Pseudocharacters of Homomorphisms into Classical Groups

M. Weidner∗

Computer Science DepartmentCarnegie Mellon UniversityPittsburgh, PA 15213, USAmaweidne@andrew.cmu.edu

April 8, 2020

Abstract

A GLd-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relationsthat make it “look like” the character of a representation. When k is an algebraically closedfield of characteristic 0, Taylor proved that GLd-pseudocharacters of Γ are the same as degree-dcharacters of Γ with values in k, hence are in bijection with equivalence classes of semisimplerepresentations Γ → GLd(k). Recently, V. Lafforgue generalized this result by showing that, forany connected reductive group H over an algebraically closed field k of characteristic 0 and forany group Γ, there exists an infinite collection of functions and relations which are naturally inbijection with H(k)-conjugacy classes of semisimple homomorphisms Γ → H(k). In this paper,we reformulate Lafforgue’s result in terms of a new algebraic object called an FFG-algebra. Wethen define generating sets and generating relations for these objects and show that, for all Has above, the corresponding FFG-algebra is finitely presented up to radical. Hence one canalways define H-pseudocharacters consisting of finitely many functions satisfying finitely manyrelations. Next, we use invariant theory to give explicit finite presentations up to radical of theFFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonalgroups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of homomorphisms, following Larsen.

Introduction

Pseudocharacters were originally introduced for GL2 by Wiles [Wil] and generalized to GLn byTaylor [Tay]. Taylor’s result on GLn-pseudocharacters is as follows. Let Γ be a group and k be acommutative ring with identity. Define a GLn-pseudocharacter of Γ over k to be a set map T : Γ→ ksuch that

• T (1) = n

• For all γ1, γ2 ∈ Γ, T (γ1γ2) = T (γ2γ1)

• For all γ1, . . . , γn+1 ∈ Γ, ∑σ∈Sn+1

sgn(σ)Tσ(γ1, . . . , γn+1) = 0, (1)

where Sn+1 is the symmetric group on n + 1 letters, sgn(σ) is the permutation sign of σ, andTσ is defined by

Tσ(γ1, . . . , γn+1) = T (γi(1)1

· · · γi(1)r1

) · · ·T (γi(s)1

· · · γi(s)rs

)

where σ has cycle decomposition (i(1)1 . . . i

(1)r1 ) · · · (i(s)1 . . . i

(s)rs ).

∗Partially supported by Caltech’s Samuel P. and Frances Krown SURF Fellowship and a Churchill Scholarship fromthe Winston Churchill Foundation of the USA.

1

If T is a GLn-pseudocharacter, then define the kernel of T by

ker(T ) = η ∈ Γ | T (γη) = T (γ) for all γ ∈ Γ.

Then:

Theorem 1 ([Tay, Theorem 1]). 1. Let ρ : Γ → GLn(k) be a representation. Then tr(ρ) is aGLn-pseudocharacter.

2. Suppose k is a field of characteristic 0, and let ρ : Γ → GLn(k) be a representation. Thenker(tr(ρ)) = ker(ρss), where ρss denotes the semisimplification of ρ.

3. Suppose k is an algebraically closed field of characteristic 0. Let T : Γ → k be a GLn-pseudocharacter. Then there is a semisimple representation ρ : Γ→ GLn(k) such that tr(ρ) =T , unique up to conjugation.

4. If Γ and k are taken to be topological, then the above statements hold in topological/continuousform.

Taylor used GLn-pseudocharacters to construct Galois representations having certain properties[Tay, §2].

Recently, V. Lafforgue formulated an analogue of GLn-pseudocharacters that work with GLnreplaced by any reductive group under conjugation by its identity component. However, instead ofconsisting of one function T : Γ→ k satisfying a finite number of relations, these “pseudocharacters”consist of an infinite sequence of algebra morphisms satisfying certain properties. These sequencesof morphisms are essentially equivalent to specifying an infinite number of functions Γm → k, withm ranging over all natural numbers, satisfying an infinite number of relations. To state Lafforgue’stheorem, we adopt the convention that reductive groups are not necessarily connected, with H0

denoting the identity component of H.

Theorem 2 ([Laf, Proposition 11.7]). Let Γ be a topological group, k be a topological field of char-acteristic 0 such that k has a topology extending the topology on k, and H be a reductive group overk such that H0 is split over k.

For n ∈ N, let k[Hn]AdH0denote the k-algebra of regular functions on Hn that are invariant

under the action of H0 on Hn by diagonal conjugation, and let C(Γn, k) denote the k-algebra ofcontinuous set maps Γn → k.

Assume that we have for any n ∈ N a k-algebra morphism

Ξn : k[Hn]AdH0 → C(Γn, k)

such that

(a) For any m,n ∈ N, set map ζ : 1, . . . ,m → 1, . . . , n, f ∈ k[Hm]AdH0, and γ1, . . . , γn ∈ Γ,

Ξn(f ζ)(γ1, . . . , γn) = Ξm(f)(γζ(1), . . . , γζ(m)),

where f ζ ∈ k[Hn]AdH0is defined by

f ζ(A1, . . . , An) = f(Aζ(1), . . . , Aζ(m))

(b) For any n ∈ N, f ∈ k[Hn]AdH0, and γ1, . . . , γn+1 ∈ Γ,

Ξn+1(f)(γ1, . . . , γn+1) = Ξn(f)(γ1, . . . , γn−1, γnγn+1),

where f ∈ k[Hn+1]AdH0is defined by

f(A1, . . . , An+1) = f(A1, . . . , An−1, AnAn+1).

2

Then there exists a continuous group homomorphism ρ : Γ → H(k′) for some finite extension k′ ofk (with H(k′) inheriting its topology from k′), such that the Zariski closure of Im(ρ) is a reductivesubgroup of H(k′), and such that for any n ∈ N, f ∈ k[Hn]AdH0

, and γ1, . . . , γn ∈ Γn, we have

f(ρ(γ1), . . . , ρ(γn)) = Ξn(f)(γ1, . . . , γn).

Moreover, ρ is unique up to conjugation by H0(k).

Remark 1. Lafforgue’s original statement requires Γ to be profinite and k to be a finite extension ofQl for some l, with the standard topologies. These conditions are not used in the proof, so we omitthem.

Lafforgue also shows how to derive Taylor’s result from the above theorem [Laf, Remark 11.8],using results of Procesi [Pro2] that state that the trace function “generates” all of the algebrask[GLmn ]AdGLn and that explicitly describe all of the relations between these trace functions.

Outline

In Section 1, we reformulate Lafforgue’s result in terms of a new algebraic structure called an FFG-algebra. Collections of morphisms Ξn as above are recast as morphisms between certain FFG-algebras. We then use the finiteness theorems of classical invariant theory and facts about reductivegroups to show that, for any connected reductive group H defined over a field of characteristic 0, theFFG-algebra derived from the invariants of H is finitely presented up to radical. Hence it is alwayspossible to define H-pseudocharacters consisting of finitely many functions Γm → k satisfying finitelymany relations.

In Section 2, we use invariant theoretic-results of Procesi and others [Pro2, ATZ, Rog] to giveexplicit finite presentations up to radical of the FFG-algebras corresponding to the general andordinary orthogonal groups GOn and On, the general and ordinary symplectic groups GSp2n andSp2n, and the special orthogonal group SOn. By extension, we define explicit pseudocharacters forthese groups.

Finally, in Section 3, we use our pseudocharacters to investigate the problem of conjugacy vs.element-conjugacy for semisimple homomorphisms Γ→ H(k), where H is a linear algebraic group forwhich one can define pseudocharacters and k is an algebraically closed field of characteristic 0. Weformulate a general condition in terms of FFG-algebras under which element-conjugacy implies con-jugacy. We then use our explicit pseudocharacters for GOn(k), On(k), GSp2n(k), Sp2n(k), SO2n+1(k)and SO4(k) to prove that for any group Γ, element-conjugate semisimple homomorphisms from Γ toone of those groups are automatically conjugate. Previous results of this form were only known forOn(C), Sp2n(C), SO2n+1(C), and SO4(C), and only for compact Γ. We also give a counterexampleto the corresponding claim for SO2n(k) (n ≥ 3) that is simpler than the one in [Lar1, Proposition3.8], and which extends that result to SO6(k).

1 General Results on Pseudocharacters

1.1 F-, FFS-, and FFG-algebras.

We begin by defining new algebraic objects which we call F-, FFS-, and FFG-algebras, modeled afterthe FI-modules defined in [CEF]. For every nonempty finite set I, let FS(I) (resp. FG(I)) denotethe free semigroup (resp. group) generated by I. We denote by F, FFS, and FFG the categorieswhose objects are finite sets and whose morphisms I → J are: for F, set functions I → J ; for FFS,semigroup homomorphisms FS(I)→ FS(J); and for FFG, group homomorphisms FG(I)→ FG(J).

The following lemma is easy.

Lemma 3. The category FFS is generated by the following two types of morphisms:

• morphisms FS(I) → FS(J) that sends generators to generators, i.e., those induced by mapsbetween finite sets I → J

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• morphisms

FS(x1, . . . , xn)→ FS(y1, . . . , yn+1), xi 7→ yi(i < n), xn 7→ ynyn+1.

The category FFG is generated by the above two types of morphisms (with FS replaced by FG) togetherwith:

• morphisms

FG(x1, . . . , xn)→ FG(y1, . . . , yn), xi 7→ yi(i < n), xn 7→ y−1n .

Definition 1. Fix a commutative ring k. An F-algebra (resp. FFS-algebra, FFG-algebra) is acovariant functor from F (resp. FFS, FFG) to the category of k-algebras. Morphisms between F-algebras (resp. FFS-algebras, FFG-algebras) are natural transformations of functors.

If A• is an F-algebra (resp. FFS-algebra, FFG-algebra) and I is a finite set, we use AI to denotethe k-algebra corresponding to I under A•, and similarly for morphisms Θ• : A• → B•. For n ∈ N,we use An to denote A1,...,n. If φ : I → J (resp. FS(I)→ FS(J), FG(I)→ FG(J)) is a morphism,then we use Aφ to denote the corresponding k-algebra morphism AI → AJ .

We can define kernels, subobjects, quotients, and tensor products over k in the category of F-algebras (resp. FFS-algebras, FFG-algebras) by using the analogous constructions in the category ofk-algebras, applying those constructions to each k-algebra in the image of an F-algebra. We say thata morphism Θ• is surjective if each ΘI is surjective.

Remark 2. Any FFG-algebra is naturally an FFS-algebra, and any FFS-algebra is naturally an F-algebra, due to the clear functors F→ FFS→ FFG. A morphism of FFG-algebras is also a morphismof FFS-algebras, and a morphism of FFS-algebras is also a morphism of F-algebras.

Example 1. Let Γ be a group and R be a k-algebra. We define an FFG-algebra Map(Γ•, R)as follows. To the finite set I, we associate Map(ΓI , R), the k-algebra of all set maps ΓI → R.Next, recall that for any finite set I, ΓI = Hom(FG(I),Γ). Thus for any group homomorphismφ : FG(I) → FG(J), we have a natural set map ΓJ → ΓI , which induces a k-algebra morphismMap(ΓI , R)→ Map(ΓJ , R); we associate this morphism to φ. When Γ and R are topological, we cananalogously define an FFG-algebra C(Γ•, R) by restricting to continuous maps.

Example 2. Let V be an affine variety over k, and let H be a group which acts on V . We define theF-algebra k[V •]H by the association I 7→ k[V I ]H , where H acts diagonally on V I . For any set mapφ : I → J , we get a variety map V J → V I defined over k, and this induces a k-algebra morphismk[V I ]H → k[V J ]H , which we associate to φ. If V is also an algebraic semigroup (resp. group) whosemultiplication is compatible with the action of H, then we can similarly give k[V •]H a structure ofFFS-algebra (resp. FFG-algebra). Specifically, given a semigroup homomorphism φ : FS(I)→ FS(J)(resp. group homomorphism φ : FG(I)→ FG(J)), letting I = x1, . . . , xn and J = y1, . . . , ym, wedefine a variety map

V (φ) : V J → V I

(A1, . . . , Am) 7→ (φ(x1)yi 7→ Ai, . . . , φ(xn)yi 7→ Ai) ,

where φ(xj)yi 7→ Ai denotes the point of V obtained by substituting each Ai for yi in φ(xj) andmultiplying using the semigroup (resp. group) operation. Then V (φ) induces a k-algebra morphismk[V I ]H → k[V J ]H , which we associate to φ.

For the remainder of this section, we state definitions and claims for F-algebras, but they easilygeneralize to FFS- and FFG-algebras.

Definition 2. Let A• be an F-algebra. The arity of an element f ∈ AI is |I|. We use this termbecause in our examples, generally elements of AI are functions of arity |I|.

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Definition 3. Let A• be an F-algebra. Given a subset Σ ⊂ tIAI , the F-algebra span of Σ in A• isdefined to be the minimal sub-F-algebra of A• containing each element of Σ. An F-algebra is finitelygenerated if it equals the span of some finite set.

There is another way to characterize finite generation, in terms of free F-algebras.

Definition 4. Let m ∈ N. The free F-algebra of arity m, denoted FF(m)•, is defined by

FF(m)I = k[xψ | ψ ∈ HomF(1, . . . ,m, I)]FF(m)φ = (xψ 7→ xφψ).

In the case of FFS-algebras (resp. FFG-algebras), we replace HomF(1, . . . ,m, I) withHomFFS(FS(1, . . . ,m),FS(I)) (resp. HomFFG(FG(1, . . . ,m),FG(I))).

It is easy to see that FF(m)• has the universal property: if A• is an F-algebra and a ∈ Am, thenthere is a unique F-algebra morphism FF(m)• → A• mapping xid1,...,m to a. Furthermore, the imageof this morphism is precisely the span of a in A•. Thus:

Proposition 4. An F-algebra A• is finitely generated iff it admits a surjective morphism⊗

i FF(mi)→A• for some finite sequence of integers (mi).

Definition 5. Let A• be an F-algebra. An F-ideal of A• is an association a• taking each finite setI to an ideal aI of AI , such that for all morphisms φ ∈ HomF(I, J), we have Aφ(aI) ⊂ aJ . Givena morphism of F-algebras Θ• : A• → B•, we define the kernel of Θ• to be the association ker(Θ•)taking each finite set I to the ideal ker(ΘI : AI → BI) of AI . We define the radical of an F-ideala• to be the association

√a• : I 7→

√aI , where the radical of aI is taken in AI . Easily kernels and

radicals are F-ideals.

Definition 6. Let A• be an F-algebra. Given a subset Σ ⊂ tIAI , we define the F-ideal generatedby Σ to be the minimal F-ideal of A• containing each element of Σ. We define an F-ideal to befinitely generated if it is generated by some finite set. We call an F-algebra A• finitely presented ifit admits a surjective morphism π• :

⊗i FF(mi)→ A• for some finite sequence of integers (mi) such

that ker(π•) is finitely generated. We call A• finitely presented up to radical if ker(π•) =√I• for

some finitely generated F-ideal I•.

1.2 Pseudocharacters from Lafforgue’s Result.

Let H be a reductive group defined over a topological field k, and let Γ be a topological group. LetH0 denote the identity component of H (in the Zariski topology). For any finite set I, let AdH0

denote the diagonal conjugation action of H0 on HI , and let k[H•]AdH0denote the FFG-algebra

in Example 2 corresponding to this action. Call a homomorphism ρ : Γ → H(k) semisimple if theZariski closure of Im(σ) in H(k) is reductive. Then from V. Lafforgue’s result, we derive the followinggeneralization of Taylor’s pseudocharacters.

Theorem 5. (1) Let ρ : Γ → H(k) be a continuous (with the k-topology on H(k)) homomorphism.Then we have an FFG-algebra morphism

Θ• : k[H•]AdH0 → C(Γ•, k)

given byΘn(f)(γ1, . . . , γn) = f(ρ(γ1), . . . , ρ(γn)).

(2) Conversely, suppose k is a field of characteristic 0, and let k have a topology extending thetopology on k. Let

Θ• : k[H•]AdH0 → C(Γ•, k)

5

be an FFS-algebra morphism. Then there is a finite extension k′ of k and a continuous semisimplehomomorphism ρ : Γ→ H(k′) such that

Θn(f)(γ1, . . . , γn) = f(ρ(γ1), . . . , ρ(γn)).

Moreover, ρ is unique up to conjugation by H0(k). Note that by (1), Θ• is also an FFG-algebramorphism.

(3) Suppose k is a field of characteristic 0 and H is connected. Let ρ : Γ → H(k) be a semisimplehomomorphism. Then

ker(ρ) =η ∈ Γ | for all n ∈ N, f ∈ k[Hn]AdH , 1 ≤ i ≤ n, and γ1, . . . , γn ∈ Γ,

f(ρ(γ1), . . . , ρ(ηγi), . . . , ρ(γn)) = f(ρ(γ1), . . . , ρ(γn)).

Remark 3. When Γ and k are not topological, we can give them the discrete topology and then applythe above theorem, giving an analogous result with C(Γ•, k) replaced by Map(Γ•, k).

Proof. (1) This is easily checked.

(2) It suffices to prove the claim with k replaced by a finite extension, so without loss of generalityH0 is split over k. Then this follows from Lafforgue’s result [Laf, Proposition 11.7], as statedin Theorem 2 above, noting that the conditions on the Ξn in that result are precisely the FFS-algebra morphism conditions for the generators in Lemma 3.

(3) Let ∆ be the right-hand set. Easily ∆ is a normal subgroup of Γ, so we can form the quotientΓ = Γ/∆. Let Θ• be the FFG-algebra morphism corresponding to ρ in (1). Then by definitionof ∆, Θ• restricts to give a well-defined FFG-algebra morphism

Θ•

: k[H•]AdH → Map(Γ•, k).

Hence by (2), we have a corresponding semisimple homomorphism ψ : Γ → H(k). Composingwith the quotient map gives a continuous semisimple homomorphism ψ : Γ → H(k). From thedefinition of ∆, ψ has trivial kernel, so ker(ψ) = ∆. Finally, by uniqueness in (2), ψ = ρ.

In Remark 5 below, we prove that this result still holds if we replace H0 by an arbitrary connectedreductive group K such that H0 ⊂ K and K normalizes H, i.e., K ⊂ NG(H) for some ambientreductive group G containing H and K.

1.3 Explicit Descriptions of Pseudocharacters.

Let k be a field of characteristic 0, and let H be a reductive group over k. We now show thatthe FFG-algebra k[H•]AdH0

is finitely presented up to radical. More generally, k[H•]AdK is finitelypresented up to radical whenever K is a reductive group normalizing H (Remark 5).

As a consequence, when H is connected, i.e., H = H0, it is always possible to define pseu-docharacters for H very explicitly. Indeed, if k[H•]AdH has a finite presentation up to radicalwith generators f1, . . . , fa of arities n1, . . . , na and generating relations R1, . . . , Rb, then to spec-ify an FFG-algebra morphism k[H•]AdH → C(Γ•, k), it is equivalent to specify continuous set mapsF1 : Γn1 → k, . . . , Fa : Γna → k satisfying the relations R1, . . . , Rb. Hence we can define explicit pseu-docharacters for H by finding a finite presentation up to radical of k[H•]AdH . This technique was firstdemonstrated in [Laf, Remark 11.8], in which V. Lafforgue implicitly gives a finite presentation up toradical of k[GL•n]AdGLn and explains how it implies Taylor’s original result on GLn-pseudocharacters.We further illustrate this technique with examples in Section 2 below.

Lemma 6. Let H0 act linearly on a finite-dimensional k-vector space V . Then the F-algebra k[V •]H0

is finitely presented.

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Proof. Finite generation follows from a strong form of the first fundamental theorem of invarianttheory, which is classical; see, e.g., [PV, Corollary on p. 253]. Finite presentation follows from astrong form of the second fundamental theorem of invariant theory proven by Schwarz [Sch, Theorem2.5(2)]1.

From this result, we easily deduce that k[H•]AdH0is finitely generated (see the proof of Theorem

9). It remains to prove a finiteness result for the relations between the generators. In the classical caseof a reductive group acting on a single vector space, finite generation of relations follows immediatelyfrom the Noetherian property of finitely generated k-algebras. However, such a Noetherian propertydoes not hold for finitely generated F-algebras. Nagel and Romer [NR, Proposition 4.8] show thatthe free FI-algebra FFI(m) is not Noetherian for all m ≥ 2, where FI denotes the category of finitesets with injective maps, and their proof easily generalizes to F-, FFS-, and FFG-algebras.

Instead, we reason about k[H•]AdH0in particular, using properties of reductive groups. We start

with two lemmas.

Lemma 7. Assume k is algebraically closed. Then for all d ∈ N, there exists qd ∈ N such that anyalgebraic subgroup G ⊂ GLd(k) is generated by at most qd elements as an algebraic group, i.e., thereexist g1, . . . gqd ∈ G such that G = 〈g1, . . . , gqd〉, where the closure is taken in the Zariski topology.

Proof. By a result of Mostow, G is the semidirect product of a reductive group and a unipotentgroup, namely, a Levi subgroup and the unipotent radical of G [Hoc, Theorem VIII.4.3]. Hence itsuffices to prove the lemma when G is reductive or unipotent.

First, consider the case that G is reductive. By [Vin, Propositions 2 and 7], we reduce to thecase that G is finite. By Jordan’s theorem on finite linear groups, we reduce to the case that G isfinite and abelian. Then the faithful representation of G on kd decomposes as a sum of 1-dimensionalrepresentations, so after conjugating, we can assume G is a subgroup of the diagonal group (k×)d.Since G is finite, it is isomorphic to a subgroup of µd ∼= (Q/Z)d, hence to a subgroup of (Z/nZ)d forsome n. Any such subgroup is generated by at most d elements.

Now consider the case that G is unipotent. Then G has a composition series (as an algebraicgroup) G = U1 ⊃ · · · ⊃ Us = e in which all of the quotients Ui/Ui+1 are isomorphic to Ga, theadditive group of k. Any non-identity element of Ga(k) topologically generates Ga, since such anelement generates an infinite subgroup of Ga(k) and all proper Zariski closed subgroups of Ga(k)are finite. Thus G is topologically generated by a set containing one element from each Ui \ Ui+1,which has size s− 1 = dimG. But dimG is bounded by dimGLd.

Let d be such that H is an affine subgroup variety of GLd. Let GLnd//AdH0 denote the variety

Spec(k[GLnd ]AdH0), and similarly for Hn//AdH0. Let πnH0 : GLnd → (GLnd//AdH0) be the natural

projection. For any group homomorphism φ : FG(x1, . . . , xm) → FG(y1, . . . , yn), we get a mapV (φ) : (GLnd//AdH0) → (GLmd //AdH0) from the k-algebra morphism (k[GL•d]

AdH0)φ. Concretely,

for A1, . . . , An ∈ GLd(k), V (φ) sends πnH0(A1, . . . , An) to πmH0(φ(x1)yi 7→ Ai, . . . , φ(xm)yi 7→ Ai),where φ(xj)yi 7→ Ai denotes the element of GLd(k) obtained by substituting each Ai for yi in φ(xj).

Lemma 8. Assume k is algebraically closed. Let d be such that H is an affine subgroup variety ofGLd, and let qd be the constant from Lemma 7 for GLd. Suppose x ∈ GLnd//AdH0 is such that forall group homomorphisms φ : FG(x1, . . . , xqd) → FG(y1, . . . , yn), V (φ)(x) ∈ (Hqd//AdH0) ⊂(GLqdd //AdH0). Then x ∈ Hn//AdH0.

Proof. It is a well-known property of πnH0 that the preimage of any element of GLnd//AdH0 in GLndcontains a unique closed orbit under conjugation by H0. Thus we can find a preimage (A1, . . . , An) ∈GLnd (k) of x whose orbit under conjugation by H0 is closed in GLnd . By the previous lemma, we

can find B1, . . . , Bqd ∈ GLd(k) such that 〈A1, . . . , An〉 = 〈B1, . . . , Bqd〉. Each Bi is in the closure of〈A1, . . . , An〉, so πqd

H0(B1, . . . , Bqd) is in the closure of πqdH0(C1, . . . , Cqd) | C1, . . . , Cqd ∈ 〈A1, . . . , An〉.

Then πqdH0(B1, . . . , Bqd) ∈ Hqd//AdH0 because all πqd

H0(C1, . . . , Cqd) ∈ Hqd//AdH0 by assumption.

1I thank C. Procesi and G. Schwarz for their assistance in locating this result.

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Next, the orbit of (B1, . . . , Bqd) is closed because the orbit of (A1, . . . , An) is closed: indeed, by[Ric1], the orbit of a tuple under conjugation by H0 is closed iff its stabilizer in H0 is reductive, andthe stabilizer of a tuple only depends on the algebraic subgroup it generates in GLd. This closedorbit must coincide with the unique closed orbit in the preimage of πqd

H0(B1, . . . , Bqd) in Hqd , so allBi ∈ H(k). Hence all Aj ∈ H(k), proving the lemma.

Theorem 9. The FFG-algebra k[H•]AdH0is finitely presented up to radical.

Proof. Let d be such that H is an affine subgroup variety of GLd over k. Then we have closedembeddings H → GLd →Md×A1, where Md is the variety of d× d matrices, A1 is one-dimensionalaffine space, and the last embedding is given byA 7→ (A,det(A)−1). These embeddings are compatiblewith the conjugation action by H. Then we get an F-algebra morphism π•1 : k[(Md × A1)•]AdH0 →k[GL•d]

AdH0and an FFG-algebra morphism π•2 : k[GL•d]

AdH0 → k[H•]AdH0. It is a standard fact

(see, e.g., [PV, p. 188]) that each πI1 and πI2 are surjective, so π•1 and π•2 are surjective.By Lemma 6, k[(Md×A1)•]AdH0

is finitely presented as an F-algebra. Next, ker(π•1) is generatedas an F-ideal by the relation det(matrix coordinate) · (affine coordinate) = 1 in k[Md × A1]AdH0

,so k[GL•d]

AdH0is finitely presented as an F-algebra. From such a finite presentation, together with

relations expressing f(A1, . . . , An−1, AnAn+1) and f(A1, . . . , An−1, A−1n ) in terms of the F-algebra for

each of the finitely many generators f , we get a finite presentation of k[GL•d]AdH0

as an FFG-algebra.Now to prove the theorem, it suffices to show that ker(π•2) is the radical of a finitely generated

FFG-ideal in k[GL•d]AdH0

. This is more difficult to show; while the kernel of the natural mapk[GL•d] → k[H•] is generated by ker(k[GLd] → k[H]), the same is not necessarily true once werestrict to the algebras of invariants.

It suffices to show this for k, so without loss of generality k is algebraically closed. Let qd be theconstant from Lemma 7 for GLd. Let I• be the FFG-ideal of k[GL•d]

AdH0generated by ker(πqd2 ). I•

is finitely generated, so it suffices to prove ker(π•2) =√I•. That is, we must show ker(πn2 ) =

√In for

all n. By the Nullstellensatz, it is equivalent to show that ker(πn2 ) and In define the same subvarietyof Spec(k[GLnd ]AdH0

). This follows from Lemma 8.

Remark 4. The same proof shows that k[H•]AdH0is finitely presented up to radical as an FFS-algebra.

Remark 5. By modifying the above proofs, we can generalize our main result on pseudocharacters(Theorem 5) to the case when H0 is replaced by an arbitrary connected reductive group K suchthat H0 ⊂ K and K normalizes H, i.e., K ⊂ NG(H) for some ambient reductive group G containingH and K. Furthermore, the FFG-algebra k[H•]AdK is finitely presented up to radical; in fact, thisholds for any reductive group K normalizing H (not necessarily connected). Thus there exist explicitpseudocharacters for semisimple homomorphisms Γ→ H(k) considered up to conjugation by K(k).

To prove these results, first let K be any reductive group normalizing H (not necessarily con-nected). Let d be such that H and K are affine subgroup varieties of GLd. Observe the fol-lowing modification of Lemma 8: if x ∈ GLnd//AdK is such that for all group homomorphismsφ : FG(qd) → FG(n), V (φ)(x) ∈ Hqd//AdK, then x ∈ Hn//AdK. The proof is the same, notingthat Hqd//AdK is closed in GLnd//AdK because K normalizes H.

As in the proof of Theorem 9, it follows that the kernel of the natural surjective FFG-algebramorphism π• : k[GL•d]

AdK k[H•]AdK is generated by ker(πqd) up to radical. Hence k[H•]AdK isfinitely presented up to radical because k[GL•d]

AdK is, noting that the proof of Lemma 6 still holdswith K in place of H0.

Now let K be a connected reductive group such that H0 ⊂ K and K normalizes H. It remainsto prove Theorem 5 with K in place of H0. Claim (1) is easily checked, and claim (3) is unchanged.To prove claim (2), let L = KH, so that L0 = K, and let ψ• : k[L•]AdK k[H•]AdK be the naturalmap. From an FFG-algebra morphism Θ• : k[H•]AdK → C(Γ•, k), by Theorem 5 applied to L andΘ•ψ•, we get a continuous semisimple homomorphism ρ : Γ→ L(k′) for some finite extension k′ of k,unique up to conjugation by K(k), with K-invariants given by Θ•ψ•. In particular, the K-invariantssatisfy all relations in ker(ψqd). Then as in the proof of Lemma 8, a qd-tuple (B1, . . . , Bqd) generatingIm(ρ) in the Zariski topology projects to an element of Hqd//AdK. Also, since ρ is semisimple,(B1, . . . , Bqd) is semisimple in the sense of [Ric2], so the K-orbit of (B1, . . . , Bqd) is closed by [Ric2,

8

Theorem 3.6]. Then (B1, . . . , Bqd) ∈ Hqd(k) as in the proof of Lemma 8. Thus Im(ρ) ⊂ H(k′),proving claim (2).

2 Explicit Pseudocharacters for Classical Groups

2.1 (General) Orthogonal Group

We now present new results that establish pseudocharacters for the orthogonal and general orthogonalgroups. Let k be a field of characteristic 0.

Let GOn(k) = A ∈ Mn(k) | for some λ ∈ k×, AAt = λI be the n-dimensional general orthog-onal group. It is a connected reductive algebraic group. Define a function λ : GOn(k) → k× byAAt = λ(A)I. Then λ ∈ k[GOn]AdGOn .

Proposition 10. k[GO•n]AdGOn is generated as an FFG-algebra by the arity 1 functions tr and λ.

Proof. Since GOn(k) ⊃ On(k), k[M•n]AdGOn ⊂ k[M•n]AdOn . By Procesi’s results on the invariants ofOn(k) acting on matrices by conjugation [Pro2, Theorem 7.1], for all m, k[Mm

n ]AdOn is generated as ak-algebra by invariants tr(M), where M ∈ FS(A1, A

t1, . . . , Am, A

tm). The tr(M) are obviously also

GOn(k)-invariants, so k[Mmn ]AdGOn has the same generators. Then k[(Mn×A1)m]AdGOn is generated

as a k-algebra by the tr(M) and by the coordinate functions for the m copies of A1, which we willdenote det−1(A1), . . . ,det−1(Am).

Next, k[GOmn ]AdGOn is a quotient of k[(Mn × A1)m]AdGOn for all m because GOn is an affinesubvariety of GLn, so k[GOmn ]AdGOn is also generated by the invariants tr(M) and det−1(Ai). Thenusing the identity At = λ(A)A−1 for A ∈ GOn(k), we see that any invariant tr(M) is in the FFG-algebra generated by tr and λ. Also, using the identity det−1(A) = det(A−1) and the fact that wecan express det(A−1) in terms of tr(A−1), . . . , tr(A−n), we see that any invariant det−1(Ai) is in theFFG-algebra generated by tr.

The relations between the invariants are more complicated to describe. We first summarizeProcesi’s result on relations between the generators tr(M) of k[Mm

n ]AdGOn = k[Mmn ]AdOn .

LetR be the polynomial ring over k with indeterminates TM asM varies over FS(A1, At1, . . . , Am, A

tm),

except that we make the identifications TMN = TNM and TM = TMt for all words M and N ,where M t is defined by reversing the order of letters in M and swapping each Ai ↔ Ati. Letπ : R → k[Mm

n ]AdGOn be the k-algebra homomorphism sending each TM to tr(M), which by [Pro2,Theorem 7.1] is surjective.

Given M1,M2, . . . ,Mn+1 ∈ FS(A1, At1, . . . , Am, A

tm) and an integer 0 ≤ j ≤ (n + 1)/2, define

Fj,n+1(M1,M2, . . . ,Mn+1) ∈ R as follows. Let s be given by n + 1 = 2j + s. Let S be a set offormal symbols (a, b), where each a and b is one of the formal symbols u1, . . . , un+1, v1, . . . , vn+1. LetDj =

∑σ∈Sn+1

sgn(σ)Djσ be the following (n+ 1)× (n+ 1) determinant, as a function of symbols in

S: ∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(u1, uj+s+1) · · · (u1, un+1) (u1, vj+1) · · · (u1, vn+1)...

......

...(uj+s, uj+s+1) · · · (uj+s, un+1) (uj+s, vj+1) · · · (uj+s, vn+1)(v1, uj+s+1) · · · (v1, un+1) (v1, vj+1) · · · (v1, vn+1)

......

......

(vj , uj+s+1) · · · (vj , un+1) (vj , vj+1) · · · (vj , vn+1)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣Next, using the formal identities (a, b) = (b, a) and allowing the symbols (a, b) to commute with eachother, write each monomial Dj

σ of Dj in the form

Djσ = (w

i(1)1

, wi(1)2

)(wi(1)2

, wi(1)3

) · · · (wi(1)r1

, wi(1)1

) · (wi(2)1

, wi(2)2

)(wi(2)2

, wi(2)3

) · · · (wi(2)r2

, wi(2)1

) · · ·

where wa stands for either ua or va and we define ua = va and va = ua. Now define T jσ(M1, . . . ,Mn+1)by

T jσ(M1, . . . ,Mn+1) = TNi(1)1

Ni(1)2

...Ni(1)r1

TNi(2)1

Ni(2)2

...Ni(2)r2

· · ·

9

where Na = Ma or Na = M ta, according to the inductively defined rules:

• Ni(k)1

= Mi(k)1

, if wi(k)1

= vi(k)1

; else Ni(k)1

= M t

i(k)1

• Set Ni(k)t+1

to be the same type as Ni(k)t

(transposed or not transposed) if and only if wi(k)t

and

wi(k)t+1

stand for instances of the same letter (u or v).

ThenFj,n+1(M1, . . . ,Mn+1) =

∑σ∈Sn+1

sgn(σ)T jσ(M1, . . . ,Mn+1)

is the result of replacing each Djσ with T jσ in Dj . Because TMN = TNM and TM = TMt by assumption,

the functions T jσ are well-defined, hence so is Fj,n+1. Note that F0,n+1(M1, . . . ,Mn+1) reduces to (1),the non-trivial relation for GLn-pseudocharacters.

Theorem 11 ([Pro2, Theorem 8.4(a)]). ker(π) is the ideal of R generated by the Fj,n+1(M1, . . . ,Mn+1),0 ≤ j ≤ (n+ 1)/2, as the Mi vary over FS(A1, A

t1, . . . , Am, A

tm).

Now let ψ : R→ k[GOmn ]AdGOn be given by

ψ : Rπ−→ k[Mm

n ]AdGOn → k[(Mn × A1)m]AdGOn k[GOmn ]AdGOn .

Note that ψ is surjective by the proof of Proposition 10. Intuitively, one should expect ker(ψ) to beker(π) plus the relations of the form TNNtP = 1

nTNNtTP , since GOn is defined by the condition thatNN t is a scalar matrix for all N ∈ GOn(k). The next proposition shows that this is indeed the case,at least up to radical.

Proposition 12. ker(ψ) is the radical of the ideal generated by ker(π) and the relations TNNtP −1nTNNtTP for N,P ∈ FS(A1, A

t1, . . . , Am, A

tm).

Proof. It suffices to show this for k, so without loss of generality k is algebraically closed. Let J ⊂ Rbe the ideal generated by ker(π) and the TNNtP − 1

nTNNtTP . It suffices to prove that π(ker(ψ)) =√π(J). Now

√π(ker(ψ)) = π(ker(ψ)) because k[Mm

n ]AdGOn/π(ker(ψ)) ∼= k[GOmn ]AdGOn is reduced,so by the Nullstellensatz, it suffices to prove that π(ker(ψ)) and π(J) define the same subvariety ofSpec(k[Mm

n ]AdGOn). Using the tr(M) as coordinate functions for Spec(k[Mmn ]AdGOn), the subvariety

associated to π(ker(ψ)) is the set of all points of the form (tr(MAi 7→ Bi))M∈FSA1,At1,...,Am,At

mfor some B1, . . . , Bm ∈ GOn(k), where MAi 7→ Bi denotes the element of GOn(k) obtained bysubstituting each Bi for Ai in M . Meanwhile, the subvariety associated to π(J) is the set of allpoints of the form (tr(MAi 7→ Ci))M∈A1,At

1,...,Am,Atm where C1, . . . , Cm ∈ Mn(k) are such that

tr(NN tP ) = 1ntr(NN t)tr(P ) whenever N and P are semigroup words in the Ci and Cti . The following

lemma shows that these two subvarieties are equal, proving the proposition.

Lemma 13. Let C1, . . . , Cm ∈ Mn(k) be such that tr(NN tP ) = 1ntr(NN t)tr(P ) whenever N and

P are semigroup words in the Ci and Cti . Then there exist B1, . . . , Bm ∈ GOn(k) such that for allM ∈ FS(A1, A

t1, . . . , Am, A

tm), tr(MAi 7→ Bi) = tr(MAi 7→ Ci).

Proof. Let (V,B) be the bilinear space with V ∼= kn

and B the standard nondegenerate symmetricbilinear form, i.e., the dot product. Let A be the noncommutative k-algebra

A = k[C1, . . . , Cm, Ct1, . . . , C

tm] ⊂Mn(k),

which has the natural involution (−)t. Then the natural representation ρ : A →Mn(k) ∼= End(V,B)is orthogonal, i.e., it preserves involutions.

Then by [Pro2, Theorem 15.2(b)(c)] and the fact that all nondegenerate bilinear forms on Vare equivalent, there exists a semisimple orthogonal representation ρss : A → End(V ) such that

10

tr(ρ) = tr(ρss). Thus setting Bi = ρss(Ci), we will be done once we prove that Bi ∈ GOn(k). Nowfor any D ∈ A, we have

tr((BiBti −

1

ntr(BiB

ti)I)ρss(D)) = tr(ρss((CiC

ti −

1

ntr(CiC

ti )I)D))

= tr(ρ((CiCti −

1

ntr(CiC

ti )I)D))

= tr((CiCti )D)− 1

ntr(CiC

ti )tr(D)

= 0

by assumption. Since ρss is semisimple, tr is a nondegenerate bilinear form on Im(ρss), so this showsthat BiB

ti − 1

ntr(BiBti)I = 0, proving the lemma.

Next, let S be the polynomial ring over k with indeterminates:

• UQ for Q ∈ FG(A1, . . . , Am), with the identifications U1 = n and UQR = URQ for all wordsQ, R

• lQ for Q ∈ FG(A1, . . . , Am), with the identifications l1 = 1 and lQR = lQlR for all words Q,R.

We have a surjective map ρ : S → k[GOmn ]AdGOn defined by ρ(UQ) = tr(Q) and ρ(lQ) = λ(Q).

Proposition 14. ker(ρ) is the radical of the ideal generated by the relations:

• UQ − lQUQ−1 for Q ∈ FG(A1, . . . , Am)

• Gj,n+1(Q1, . . . , Qn+1), 0 ≤ j ≤ (n + 1)/2, as the Qi vary over words in FG(A1, . . . , Am),defined as follows. First, define G′j,n+1(X1, . . . , Xn+1) to be the same as Fj,n+1(X1, . . . , Xn+1)except that we replace each TM , M ∈ FS(X1, X

t1, . . . , Xn+1, X

tn+1), with lM ′UM ′, where M ′ ∈

FG(X1, . . . , Xn+1) is the result of substituting all transposed letters Xti in M with X−1i . Then

Gj,n+1(Q1, . . . , Qn+1) = (G′j,n+1(X1, . . . , Xn+1))Xi 7→ Qi.

Proof. Let J ⊂ S be the ideal generated by relations of the form UQ − lQUQ−1 . Then ρ inducesa surjective map ρ : S/J k[GOmn ]AdGOn . Easily ψ = ρ τ where τ : R → S/J is defined by:τ(TM ) = lM ′′UM ′ , where M ′ is the result of substituting all transposed letters Ati in M with A−1i ,and M ′′ is the product (with multiplicity) of all letters A1, . . . , Am that appear transposed in M .Then ker(ρ) = J + ker(ρ) = J + τ(ker(ψ)).

The ideal J corresponds to the relations UQ − lQUQ−1 . Applying τ to the relations TNNtP −1nTNNtTP from Proposition 12 yields 0.

It remains to show that τ(ker(π)) is the ideal generated by the relations Gj,n+1(Q1, . . . , Qn+1). Let0 ≤ j ≤ (n+ 1)/2 and let M1, . . . ,Mn+1 ∈ FS(A1, A

t1, . . . , Am, A

tm), so that Fj,n+1(M1, . . . ,Mn+1)

is one of the generators of ker(π) in Theorem 11. Then τ(Fj,n+1(M1, . . . ,Mn+1)) is the sameas Fj,n+1(M1, . . . ,Mn+1) except that we replace each TM with lM ′′UM ′ , where M ′ and M ′′ are

as in the definition of τ . From the definition of Fj,n+1, in each monomial T jσ(M1, . . . ,Mn+1) ofFj,n+1(M1, . . . ,Mn+1), all subscripts of T are in FS(M1,M

t1, . . . ,Mn+1,M

tn+1) and every Mi ap-

pears exactly once (possibly transposed). Thus for each i, τ(T σj (M1, . . . ,Mn+1)) gets a factor ofeither lM ′′i or l(Mt

i )′′ depending on whether Mi or M t

i appears. Now using the identities lNP = lPN

and lN−1 = l−1N , easily l(Mti )′′/lM ′′i = lM ′i , so

τ(Fj,n+1(M1, . . . ,Mn+1))/

n+1∏i=1

lM ′′i = Gj,n+1(M′1, . . . ,M

′n+1).

This proves the proposition as∏n+1i=1 lM ′′i is an invertible element of S.

11

From this proposition, we immediately deduce the following finite presentation up to radical ofk[GO•n]AdGOn as an FFG-algebra.

Corollary 15. Let A• = FFFG(1) ⊗ FFFG(1) be an FFG-algebra with two free generators of ar-ity 1, denoted by T and l. Then the FFG-algebra map Θ• : A• → k[GO•n]AdGOn sending T totr(A1) and l to λ(A1) is surjective. For g ∈ FG(g1, . . . , gn), let φg denote some fixed mapFG(g1, . . . , gn) → FG(g1, . . . , gn) sending g1 to g. Note that Θ•(Aφg(T )) = tr(ggi 7→ Ai)and Θ•(Aφg(l)) = λ(ggi 7→ Ai). Then the kernel of Θ• is the radical of the FFG-ideal generated bythe relations:

• Aφ1(T )− n

• Aφg1g2 (T )−Aφg2g1 (T )

• Aφ1(l)− 1

• Aφg1g2 (l)−Aφg1 (l)Aφg2 (l)

• T − lAφg−11 (T )

• Hj,n+1(g1, . . . , gn+1), 0 ≤ j ≤ (n+ 1)/2, which we define to be the same as Gj,n+1(g1, . . . , gn+1)except that we replace each variable lg with Aφg(l) and each Ug with Aφg(T ).

We are now ready to define pseudocharacters for GOn and On.

Definition 7. Let Γ be a group. A GOn-pseudocharacter of Γ over k is a pair (T, l), consisting of aset map T : Γ→ k and a group homomorphism l : Γ→ k×, such that

• T (1) = n

• For all γ1, γ2 ∈ Γ, T (γ1γ2) = T (γ2γ1)

• For all γ ∈ Γ, T (γ) = l(γ)T (γ−1)

• For all integers 0 ≤ j ≤ (n+ 1)/2 and for all γ1, . . . , γn+1 ∈ Γ, T and l satisfy the relation

Ij,n+1(l, T, γ1, . . . , γn+1) = 0,

where we define Ij,n+1(l, T, γ1, . . . , γn+1) to be the same as Gj,n+1(γ1, . . . , γn+1) except that wereplace each variable lγ with l(γ) and each Uγ with T (γ).

Definition 8. An On-pseudocharacter of Γ over k is a set map T : Γ → k such that (T, 1) is aGOn-pseudocharacter.

Theorem 16. Assume k is a topological field of characteristic 0.

(1) Let ρ : Γ → GOn(k) be a continuous (with the k-topology on GOn(k)) homomorphism. Then(tr(ρ), λ(ρ)) is a GOn-pseudocharacter.

(2) Conversely, let k have a topology extending the topology on k. Let (T, l) be a GOn-pseudocharacter.Then there is a finite extension k′ of k and a continuous semisimple homomorphism ρ : Γ →GOn(k′) such that tr(ρ) = T and λ(ρ) = l. Moreover, ρ is unique up to conjugation by GOn(k).

(3) Let ρ : Γ→ GOn(k) be a semisimple homomorphism. Then

ker(ρ) = η ∈ Γ | λ(η) = 1 and T (γη) = T (γ) for all γ ∈ Γ .

The same result holds with GOn replaced by On and GOn-pseudocharacters (T, l) replaced by On-pseudocharacters T .

12

Proof. This is a direct consequence of our general result on pseudocharacters (Theorem 5), using thefinite presentation up to radical of k[GO•n]AdGOn described in Corollary 15.

Remark 6. The above results can also be proven by modifying Taylor’s proof forGLn-pseudocharacters[Tay, Theorem 1]. In fact, one can generalize the above result to algebras, as follows. First define a∗-algebra to be a (possibly noncommutative) k-algebra with an involution ∗. Define an orthogonaln-dimensional representation of a ∗-algebra R to be a k-algebra morphism R → Mn(k) mapping ∗to the transpose. Then one can define n-dimensional orthogonal pseudocharacters of a ∗-algebra Rsimilarly to the definition of On-pseudocharacters above. Using [Pro2, Theorem 15.3] in place of[Tay, Lemma 2] in Taylor’s proof, one can prove that these are in bijection with On(k)-conjugacyclasses of semisimple orthogonal representations of R. By taking R to be the group algebra k[Γ] withinvolution determined by (γ)∗ = l(g)(g−1) for γ ∈ Γ, one recovers Theorem 16.

2.2 (General) Symplectic Group

Again let k be a field of characteristic 0. Let GSp2n(k) = A ∈M2n(k) | for some λ ∈ k×, AA∗ = λIbe the n-dimensional general symplectic group; here ∗ is the symplectic involution

A∗ = Ω−1ATΩ

where

Ω =

(0 I−I 0

)is the matrix of the standard symplectic form. It is a connected reductive algebraic group.

The results and proofs for GSp2n are exactly analogous to those for GOn, except that instead ofstarting with the relations Fj,n+1 defined above, we start with the relations F ih,n, for 1 ≤ i ≤ n + 1and 0 ≤ h < i, defined in [Pro2, Theorem 10.2(a)]. For convenience, we state the analogue ofTheorem 16; from this and the original proof, it is easy to read off a finite presentation up to radicalof k[GSp•2n]AdGSp2n as an FFG-algebra.

Define a function λ : GSp2n(k)→ k× by AA∗ = λ(A)I. Note that λ ∈ k[GSp2n]AdGSp2n .

Definition 9. Let Γ be a group. A GSp2n-pseudocharacter of Γ over k is a pair (T, l), consisting ofa set map T : Γ→ k and a group homomorphism l : Γ→ k×, such that

• T (1) = 2n

• For all γ1, γ2 ∈ Γ, T (γ1γ2) = T (γ2γ1)

• For all γ ∈ Γ, T (γ) = l(γ)T (γ−1)

• For all integers 1 ≤ i ≤ n + 1 and 0 ≤ h < i, and for all γ1, . . . , γn+i ∈ Γ, T and l satisfy therelation

Iih,n+1(l, T, γ1, . . . , γn+i) = 0,

where Iih,n+1(l, T, γ1, . . . , γn+i) is defined as follows:

– Taking X1, . . . , Xn+i to be matrix variables, define Gih,n+1(X1, . . . , Xn+i) to be the same

as F ih,n+1(X1, . . . , Xn+i), except that we replace each variable TM with formal symbolslM ′UM ′ , where M ′ ∈ FG(X1, . . . , Xn+i) is the result of substituting all transposed lettersXtj in M with X−1j .

– Define Iih,n+1(l, T, γ1, . . . , γn+i) to be the same as Gih,n+1(γ1, . . . , γn+i) except that wereplace each symbol lγ with l(γ) and each Uγ with T (γ).

Definition 10. An Sp2n-pseudocharacter of Γ over k is a set map T : Γ → k such that (T, 1) is aGSp2n-pseudocharacter.

Theorem 17. Assume k is a topological field of characteristic 0.

13

(1) Let ρ : Γ→ GSp2n(k) be a continuous (with the k-topology on GSp2n(k)) homomorphism. Then(tr(ρ), λ(ρ)) is a GSp2n-pseudocharacter.

(2) Conversely, let k have a topology extending the topology on k. Let (T, l) be a GSp2n-pseudocharacter.Then there is a finite extension k′ of k and a continuous semisimple homomorphism ρ : Γ →GSp2n(k′) such that tr(ρ) = T and λ(ρ) = l. Moreover, ρ is unique up to conjugation byGSp2n(k).

(3) Let ρ : Γ→ GSp2n(k) be a semisimple homomorphism. Then

ker(ρ) = η ∈ Γ | λ(η) = 1 and T (γη) = T (γ) for all γ ∈ Γ .

The same result holds with GSp2n replaced by Sp2n and GSp2n-pseudocharacters (T, l) replaced bySp2n-pseudocharacters T .

2.3 Special Orthogonal Group

Odd Dimension

When the dimension is 2n + 1 for some n, we have k[GO•2n+1]AdSO2n+1 = k[GO•2n+1]

AdO2n+1 , sinceevery orthogonal matrix is ±1 times a special orthogonal matrix. By the same reasoning as inthe proof of Proposition 10, this equals k[GO•2n+1]

AdGO2n+1 . Next, the kernel of the natural sur-jective map k[GO•2n+1]

AdSO2n+1 k[SO•2n+1]AdSO2n+1 is generated up to radical by the relation

det(A1) − 1 (expressed in terms of tr(A1), tr(A21), etc.). Hence k[SO•2n+1]

AdSO2n+1 is generated bytr as an FFG-algebra, and the relations between tr are generated, up to radical, by the relations fork[GO•2n+1]

AdGO2n+1 with λ = 1 and the relation det = 1 expressed in terms of tr.

Definition 11. An (odd-dimensional) SO2n+1-pseudocharacter of G over k is anO2n+1-pseudocharacterT : G→ k which additionally satisfies the relation det(T )(g) = 1 for all g ∈ G, where det(T )(g) is apolynomial in the T (gi) such that det(tr)(B) = det(B) for all matrices B.

Then the usual result holds by our general result on pseudocharacters (Theorem 5) and the abovediscussion.

Even Dimension

When the dimension is 2n for some n, the invariant theory of SO2n is more complicated. Aslaksen,Tan, and Zhu [ATZ, Theorem 3] show that for all m, k[Mm

2n]AdSO2n is generated as a k-algebra by trand the n-ary linearized Pfaffian pl, defined as the full polarization of the function

pf(W ) = pf(W −W t)

where pf is the usual Pfaffian. Here the inputs to tr and pl are again drawn from FS(A1, At1, . . . , Am, A

tm).

Then k[SO•2n]AdSO2n is generated as an FFG-algebra by tr and pl.A result due to Rogora [Rog] allows us to determine the relations between these generators up to

radical, as follows.

Lemma 18. The FFG-ideal of relations between the generators tr and pl of k[SO•2n]AdSO2n is theradical of the FFG-ideal generated by the relations between tr for k[SO•2n]AdGO2n and the relationdescribed in [Rog, Theorem 3.2].

Proof. Let R be a polynomial in terms of the given generators (i.e., in terms of their images underthe internal morphisms in the free FFG-algebra) which maps to 0 in k[SO•2n]AdSO2n . Note thatconjugating all inputs to R by an element of O2n(k) \ SO2n(k) preserves the value of any generatortr(M) or λ(M) while negating the value of any generator pl(M1, . . . ,Mn). Thus conjugating all inputsof any monomial in R sends that monomial to either itself or its negation; we call the monomial “even”in the former case and “odd” in the latter case. Let Re and Ro be the sums of all even and odd

14

monomials in R, respectively. Then Re and −Ro are mapped to the same image in k[SO•2n]AdSO2n .Then conjugating all of their image’s inputs by an element of O2n(k) \ SO2n(k), we see that Re andRo also map to the same image in k[SO•2n]AdSO2n . Hence Re and Ro both map to 0, so that they areboth in the FFG-ideal of relations.

It now suffices to show that the even and odd relations are in the given FFG-ideal. If Re is aneven relation, then each of its monomials consists of traces and pairs of linearized Pfaffians. Afterreplacing each pair of linearized Pfaffians with a polynomial in traces using the relations describedin [Rog, Theorem 3.2], we get a polynomial in terms of traces which is a GO2n-invariant. Hence Reis in the given FFG-ideal. Next, if Ro is an odd relation, then R2

o is an even relation, hence is in thegiven FFG-ideal. Then Ro is in the radical of the given FFG-ideal.

Definition 12. An (even-dimensional) SO2n-pseudocharacter of G over k is a pair of functionsT : G→ k, P : Gn → k, such that

• T is an O2n-pseudocharacter of G over k

• For all g ∈ G, det(T )(g) = 1

• For all g1, . . . , gn, h1, . . . , hn ∈ G, P (g1, . . . , gn)P (h1, . . . , hn) satisfies the relation in [Rog, The-orem 3.2] with P in place of Q and T in place of tr.

Then we have the usual result.

3 Application: Conjugacy vs. Element-Conjugacy

In this section, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of group homomorphisms Γ → H(k) for H a linear algebraic group, following Larsen[Lar1, Lar2].

Definition 13. Fix a linear algebraic group H over a field k, and let Γ be an abstract group. Twohomomorphisms ρ1, ρ2 : Γ → H(k) are called globally conjugate if there exists h ∈ H(k) such thatρ1 = hρ2h

−1. They are called element-conjugate if for all γ ∈ Γ, there exists hγ ∈ H(k) such thatρ1(γ) = hγρ2(γ)h−1γ .

Recall that we call a homomorphism ρ : Γ → H(k) semisimple if the Zariski closure of Im(σ)in H(k) is reductive. The conjugacy vs. element-conjugacy question for H(k) asks whether or notelement-conjugate semisimple homomorphisms Γ→ H(k) are automatically globally conjugate.

Definition 14. A linear algebraic group H(k) is acceptable if element-conjugacy implies globalconjugacy for all semisimple homomorphisms of arbitrary groups Γ. We call H(k) finite-acceptableif element-conjugacy implies global conjugacy for all finite groups Γ, and compact-acceptable if k istopological and element-conjugacy implies conjugacy for all continuous semisimple homomorphismsof compact groups Γ.

In [Lar1, Lar2], Larsen mostly classifies the complex and compact simple Lie groups as finite-acceptable or finite-unacceptable (which implies unacceptable). Recent results by Fang, Han, andSun [FHS] show that GLn(C), On(C), Sp2n(C), and their real compact forms are in fact compact-acceptable.

In this section, we give a simple sufficient condition for the acceptability of a connected reduc-tive group H over an algebraically closed field k of characteristic 0, in terms of the FFG-algebrak[H•]AdH . This condition and the results of Section 2 immediately imply that GOn(k), On(k),GSp2n(k), Sp2n(k), and SO2n+1(k) are acceptable (not just finite- or compact-acceptable). By[Lar1, Proposition 1.7], it follows that the maximal compact subgroups of these groups over C arecompact-acceptable. Previous results of this form were only known for On(C), Sp2n(C), SO2n+1(C),and SO4(C), and only for compact-acceptability.

We also use our pseudocharacters for SO2n to give a criterion for when a semisimple homomor-phism ρ : Γ→ SO2n(k) is a counterexample to acceptability for SO2n(k), at least when Γ is torsion.

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Using this criterion, we prove that SO4(k) is acceptable, improving a result due to Yu [Yu1] show-ing that SO4(C) is compact-acceptable. We also construct a counterexample to acceptability forSO2n(k) (n ≥ 3) with domain group Γ = Z/4Z × Z/4Z; this gives a simpler example than the onein [Lar1, Proposition 3.8], and it additionally shows that SO6(k) is unacceptable, a result which wasnot previously known.

3.1 General Principles

LetH be a linear algebraic group. Suppose thatH has pseudocharacters consisting of arity 1 functionsonly. More formally, let k be an algebraically closed field of characteristic 0, and suppose that thereexist invariants f1, . . . , fn ∈ k[H]AdH such that for any group Γ, the map ρ 7→ (f1(ρ), . . . , fn(ρ))induces a bijection between

H(k)-conjugacy classes of semisimple homomorphisms ρ : Γ→ H(k)

andmaps F1, . . . , Fn : Γ→ k satisfying certain fixed relations.

Then H(k) is acceptable: indeed, if ρ1, ρ2 : Γ → H(k) are semisimple element-conjugate homomor-phisms, then for all γ ∈ Γ, we have fi(ρ1(γ)) = fi(ρ2(γ)) for 1 ≤ i ≤ n, so ρ1 and ρ2 have the sameH-pseudocharacters, hence they are conjugate.

When H is a connected reductive group (or, more generally, when the conclusion of Theorem 5holds for H, such as for H = On), we can restate this result as follows.

Theorem 19. Let H be an algebraic group over an algebraically closed field k of characteristic 0such that Theorem 5 holds for H with the action of AdH (e.g., H is a connected reductive group).Suppose that k[H•]AdH is generated by k[H]AdH as an FFG-algebra, i.e., by functions of arity 1.Then H(k) is acceptable.

In particular, GOn(k), On(k), GSp2n(k), Sp2n(k), and SO2n+1(k) are acceptable.Although it would be convenient if the converse to this theorem were true, it is not. We show

below that SO4(k) is acceptable, while Yu [Yu2] has shown that the arity 2 function pl in k[SO•4]AdSO4

is not generated by k[SO4]AdSO4 .

Even when H is not acceptable, so that a homomorphism can have element-conjugate but notconjugate homomorphisms, the number of such homomorphisms is uniformly bounded, as follows.2

Proposition 20. Let H be a connected reductive algebraic group over an algebraically closed fieldk of characteristic 0. Then there exists NH ∈ N such that for any semisimple homomorphismρ : Γ → H(k), the number of H(k)-conjugacy classes of semisimple homomorphisms H(k)-element-conjugate to ρ is at most NH .

Proof. Let d be such that H is an affine subgroup variety of GLd. Then any semisimple homo-morphism H(k)-element-conjugate to ρ is also GLd(k)-element-conjugate, hence GLd(k)-conjugate.We will show that the number of H(k)-conjugacy classes of homomorphisms into H(k) that areGLd(k)-conjugate to ρ is bounded by some NH .

Replacing Γ by the Zariski closure of Im(ρ), it suffices to prove the proposition when Γ is analgebraic group and we only consider homomorphisms that are also algebraic maps. By Lemma 7,there exists qd ∈ N depending only on d such that Γ = 〈γ1, . . . , γqd〉 for some γ1, . . . , γqd ∈ Γ.

If ρ′ is such that f(ρ(γ1), . . . , ρ(γqd)) = f(ρ′(γ1), . . . , ρ′(γqd)) for all f ∈ k[Hqd ]AdH , then ρ′ is

H-conjugate to ρ, as follows. For any m ∈ N, g ∈ k[Hm]AdH , and η1, . . . , ηm ∈ 〈γ1, . . . , γqd〉, thereis some g ∈ k[Hqd ]AdH such that g(ρ(η1), . . . , ρ(ηm)) = g(ρ(γ1), . . . , ρ(γqd)) and likewise for ρ′, henceg(ρ(η1), . . . , ρ(ηm)) = g(ρ′(η1), . . . , ρ

′(ηm)). Thus the H-pseudocharacters of ρ and ρ′ are identicalon 〈γ1, . . . , γqd〉, from which they are identical on all of Γ. Then ρ′ is H(k)-conjugate to ρ by theuniqeness claim in Theorem 5(2).

2I thank one of the anonymous reviewers for bringing to my attention this question and its relation to [Vin].

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Now by a result of Vinberg [Vin, Theorem 1], the natural map Spec(k[Hqd ]AdH)→ Spec(k[GLqdd ]AdGLd)is finite. Hence for a semisimple algebraic homomorphism ρ′ that is GLd(k)-conjugate to ρ, thenumber of possible values for the f(ρ′(γ1), . . . , ρ

′(γqd)), f ∈ k[Hqd ]AdH , is bounded by a constantdepending only on H.

3.2 Element-conjugacy vs. Conjugacy for SO2n

Let k be an algebraically closed field of characteristic 0, and let n be an integer. We wish tocharacterize all pairs of semisimple homomorphisms ρ1, ρ2 : Γ→ SO2n(k) that are element-conjugatebut not globally conjugate, at least when Γ is torsion. Let pl denote the linearized antisymmetrizedPfaffian (see Section 2.3 above). Our first result is as follows.

Proposition 21. Let Γ be a group, and let ρ1 : Γ → SO2n(k) be a semisimple homomorphism. Ifthere exists a semisimple homomorphism ρ2 : Γ→ SO2n(k) that is element-conjugate but not globallyconjugate to ρ1, then:

• For all γ ∈ Γ, pf(ρ1(γ)− ρ1(γ)t) = 0

• There exist γ1, . . . , γn ∈ Γ such that pl(ρ1(γ1), . . . , ρ1(γn)) 6= 0.

If Γ is torsion, then the converse holds as well.When such a ρ2 exists, it is unique up to conjugation by SO2n(k), and it is given by

ρ2 = Xρ1X−1

for some X ∈ O2n(k) \ SO2n(k).

Proof. Uniqueness: Let ρ2 be element-conjugate but not globally conjugate to ρ1 in SO2n. Then ρ1and ρ2 are element-conjugate in O2n, hence globally conjugate in O2n. Thus there is an X ∈ O2n(k)such that ρ2 = Xρ1X

−1, and necessarily X /∈ SO2n(k). Since SO2n(k) has index 2 in O2n(k), anyother choice of X gives a homomorphism that is globally conjugate to ρ2 in SO2n(k).

Existence, (=⇒): Let ρ2 be a semisimple homomorphism that is element-conjugate but notglobally conjugate to ρ1. The invariant pl is an “odd” invariant in the sense that

pl(ρ1(γ1), . . . , ρ1(γn)) = −pl(Xρ1(γ1)X−1, . . . , Xρ1(γn)X−1)

= −pl(ρ2(γ1), . . . , ρ2(γn))(2)

for all γ1, . . . , γn ∈ Γ. Since ρ1 and ρ2 are not globally conjugate, they must have different pseu-docharacters, and since tr(ρ1) = tr(ρ2) by element-conjugacy, there must exist γ1, . . . , γn ∈ Γ suchthat

pl(ρ1(γ1), . . . , ρ1(γn)) 6= pl(ρ2(γ1), . . . , ρ2(γn)).

Then by (2), pl(ρ1(γ1), . . . , ρ1(γn)) 6= 0.Next, since ρ1 and ρ2 are element-conjugate, ρ1|〈γ〉 is SO2n-conjugate to ρ2|〈γ〉 for each γ ∈ Γ, so

pl(ρ1(γm1), . . . , ρ1(γ

mn)) = pl(ρ2(γm1), . . . , ρ2(γ

mn))

for all γ ∈ Γ and m1, . . . ,mn ∈ Z. Then by (2), pl(ρ1(γm1), . . . , ρ1(γ

mn)) = 0. In particular,pf(ρ1(γ)) = 1

n!pl(ρ1(γ), . . . , ρ1(γ)) = 0 for all γ ∈ Γ. Hence

pf(ρ1(γ)− ρ1(γ)t) = pf(ρ1(γ)) = 0.

Existence, (⇐=): Assume Γ is torsion. Let X ∈ O2n(k)\SO2n(k), and set ρ2(γ) = Xρ1(γ)X−1.Then by assumption, there exist γ1, . . . , γn such that

pl(ρ1(γ1), . . . , ρ1(γn)) 6= −pl(ρ1(γ1), . . . , ρ1(γn)) = pl(ρ2(γ1), . . . , ρ2(γn)),

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so ρ1 and ρ2 are not globally conjugate.Now fix γ ∈ Γ. Since Γ is torsion, Maschke’s theorem implies that both ρ1|〈γ〉 and ρ2|〈γ〉 are

semisimple. Thus to show that ρ1|〈γ〉 and ρ2|〈γ〉 are conjugate in SO2n, it suffices to show that theyhave the same SO2n-pseudocharacters. They have the same traces because ρ1 and ρ2 are conjugatein O2n. To show that they have the same values of pl, we must show

pl(ρ1(γm1), . . . , ρ1(γ

mn)) = 0

for all m1, . . . ,mn ∈ Z, since the corresponding value for ρ2 is the negative of that for ρ1. Bydefinition, pl(ρ1(γ

m1), . . . , ρ1(γmn)) is the multilinear term in

pf (t1ρ1(γm1) + · · ·+ tnρ1(γ

mn))

= pf(t1(ρ1(γ

m1)− ρ1(γm1)t) + · · ·+ tn(ρ1(γmn)− ρ1(γmn)t

).

But ρ1(γ) − ρ1(γ)t = ρ1(γ) − ρ1(γ)−1 divides ρ1(γ)mi − ρ1(γ)−mi = ρ1(γmi) − ρ1(γmi)t for all i, so

the assumption det(ρ1(γ)− ρ1(γ)t) = pf(ρ1(γ)− ρ1(γ)t)2 = 0 implies that

det(t1(ρ1(γ

m1)− ρ1(γm1)t) + · · ·+ tn(ρ1(γmn)− ρ1(γmn)t)

)= 0.

Hence taking the square root, the Pfaffian is zero as well for all values of t1, . . . , tn. Thuspl(ρ1(γ

m1), . . . , ρ1(γmn)) = 0, proving the proposition.

We now use this result to determine the acceptability or unacceptability of SO2n(k) for all n.Previous results are as follows:

• SO2(k) is acceptable because it is abelian.

• SO4(C) is compact-acceptable. Yu [Yu1, Theorem 4.1(3)] recently showed that SO4(R) iscompact-acceptable, so SO4(C) is compact-acceptable by [Lar1, Proposition 1.7]. Yu’s proofuses the notion of strongly controlling fusion to show that the exceptional Lie group G2(R) iscompact-acceptable and then derives compact-acceptability of SO4(R) as a consequence.

• SO2n(C) is unacceptable for n ≥ 4. Larsen [Lar1, Proposition 3.8] shows this by constructinga counterexample with domain group Γ = SL3(Z/2Z).

We complete this program by proving that SO4(k) is acceptable and SO6(k) is unacceptable. Ourcounterexample to the acceptability of SO6(k), which extends to a counterexample to the accept-ability of SO2n(k) for all n ≥ 3, is especially simple, with Γ = Z/4Z× Z/4Z.

Theorem 22. SO4(k) is acceptable.

Proof. Let Q1, Q2 ∈ SO4(k). We claim that if det(Q−Qt) = 0 for all Q ∈ 〈Q1, Q2〉, then pl(Q1, Q2) =0. The theorem then follows from the first statement in Proposition 21.

It suffices to prove the claim when k = C. To simplify our computations, we use a variant ofthe special isomorphism (SL2(C) × SL2(C))/〈(−I,−I)〉 ∼= SO4(C) corresponding to the isoclinicdecomposition of 4-dimensional rotations. Let HC = H ⊗R C denote the complex quaternions, andlet (HC)∗ ∼= SL2(C) denote its unit group. For q = q1 + q2i + q3j + q4k ∈ HC, left (resp. right)multiplication by q on HC defines matrices L(q) (resp. R(q)) given by

L(q) =

q1 −q2 −q3 −q4q2 q1 −q4 q3q3 q4 q1 −q2q4 −q3 q2 q1

R(q) =

q1 −q2 −q3 −q4q2 q1 q4 −q3q3 −q4 q1 q2q4 q3 −q2 q1

.

Then we have an isomorphism

ϕ : ((HC)∗ × (HC)∗) /〈(−1,−1)〉 ∼−→ SO4(C)

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given by ϕ(q, r) = L(q)R(r).Write Q1 = ϕ(u, v), Q2 = ϕ(w, x), and define e = u2w2+u3w3+u4w4, f = v2x2+v3x3+v4x4. One

computes pf(ϕ(q, r)−ϕ(q, r)t)/4 = r21− q21. Then since Q1Q2 = ϕ(uw, vx) and Q21Q2 = ϕ(u2w, v2x),

pf(Q1 −Qt1)/4 = v21 − u21pf(Q2 −Qt2)/4 = x21 − w2

1

pf(Q1Q2 − (Q1Q2)t)/4 = (v1x1 − f)2 − (u1w1 − e)2

pf(Q21Q2 − (Q2

1Q2)t)/4 = (2v21x1 − 2v1f − x1)2 − (2u21w1 − 2u1e− w1)

2.

By assumption, all of these Pfaffians are 0. It remains to show that pl(Q1, Q2)/8 = v1x1e−u1w1fis also 0. Applying the relations x21 = w2

1 and (v1x1 − f)2 = (u1w1 − e)2 to the relation, we get

x1(v21x1 − v1f) = w1(u

21w1 − u1e).

Then v1x1f = u1w1e because x21 = w21 and u21 = v21. Hence x1 = ±w1, u1 = ±v1, and v1x1f = u1w1e.

Regardless of the choice of signs, v1x1e = u1w1f , proving the claim.

Lemma 23. SO6(k) is unacceptable.

Proof. Let Γ = Z/4Z× Z/4Z, with generators (1, 0) and (0, 1). Let

A =

(0 1−1 0

)∈ SO2(k).

Define a homomorphism ρ6 : Γ→ SO6(k) by

ρ6(1, 0) = A⊕A⊕ I,ρ6(0, 1) = I ⊕A⊕A.

Then one can check that det(ρ6(γ)− ρ6(γ)t) = 0 for all γ ∈ Γ, while

pl(ρ6(1, 0), ρ6(0, 1), ρ6(0, 1)) = 16.

Hence by Proposition 21, ρ6 is a counterexample to acceptability for SO6(k).

More generally, we have:

Theorem 24. Let Γ and A be as in the proof of the previous lemma. For any n ≥ 3, the homomor-phism ρ2n : Γ→ SO2n(k) defined by

ρ2n(1, 0) = A⊕A⊕ I ⊕n⊕i=4

A,

ρ2n(0, 1) = I ⊕A⊕A⊕n⊕i=4

A

satisfies det(ρ2n(γ)− ρ2n(γ)t) = 0 for all γ ∈ Γ and pl(ρ2n(1, 0), ρ2n(0, 1), . . . , ρ2n(0, 1)) 6= 0. Henceby Proposition 21, ρ2n gives a counterexample to acceptability for SO2n(k).

Proof. Let γ ∈ Γ, and write ρ(γ) =⊕n

i=1B(i). We have

det(ρ(γ)− ρ(γ)t

)= det

(n⊕i=1

(B(i) − (B(i))t)

)

=n∏i=1

det(B(i) − (B(i))t).

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Hence to show det(ρ(γ)− ρ(γ)t) = 0, it suffices to prove that some 2× 2 diagonal block B(i) of ρ(γ)satisfies det(B(i) − (B(i))t) = 0. But one can check that for all γ ∈ Γ, one of the first three 2 × 2diagonal blocks is ±I.

Next, recall that for matrices C1, . . . , Cn, pl(C1, . . . , Cn) is defined to be the coefficient of t1 · · · tnin pf(t1(C1 − Ct1) + · · · + tn(Cn − Ctn)). Letting each Cj =

⊕ni=1C

(i)j for some 2 × 2 matrices C

(i)j ,

we have

pf(t1(C1 − Ct1) + · · ·+ tn(Cn − Ctn)) =n∏i=1

pf(t1(C(i)1 − (C

(i)1 )t) + · · ·+ t1(C

(i)n − (C(i)

n )t).

Now pf is a linear function of 2× 2 antisymmetric matrices, so this equals

n∏i=1

n∑j=1

tjpf(C(i)j − (C

(i)j )t).

Taking the coefficient of t1 · · · tn in this formula, we find that

pl(C1, . . . , Cn) =∑σ∈Sn

n∏i=1

pf(C(i)σ(i) − (C

(i)σ(i))

t).

Finally, note that pf(A−At) = 2 and pf(I − It) = 0. Thus

pl (D1 = ρ2n(1, 0), D2 = ρ2n(0, 1), . . . , Dn = ρ2n(0, 1))

will be positive so long as for some σ ∈ Sn, for all i, D(i)σ(i) = A. Taking σ to be the identity

permutation works.

Acknowledgments: I would like to heartily thank Xinwen Zhu for proposing the idea for thisproject and for mentoring me throughout it. I also thank the anonymous reviewers for numeroushelpful suggestions.

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