid:77) C (cid:54) G cyclic R ( C ) , the natural transformation Φ is determined by its behavior on cyclic groups. For a cyclicgroup C of order n , the representation ring R ( C ) is isomorphic to the group ring Z [ C ∨ ] of thecharacter group C ∨ . This is generated (as a ring) by any embedding γ : C → K × of groups (andit holds that γ n = Φ ( γ ) is also a unit that is torsion. By Higman’s theorem [Hig40,Thm. 3], for instance, every unit of Z [ C ∨ ] that is torsion is trivial in Whitehead’s sense: wehave Φ ( γ ) = ± γ k n for some integer k n . The restriction to the trivial subgroup shows that Φ preserves the augmentation R ( G ) → Z of the representation ring R ( G ) given by the dimension,so that the sign has to be positive, and Φ C = Ψ k n C for all cyclic groups C of order n . Restrictionto subgroups shows that k n is congruent to k m modulo m whenever m divides n , so that the fam-ily ( k n ∈ Z / n | n (cid:62) ) describes a pro-finite integer k such that Φ C = Ψ kC for all cyclic groups C .As mentioned at the beginning, the equation Φ G = Ψ kG for all groups G follows. Conversely,this argument also shows how to read off k from Ψ k . Remark 2.3.
The reduction to cyclic groups is omnipresent in the representation theory of finitegroups, because representations are determined by their characters, which are functions on the5roup, and every group element lies in a cyclic group. In the other direction, we have Artin’stheorem, which says that each character of a finite group is a rational linear combination ofcharacters of representations induced from cyclic subgroups. See the work of Hopkins, Kuhn,and Ravenel [Hop89, Kuh89, HKR92, HKR00] for generalizations.
We will now explain how to find an expression for the representative matrix of the k –th Adamsoperation Ψ k : R ( G ) → R ( G ) on the representation ring R ( G ) of a finite group G in termsof the basis given by irreducible characters. To do so we write as before χ , . . . , χ r for theirreducible characters of the group G , and write the class function Ψ k ( χ i ) : g (cid:55)→ χ i ( g k ) as alinear combination Ψ k ( χ i ) = ∑ j Ψ ki , j χ j of the irreducible characters. This gives the matrix ( Ψ ki , j | (cid:54) i , j (cid:54) r ) of Ψ k with respect to thechosen basis—at least in theory. The scalars Ψ ki , j can be computed directly, using representationtheory, as follows.We have the standard inner product (cid:104) α | β (cid:105) = | G | ∑ g α ( g ) β ( g ) (2.1)for class functions. The irreducible characters form an orthogonal basis, so that we can compute Ψ ki , j = (cid:104) χ j | Ψ k ( χ i ) (cid:105) = | G | ∑ g χ j ( g − ) χ i ( g k ) . (2.2)Let V i be an irreducible representation that corresponds to the irreducible character χ i . Wechoose the indexing so that χ is the trivial character, and V is the trivial representation.For a given integer k , the k –th Frobenius–Schur indicator of a G –representation V is the traceof the G –linear endomorphism 1 | G | ∑ g g k : V −→ V . Equation (2.2) shows that Ψ k , j is the k –th Frobenius–Schur indicator of V j .Let σ k be the linear operator on the k –th tensor power V ⊗ ki that cyclically permutes the factors.Since this is K G –linear, we have an induced map ( σ k ) ∗ on the space Hom G ( V j , V ⊗ ki ) by post-composition. In order to describe Ψ ki , j explicitly, we need the following lemma, whose proofcan be found in [KSZ06, Sec. 2.3]: Lemma 2.4.
Given any K –vector space V , let σ k : V ⊗ k → V ⊗ k denote the cyclic permutation.Given K –linear endomorphisms f , . . . , f k : V → V , we have an equality tr ( σ k ( f ⊗ f ⊗ · · · ⊗ f k ) | V ⊗ k ) = tr ( f f · · · f k | V ) of traces. roposition 2.5. Let V , . . . , V r be representatives of the isomorphism classes of irreducible G–representations of a finite group G. Then we have Ψ k ( χ i ) = r ∑ j = tr (cid:16) σ ∗ k | Hom G ( V j , V ⊗ ki ) (cid:17) χ j where V i is the irreducible representation which corresponds to the character χ i in the repre-sentation ring R ( G ) , where σ k is the cyclic permutation of the tensor factors of V ⊗ ki . See Prop. 2.5 in [Ati66] and the remarks following it for a similar statement when k = p is aprime (and in the context of vector bundles). Proof.
Consider the vector space W = Hom G ( V j , V ⊗ ki ) . Notice first that W is the subspaceof G –invariants in the representation U = Hom ( V j , V ⊗ ki ) where the action of the group G is theusual diagonal action. Also, the G –representation U is naturally isomorphic with V ∗ j ⊗ V ⊗ ki .The projection U → W is given by the action of the idempotent (cid:15) = | G | − ∑ g ∈ G g . The opera-tor σ ∗ commutes with the action of (cid:15) , and we have tr ( σ ∗ | W ) = tr ( (cid:15)σ ∗ | U ) . This is in fact truewhenever we have an operator commuting with a projection. As in Lemma 2.4 we computetr ( σ ∗ | W ) = tr ( (cid:15)σ ∗ | U )= tr (cid:32) | G | ∑ g ∈ G g ⊗ g k | V ∗ j ⊗ V i (cid:33) = | G | ∑ g ∈ G χ j ( g − ) χ i ( g k )= Ψ ki , j to finish the proof. Remark 2.6.
We have shown that Ψ ki , j is the character value of a k –cycle for some representa-tion of the permutation group Σ k . This makes it also clear that the coefficients Ψ ki , j are integers:all irreducible representations of Σ k are already realizable over the prime field Q , and charactervalues are always algebraic and contained in the field of definition. To end this section, we explain how to extract some basic numerical information about a groupfrom its representation ring (as a λ –ring). The results here will not be used in the rest of thepaper.Recall that the rank of the representation ring R ( G ) (as an abelian group) is the number ofconjugacy classes of elements in G , so that this numerical invariant of G is determined by R ( G ) .We include here a proof that R ( G ) determines also the order of G , as we are unaware of anappropriate reference for that. Proposition 2.7.
The representation ring R ( G ) determines the order of G.Proof. We can identify R ( G ) with a subring of the ring C ⊗ Z R ( G ) and think of this as thering of complex class functions on G . The primitive central idempotents in C ⊗ Z R ( G ) are the7haracteristic functions of the conjugacy classes. Let us write (cid:15) g for the one corresponding tothe conjugacy class of the element g . The inner product (2.1) evaluates to (cid:104) χ j | (cid:15) g (cid:105) = | G | ∑ h χ j ( h − ) (cid:15) g ( h ) , which is χ j ( g − ) / | G | times the number of conjugates of g in G , or | G / C G ( s ) | . Since theirreducible characters form an orthonormal basis, we get the formula (cid:15) g = | C G ( g ) | ∑ j χ j ( g − ) χ j . For instance, we have (cid:15) = ρ / | G | . In general, the coefficient of the trivial representationis 1 / | C G ( g ) | . Let n g be the minimal positive integer such that n g (cid:15) g is contained in R ( G ) or 0 ifsuch an integer does not exist. Then the maximum max { n g | g ∈ G } is achieved at the identityelement, and it is n e = | G | . In this way we recover | G | just from the ring R ( G ) .A little more work, and the Adams operations, give the exponent: Proposition 2.8.
The λ –ring structure on the representation ring R ( G ) determines the exponentof G.Proof. Let e = exp ( G ) be the exponent of G . Then g k + e = g k for all elements g of G , so thatwe have Ψ k + e = Ψ k for all integers k . This shows that the family Ψ = ( Ψ k | k ∈ Z ) is periodic,and we claim that e is its period.We can use that Ψ = dim is the dimension function R ( G ) → Z → R ( G ) . Assume that wehave Ψ k + f = Ψ k for all integers k . If ρ denotes the character of the regular representation,then Ψ f ( ρ ) = Ψ ( ρ ) = dim ( ρ ) is constant and equals the order of G . But, on the other hand,we have ( Ψ f ρ )( g ) = ρ ( g f ) . Since the character of the regular representation vanishes awayfrom the neutral element of G , we deduce g f is the neutral element of the group G for all g sothat f is a multiple of the exponent of G , as claimed. Remark 2.9.
It follows from Shimizu’s work [Shi10] that isocategorical groups have the sameexponent. This shows that the exponent is an invariant which is also computable from themonoidal category
Rep ( KG ) alone, not using its symmetry. It is arguably much easier to usethe λ –ring structure on R ( G ) , though. We have already mentioned in the course of the introduction that the representation cate-gory
Rep ( K G ) of a finite group G , as a symmetric monoidal category, determines the group G up to isomorphism. Let us be more precise now.Deligne [Del90] showed that every symmetric monoidal category C that satisfies certainfiniteness conditions admits a unique symmetric fiber functor F : C → Rep ( K ) . (See alsoBreen’s exposition in [Bre94].) If G is the group of monoidal autoequivalences of the func-tor F , then Tannaka reconstruction gives us an equivalence of symmetric monoidal cate-gories C (cid:39) Rep ( K G ) . Under this equivalence, the functor F gives us the forgetful func-tor Rep ( K G ) → Rep ( K ) . The symmetric monoidal structure is crucial here: it is possi-ble that Rep ( K G ) and Rep ( K G (cid:48) ) will be equivalent as monoidal categories without the two8roups G and G (cid:48) being isomorphic. Stated differently, it is possible that a representation cate-gory Rep ( K G ) admits a non-canonical symmetry for its monoidal structure which will make itequivalent, as a symmetric monoidal category, to Rep ( K G (cid:48) ) (with its canonical symmetry) fora group G (cid:48) that is not isomorphic to G .Two groups G and G (cid:48) are called isocategorical in case Rep ( K G ) and Rep ( K G (cid:48) ) are equiva-lent as monoidal categories. In [EG01], Etingof and Gelaki constructed all examples of non-isomorphic isocategorical finite groups, up to isomorphism. What we add to this here is anexplicit description of monoidal equivalences, which we dub Etingof–Gelaki equivalences, seeDefinition 3.3 below.
Example 3.1.
It is shown in [GM] that the smallest examples of isocategorical groups haveorder 64. From the 267 isomorphism classes of groups of that order, these are the two groupsthat are named Γ a and Γ a by Hall and Senior [HS64], or SmallGroup(64,136) and
SmallGroup(64,135) in [GAP], respectively. They both have exponent 8 and rep-resentations rings of rank 16, but the ranks of their Burnside rings differ: they are 76 and 73,respectively. These groups can also be distinguished by looking at their homology H ( G ; Z ) or cohomology H ( G ; Z / ) .We now review the work [EG01] of Etingof and Gelaki that describes a construction of allgroups, up to isomorphism, that are isocategorical to a given group G First of all, we consider a presentation1 −→ A −→ G −→ Q −→ G as an extension of some group Q by an abelian group A . Let us agree towrite A ∨ = Hom ( A , K × ) for the character group of the kernel. The quotient group Q acts onthe groups A and A ∨ , and we can form the split extension1 −→ A ∨ −→ A ∨ (cid:111) Q −→ Q −→ . The first differential on the second page of the Lyndon–Hochschild–Serre spectral sequence forthat extension is a homomorphismd : H ( Q ; H ( A ∨ ; K × )) −→ H ( Q ; H ( A ∨ ; K × )) = H ( Q ; A ) , and we need to make this homomorphism explicit.Let α : A ∨ × A ∨ −→ K × be a non-degenerate 2–cocycle for the character group such that the cohomology class [ α ] ofthe 2–cocycle α is Q –invariant. In formulas, this means [ q ( α )] = [ α ] for all elements q inthe quotient group Q . Therefore, we can choose, for each element q in the quotient group Q ,a 1–cochain z ( q ) : A ∨ → K × such that d z ( q ) = q ( α ) α in the standard cochain complex C ( A ∨ ; K × ) → C ( A ∨ ; K × ) . We will work with one suchchosen family ( z ( q ) | q ∈ Q ) from now on. We define, for all elements p , q in Q , b ( p , q ) = z ( pq ) z ( p ) p ( z ( q )) : A ∨ → K × . (3.2)9uch a function is a 1–cochain, and a direct calculation shows that the differential vanisheson it: d ( b ( p , q ))( ϕ ) = ϕ ∈ A ∨ , so that b ( p , q ) is a 1–cocycle, that is a 1–dimensionalrepresentation A ∨ → K × of A ∨ . In other words, the function b assigns to any pair of elementsin Q an element in ( A ∨ ) ∨ = A . It turns out that b is a 2–cocycle of Q , and we can think of it ashaving values in A , that is b ∈ Z ( Q ; A ) . The differential is given by d [ α ] = [ b ] , see [ENO10,Appendix], for instance.Now that we have described a cocycle b ∈ Z ( Q ; A ) , we can use it to define an isocategoricalgroup G b . It has the same underlying set as the group G , but its multiplication · b is defined bythe rule g · b h = b ( ¯ g , ¯ h ) gh , (3.3)where g (cid:55)→ ¯ g is the projection G → Q . Theorem 3.2. (Etingof–Gelaki [EG01, Theorem 1.3])
Two finite groups G and G (cid:48) are isocat-egorical if and only if the group G (cid:48) is isomorphic to a group of the form G b for some b, wherethe group A has order m for some m. Etingof and Gelaki classified all pairs of isocategorical groups, but they did not describe explic-itly the categorical equivalences. We do this here; we describe explicitly the monoidal equiva-lence F b : Rep ( K G ) → Rep ( K G b ) . (3.4)arising from their construction. For this, the normal subgroup A does not have to be a 2–group,and the cocycle α does not have to be non-degenerate. However, we will assume that α isnon-degenerate, since if α is degenerate we can always reduce to a subgroup B of A that willadmit such a non-degenerate 2–cocycle. Without further ado, we describe (3.4). If V is a G –representation, then we set F b ( V ) = V , the same underlying vector space, but with the action ofan element g in G b given as follows: if V = (cid:77) ϕ ∈ A ∨ V ( ϕ ) (3.5)is the isotypical decomposition of V as an A –representation, then g · b v = z ( ¯ g )( ¯ g ( ϕ )) gv (3.6)for the elements v ∈ V ( ϕ ) . Then F b is an equivalence of categories. We can define a monoidalisomorphism F b ( V ⊗ W ) → F b ( V ) ⊗ F b ( W ) by v ⊗ w (cid:55)−→ α ( ϕ , ϕ ) v ⊗ w where v ∈ V ( ϕ ) and w ∈ W ( ϕ ) . This ends our description of the monoidal equiva-lence F b (see also [GM]). Definition 3.3.
We will call a functor of the form F b an Etingof–Gelaki equivalence . Remark 3.4.
In order to have an equivalence, it is not necessary that A is a 2–group. If b (cid:48) is theresulting cocycle from a different choice of the family ( z ( q ) | q ∈ Q ) , then [ b ] = [ b (cid:48) ] and thereexists a canonical isomorphism of groups ρ : G b (cid:48) ∼ = G b . The functor F b (cid:48) will then be isomorphicto the composition Rep ( K G ) F b −→ Rep ( K G b ) ρ ∗ −→ Rep ( K G b (cid:48) ) .
10n particular, if b is trivial (which is always the case, for instance, when the order of A is odd),we will get in this way an autoequivalence of Rep ( K G ) . However, in case A is non-trivial, theresulting autoequivalence will be monoidal but not symmetric.The induced symmetry on the category Rep ( K G ) that is obtained from conjugating the sym-metry on Rep ( K G b ) with the equivalence Rep ( K G ) → Rep ( K G b ) will be v ⊗ w (cid:55)→ α ( ϕ , ϕ ) α ( ϕ , ϕ ) w ⊗ v . (3.7)Notice that even if b is a trivial cocycle we might get a new non-trivial symmetry on the cate-gory Rep ( K G ) .In the next section, we prove that these are all possible symmetric monoidal structures that arisefrom equivalences between representation categories of finite groups. We shall also classify allmonoidal autoequivalences of representation categories of finite groups. In order to state our results on the classification of symmetries, we will recall here first somenotions and results from the theory of fusion categories. Recall that a fusion category over K isa semi-simple rigid monoidal K –linear category with finitely many simple objects and finite-dimensional Hom spaces [ENO05]. A guiding example to keep in mind is the representationcategory Rep ( K G ) of a finite group G . A fiber functor on a fusion category is a K –linear exactfaithful monoidal functor T : D → Rep ( K ) . (The target category Rep ( K ) is just the categoryof vector spaces over the ground field K .) The endomorphism algebra End K ( T ) has a canonicalstructure of a Hopf algebra, and the functor T induces an equivalence of monoidal categoriesbetween D and Rep ( End K ( T )) . This is what is known as Tannaka reconstruction for Hopfalgebras.If now D and D are two fusion categories, and F : D → D is a monoidal functor betweenthese, and T : D → Rep ( K ) is a K –linear monoidal functor, then F induces a Hopf algebramorphism End K ( T ) → End K ( T F ) and the diagram D (cid:47) (cid:47) (cid:15) (cid:15) D (cid:15) (cid:15) Rep ( End ( T F )) (cid:47) (cid:47) Rep ( End ( T )) is commutative up to a natural equivalence, where the lower horizontal arrow is restriction ofrepresentations.Let us specialize these considerations to the case where H is a finite-dimensional semi-simpleHopf algebra over the field K , and where T : Rep ( H ) → Rep ( K ) is the forgetful functor. Thena monoidal autoequivalence L : Rep ( H ) → Rep ( H ) is given by a pair ( T (cid:48) , Φ ) , where the firstentry T (cid:48) : Rep ( H ) → Rep ( K ) is a K –linear monoidal functor (which will become the compo-sition T L ) and the second entry is an isomorphism Φ : H ∼ = End K ( T (cid:48) ) of Hopf algebras. Onecan show that ( T (cid:48) , Φ ) will define the identity autoequivalence if and only if T (cid:48) ∼ = T , and underthis isomorphism it holds that Φ corresponds to an automorphism of H which arises from con-jugation by a group-like element. 11onsider now the case where the Hopf algebra H = K G is the group algebra of a finite group G .Following the works of Movshev [Mov93], Davydov [Dav01], Ostrik [Ost03a, Ost03b], andNatale [Nat03], we know that fiber functors F : Rep ( K G ) → Rep ( K ) are in one-to-one corre-spondence with pairs of the form ( A , [ ψ ]) where A is a subgroup of G and [ ψ ] ∈ H ( A ; K × ) isa cohomology class of a non-degenerate 2–cocycle. Two pairs ( A , [ ψ ]) and ( A , [ ψ ]) defineisomorphic functors if and only if they differ by conjugation in G . For instance, the pair ( , [ ]) consisting of the trivial subgroup and the trivial 2–cocycle on it corresponds to the forgetfulfunctor Rep ( K G ) → Rep ( K ) . Furthermore, the Hopf algebra that one receives from Tannakareconstruction is again a group algebra if and only if the subgroup A is abelian and normal, andthe cohomology class of the 2–cocycle ψ is invariant under the action of the group G .In the case of abelian groups the cocycle ψ amounts to a skew-symmetric pairing on A , givenby ( a , b ) (cid:55)−→ ψ ( a , b ) ψ ( b , a ) . In case ψ is non-degenerate, this gives us an isomorphism S : A ∼ = A ∨ . The 2–cocycle ( S − ) ∗ ( ψ ) is then exactly the 2–cocycle which appears in the Etingof–Gelaki classification of isocategor-ical groups mentioned in the previous section. This discussion leads to the following results: Theorem 4.1.
Assume that σ is a symmetric monoidal structure on Rep ( K G ) . Assume alsothat the exterior powers Λ n V (defined using the symmetry) satisfy Λ n V = for all V and largeenough n (depending on V ). Then there exists a normal abelian subgroup A of G and a G / A–invariant non-degenerate cohomology class of a –cocycle α : A ∨ × A ∨ → K × such that thesymmetry is given by the formula (3.7) :v ⊗ w (cid:55)−→ α ( ϕ , ϕ ) α ( ϕ , ϕ ) w ⊗ v , where, in the notation of (3.5) , the vector v is in V ( ϕ ) and w is in V ( ϕ ) .Proof. Let us first assume that the symmetry σ satisfies the condition on exterior powers.From Deligne’s work [Del90] we know that there exists an equivalence of symmetric monoidalcategories ( Rep ( K G ) , σ ) → ( Rep ( K G (cid:48) ) , σ (cid:48) ) where σ (cid:48) is the canonical symmetry on the cat-egory Rep ( K G (cid:48) ) for some finite group G (cid:48) . By the above discussion and the classificationof fiber functors on Rep ( K G ) we know that every monoidal equivalence between Rep ( K G ) and Rep ( K G (cid:48) ) is an Etingof–Gelaki equivalence. This implies that the symmetric monoidalstructure is of the claimed form. Theorem 4.2.
Any monoidal autoequivalence of
Rep ( K G ) is given by a triple ( A , α , ϕ ) con-sisting of an abelian normal subgroup A of G, a G / A–invariant non-degenerate –cocycle α on A ∨ and an isomorphism ϕ : G → G b , where b is the –cocycle from the Etingof–Gelaki con-struction. Moreover, two tuples ( A , α , ϕ ) and ( A (cid:48) , α (cid:48) , ϕ (cid:48) ) will define isomorphic equivalences ifand only if A = A (cid:48) , α = α (cid:48) and ϕ differs from ϕ (cid:48) by conjugation with an element of the group G.Proof. The proof of this is similar to the one before.
Remark 4.3.
There is a connection between the group of monoidal autoequivalences of thefusion category
Rep ( K G ) that appears here and the Brauer–Picard group BrPic ( Rep ( K G )) that has been introduced in [ENO10]. We have a natural homomorphism Φ : Aut ⊗ ( Rep ( K G )) −→ BrPic ( Rep ( K G ))
12f groups which sends an automorphism ψ to the quasi-trivial bimodule Rep ( K G ) ψ . Thekernel of this homomorphism contains all autoequivalences that are given (up to isomor-phism) by conjugation with an invertible object in the category Rep ( K G ) . Since the cate-gory Rep ( K G ) is symmetric, the homomorphism Φ is in fact injective, and we can considerthe group Aut ⊗ ( Rep ( K G )) as a subgroup of the Brauer–Picard group BrPic ( Rep ( K G )) . Theimage of the homomorphism Φ is denoted by Out ( Rep ( K G )) . An element in the Brauer–Picard group is given by a module category M over Rep ( K G ) together with an equiva-lence between Rep ( K G ) and the dual Rep ( K G ) M . Equivalence classes of module categoriesover Rep ( K G ) for which the dual is equivalent to Rep ( K G (cid:48) ) for some group G (cid:48) are given bypairs ( A , ψ ) where A is an abelian normal subgroup, and ψ is a 2–cocycle, not necessarilynon-degenerate. We see that the image of Φ contains all the bimodule categories in which thecocycle ψ is non-degenerate (and for which the resulting group is isomorphic to G ). For adeeper study of the Brauer–Picard group of Rep ( K G ) we refer the reader to the paper [NR14].Note also that we only describe the elements of the (categorical) automorphism groupof Rep ( K G ) . We do not explain the composition. That is a different story, and it is the coredifficulty in many calculations done for the Brauer–Picard group. See [LP18], which is relatedto that problem. The purpose of this section is to prove Theorem 1.1.As we have already noted in the proof of Theorem 4.1, the discussion in Section 4 shows thatevery monoidal equivalence between representation catgegories is an Etingof–Gelaki equiva-lence. Let us, therefore, begin with the following statement.
Lemma 5.1.
Assume that we have a short exact sequence → A → G → Q → , a non-degenerate –cocycle [ α ] ∈ H ( A ∨ ; K × ) Q , and b : Q × Q → A the resulting –cocycle fromthe Etingof–Gelaki construction. If b (cid:48) : Q × Q → A is the resulting –cocycle from a differentchoice of family ( z ( q ) | q ∈ Q ) , then for every integer k the functor F b preserves Ψ k if and onlyif the functor F b (cid:48) preserves Ψ k .Proof. Recall from Remark 3.4: if b (cid:48) is the resulting cocycle from a different choice of the fam-ily ( z ( q ) | q ∈ Q ) , then [ b (cid:48) ] = [ b ] , there exists a canonical isomorphism of groups ρ : G b (cid:48) ∼ = G b ,and the functor F b (cid:48) is isomorphic to the composition F b (cid:48) : Rep ( K G ) F b −→ Rep ( K G b ) ρ ∗ −→ Rep ( K G b (cid:48) ) . The result follows from this and the fact that the symmetric monoidal equivalence ρ ∗ preservesall Adams operations: the Adams operations are determined by the λ –ring structure, and thisstructure in turn is determined by the symmetric monoidal structure.We can (and will) therefore assume that we are dealing with an Etingof–Gelaki equivalencebetween the finite groups G and G b which arises from an extension A → G → Q , a Q –invariantcohomology class [ α ] in H ( A ∨ ; K × ) , and a 2–cocycle b as above. The group A is a 2–group. Lemma 5.2.
We can choose the functions z ( q ) : A ∨ → K × so that we have z ( q )( ϕ ) = when-ever q ( ϕ ) = ϕ . roof. For an element q ∈ Q , pick any function z ( q ) that satisfies the equation d z ( q ) = q ( α ) / α .The restriction of the function z ( q ) to the q –invariant subgroup B ( q ) = ( A ∨ ) q is then a 1–dimensional representation of the abelian group B ( q ) : we calculated ( z ( q ) | B ( q )) = q ( α | B ( q )) α | B ( q ) = , by invariance. Since the field K is assumed to be algebraically closed, its group of units isdivisible, so that any homomorphism B ( q ) → K × can be extended to A ∨ . By multiplying z ( q ) with the inverse of an extension, we do not change the 2–cocycle d z ( q ) , but we assure that wehave z ( q ) | B ( q ) =
1, as desired.From now on we assume that z ( q )( ϕ ) = q ( ϕ ) = ϕ . Under this assumption, we canprove the following lemma: Lemma 5.3.
The isomorphism R ( G ) ∼ = R ( G b ) induced by F b is the identity: if χ is the characterof a representation V , and χ b is the character of the representation F b ( V ) , then χ b = χ . Since G b is equal to G as a set, this formulation makes sense. Proof.
Let V be a representation of the group G with character χ . Recall our nota-tion B ( q ) = ( A ∨ ) q for elements q in the quotient Q = G / A . When we look at the way thatan element g of G permutes the isotypical summands V ( ϕ ) of V , the only places where wewill get a contribution to the character are those V ( ϕ ) which are stable under the action of ¯ g .Therefore, we can write χ ( g ) = ∑ ϕ ∈ B ( ¯ g ) tr ( g | V ( ϕ ) ) . (5.1)Let χ b be the character of the G b –representation F b ( V ) . We then have χ b ( g ) = ∑ ϕ ∈ B ( ¯ g ) z ( ¯ g )( ¯ g ( ϕ )) tr ( g | V ( ϕ ) ) by the definition (3.6) of the new action on V when restricted to the subspace V ( ϕ ) . Since wecan assume, by Lemma 5.2, that we have z ( ¯ g )( ϕ ) = g fixes ϕ , we deduce χ = χ b as functions on G = G b and we are done.Now that we have an explicit model for an isomorphism between the representation rings R ( G ) and R ( G b ) , we can check that it respects the odd Adams operations: Proof of Theorem 1.1.
Let k be an odd integer. We have ( Ψ k χ )( g ) = χ ( g k ) by definition ofthe Adams operations and similarly for Ψ k χ b . We already know that χ b = χ by the previousLemma 5.3. Therefore, the only possible difference is in the k –th power g k of g : it is calculatedonce in the group G and once in the group G b , respectively. We can assume that k is a positiveinteger. When we calculate g k using the multiplication (3.3) in the group G b , we get b ( ¯ g , ¯ g ) · b ( ¯ g , ¯ g ) · . . . · b ( ¯ g k − , ¯ g ) · g k
14n terms of G . When we use formula (3.2), we can cancel repeating terms, and the result is z ( ¯ g k ) z ( ¯ g ) · g ( z ( ¯ g )) · . . . · g k − ( z ( ¯ g )) · g k . (5.2)The fractional part here, which we will denote by a = a / a , is an element of ( A ∨ ) ∨ ∼ = A ,described as a 1–dimensional representation A ∨ → K × , and to finish the proof it suffices toshow that this element does not change the value of the character.Using the formula (5.1) from the proof of the previous Lemma 5.3, we know that ( Ψ k χ )( g ) = χ ( g k ) = ∑ ϕ ∈ B ( ¯ g k ) tr ( g k | V ( ϕ ) ) , and we have a similar expression for Ψ k ( χ b ) . Therefore, we only need to consider the val-ues a ( ϕ ) = a ( ϕ ) / a ( ϕ ) of the element a on ϕ ∈ B ( ¯ g k ) . Recall that these ϕ ∈ B ( ¯ g k ) arehomomorphisms ϕ : A → K × that are fixed by g k .By the assumption that we made (after Lemma 5.2 and its proof) on the functions z ( q ) , andbecause ϕ ∈ A ∨ is fixed by g k , we get a ( ϕ ) = z ( ¯ g k )( ϕ ) = a ( ϕ ) = z ( ¯ g )( ϕ ) z ( ¯ g )( g ( ϕ )) · · · z ( ¯ g )( g k − ( ϕ )) . (5.3)of (5.2) needs more effort. We shall prove a ( ϕ ) = a ( ϕ ) is 2 m for some m . This follows from the fact that A isa 2–group and that all the values of α and of z ( q ) can be chosen to have values which are 2 l –throots of unity, for some l . (In case the group A has odd order we can choose a representativeof [ α ] which is Q –invariant as a function. It is then easy to prove that in this case all Adamsoperations are being preserved.) On the other hand, we will show now that a ( ϕ ) k =
1. Since k is odd, this will finish the proof.To prove a ( ϕ ) k =
1, we first extend the 2–cocycle α in the following way: if ϕ , ϕ , . . . ϕ k alllie in A ∨ , then we define α ( ϕ , ϕ , . . . , ϕ k ) = α ( ϕ , ϕ ) α ( ϕ ϕ , ϕ ) . . . α ( ϕ ϕ · · · ϕ k − , ϕ k ) . In other words, if we think of the 2–cocycle α as realizing an extension1 −→ K × −→ Γ −→ A ∨ −→ s ( ϕ ) · · · s ( ϕ k ) = α ( ϕ , . . . , ϕ k ) s ( ϕ ϕ · · · ϕ k ) where the map ϕ (cid:55)→ s ( ϕ ) isa set-theoretical section of the projection Γ → A ∨ .We have the formula z ( ¯ g )( ϕ ) z ( ¯ g )( ψ ) z ( ¯ g )( ϕψ ) = α ( ϕ , ψ ) α ( g ( ϕ ) , g ( ψ )) . By using it repeatedly, we get a ( ϕ ) = α ( ϕ , g ( ϕ ) , . . . , g k − ( ϕ )) α ( g ( ϕ ) , . . . , g k − ( ϕ ) , ϕ ) . We have also used here the fact that z ( ¯ g )( ϕ g ( ϕ ) · · · g k − ( ϕ )) =
1. This is because the homo-morphism ϕ g ( ϕ ) · · · g k − ( ϕ ) is g –invariant, and by Lemma 5.2 we can assume that the valueof z ( ¯ g ) on it is 1. 15or every j = , . . . , k − a ( ϕ ) = z ( ¯ g )( g j ( ϕ ) · · · z ( ¯ g )( g j + k − ( ϕ ))) . By doing the same calculation again, we get a ( ϕ ) = α ( g j ( ϕ ) , . . . , g j + k − ( ϕ )) α ( g j + ( ϕ ) , . . . , g j + k ( ϕ )) for every j . We multiply all these expression for a ( ϕ ) . They cancel each other, and we areleft with a ( ϕ ) k =
1. This finishes the proof.
The following examples illustrate the subtlety of the situation.
Example 6.1.
Let D again denote the dihedral group of order 8. We will describe a (non-symmetric) monoidal autoequivalence of Rep ( D ) which does not preserve Ψ . We will usethe following presentation:D = (cid:104) x , y , q | x = y = q = , [ x , y ] = , [ q , y ] = , [ q , x ] = y (cid:105) . The subgroup A = (cid:104) x , y (cid:105) is a normal subgroup isomorphic to the Klein group. Let us denotea dual pair of generators of A ∨ by ν , µ . The group H ( A ∨ ; K × ) has order 2. A gener-ator is given by α ( ν a µ b , ν c µ d ) = ( − ) bc . This cohomology class must be G / A –invariant.We choose z ( q )( ν a µ b ) = i b . We then get that z satisfies the condition of Lemma 5.2.The resulting 2–cocycle b is given by b ( q , q ) = y . Notice that b is a trivial cocycle sincein G the equation ( qx ) = y holds and therefore the function G → G b given by send-ing x i y j q k to x i + k y j q k is an isomorphism of groups. However, the resulting monoidal equiv-alence Rep ( K G ) → Rep ( K G b ) → Rep ( K G ) is not symmetric and it does not preserve thesecond Adams operation. Indeed, the group G has four 1-dimensional irreducible represen-tations V , V , V , V and a unique two dimensional irreducible representation W . The ele-ment x i y j q k acts on V ab by the scalar ( − ) ai + kb . A direct calculation shows that Ψ ( W ) = V + V + V − V while F ( W ) = W and F ( Ψ ( W )) = V − V + V + V is different from Ψ ( F ( W )) . Example 6.2.
There are examples of monoidal equivalences which preserve all Adams opera-tions, but which are not symmetric. Indeed, there are Etingof–Gelaki equivalences that preserveall Adams operations without being symmetric. To see this, consider the case where the quo-tient Q is trivial but the subgroup A is not. The cocycle b is then of course trivial, and thefunctor F b preserves trivially all the Adams operations. However, since the abelian subgroup A is not trivial, α is not trivial, and the functor F b is not symmetric. The smallest such exampleis the Klein 4–group C × C . This group has four 1–dimensional irreducible representations,which we shall denote by V i j for i , j ∈ { , } . On the category Rep ( K ( C × C )) we have twodifferent symmetries. The canonical symmetry V i j ⊗ V kl ∼ = V kl ⊗ V i j is given by x ⊗ y (cid:55)−→ y ⊗ x , x ⊗ y (cid:55)−→ ( − ) il + jk y ⊗ x . However, with respect to both symmetries, the Adams operation Ψ k is the identity for everyodd k , and Ψ k ( V i j ) = V for every even k . Remark 6.3.
In all examples of monoidal equivalences F : Rep ( G ) → Rep ( G (cid:48) ) that preserveall Adams operations that we have seen so far, the groups G and G (cid:48) are isomorphic. If F issymmetric, then this is clear: an isomorphism is induced by the functor F , because the groups G and G (cid:48) are isomorphic to the automorphism groups of the fiber functors. In the Examples 6.2,we have G ∼ = G (cid:48) by construction.The preceding remark may be taken as evidence for an affirmative answer to the following. Question 6.4.
Suppose that G and G (cid:48) are two isocategorical groups, and assume that theinduced isomorphism R ( G ) ∼ = R ( G (cid:48) ) of representation rings arising from a monoidal equiv-alence Rep ( G ) → Rep ( G (cid:48) ) of representation categories preserves (not only the odd Adamsoperations but also) the second Adams operation Ψ . Do the groups G and G (cid:48) have to beisomorphic?Here is more that may be considered in support of an affirmative answer: Remark 6.5.
Let G be a group, and let V and W be two G –representations. We will usethe notation of Section 3. If G b is a group such that G and G b are isocategorical we havetwo corresponding G b –representations V b and W b . A slight variation of the results of Sec-tion 2.3 allows us to deduce the following: for every permutation σ in the symmetric group Σ n ,we have χ U ( σ ) = tr ( σ | U ) = tr ( σ | U b ) = χ U b ( σ ) when we consider the action of σ on thespaces U = Hom G ( V , W ⊗ n ) and U b = Hom G b ( V b , W ⊗ nb ) . Since the permutation groups Σ n arefinite and the ground field is of characteristic zero, this already means that U and U b are actuallyisomorphic as Σ n –representations. Acknowledgment
Both authors were supported by the Danish National Research Foundation through the Centrefor Symmetry and Deformation (DNRF92). The first author was also supported by the ResearchTraining Group 1670 “Mathematics Inspired by String Theory and Quantum Field Theory.” Wethank Alexei Davydov, Lars Hesselholt, Victor Ostrik, and Bj¨orn Schuster for discussions, andthe referee for their valuable comments on the exposition.
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