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On Higher Frobenius-Schur Indicators 5 0 0 2 Yevgenia Kashina Yorck Sommerh¨auser n a Yongchang Zhu J 0 3 Abstract ] A We study the higher Frobenius-Schur indicators of modules over semi- R simpleHopfalgebras,andrelatethemtootherinvariantsastheexponent, . theorder, and theindex.Weprovevariousdivisibility and integrality re- h t sults for these invariants. In particular, we prove a version of Cauchy’s a theorem for semisimple Hopf algebras. Furthermore,we give some exam- m ples that illustrate thegeneral theory. [ 2 v Introduction 9 9 1 For a finite group, one can evaluate a character on the sum of all m-th powers 1 ofthegroupelements.Theresultingnumber,dividedbytheorderofthegroup, 1 3 is called the m-th Frobenius-Schur indicator of the character. The first use of 0 these indicators was made by F. G. Frobenius and I. Schur (cf. [13]) to give a / criterionwhenarepresentationofafinitegroupcanberealizedbymatriceswith h t realentries—forthis question,it isthe secondindicatorthatis relevant.This is a also meaningful for other fields than the complex numbers: Here the indicator m tells whether or not a given module is self-dual. : v i Higherindicators,i.e.,indicatorswithm>2,arisewhenoneconsidersthe root X number function in a finite group.This function assignsto a groupelement the r number of its m-th roots,i.e., the number of groupelements whosem-thpower a is equal to the given element. It is clear that this number depends only on the conjugacy class, and therefore defines a class function that can be expanded in terms of the irreducible characters. Using the orthogonality relations for char- acters,itisnothardtoseethatthecoefficientofanirreduciblecharacterinthis expansion is its m-th Frobenius-Schur indicator (cf. [19], Lem. (4.4), p. 49). For Hopf algebras, Frobenius-Schur indicators were first considered by V. Lin- chenko and S. Montgomery on the one hand (cf. [27]) and by J. Fuchs, A. Ch. Ganchev,K.Szlacha´nyi,andP.Vecserny´esontheotherhand(cf.[14]).Here,the sumofthe m-thpowersofthe groupelements is replacedby the m-thSweedler 1 poweroftheintegral.Theauthorsthenusetheindicators,oratleastthesecond indicator,to provean analogueof the criterionofFrobenius and Schurwhether or not a representation is self-dual: The Frobenius-Schur theorem asserts that this depends on whether the second indicator is 0, 1, or −1. The topic of the present writing are the higher Frobenius-Schur indicators for semisimple Hopf algebras and their relation to other invariants of irreducible characters.These other invariants are the order,the multiplicity, the exponent, andthe index.Letusbriefly describethe natureoftheseinvariants.Thenotion of the order of an irreducible character is a generalization of the notion of the orderofanelementinafinitegroup:Itisthesmallestintegersuchthatthecor- responding tensor power contains a nonzero invariantsubspace. The dimension of this invariant subspace is called the multiplicity of the irreducible character. An irreducible character has order 1 if and only if it is trivial, and has order 2 if andonly if itis self-dual.Inthese cases,the multiplicity ofthe characteris 1. The exponent of a semisimple Hopf algebra is another invariant, which gener- alizes the exponent of a group (cf. [21]). The exponent of a module is a slight generalizationofthis concept:Inthe groupcase,itistheexponentofthe image of the group in the representation. There are various ways to generalize this concept to semisimple Hopf algebras; we will use Sweedler powers on the one hand and a certain canonical tensor on the other hand. Thenextinvariantthatwestudy,theindexofimprimitivity,arisesfromPerron- Frobenius theory. It is clear that the matrix representation of the left multipli- cation by the character of a module with respect to the basis consisting of all irreducible characters has nonnegative integer entries. As we will explain be- low,thecorrespondingPerron-Frobeniuseigenvalue,i.e.,thepositiveeigenvalue thathasthelargestpossibleabsolutevalue,isthe degreeofthecharacter.How- ever, since the entries of the above matrix are in general not strictly positive, this eigenvalue is notnecessarilystrictly greaterthanthe absolutevalues of the other characters, so that there can be other eigenvalues which are not positive, but have the same absolute value. As we will see, in the most interesting cases theabovematrixisindecomposable;inthiscase,thenumberofsucheigenvalues is called the index of imprimitivity. The text is organized as follows: In Section 1, we discuss the formalism of Sweedler powers in an arbitrary bialgebra. A Sweedler power of an element in a bialgebra is constructed by applying the comultiplication several times, permuting the arisingtensor factorsand multiplying them together afterwards. This notion is a slight modification of the original notion (cf. [21]), where the tensorfactorswerenotpermuted,andhastheadvantagethatiteratedSweedler powers are still Sweedler powers. We then consider the Sweedler powers that arise from a certain special kind of permutations. This is motivated by the fact thatthevaluesofcharactersontheSweedlerpowersoftheintegrallieincertain cyclotomic fields, and this special kind of Sweedler powers is well adapted to describe the action of the Galois group on these values. 2 From Section 2 on, we consider semisimple Hopf algebras over algebraically closed fields of characteristic zero. We prove a first formula for the higher Frobenius-Schur indicators that should be understood as a generalization of the Frobenius-Schur theorem for these indicators—in particular, it implies the Frobenius-Schur theorem for the second indicators. This first formula describes Frobenius-Schur indicators in terms of a certain operator on the corresponding tensor power of the module, and we establish several other properties of this operator as well. In Section 3, we then consider the exponent of a module and prove a second formula for the higher Frobenius-Schur indicators that uses a certain canonical tensor. Combining this with the first formula, we prove a version of Cauchy’s theorem for Hopf algebras:A prime that divides the dimension of a semisimple Hopf algebra must also divide its exponent. This result was conjectured by P. Etingof and S. Gelaki (cf. [10]); it was known in the case of the prime 2 (cf. [23]). Furthermore, we prove that the higher indicators are integers if the exponent is squarefree. In Section 4, we define the notion of the order and the multiplicity of a module andprovethattheorderofamoduledividesitsmultiplicitytimesthedimension ofthe Hopfalgebra.This resultgeneralizesthe theoremthatasemisimple Hopf algebrathathasanontrivialself-dualsimple modulemusthaveevendimension (cf.[23])tomodulesofarbitraryorders—nontrivialself-dualsimplemodulesare oforder2,andthemultiplicityis1inthiscase.Inparticular,wegetinthisway a fully independent new proof of the old theorem. In Section 5, we study the index of imprimitivity, or briefly the index, of the matrix that represents the left multiplication by a character with respect to the canonical basis that we have in the character ring—the basis consisting of the irreducible characters. The main result of this section is a precise formula for the index in terms of central grouplike elements. Essentially, the result says that the eigenvalues of the above matrix that have the same absolute value as the degreeare obtainedby evaluating the characteratcertaincentralgrouplike elements. As a consequence of this formula, we see that the index divides the order as well as the exponent. InSection6,weapplyanewtool—theDrinfel’ddoubleoftheHopfalgebra.We provethat,byrestrictingmodulesovertheDrinfel’ddoubletotheHopfalgebra, we get a map from the characterring of the Drinfel’d double onto the center of the character ring of the Hopf algebra. From this, we deduce that the center of the rational character ring of the Hopf algebra, i.e., the span of the irreducible characters over the rational numbers, is isomorphic to a product of subfields of thecyclotomicfieldwhoseorderistheexponentoftheHopfalgebra.Finally,we deduceathirdformulafortheFrobenius-Schurindicatorsintermsoftheaction of the Drinfel’d element on the induced module over the Drinfel’d double. In Section 7, we finally compute explicitly a number of examples. In this way, we can limit the possible generalizations of the results that we have obtained. 3 The class of examples that we study are certain extensions of group rings by dual group rings. In particular, the Drinfel’d doubles of finite groups belong to this class. Throughoutthewholeexposition,weconsiderabasefieldthatisdenotedbyK. AllvectorspacesconsideredaredefinedoverK,andalltensorproductswithout subscriptsaretakenoverK.Unlessstatedotherwise,amoduleisaleftmodule. The set of naturalnumbers is the set N:={1,2,3,...}; in particular,0 is not a natural number. The symbol Q denotes the n-th cyclotomic field, and not the n fieldofn-adicnumbers,andZ denotesthe setZ/nZofintegersmodulon,and n not the ring of n-adic integers. From Section 2 on, except for Corollary3.4, we assume that the base field K is algebraicallyclosedofcharacteristiczero.H denotes a semisimple Hopfalgebra with coproduct ∆, counit ε, and antipode S. We will use the same symbols to denote the corresponding structure elements of the dual Hopf algebra H∗. NotethatasemisimpleHopfalgebraisautomaticallyfinite-dimensional(cf.[41], Chap.V,Exerc.4,p.108).ByresultsofR.G.LarsonandD.E.Radford(cf.[26], Thm. 3.3, p. 276; [25], Thm. 4, p. 195), H is also cosemisimple and involutory, i.e.,H∗issemisimpleandtheantipodeisaninvolution.FromMaschke’stheorem (cf. [28], Thm. 2.2.1, p. 20), we get that there is a unique two-sided integral Λ∈H such that ε(Λ)=1; this element will be used heavily throughout. Furthermore, we use the convention that propositions, definitions, and simi- lar items are referenced by the paragraph in which they occur; they are only numbered separately if this reference is ambiguous. 1 The Calculus of Sweedler Powers 1.1 Let us begin by introducing some notation that will be used throughout thissection.Foranaturalnumbern,whichisbydefinitionatleast1,weusethe notationI :={1,...,n}.Supposenowthatm ,m ,...,m areseveralnatural n 1 2 k numbers. On the product I ×I × ...×I , we introduce the so-called m1 m2 mk lexicographical ordering. This means that, for two k-tuples (i ,i ,...,i ) and 1 2 k (j ,j ,...,j ) in this set, we define (i ,i ,...,i ) < (j ,j ,...,j ) if and only 1 2 k 1 2 k 1 2 k if there is an index l ≤k such that i =j ,i =j ,...,i =j , but i <j . 1 1 2 2 l−1 l−1 l l As this is a total ordering, there is a unique strictly monotone map ϕ :I ×I ×...×I −→I m1,...,mk m1 m2 mk n where n:=m ·m ·...·m . Explicitly, this map is given by the formula 1 2 k ϕ (i ,...,i )=(i −1)n +(i −1)n +...+(i −1)n +i m1,...,mk 1 k 1 2 2 3 k−1 k k where n :=m ·m ·...·m , so that n =m and n =n. This holds since i i i+1 k k k 1 if i =j for r=1,...,l−1, but i <j , then we have r r l l (i −1)n +...+(i −1)n +i <(j −1)n +...+(j −1)n +j l l+1 k−1 k k l l+1 k−1 k k 4 as the maximal value that (i −1)n +...+(i −1)n +i can attain is l+1 l+2 k−1 k k (m −1)n +...+(m −1)n +m =m n =n l+1 l+2 k−1 k k l+1 l+2 l+1 and j ≥ 1. Considering this calculation for l = 0, we also see that the map is k well-defined. Obviously, we have ϕ =id if k=1. n In Now suppose that, for a different l, we have another set of natural numbers m′,m′,...,m′, and let n′ := m′ ·m′ ·...·m′. On the one hand, we can then 1 2 l 1 2 l look at the composition (I ×...×I )×(I ×...×I )−ϕ−m−1−,.−..,−m−k−×−ϕ−m−′1−,.−..,−m→′l I ×I −ϕ−n−,n→′ I m1 mk m′1 m′l n n′ nn′ On the other hand, we have the map ϕ :I ×...×I ×I ×...×I −→I m1,...,mk,m′1,...,m′l m1 mk m′1 m′l nn′ It is clear from the definition of the lexicographical ordering that we have (i ,...,i ,i′,...,i′)<(j ,...,j ,j′,...,j′) 1 k 1 l 1 k 1 l if and only if we have (i ,...,i ) < (j ,...,j ) or (i ,...,i ) = (j ,...,j ) 1 k 1 k 1 k 1 k and (i′,...,i′)<(j′,...,j′). This shows that, if all appearing product sets are 1 l 1 l ordered lexicographically, both maps considered above are strictly monotone, and therefore must be equal. This proves the following statement: Lemma For natural numbers m ,...,m and m′,...,m′, we have 1 k 1 l ϕ =ϕ ◦(ϕ ×ϕ ) m1,...,mk,m′1,...,m′l n,n′ m1,...,mk m′1,...,m′l where n:=m ·m ·...·m and n′ :=m′ ·m′ ·...·m′. 1 2 k 1 2 l Of course, this can also be seen from the explicit formulas, as we have ϕ (ϕ (i ,...,i ),ϕ (i′,...,i′)) n,n′ m1,...,mk 1 k m′1,...,m′l 1 l =n′(ϕ (i ,...,i )−1)+ϕ (i′,...,i′) m1,...,mk 1 k m′1,...,m′l 1 l k l = (i −1)n′n + (i′ −1)n′ +1 X r r+1 X s s+1 r=1 s=1 =ϕ (i ,...,i ,i′,...,i′) m1,...,mk,m′1,...,m′l 1 k 1 l where we have set n =n′ =1. k+1 l+1 1.2 We now consider the union ∞ Sˆ:= S [ n n=1 5 of all permutation groups. Note that the different permutation groups are dis- joint, which implies that for an element σ ∈ Sˆ we can say exactly to which permutation group S it belongs. This unique n will be called the degree of σ. n On the set Sˆ, we introduce a product as follows: For σ ∈ S and τ ∈ S , n m we define σ·τ ∈ S to be the unique permutation that makes the following mn diagram commutative: ϕm,n // I ×I I m n mn (i,j)7→ σ·τ (σ(j),τ(i)) (cid:15)(cid:15) (cid:15)(cid:15) // I ×I I n m ϕn,m mn Explicitly, this permutation is given by the formula σ·τ((i−1)n+j)=(σ(j)−1)m+τ(i) This product turns Sˆ into a monoid: Proposition Sˆ is a monoid with unit element id ∈S . I1 1 Proof. Suppose that n, m, and p are natural numbers. For ρ ∈ S , σ ∈ S , p n and τ ∈S , we have to show that (ρ·σ)·τ =ρ·(σ·τ). For this, note that the m following diagram is commutative: I ×I ×I ϕm,n×id //I ×I ϕmn,p //I m n p mn p mnp (i,j,k)7→ (l,k)7→ ρ·(σ·τ) (ρ(k),σ(j),τ(i)) (ρ(k),(σ·τ)(l)) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) I ×I ×I // I ×I //I p n m id×ϕn,m p mn ϕp,mn mnp In addition, the following diagram is also commutative: I ×I ×I id×ϕn,p // I ×I ϕm,np //I m n p m np mnp (i,j,k)7→ (i,l)7→ (ρ·σ)·τ (ρ(k),σ(j),τ(i)) ((ρ·σ)(l),τ(i)) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) I ×I ×I // I ×I //I p n m ϕp,n×id np m ϕnp,m mnp Since the horizontal compositions in both diagrams are equal by Lemma 1.1, the associative law follows. 6 For the discussion of the unit element, note that the diagrams I ×I ϕm,1 // I I ×I ϕ1,n // I m 1 m 1 n n (i,1)7→ (1,j)7→ τ σ (1,τ(i)) (σ(j),1) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) // // I ×I I I ×I I 1 m ϕ1,m m n 1 ϕn,1 n commute, since ϕ (i,1) = i = ϕ (1,i). This shows that id ·τ = τ and n,1 1,n I1 σ·id =σ. 2 I1 1.3 Suppose now that H is a bialgebra with coproduct ∆ and counit ε. We define certain powers that will play an important role in the sequel: Definition For σ ∈S and h∈H, we define the σ-th Sweedler power of h to n be hσ :=h ·h ·...·h (σ(1)) (σ(2)) (σ(n)) Here,wehaveusedavariantoftheso-calledSweedlernotation(cf.[41],Sec.1.2, p. 10)to denote the images under the iterated comultiplication∆n :H →H⊗n as ∆n(h)=h ⊗h ⊗...⊗h (1) (2) (n) If n is the degreeof σ, we shall alsosay that hσ is an n-thSweedler power ofh. This definition of Sweedler powers deviates slightly from the one in [20], [21], where the permutation σ is always the identity. Of course, this permutation is not relevant if H is commutative or cocommutative, which is the case in the theory of algebraic groups, the setting in which Sweedler powers were first considered (cf. [15], Par. 8.5, p. 474; [42], Sec. 1, p. 3). But the introduction of theadditionalpermutationcausesSweedlerpowerstobeclosedunderiteration, as the following power law for Sweedler powers asserts: Proposition For σ ∈S , τ ∈S , and h∈H, we have (hσ)τ =hσ·τ. n m Proof. We givea simplificationofthe originalargumentthatwaspointedout by P. Schauenburg. Recall that σ·τ satisfies by definition the equation σ·τ((i−1)n+j)=(σ(j)−1)m+τ(i) We have in general that n n m ∆m(h )=h ⊗...⊗h = h O (j) (1) (mn) OO ((j−1)m+i) j=1 j=1 i=1 7 We therefore get n n n m ∆m(hσ)=∆m( h )= ∆m(h )= h Y (σ(j)) Y (σ(j)) YO ((σ(j)−1)m+i) j=1 j=1 j=1i=1 m n = h OY ((σ(j)−1)m+i) i=1j=1 Permuting the tensor factors and multiplying, we get m n m n mn (hσ)τ = h = h = h =hσ·τ YY ((σ(j)−1)m+τ(i)) YY (σ·τ((i−1)n+j)) Y (σ·τ(i)) i=1j=1 i=1j=1 i=1 as asserted. 2 1.4 We proceed to construct certain Sweedler powers from special permuta- tions that are based on finite sequences of positive integers. We denote such sequences, using square brackets, in the form [n ,n ,...,n ]. We call such a 1 2 k sequence normalized if every entry divides its predecessor, so that we have n /n /.../n /n . Given a sequence [n ,n ,...,n ], we define its normal- k k−1 2 1 1 2 k ization [n′,n′,...,n′] recursively as follows: We set n′ := n and n′ := 1 2 k 1 1 i+1 gcd(n ,n′), the greatest common divisor of n and n′. i+1 i i+1 i We define a product of two such sequences by the formula [n ,n ,...,n ][m ,m ,...,m ]=[m n ,m n ,...,m n ,m ,m ,...,m ] 1 2 k 1 2 l 1 1 1 2 1 k 1 2 l In addition, we introduce a unique element, called the empty sequence and denotedby[],that isby definitionnormalizedandaunit forthis product.Ifwe denotethesetofallsuchsequencesbyM andthesetofallnormalizedsequences by M , we have the following result: N Proposition M is a monoid and M is a submonoid. Normalization defines N a monoid homomorphism from M to M . N Proof. The product is associative since we have ([n ,n ,...,n ][m ,m ,...,m ])[p ,p ,...,p ]= 1 2 k 1 2 l 1 2 q [p m n ,p m n ,...,p m n ,p m ,p m ,...,p m ,p ,p ,...,p ]= 1 1 1 1 1 2 1 1 k 1 1 1 2 1 l 1 2 q [n ,n ,...,n ]([m ,m ,...,m ][p ,p ,...,p ]) 1 2 k 1 2 l 1 2 q It is obvious that M is a submonoid. If [n ,...,n ] and [m ,...,m ] are two N 1 k 1 l sequences with normalizations [n′,...,n′] and [m′,...,m′], then the normal- 1 k 1 l ization of the product sequence [m n ,m n ,...,m n ,m ,m ,...,m ] is 1 1 1 2 1 k 1 2 l [m′n′,m′n′,...,m′n′,m′,m′,...,m′] 1 1 1 2 1 k 1 2 l sincem′n′ =gcd(m′n ,m′n′).Thisprovesthatnormalizationisamonoid 1 i+1 1 i+1 1 i homomorphism. 2 8 1.5 Suppose now that [n ,n ,...,n ] is a finite sequence of positive integers, 1 2 k and denote its normalization by [n′,n′,...,n′], so that n′ := gcd(n ,n′). 1 2 k i+1 i+1 i We define the numbers m ,m ,...,m by n′ = m n′ and put m := n′. 1 2 k−1 i i i+1 k k Similarly, we define l ,l ,...,l by n = l n′ and put l := 1. Note that 1 2 k−1 i+1 i i+1 k m and l are relatively prime by construction, which implies that the map i i ρ :I →I , j 7→l (j−1)+1 i mi mi i isbijective,wheretherighthandsidereallyshouldbeunderstoodastheunique element of I that is congruent to l (j −1)+1 modulo m . The subtraction mi i i andadditionof1inthedefinitionofρ stems fromthe factthatwehavechosen i the set I to consist of the numbers from 1 to m , and not from 0 to m −1. mi i i Up to this shift, ρ is just multiplication by l . Note that ρ =id . i i k Imk Definition We define P(n ,...,n )∈S to be the unique permutation that 1 k n1 makes the following diagram commutative: I ×...×I ϕmk,...,m1 //I mk m1 n1 (ρ1((iik1,).,....,.i,1ρ)k7→(ik)) P(n1,...,nk) (cid:15)(cid:15) (cid:15)(cid:15) I ×...×I // I m1 mk ϕm1,...,mk n1 For the empty sequence [], we define P():=id . I1 Sincewehaven =m ·...·m ,thisdefinitionmakessense.Therelationbetween 1 1 k sequences and permutations becomes clear through the following fact: Proposition The map M →Sˆ, [n ,...,n ]7→P(n ,...,n ) 1 k 1 k is a monoid homomorphism. Proof. Suppose that, besides the sequence [n ,n ,...,n ] considered above, 1 2 k we have another sequence [p ,p ,...,p ] with normalization [p′,p′,...,p′], so 1 2 l 1 2 l that p′ =gcd(p ,p′). For i=1,...,l−1,define q by p′ =q p′ and r by i+1 i+1 i i i i i+1 i p =r p′ . In addition, we set q :=p′ and r :=1. For i=1,...,l, we then i+1 i i+1 l l l have a permutation π :I →I , j 7→r (j−1)+1 i qi qi i 9 and P(p ,...,p ) is defined via the commutative diagram 1 l I ×...×I ϕql,...,q1 //I ql q1 p1 (π1((jjl1,).,....,.j,1π)l7→(jl)) P(p1,...,pl) (cid:15)(cid:15) (cid:15)(cid:15) I ×...×I //I q1 ql ϕq1,...,ql p1 Consider now the sequence [n ,n ,...,n ][p ,p ,...,p ]=[n p ,n p ,...,n p ,p ,p ,...,p ] 1 2 k 1 2 l 1 1 2 1 k 1 1 2 l By Proposition 1.4, its normalization is [n′p′,n′p′,...,n′p′,p′,p′,...,p′]. 1 1 2 1 k 1 1 2 l Therefore, forming quotients of adjacent elements as above, we arrive at the sequence [m ,m ,...,m ,q ,q ,...,q] 1 2 k 1 2 l By dividing every element of the sequence by its normalized counterpart and shifting the result by 1, we arrive at the sequence [l ,l ,...,l ,l ,r ,r ,...,r ,r ] 1 2 k−1 k 1 2 l−1 l We then have the commutative diagram I ×...×I ×I ×...×I ϕql,...,q1×ϕmk,...,m1//I ×I ϕp1,n1 //I ql q1 mk m1 p1 n1 n1p1 (jl,...,j1,ik,...,i1)7→ (j,i)7→ (ρ1(i1),...,ρk(ik), (σ(i),σ′(j)) σ′′ π1(j1),...,πl(jl)) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) I ×...×I ×I ×...×I // I ×I // I m1 mk q1 qlϕm1,...,mk×ϕq1,...,ql n1 p1 ϕn1,p1 n1p1 where we have used the abbreviations σ :=P(n ,...,n ) σ′ :=P(p ,...,p ) σ′′ :=P(n p ,...,n p ,p ,...,p ) 1 k 1 l 1 1 k 1 1 l In this diagram, the left rectangle is commutative by definition, and the large rectangleiscommutativesincethehorizontalarrowsare,byLemma1.1,equalto ϕ andϕ respectively.Therefore,therightrectangle ql,...,q1,mk,...,m1 m1,...,mk,q1,...,ql isalsocommutative,which,bycomparisonwiththedefinitioninParagraph1.2, shows that σ′′ =σ·σ′. 2 1.6 Suppose now again that H is a bialgebra and that [n ,n ,...,n ] is a 1 2 k finite sequence of positive integers. For h∈H, we define h[n1,n2,...,nk] :=hP(n1,n2,...,nk) 10

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