Fourier matrices for G(d,1,n) from quantum general linear groups
aa r X i v : . [ m a t h . QA ] N ov FOURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERALLINEAR GROUPS
ABEL LACABANNE
Abstract.
We construct a categorification of the modular data associated with every familyof unipotent characters of the spetsial complex reflection group G ( d, , n ). The constructionof the category follows the decomposition of the Fourier matrix as a Kronecker tensor productof exterior powers of the character table S of the cyclic group of order d . The representationof the quantum universal enveloping algebra of the general linear Lie algebra gl m , withquantum parameter an even root of unity of order 2 d , provides a categorical interpretationof the matrix V m S . We also prove some positivity conjectures of Cuntz at the decategorifiedlevel. In the theory of representations of finite groups of Lie type, unipotents characters are animportant object of study. Indeed, these characters are the building blocks for the irreduciblecharacters. They have been classified by Lusztig and are partitioned into families. To eachfamily of unipotent characters, Lusztig has associated a modular datum consisting of a Fouriermatrix and of the eigenvalues of the Frobenius.One important observation is that the classification of unipotents characters does not de-pend on the finite field of definition of the reductive group, and only depends on the structureof the Weyl group W , neither do the modular data. It has been later realized by Lusztig [16],that one can define a similar set of “unipotent characters” for a finite Coxeter group which isnot a Weyl group, together with polynomials which share similar properties with the degreesof unipotent characters of a a finite group of Lie type. Thereafter, modular data associatedwith families of unipotent characters of finite Coxeter groups have also been constructed [17].As Weyl groups are rational reflection groups and Coxeter groups are real reflection groups,similar combinatorial version of unipotent characters for a certain class of complex reflectiongroups, called “spetsial groups”, have been studied by Brou´e, Malle and Michel [4, 5]. Malle[18] has tackled the case of spetsial imprimitive complex reflection groups, by defining unipo-tent characters and their degrees, families of characters and a modular data for each family.The combinatorics developed by Malle is a generalization of Lusztig’s combinatorics for theclassification of unipotent characters of a group of Lie type with Weyl group of type B .In this article, we will study the case of the complex reflection group G ( d, , n ) and theproblem of categorifying the modular data associated to the families of unipotent characters.Indeed, modular categories are known to produce modular data, and given a modular datum,it is a classical problem to determine whether it arises from a category or not. As the modulardata associated with a family of unipotent characters of G ( d, , n ) does not satisfy a positivityproperty, one needs to use triangulated or super categories.A first step in this direction has been achieved by Bonnaf´e and Rouquier [3]: they gave acategorical interpretation of the modular data associated with the unique non trivial familyof unipotent characters of the cyclic group G ( d, , A.L. is a Postdoctoral Researcher of the Fonds de la Recherche Scientifique-FNRS. the Drinfeld double of the Taft algebra, which is a finite dimensional version of the quantumenveloping algebra of the standard Borel of sl . In [14], the author explained how to reinterpretthe category of Bonnaf´e and Rouquier into the framework of slightly degenerate categories.This framework turned out to be well adapted for the problem of categorifying modular data:the modular data of some families of the complex reflection group G ( d, , n ) arise from therepresentation of the Drinfeld double of the quantum enveloping algebra of the standard Borelof sl m [15].The aim of this paper is to give a categorical interpretation of every modular data arisingfrom a family of unipotent characters of G ( d, , n ). More precisely, Cuntz [9] has noticed thatthe Fourier matrix of any family of unipotent characters of G ( d, , n ) is obtained from theKronecker tensor product of some elementary building blocks of the form V m S with S beingthe renormalized character table of the cyclic group of order d . This matrix V m S , togetherwith a diagonal matrix V m T define a modular datum. Therefore, we will first construct acategorification of these building blocks which will then give a categorification of the Fouriermatrix of any family of unipotent characters of G ( d, , n ).The categorical interpretation of V m S will be obtained using representations of the quan-tum group U ξ ( gl m ) where ξ is an even root of unity of order 2 d . More precisely, we considerthe semisimplification of the category of tilting modules, as in the classical construction forsimple Lie algebra. But this will produce a category with an infinite number of non-isomorphicsimple objects. Fortunately, many of the invertible objects of this category lie in the sym-metric center and one can modularize this category in order to obtain a category that wedenote by e D m,ξ . The category e D m,ξ admits many pivotal structure and the choice of a piv-otal structure is related to the choice of the unit element in the fusion algebra defined by V m S . Theorem (Theorem 3.5) . The category e D m,ξ is a categorification of the modular datumdefined by V m S and V m T .Each family F of unipotent characters of G ( d, , n ) is defined by the choice of non-negativeintegers w , . . . , w r and integers 0 < n , . . . , n r ≤ d . Up to some constant, the Fourier matrixof the corresponding family of unipotent characters is a submatrix of the complex conjugateof V n S ⊗ · · · ⊗ V n r S , the product being the Kronecker tensor product of matrices; there isa similar construction for the eigenvalues of the Frobenius using the various matrices V n i T .It is then natural to consider the Deligne tensor product e D n,ξ − = e D n ,ξ − ⊠ · · · ⊠ e D n r ,ξ − . Theorem (Theorem 4.5) . There exists a non-degenerate subcategory e E n,ξ of e D n,ξ − which isa categorification of the modular datum associated to the family of unipotent characters F .The first section of this paper is a recollection of known results concerning fusion algebras,modular data and their categorification. Then, in a second section, we study thoroughly therepresentations of the quantum group U q ( gl m ) at an even root of unity. The semisimplificationof the category of tilting modules gives rises to a semisimple braided pivotal category, witha non-trivial symmetric center. We then explain the effect of killing the symmetric centerand describe the modular datum that one can extract from the category e D m,ξ . The thirdpart is dedicated to the comparison of this modular datum with the m -th exterior powerof the renormalized character table of the cyclic group of order d . Using the categoricalinterpretation of the matrix V m S , we prove several conjectures of Cuntz related to positivityquestions. Finally, in the fourth section, we consider the modular data defined by Malle, which OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 3 are associated to families of unipotent characters of the complex reflection group G ( d, , n ).These modular data are shown to be obtained from a variant of the categorical constructionof the second section. 1. Categorical prolegomena
Our base field is the field of complex numbers C , but most of the materials of this sectionremains true over an algebraically closed field of characteristic 0.1.1. Fusion algebras from S -matrices. We start by recollection a some basic facts on thenotion of a fusion algebra.Let I be a finite set and S a square matrix with complex entries indexed by I . We supposethat S is symmetric and unitary, and that there exists i ∈ I such that S i ,i = 0 for every i ∈ I . We also suppose that for every i, j, k ∈ I , the number(1) N ki,j = X l ∈ I S i,l S j,l S k,l S i ,l ∈ Z . is an integer.To such a matrix S , we associate a Z -algebra A S , which is free as a Z -module with basis( b i ) i ∈ I . The product is defined on the basis by b i · b j = X k ∈ I N ki,j b k and is linearly extended to A S = L i ∈ I Z b i . It is easily checked that this multiplication isassociative and that b i is the unit element. The algebra A S is the fusion algebra associatedwith the matrix S and the integers ( N ki,j ) i,j,k ∈ I are the structure constants with respect tothe basis ( b i ) i ∈ I . Note that multiplying S by any complex number ω of module 1 leads to anisomorphic fusion algebra A ω S ≃ A S . Lemma 1.1.
Let Σ be a diagonal matrix with entries ( σ i ) i ∈ I with σ i ∈ {± } . Let S ′ bethe matrix Σ S Σ − . Denote by ( b i ) i ∈ I the basis of A S and by ( b ′ i ) i ∈ I the basis of A S ′ . Then b i σ i σ i b ′ i is an algebra isomorphism between A S and A S ′ .Proof. If we denote by ( N ki,j ) i,j,k ∈ I (resp. ( N ′ ki,j ) i,j,k ∈ I ) the structure constants of A S (resp. A ′ S ), we have that N ′ ki,j = σ i σ j σ k σ i N ki,j , for any i, j, k ∈ I . The lemma follows easily from this equality. (cid:3) Therefore, conjugation by a diagonal matrix of signs translates into a change of signs ofthe basis of the fusion algebra. Given a matrix S we are interested in the following question: Question 1.2.
Does it exist a collection of signs such that the algebra A S ′ , obtained from S ′ as in Lemma 1.1, has non-negative structure constants? It is usually not easy to give an answer to this question since the structure constants mightbe tedious to compute. Nevertheless, we will later give some examples of such matrices S andanswer to this question via categorical methods. ABEL LACABANNE
Modular data and fusion algebras.
We now define the notion of a modular datum,which is inspired from [12, 17].
Definition 1.3. A modular datum is a quadruple ( I, i , S , T ), where I is a finite set, i isan element of I called special or distinguished , S is a complex matrix with entries indexedby I , and T is a complex diagonal matrix with entries indexed by I satisfying the followingconditions: • S is symmetric and unitary, • S and T define a projective representation of SL ( Z ): there exists ξ ∈ C ∗ such that S = id , ( ST ) = ξ id , and S T = TS , • for all i ∈ I , S i ,i = 0, • for all i, j, k ∈ I , we have N ki,j = X l ∈ I S i,l S j,l S k,l S i ,l ∈ Z . Note that by renormalizing T by a third root of ξ , one can obtain a genuine representationof SL ( Z ).Since the matrix S of a modular datum ( I, i , S , T ) satisfies the conditions of Section 1.1,we have at our disposal the fusion algebra A S . Even if this algebra depends only on S , wewill call it the fusion algebra associated with the modular datum ( I, i , S , T ).1.3. Non-degenerate and slightly degenerate categories.
Using modular categories,one can try to categorify a modular datum whose fusion ring has non-negative structure con-stants. In [14], the author explains how slightly degenerate pivotal fusion categories providesa broader framework for the categorifications of modular data where the fusion ring may havenegative structure constants. We quickly recall these categorical notions, and the main resultsof [14].1.3.1.
Pivotal fusion categories.
For the definition of a fusion category, we refer to [10, Def-inition 4.1.1]. The tensor product will be denoted by ⊗ , the unit object by , and theassociativity and unit constraints will be omitted. The set of isomorphism classes of simpleobjects of a fusion category C is denoted by Irr( C ) and its Grothendieck ring by Gr( C ). Thelatter is a free abelian group with generators given by ([ X ]) ∈ Irr( C ) and the multiplication isgiven by the tensor product: [ X ][ Y ] = X Z ∈ Irr( C ) N ZX,Y [ Z ] , where N ZX,Y denotes the multiplicity of the simple object Z in the tensor product X ⊗ Y .The left dual (resp. right dual ) ( X ∗ , ev X , coev X ) (resp. ( ∗ X, ev ′ X , coev ′ X )) of an object X consists of the datum of an object X ∗ , a evaluation map ev X : X ∗ ⊗ X → (resp. ev ′ X : X ⊗ ∗ X → ) and a coevaluation map coev X : → X ⊗ X ∗ (resp. coev ′ X : → ∗ X ⊗ X ) satisfying(id X ⊗ ev X ) ◦ (coev X ⊗ id X ) = id X and (ev X ⊗ id X ∗ ) ◦ (id X ∗ ⊗ coev X ) = id X ∗ (resp.(ev ′ X ⊗ id X ) ◦ (id X ⊗ coev ′ X ) = id X and (id X ∗ ⊗ ev ′ X ) ◦ (coev ′ X ⊗ id X ∗ ) = id X ∗ ) . OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 5 Left (resp. right) duals are unique up to unique isomorphism and a monoidal category is saidto be rigid if every object admits a left and a right dual. Recall that, by definition, a fusioncategory is rigid.If X and Y have left duals, we also have the notion of a left dual map for any f ∈ Hom C ( X, Y ). It is a map f ∗ ∈ Hom C ( Y ∗ , X ∗ ) and is defined by f ∗ = (ev Y ⊗ id X ∗ ) ◦ (id Y ∗ ⊗ f ⊗ id X ∗ ) ◦ (id Y ∗ ⊗ coev X ) . A rigid monoidal category is said to be pivotal if there exists a natural isomorphism a X : X → X ∗∗ compatible with the tensor product, that is a X ⊗ Y = a X ⊗ a Y , up to theusual identification between ( X ⊗ Y ) ∗∗ and X ∗∗ ⊗ Y ∗∗ .In a pivotal fusion category we have at our disposal the right quantum trace of an endo-morphism. Given f ∈ End C ( X ), its right quantum trace is the unique scalar Tr( f ) such thatthe composition X ⊗ X ∗ X ∗∗ ⊗ X ∗ coev X ( a X ◦ f ) ⊗ id X ev X ∗ is equal to Tr( f ) id . The right quantum dimension dim( X ) of an object X is simply the rightquantum trace of the identity morphism. There also exists a notion of left quantum traceand left quantum dimension. A pivotal structure is said to be spherical if dim( X ) = dim( X ∗ )for every object, or equivalently if the left and right quantum traces coincide. Most of thepivotal structures we will consider in Section 2 are not spherical. Remark 1.4.
We choose the above convention for right quantum traces which is differentto [10, Definition 4.7.1]. Our convention follows from graphical calculus, where the rightquantum trace of an endomorphism is obtained by closing a diagram on the right.Simple objects of a fusion category have a non-zero quantum dimension and we define thecategorical dimension of such a fusion category C bydim( C ) = X X ∈ Irr( C ) | dim( X ) | . It is a positive real number.1.3.2.
Braided categories, degeneracy and twist. A braiding on a monoidal category is thedatum of a binatural isomorphism c X,Y : X ⊗ Y → Y ⊗ X such that the hexagon axioms aresatisfied: c X ⊗ Y,Z = ( c X,Z ⊗ id Y ) ◦ (id X ⊗ c Y,Z ) and c X,Y ⊗ Z = (id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) . As an immediate consequence, the Grothendieck ring of a braided fusion category is commu-tative.A simple object X of a braided category is said to be transparent if c Y,X ⊗ c X,Y = id X ⊗ Y forevery object Y . The symmetric center Z sym ( C ) of a braided category C is the full subcategoryof C whose objects are the transparent objects of C .We say that a braided fusion category is non-degenerate if its symmetric center is tensorgenerated by the unit object . We say that a braided fusion category is slightly degenerate ifits symmetric center is equivalent to the symmetric category sVect of finite dimensional supervector spaces, with braiding c V,W ( v ⊗ w ) = ( − | v || w | w ⊗ v for any super vector spaces V, W , v ∈ V and w ∈ W .We now define the S -matrix of a braided fusion category which will play a prominent role. ABEL LACABANNE
Definition 1.5.
The S -matrix of a braided pivotal fusion category C is the matrix S =( S X,Y ) X,Y ∈ Irr( C ) indexed by Irr( C ) with entries given by the left quantum trace of the doublebraiding: S X,Y = Tr( c Y,X ◦ c X,Y ) . If a simple object X is transparent, then for all Y ∈ Irr( C ) one have S X,Y = dim( X ) dim( Y ).The converse is also true. Proposition 1.6 ([10, Proposition 8.20.5]) . Let C be a braided fusion category. Thenan object X ∈ Irr ( C ) is transparent if and only if for all Y ∈ Irr( C ) one has S X,Y =dim( X ) dim( Y ) . In a rigid braided category, there always exists an isomorphism u X : X → X ∗∗ , called the Drinfeld morphism which is given by the following composition
X X ⊗ X ∗ ⊗ X ∗∗ X ∗ ⊗ X ⊗ X ∗∗ X ∗∗ . id X ⊗ coev X ∗ c X,X ∗ ⊗ id X ∗∗ ev X ⊗ id X ∗∗ However, this morphism is not a pivotal structure, but satisfies u X ⊗ u Y = u X ⊗ Y ◦ c Y,X ◦ c X,Y . We now suppose that C is a braided pivotal fusion category. The composition of theDrinfeld morphism and of the pivotal structure give rise to an endofunctor θ = a ◦ u − of theidentity. This endofunctor is a twist , that is satisfies θ X ⊗ Y = θ X ⊗ θ Y ◦ c Y,X ◦ c X,Y and the pivotal structure a is spherical if and only if θ X ∗ = ( θ X ) ∗ for every object X , that is a is spherical structure if and only if θ is a ribbon .1.3.3. Non-degenerate and slightly degenerate categories.
Under some assumptions on thesymmetric center of a braided pivotal fusion category, the S -matrix and the twist give rise toa modular datum, with fusion algebra related to the Grothendieck ring of the category. Hypothesis:
We assume that the category C is non-degenerate.There exists an invertible object ¯ such that S ¯ ,Y = dim(¯ ) dim( Y ∗ ) for every simple object Y , see [14, § T the diagonal matrix with entries ( δ X,Y θ X ) X,Y ∈ Irr( C ) , where wehave identified θ X and the unique scalar λ such that θ X = λ id X .We define the renormalized matrix S as S = S p dim( C ) p dim(¯ ) , where p dim(¯ ) is a square root of dim(¯ ). Proposition 1.7 ([14, Theorem 2.22]) . Let C be a non-degenerate braided pivotal fusioncategory. Then (Irr( C ) , , S , T ) is a modular datum. The associated fusion algebra A S isisomorphic to the Grothendieck ring of C . Hypothesis:
We assume that the category C is slightly degenerate. Wealso suppose that the twist of the simple transparent non-unit object of C is of quantum dimension − OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 7 Denote by ε the unique simple transparent of C such that ε . Then ε ⊗ ε ≃ ,dim( ε ) = − θ ε = 1.Tensoring by ε has no fixed points on Irr( C ) and then we choose a subset J ⊆ Irr( C )containing one element for each orbit of simple object under tensorization by ε . We will makethe assumption that ∈ J (hence ε J ). There again exists an invertible object ¯ ∈ J suchthat S ¯ ,Y = dim(¯ ) dim( Y ∗ ) for every simple object Y . We consider the submatrix e S of S whose entries are indexed by J . Since Irr( C ) = J ⊔ J ⊗ ε , we have S = (cid:18) e S − e S − e S e S (cid:19) . Denote by e T the diagonal matrix with entries ( δ X,Y θ X ) X,Y ∈ J , where we have once againidentified θ X and the unique scalar λ such that θ X = λ id X .We define the renormalized matrix e S as e S = e S q dim( C ) p dim(¯ ) , where p dim(¯ ) is a square root of dim(¯ ). Proposition 1.8 ([14, Theorem 3.7]) . Let C be a slightly degenerate braided pivotal fusioncategory. Then ( J, , e S , e T ) is a modular datum. The associated fusion algebra A e S is isomorphicto the quotient Gr( C ) / ([ ε ] + [ ]) of the Grothendieck ring of C . The two quotients Gr( C ) / ([ ε ] + [ ]) and Gr( C ) / ([ ε ] − [ ]) have a basis indexed by J andare both quotients of Gr( C ): Gr( C )Gr( C ) / ([ ε ] + [ ]) Gr( C ) / ([ ε ] − [ ]) [ ε ]= − [ ] [ ε ]=[ ] One can easily describe their structure constants using the structure constants of Gr( C ): for X, Y, Z ∈ J , the structure constant of Gr( C ) / ([ ε ] + [ ]) are given by N ZX,Y − N Z ⊗ εX,Y and thestructure constant of Gr( C ) / ([ ε ] − [ ]) are given by N ZX,Y + N Z ⊗ εX,Y . Hence, if N ZX,Y N ε ⊗ ZX,Y = 0for every
X, Y, Z ∈ Irr( C ), the structure constants of Gr( C ) / ([ ε ] − [ ]) are the absolutevalues of the structure constant of Gr( C ) / ([ ε ] + [ ]). The condition N ZX,Y N ε ⊗ ZX,Y = 0 for every
X, Y, Z ∈ Irr( C ) follows often from a grading on the category C such that ε sits in non-trivialdegree.We can now easily give an answer to Question 1.2 for the fusion algebra A e S ≃ Gr( C ) / ([ ε ] +[ ]). Indeed, the answer is positive if and only if the slightly degenerate category C isequivalent to C ⊠ sVect, where sVect is the category of super vector spaces and C is anon-degenerate braided category. From the categorical point of view, changing a sign of abasis element [ X ] of Gr( C ) / ([ ε ] + [ ]) amounts to pick ε ⊗ X instead of X in the set J .To a slightly degenerate category C as above, one can attach a non-degenerate braidedpivotal supercategory e C by adding an odd isomorphism between X and X ⊗ ε , see [14, Sec-tion 4] for more details. This procedure can be thought as a super version of modularization ABEL LACABANNE procedure for degenerate braided pivotal fusion categories, due to Brugui`eres [6] and indepen-dently M¨uger [19]. In this case, the ring Gr( C ) / ([ ε ] + [ ]) is seen as the super Grothendieckring of e C . 2. Quantum gl n and its representations In order to produce slightly degenerate categories, we consider categories of representationsof the universal enveloping algebra of the reductive Lie algebra gl n , with the deformationparameter being an even root of unity. The semisimplification of the category of tiltingmodules will provide a semisimple category, as in the case of a simple Lie algebra, but with aninfinite number of simple objects. Nevertheless, the symmetric center has an infinite numberof non-isomorphic simple objects, and killing the one-dimensional transparent objects willproduce a non-degenerate of a slightly degenerate category.2.1. Root system for gl n . We set up some notations for the root system of gl n . Let P = Z n be the weight lattice of gl n with standard basis ε , . . . , ε n . We equip it with the usual scalarproduct, which is given by h ε i , ε j i = δ i,j . We also define the simple roots α i = ε i − ε i +1 forevery 1 ≤ i < n , which span over Z the root lattice Q . Fro 1 ≤ j ≤ n , let ̟ j = ε + · · · + ε j be the fundamental roots which satisfy for every 1 ≤ i < n and 1 ≤ j ≤ n , h ̟ j , α i i = δ i,j .The symmetric groups in n letters W acts on P by permuting the coordinates and thescalar product h− , −i is W -equivariant. Let l be the length function of W for its usual Coxeterstructure and we denote by w the longest element of W which is given by w ( k ) = n + 1 − k .Let P + be the set of dominant integral weights P + = ( n X i =1 λ i ε i ∈ P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ≥ λ ≥ · · · ≥ λ n ) . Note that every fundamental weight is in P + . Let ρ = ̟ + . . . + ̟ n − = P ni =1 ( n − i ) ε i ,which is again an element of P + .2.2. Rational form of quantum gl n . In this section, we fix q an indeterminate over Z , let A = Z [ q, q − ] and k = Q ( q ) its field of fractions. In A , we define the following elements[ n ] = q n − q − n q − q − , [ n ]! = n Y i =1 [ i ] , (cid:20) nk (cid:21) = [ n ]![ k ]![ n − k ]! , for any n ∈ N and 0 ≤ k ≤ n . Definition 2.1.
The quantum enveloping algebra U q ( gl n ) of gl n is the Q ( q ) algebra generatedby E i , F i , L ± j for 1 ≤ i < n and 1 ≤ j ≤ n subject to the following relations L i L − i = L − i L i = 1 , L i L j = L j L i ,L i E j = q h ε i ,α j i E j L i , L i F j = q −h ε i ,α j i F j L i , [ E i , F j ] = δ i,j K i − K − i q − q − , where K i = L i L − i +1 , and subject to the quantum Serre relations E i E j = E j E i , F i F j = F j F i , if | i − j | > ,E i E j − [2] E i E j E i + E j E i = 0 , F i F j − [2] F i F j F i + F j F i = 0 , if | i − j | = 1 . OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 9 Note that U q ( sl n ) is isomorphic to the Q ( q )-subalgebra generated by E i , F i and K i for1 ≤ i < n . For any λ ∈ P , we also define L λ = Q ni =1 L λ i i so that L i = L ε i and K i = L α i . Itis trivial, but nonetheless crucial, to check that L ̟ n is a central element in U q ( gl n ).Let U q ( gl n ) < (resp. U q ( gl n ) ≤ ) be the subalgebra generated by ( F i ) ≤ i
1) = E i ⊗ K − i , Ψ( F i ⊗
1) = F i ⊗ K i , Ψ( L i ⊗
1) = L i ⊗ , Ψ(1 ⊗ E i ) = K − i ⊗ E i , Ψ(1 ⊗ F i ) = K i ⊗ F i , Ψ(1 ⊗ L i ) = 1 ⊗ L i . Then one has(2) Θ∆( x ) = (Ψ ◦ ∆ op )( x )Θ , for any x ∈ U q ( gl n ), where ∆ op denotes the opposite comultiplication. Moreover Θ is invert-ible and satisfies(3) (∆ ⊗ id)(Θ) = Ψ (Θ )Θ and (id ⊗ ∆)(Θ) = Θ (Θ )Θ . Finally, one may give an explicit form of Θ, see for example [7, § Lusztig’s restricted integral form.
Following [7, § U q ( gl n ) over A , which will be suitable for specializations at roots of unity. Definition 2.2.
The Lusztig’s restricted integral form U A q ( gl n ) of U q ( gl n ) is the A -subalgebraof U q ( gl n ) generated by E ( n ) i = E n [ n ]! , F ( n ) i = F n [ n ]! , L j and (cid:20) L j ; ct (cid:21) = t Y s =1 q c +1 − s L j − q s − c − L − j q s − q − s , for 1 ≤ i < n and 1 ≤ j ≤ n . We denote by U A q ( gl n ) ? = U q ( gl n ) ? ∩ U A q ( gl n ) for ? ∈ { < , ≤ , , ≥ , > } . The restrictedintegral form still has a triangular decomposition as an A -module U A q ( gl n ) ≃ U A q ( gl n ) < ⊗ U A q ( gl n ) ⊗ U A q ( gl n ) > . The comultiplication, counit and antipode restricts to the integral form and endow it witha structure of a Hopf algebra. Using the explicit form of the quasi- R -matrix Θ, one may showthat it lies in (a completion of) U A q ( gl n ) > ⊗ U A q ( gl n ) < .2.4. Representations.
Since U q ( gl n ) is a Hopf algebra, the tensor product of two U q ( gl n )-modules is still an U q ( gl n )-module. Using the antipode S , we also equip the dual V ∗ of an U q ( gl n )-module V with a structure of an U q ( gl n )-module:( x · ϕ )( v ) = ϕ ( S ( x ) · v ) , for any x ∈ U q ( gl n ), v ∈ V and ϕ ∈ V ∗ .2.4.1. Rational representations.
For an U q ( gl n )-module M and λ ∈ P , we define the λ -weightspace of M as M λ = n m ∈ M (cid:12)(cid:12)(cid:12) L i m = q h λ,ε i i , for all 1 ≤ i ≤ n o . We will only consider weight modules of type 1: these modules are direct sums of theirweight spaces as defined above. For any λ ∈ P , the Verma module of highest weight λ is M ( λ ) = U q ( gl n ) ⊗ U q ( gl n ) ≥ Q ( q ) v λ , where Q ( q ) v λ is the one dimensional representation of U q ( gl n ) ≥ given by E i · v λ = 0 and L j · v λ = q h ε j ,λ i v λ , for every 1 ≤ i < n and 1 ≤ j ≤ n . Proposition 2.3. If λ ∈ P + then M ( λ ) has a unique irreducible finite dimensional quotient L ( λ ) . Moreover, every irreducible finite dimensional weight module is isomorphic to a L ( λ ) for a unique λ ∈ P + : irreducible finite dimensional weight modules are parameterized by P + . For λ = k̟ n , it is easy to see that L ( λ ) is one-dimensional, and we will denote by det q themodule L ( ̟ n ). Note that det ∗ q ≃ L ( − ̟ n ) and therefore setting det ⊗ kq = L ( k̟ n ) is coherentwith the fact that det kq ⊗ det lq ≃ det k + lq .It is clear that if λ ∈ P + then − w ( λ ) ∈ P + . Moreover, we have L ( λ ) ∗ ≃ L ( − w ( λ )).2.4.2. A Z -grading. Since the element L ̟ n is central in U q ( gl n ), it induces a Z -grading on thecategory of finite dimensional representations of U q ( gl n ): a simple object L ( λ ) is of degree h λ, ̟ n i . Since L ̟ n is group-like, this grading is of course compatible with the tensor product:every simple summand of a L ( λ ) ⊗ L ( µ ) is of degree h λ, ̟ n i + h µ, ̟ n i .2.4.3. Braiding and pivotal structures.
Using the quasi- R -matrix Θ, we define a braiding onthe category of finite dimensional weight modules. For M and M ′ two finite dimensionalweight modules, we let Θ M,M ′ : M ⊗ M ′ → M ⊗ M ′ be the Q ( q )-linear isomorphism givenby the action of Θ. Since M and M ′ are finite dimensional, only a finite number of Θ λ actsnon-trivially: Θ λ ( M µ ⊗ M ′ µ ′ ) ⊂ M µ + λ ⊗ M ′ µ ′ − λ .Now consider the map f M,M ′ : M ⊗ M ′ → M ⊗ M ′ given by f M,M ′ ( m ⊗ m ′ ) = q h µ,µ ′ i m ⊗ m ′ OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 11 for every m ∈ M µ and m ∈ M ′ µ ′ . One easily check on the generators of U q ( gl n ) that for every x ∈ U q ( gl n ) ⊗ U q ( gl n ), any m ∈ M and m ∈ M ′ one has(4) x · f ( m ⊗ m ′ ) = f (Ψ( x ) · ( m ⊗ m ′ )) . We then define c M,M ′ = τ ◦ f M,M ′ ◦ Θ M,M ′ , where τ ( m ⊗ m ′ ) = m ′ ⊗ m . Combining (2)and (4), one obtains that c M,M ′ is an U q ( gl n )-equivariant map.Using (3) and (4), one shows that c − , − satisfy the hexagon axioms: c L,M ⊗ N = (id M ⊗ c L,N ) ◦ ( c L,M ⊗ id N ) and c L ⊗ M,N = ( c L,N ⊗ id M ) ◦ (id L ⊗ c M,N ) . Proposition 2.4.
The family of maps c M,M ′ endow the category of finite dimensional weightmodules over U q ( gl n ) with a structure of a braided category. We now turn to pivotal structures on the category of finite dimensional weight modulesover U q ( gl n ). Since the square of the antipode is given by conjugation by L ρ , the map a M : M → M ∗∗ sending m ∈ M to a M ( m ) ∈ M ∗∗ defined by a M ( m )( ϕ ) = ϕ ( L ρ · m )for any ϕ ∈ M ∗ is an isomorphism of U q ( gl n )-modules. But conjugation by the element L ρ L k̟ n also gives the square of the antipode since L k̟ n is central. We define thereforeanalogously an isomorphism a k,M : M → M ∗∗ . Proposition 2.5.
The family of maps a k,M endow the category of finite dimensional weightsmodules over U q ( gl n ) with a structure of a pivotal category. Moreover, the pivotal structureis spherical if and only if k = 1 − n .Proof. The twist associated to the pivotal structure a k is given on the simple object L ( λ ) bymultiplication by q h λ,λ +2 ρ + p̟ n i . Since L ( λ ) ∗ ≃ L ( − w ( λ )), the pivotal structure is sphericalif and only if for all λ ∈ P + , we have q h λ,λ +2 ρ + k̟ n i = q h− w ( λ ) , − w ( λ )+2 ρ + k̟ n i . As q is anindeterminate over Q the last condition is equivalent to h λ, λ +2 ρ + k̟ n i = h− w ( λ ) , − w ( λ )+2 ρ + k̟ n i . But − w ( ρ ) = ρ + (1 − n ) ̟ n so that h− w ( λ ) , − w ( λ ) + 2 ρ + k̟ n i = h λ, λ + 2 ρ + k̟ n i + 2((1 − n ) − k ) h λ, ̟ n i . Therefore the pivotal structure a k, − is spherical if and only if k = 1 − n . (cid:3) The right quantum trace with respect to the pivotal structure a k, − is denoted Tr k , theassociated quantum dimension by dim k and the associated twist by θ k, − .2.4.4. Integral representations.
Since we work with Lusztig’s restricted integral form, we needto adapt slightly the definition of a weight space. Given M an U A q ( gl n )-module and λ ∈ P ,the λ -weight space of M is M λ = (cid:26) m ∈ M (cid:12)(cid:12)(cid:12)(cid:12) L i · m = q h λ,ε i i m, (cid:20) L i ; 0 t (cid:21) · m = (cid:20) h λ, ε i i t (cid:21) m, for all 1 ≤ i ≤ n and t ∈ N (cid:27) . We will again only consider weight modules, that is modules which are sum of their weightspaces.There also exists an integral version of the representations M ( λ ) and L ( λ ). Denote by M A ( λ ) (resp. L A ( λ )) the U A q ( gl n )-submodule of M ( λ ) (resp. of L ( λ )) generated by v λ . Then M A ( λ ) ⊗ A Q ( q ) ≃ M ( λ ) and L A ( λ ) ⊗ A Q ( q ) ≃ L ( λ ) . The module L ( λ ) is the integral Weyl module of highest weight λ . Similarly to the braiding described above for finite dimensional weight modules over U q ( gl n )one defines a braiding structure on the category of finite dimensional weight modules over U A q ( gl n ): we have already seen that the quasi- R -matrix Θ lies in the Lusztig’s restrictedintegral form of U q ( gl n ). One also has the family of pivotal structures a k, − .2.5. Specialization, tilting modules and semisimplification.
Let d > ξ = exp( iπ/d ). We define the quantum group U ξ ( gl n ) as the specialization of Lusztig’srestricted integral form: U ξ ( gl n ) = U A q ( gl n ) ⊗ A C , where we see C as an A -algebra via the Z -linear map q ξ . Since ξ is a primitive 2 d -th rootof unity, we have extra relations in this specialization: for example E di = F di = 0.The construction of a fusion category from the quantum enveloping algebra of a simple Liealgebra extends to our situation with gl n , which is only a reductive Lie algebra. We recallquickly the main steps of this construction:(1) We have at our disposal the specialization of the Weyl module L ( λ ) ⊗ A C . We saythat a module M over U ξ ( gl n ) is tilting if both M and M ∗ have a filtration byspecializations of Weyl modules.(2) One shows that the category of tilting modules is stable under direct sum, tensorproduct and duality.(3) We semisimplify the monoidal category of tilting modules by killing negligible moduleswith respect to the pivotal structure a , − (or equivalently any a k, − ), see [11] for adescription of this procedure.When doing these steps with the quantum enveloping algebra of a simple Lie algebra, oneobtains a fusion category, with a braiding and a pivotal (even spherical) structure. Here,since we work with gl n , we do not have a finite number of simple objects. Let us denote by C ξ the category obtained with this procedure. One shows that the simple objects are givenby the images of the L ( λ ) ⊗ A C for λ in the alcove C n,d = (cid:8) λ ∈ P + (cid:12)(cid:12) h λ, ε − ε n i ≤ d − n (cid:9) . In order to distinguish the simple objects of C ξ with the Weyl modules, we denote by X ( λ )the simple object in C ξ parameterized by λ ∈ C n,d . We nonetheless use the notation det ⊗ kξ for X ( k̟ n ). The category C ξ inherits the Z -grading from Section 2.4.2. Note that this gradingis not obtained from the action of the element L ̟ n since L d̟ n = 1 and we would only obtaina grading by the group Z / d Z .Note that C ξ is non-zero if and only if d ≥ n . From now on, we will always work underthis assumption. Even if d = n , we have an infinite number of simple objects since ̟ n andits multiples are in C n,d . The tensor product of X ( ̟ i ) with X ( λ ) is given by(5) X ( ̟ i ) ⊗ X ( λ ) ≃ M ≤ j < ··· The S -matrix of C ξ is given by S X ( λ ) ,X ( µ ) = ξ h λ + µ,k̟ n i P w ∈ W ( − l ( w ) ξ h w ( λ + ρ ) ,µ + ρ i P w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i , for all λ, µ ∈ C n,d .The value of the twist associated with the pivotal structure a k, − on the simple object X ( λ ) is given by ξ h λ,λ +2 ρ + k̟ n i .Proof. It suffices to do it for the pivotal structure a , − . Indeed, the quantum trace Tr k of anelement f ∈ End( X ) is given by Tr k ( f ) = Tr( L k̟ n L ρ f ) , where Tr is the usual trace. Since L k̟ n is group-like and central, it acts by a scalar on X ( λ ) ⊗ X ( µ ), and it is easy to check on the highest weight vector v λ ⊗ v µ that L k̟ n · v λ ⊗ v µ = ξ h λ + µ,k̟ n i v λ ⊗ v µ . The formula for p = 0 is obtained using the same arguments as in [2,Theorem 3.3.20]. (cid:3) Note that since ξ is a 2 d -th root of unity, the pivotal structure a k, − is spherical if k ≡ − n [ d ]and a k, − = a l, − if k ≡ l [2 d ] since L d̟ n acts by 1 on any X ( λ ).2.6. Symmetric center and modularization. The category C ξ has an infinite numberof simple objects and we aim to produce out of it a fusion or superfusion category withinvertible S -matrix. Therefore we need to determine the symmetric center of C ξ becausehaving an invertible S -matrix is equivalent to having a trivial symmetric center. Proposition 2.7. The simple objects belonging to symmetric center of C ξ are the X (( d − n ) ̟ i + r̟ n ) with ≤ i ≤ n and r ≡ i [ d ] . The symmetric center of C ξ is then pointed andtensor generated by X (( d − n ) ̟ + ̟ n ) .Proof. We will use the same strategy as in [6, Section 4] by proving first that a transparentsimple object is invertible and then detecting the transparent objects among the invertibleones.Let X be a transparent simple object. Then it satisfies c Y,X ◦ c X,Y = id X ⊗ Y for all object Y . We now use the ribbon θ − n, − and obtain that θ − n,X ⊗ θ − n,Y = θ − n,X ⊗ Y . Now supposethat moreover Y is simple, and that X ⊗ Y ≃ L i Z i . Taking the quantum trace Tr − n , weobtain that θ − n,X θ − n,Y P i dim − n ( Z i ) = P i dim − n ( Z i ). The twist being a power of ξ , bytaking the norm we find that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i dim − n ( Z i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i θ − n,Z dim − n ( Z i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . But with the spherical structure a − n, − , the quantum dimension of any simple object is apositive real number, and therefore X i dim − n ( Z i ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i θ − n,Z i dim − n ( Z i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By the case of equality in the triangle inequality, we deduce that the argument of θ − n,Z i dim − n ( Z i )does not depend on i . Since dim − n ( Z i ) > θ − n,Z i is a root of unity, we deduce that θ − n,Z i does not depend on i so that θ − n,X ⊗ Y is a scalar multiple of id X ⊗ Y . It is then alsotrue for any twist θ k, − : we check that θ k,X ( λ ) = ξ ( k + n − h λ,̟ n i θ − n,X ( λ ) and it remains tocheck that h λ, ̟ n i only depends on X and Y if X ( λ ) is a summand of X ⊗ Y . But this isimmediate since h λ, ̟ n i is equal to the degree of X ⊗ Y for the Z -grading.Now, we show that a simple object X such that θ ,X ⊗ Y is a scalar for any simple object Y is invertible. If d = n then any simple object is invertible, so we may and will supposethat d > n . Suppose that X ≃ X ( λ ), we take Y = X ( ̟ ) and look at the decomposition of X ⊗ X ( ̟ ). Thanks to (5), it is given by X ( λ ) ⊗ X ( ̟ ) ≃ M ≤ j ≤ nλ + ε j ∈ C n,d X ( λ + ε j ) . Suppose that there exists 1 ≤ i < j ≤ n such that λ + ε i ∈ C n,d and λ + ε j ∈ C n,d . Since θ ,X ⊗ Y is a scalar, we have θ ,X ( λ + ε i ) = θ ,X ( λ + ε j ) , that is h λ + ε i , λ + ε i + 2 ρ i ≡ h λ + ε j , λ + ε j + 2 ρ i [2 d ] , which is equivalent to 2 λ i + 2( n − i ) + 1 ≡ λ j + 2( n − j ) + 1 [2 d ]. Then λ i − λ j ≡ i − j [ d ]and since d − n ≥ λ i − λ j ≥ λ i − λ j = d + i − j . As 0 > i − j > − n , this leads to acontradiction.Hence there exists a unique 1 ≤ i ≤ n such that λ + ε i ∈ C n,d and this is possible if andonly if λ = ( d − n ) ̟ i + r̟ n for some r and X is thus invertible.We finally determine the transparent objects among the invertible ones. Let 1 ≤ i ≤ n and X = X (( d − n ) ̟ i + r̟ n ) be an invertible object. Since X ⊗ X ( λ ) ≃ X (sh i ( λ )+( d − n ) ̟ i + r̟ n ),the object X is transparent if and only if θ ,X (sh i ( λ )+( d − n ) ̟ i + r̟ n ) = θ ,X θ ,X ( λ ) for all λ ∈ C n,d .This last equality is equivalent to h sh i ( λ ) + ( d − n ) ̟ i + r̟ n , sh i ( λ ) + ( d − n ) ̟ i + r̟ n + 2 ρ i ≡h λ, λ + 2 ρ i + h ( d − n ) ̟ i + r̟ n , ( d − n ) ̟ i + r̟ n + 2 ρ i [2 d ] . Going back to the definition of ρ , we find that X is transparent if and only of for all λ ∈ C n,d we have ( r − i ) h λ, ̟ n i ≡ d ]. If n = d there is no condition on r and i and then everyobject is transparent. If d > d , taking λ = ̟ , we see that r ≡ i [ d ] so that the onlytransparent objects in C ξ are the X (( d − n ) ̟ i + r̟ n ) with r ≡ i [ d ]. Finally, we need tocheck that X (( d − n ) ̟ i + r̟ n ) is a tensor power of X (( d − n ) ̟ + ̟ n ). We remark that X (( d − n ) ̟ + ̟ n ) ⊗ n ≃ det ⊗ dq , so that, by tensoring to a suitable power of det ⊗ dq , we may andwill suppose that 0 ≤ r < d . Then i = r and X (( d − n ) ̟ i + i̟ n ) ≃ X (( d − n ) ̟ + ̟ n ) ⊗ i . (cid:3) If we want to kill the symmetric center by a process of modularization, we have to checkthat every simple object in the symmetric center is of twist 1 for the chosen pivotal structure.Let us denote by ε the object X (( d − n ) ̟ + ̟ n ) which tensor generates the symmetric center. Lemma 2.8. We endow C ξ with the pivotal structure a k, − . The quantum dimension and thetwist of ε are respectively ( − k + n − and ( − d + k .Proof. Since ε is an invertible object, the quantum dimension of ε with respect to the sphericalstructure a − n, − is 1 since the quantum dimension of a simple object is positive with respect OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 15 to this spherical structure. Therefore the quantum dimension of ε with respect to the pivotalstructure a k, − is ξ h ( d − n ) ̟ + ̟ n , ( k + n − ̟ n i = ( − k + n − .Concerning the twist, we only have to check that h ( d − n ) ̟ + ̟ n , ( d − n ) ̟ + ̟ n + 2 ρ + k̟ n i ≡ d ( k + d ) [2 d ] , which is an easy computation. (cid:3) Since we want the twist of objects in the symmetric center to be equal to 1, we will alwayschoose from now on a pivotal structure of the form a p + d, − with p ∈ Z . The object ε ⊗ isalways of quantum dimension and twist equal to 1 so that we can apply the modularizationprocedure of Brugui`eres and M¨uger. We then obtain a fusion category D ξ with symmetriccenter generated by the image of ε . Tensoring by ε ⊗ does not have fixed points on theset of simple objects since this object sits in non-trivial degree for the Z -grading. Thus themodularization procedure only add isomorphism between some simple objects and the imagein D ξ of every simple object in C ξ is still a simple object.The structure of the symmetric center depends now on the parity of n + d : • if n d [2] then the symmetric center of D ξ is equivalent, as a pivotal category toRep( Z / Z ) and D ξ is then modularizable. We then obtain a non-degenerate fusioncategory e D ξ from the modularization of D ξ . • if n ≡ d [2] then the symmetric center of D ξ is equivalent, as a pivotal category tosVect and D ξ is then slightly degenerate. We then obtain a non-degenerate superfusioncategory e D ξ from the supermodularization of D ξ . The simple objects of e D ξ are then parameterized by the orbits of Irr( C ξ ) under tensorizationby ε .2.7. Modular data arising from e D ξ . We turn to the computation of the modular invariantsof the category e D ξ . First, we determine a suitable subset of C n,d for the pasteurization ofsimple objects of f C ξ . This amounts to choose a representative of each orbit of isomorphismclasses of simple objects of C ξ under tensorization by ε . We say that two weights λ ∼ elem µ in C n,d if λ = sh( µ ) + ( d − n ) ̟ + ̟ n or µ = sh( λ ) + ( d − n ) ̟ + ̟ n . We then definean equivalence relation ∼ on C n,d as the reflexive and transitive closure of ∼ elem . It isalmost immediate to see that if λ ∼ µ then there exists 0 < qk ≤ n and r ∈ Z such that µ = sh k ( λ ) + ( d − n ) ̟ k + ( rd + k ) ̟ n . Note that λ ∼ µ if and only if X ( λ ) ≃ X ( µ ) in e D ξ . Lemma 2.9. Let λ ∈ C n,d . Then there exists a unique λ ′ ∈ C n,d such that λ ′ ∼ λ and d − n ≥ λ ′ ≥ · · · ≥ λ ′ n ≥ .Proof. First notice that for every λ ∈ C n,d we have λ ∼ λ + d̟ n since ε ⊗ n ≃ det ⊗ dξ .Now fix λ ∈ C n,d . By adding or subtracting a multiple of d̟ n , we may and will supposethat d − n ≥ λ > − n . If moreover λ n ≥ 0, we have nothing to prove. Otherwise, let 1 ≤ k ≤ n be minimal such that λ k ≤ k − n − 1. Since λ n < k is well defined, and since λ > − n , we have k ≥ 2. Let λ ′ = sh n +1 − k ( λ ) + ( d − n ) ̟ n +1 − k + ( n + 1 − k ) ̟ n . Then λ ′ ∼ λ and we have λ ′ = λ k + d + 1 − k ≤ d − n be definition of k . Finally, λ ′ n = λ k − + n + 1 − k ≥ k .For the uniqueness, suppose that λ ∼ µ with 0 ≤ λ i ≤ d − n and 0 ≤ µ i ≤ d − n for all1 ≤ i ≤ n . If d = n , there is nothing to do since λ = µ = 0. Therefore, we suppose d > n .Since λ ∼ µ we have − ( d − n ) ≤ λ i − µ j ≤ d − n for all 1 ≤ i, j ≤ n . There also exists < k ≤ n and r ∈ Z such that µ = sh k ( λ ) + ( d − n ) ̟ k + ( rd + k ) ̟ n . We want to show that k = n and r = − k < n . Therefore, by choosing suitably i and j we have − ( d − n ) ≤ rd + k ≤ d − n and − ( d − n ) ≤ rd + k + d − n ≤ d − n . This implies that − ( d − n ) ≤ rd + k ≤ ≤ rd + k + d − n < d . But as 0 ≤ k + d − n < d we obtain that r = 0 and then k ≤ 0, whichis a contradiction.Then k = n and λ = µ + ( r + 1) d̟ n . As | λ i − µ i | < d , we necessarily have r = − 1, whichends the prove of uniqueness. (cid:3) Let e C n,d be the subset of C n,d consisting of dominant integral weights λ satisfying d − n ≥ λ ≥ · · · ≥ λ n ≥ 0. We will therefore use the set e C n,d to parameterize simple objects in e D ξ .This also shows that the rank of e D ξ is equal to (cid:0) dn (cid:1) . Lemma 2.10. The categorical dimension of e D ξ is dim( e D ξ ) = ( − n ( n − / d n (cid:16)Q n − i =1 ( ξ i − ξ − i ) n − i (cid:17) . Proof. Since the simple object det ξ is invertible, tensoring by det ξ does not change the squarenorm | X | of a simple object X . Note that we have a bijection between the simple objects ofthe fusion category associated with sl n and the set of λ ∈ C n,d with λ n = 0. Moreover, thesquared norm of the object X ( λ ) is the same if we restrict to sl n . Therefore, by [2, Theorem3.3.20] X λ ∈ C n,d λ n =0 | X ( λ ) | = ( − n ( n − / nd n − (cid:16)Q n − i =1 ( ξ i − ξ − i ) n − i (cid:17) . Consider now the category C ′ ξ obtained from C ξ by adding isomorphisms between powers ofdet dξ . Its objects can be parameterized by { λ ∈ C n,d | ≤ λ n < d } , so that its categoricaldimension is equal todim( C ′ ξ ) = 2 d X λ ∈ C n,d λ n =0 | X ( λ ) | = ( − n ( n − / nd n (cid:16)Q n − i =1 ( ξ i − ξ − i ) n − i (cid:17) . Note that C ′ ξ inherits of a Z / dn Z -grading since det ⊗ dξ sits in degree 2 dn . As ε is of order 2 n in C ′ ξ and none of its non-trivial tensor power is of degree 0, we find that the (super)dimensionof e D ξ is equal to dim( C ′ ξ ) / n , which ends the proof. (cid:3) Since the category e D ξ is pivotal but not necessarily not spherical, we need to determinethe invertible object ¯ among the X ( λ ), λ ∈ e C n,d and its quantum dimension. Proposition 2.11. Let p ∈ Z and we equip e D ξ with the pivotal structure a p + d, − . The object ¯ is isomorphic to det − (2 p + n − ξ .Proof. The object ¯ is determined by dim p + d ( X ( λ ) ∗ ) dim p + d (¯ ) = S ¯ ,X ( λ ) for all λ ∈ C n,d .We first compute the quantum dimension of X ( λ ) ∗ by using the fact that X ( λ ) ∗ ≃ X ( − w ( λ ))dim p + d ( X ( λ ) ∗ ) = ξ −h w ( λ ) , (2 p + d ) ̟ n i P w ∈ W ( − l ( w ) ξ h w ( − w ( λ )+ ρ ) ,ρ i P w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i . OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 17 Now we use that w ( ρ ) = − ρ + ( n − ̟ n and the fact that ̟ n is invariant under the actionof W to obtain h w ( − w ( λ ) + ρ ) , ρ i = h ww ( λ + ρ ) , w ( ρ ) i − h λ, ( n − ̟ n i . Therefore, by a change of variables in the sum at the numerator, we obtaindim p + d ( X ( λ ) ∗ ) = ξ −h λ, (2 p + d +2( n − ̟ n i P w ∈ W ( − l ( w ) ξ h w ( λ + ρ ) ,ρ i P w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i . It is now an easy calculation to show that S det − (2 p + n − ξ ,X ( λ ) dim p + d (cid:16) det − (2 p + n − ξ (cid:17) = ξ −h λ, (2 p + d +2( n − ̟ n i P w ∈ W ( − l ( w ) ξ h w ( λ + ρ ) ,ρ i P w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i = dim p + d ( X ( λ ) ∗ ) , which ends the proof. (cid:3) The unique λ ∈ e C n,d such that − (2 p + d + n − ̟ n ∼ λ is subtle to determine, but weonly need the value of its quantum dimension, which is then ± ξ − h (2 p + n − ̟ n ,ρ + p̟ n i . Theorem 2.12. We endow e D ξ with the pivotal structure a p + d, − . There exists a fourth rootof unity ω such that the renormalized S -matrix of e D ξ is given by S X ( λ ) ,X ( µ ) = ωξ h λ + µ, (2 p + d ) ̟ n i +2 pn ( p + n − P w ∈ W ( − l ( w ) ξ h w ( λ + ρ ) ,µ + ρ i √ d n , for any λ, µ ∈ e C n,d . The twist of the object X ( λ ) is given by θ p + d,X ( λ ) = ξ h λ,λ +2 ρ +(2 p + d ) ̟ n i .Proof. We need to compute the suitable renormalization of the S -matrix, which is given bythe positive square root of the categorical (super)dimension of e D ξ multiplied by a square rootof the quantum dimension of ¯ .Thanks to Lemma 2.10, the categorical (super)dimension of e D ξ is given bydim( e D ξ ) = ( − n ( n − / d n (cid:16)Q n − i =1 ( ξ i − ξ − i ) n − i (cid:17) . In order to renormalize the S -matrix, we need the positive square root of this dimension.Using Weyl’s denominator formula we obtain X w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i = ξ h ρ,ρ i n − Y i =1 (1 − ξ − i ) n − i = ξ h ( n − ̟ n ,ρ i n − Y i =1 ( ξ i − ξ − i ) n − i , so that the desired square root is i n ( n − / ξ h ( n − ̟ n ,ρ i √ d n X w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i . As the dimension of ¯ is ± ξ − h (2 p + n − ̟ n ,ρ + p̟ n i , there exists a fourth root of unity ω suchthat q dim( e D ξ ) q dim p + d (¯ ) = ω √ d n ξ −h (2 p + n − ̟ n ,ρ + p̟ n i + h ( n − ̟ n ,ρ i X w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i = ω √ d n ξ − pn ( p + n − X w ∈ W ( − l ( w ) ξ h w ( ρ ) ,ρ i . Therefore, using Proposition 2.6 the renormalized S -matrix is given by S X ( λ ) ,X ( µ ) = ωξ h λ + µ, (2 p + d ) ̟ n i +2 pn ( p + n − P w ∈ W ( − l ( w ) ξ h w ( λ + ρ ) ,µ + ρ i √ d n . The twist has already been computed in Proposition 2.6. (cid:3) If n ≡ d [2] then we have seen that the category D ξ is slightly degenerate with symmetriccenter tensor generated by ε . This category inherits a Z / d Z -grading from the Z -grading of C ξ as the object ε ⊗ is of degree 2 d in C ξ . Proposition 2.13. Suppose that n ≡ d [2] . The category D ξ is equivalent to D ξ, ⊠ sVect ,where D ξ, is a non-degenerate braided fusion category, if and only if n and d are both odd or n = d .Proof. Suppose that n and d are both odd and consider the full subcategory D ξ, of D ξ with objects of even degree in Z / d Z . Since d is odd, the object ε of degree d is not in C ξ .This shows that for each simple object X , either X or X ⊗ ε is in D ξ, and that therefore D ξ ≃ D ξ, ⊠ sVect.Suppose that n = d . Then the simple objects of D ξ are and ε and then D ξ ≃ sVect.Suppose now that both n and d are even, that n > d and that there exists a full subcategory D ξ, of D ξ such that D ξ ≃ D ξ, ⊠ sVect. Since d > n , the object X ( ̟ ) is a simple objectof D ξ , and X ( ̟ ) or X ( ̟ ) ⊗ ε is in D ξ, . Because of (5), the simple object ε is a directsummand of X ( ̟ ) ⊗ d . Since d is even, X ( ̟ ) ⊗ d ≃ ( X ( ̟ ) ⊗ ε ) ⊗ d . Therefore ε is in f D ξ which is a contradiction because the only simple transparent object of D ξ, is isomorphic tothe unit object. (cid:3) Exterior powers and Cuntz’ positivity conjecture Let d ≥ ≤ n ≤ d be another integer. We keep the notation ξ = exp( iπ/d )and let ζ = ξ = exp(2 iπ/d ).3.1. Set-up and known results. We consider the n -th exterior power V n S of the matrix S = (cid:16) ζ ij √ d (cid:17) ≤ i,j 1) if p ≤ d − n, (0 , , . . . , p + n − − d, p, p + 1 , . . . , d − 1) otherwise . OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 19 The entry ( V n S ) i ( p ) ,i is always non-zero since it is a multiple of a Vandermonde determi-nant. Therefore, we for any a, b and c ordered n -tuples, the following complex number: p N ca,b = X k ∈ I n,d ( V n S ) a,k ( V n S ) b,k ( V n S ) c,k ( V n S ) i ( p ) ,k . Proposition 3.1 ([9, Theorem 4.2]) . For any p ∈ Z and a, b, c ∈ I n,d , p N ca,b ∈ Z . Therefore,these integers are the structure constants of a Z -algebra. In [9], it is proven only for p = 0, but the same argument applies for any p ∈ Z (see [8,Wahl der Eins]). Therefore, the matrix V n S satisfies the hypothesis of Section 1.1 for anychoice of special element i ( p ) and defines a fusion algebra A n,p .There also exist a T -matrix associated with S . Let T be the diagonal matrix indexed by I ,d with entries given by T a = ζ d − ξ a + da . Then S and T satisfy S = id , ( ST ) = id and ST = T S, see [9, Proposition 5.4] and similar relations are satisfied by V n S and V n T .3.2. Cuntz’ conjectures. In [9], Cuntz has conjectured the following positivity property: Conjecture 3.2 ([9, § . Suppose that < n < d . (1) Suppose moreover that n and d are not both even. Then there exist a choice of signs ( σ a ) a ∈ I n,d ∈ {± } I n,d such that for any a, b, c ∈ I n,d , the integer p N ca,b σ a σ b σ c is non-negative. (2) Suppose moreover that both n and d are even. Then for all choices of signs ( σ a ) a ∈ I n,d ∈{± } I n,d there exists a, b, c ∈ I n,d such that p N ca,b σ i σ j σ k is negative. However, theabsolute values of p N ca,b define an associative Z -algebra. He also conjectured in his thesis [8] that the fusion ring A defined by V n S is a quotient ofa free algebra of rank doubled. Conjecture 3.3 ([8, Vermutung 5.1.6]) . Let A ′ be a free Z -module with basis { b a , b ′ a | a ∈ I n,d } and denote also by { b a | a ∈ I n,d } the basis of A . Let π : A ′ → A be the Z -module map definedby π ( b a ) = b a and π ( b ′ a ) = − b a . Define also ϕ : A → A ′ by X a ∈ I n,d λ a b a X a ∈ I n,d ( δ λ a > λ a b a − δ λ a < λ a b ′ a ) . Then the multiplication on A ′ defined by xy = ϕ ( π ( x ) π ( y )) is associative and its structureconstants lie in N . The ring A ′ has then two quotients, namely A obtained by identifying b i and − b ′ i andanother one A abs obtained by identifying b i and b ′ i . It is clear that A abs has non-negativestructure constants and that its structure constants are the absolute values of the structureconstants of A . One can depicts the situation by the following diagram A ′ A A abs b i = − b ′ i b i = b ′ i which is similar to the situation explained in Section 1.3.3. Note that if one can find a change of basis of A by changing signs such that the structureconstants are non-negative, then Conjecture 3.3 is almost trivial.3.3. Relationship with quantum gl n . We now relate the renormalized S -matrix of thecategory e D ξ for gl n introduced in Section 2.6 with the exterior power V n S , up to some signs.The twists will also correspond with the diagonal matrix V n T , up to a multiplication by aroot of unity. The pivotal structure on e D ξ will depend on the choice of the special element i ( p ) .We define for every p ∈ Z a map ι p : I n,d → C n,d by a w n X i =1 a i ε i ! − ρ − p̟ n . Since a is strictly increasing, ι p ( a ) ∈ P + . Moreover, ι p ( a ) − ι p ( a ) n = a n − a − n + 1 ≤ d − n ,that is ι p ( a ) ∈ C n,d . We then define e ι p ( a ) ∈ e C n,d as the unique element in e C n,d such that e ι p ( a ) ∼ ι p ( a ). As ∈ p is clearly an injection, so is e ι p . But | e C n,d | = (cid:0) dn (cid:1) = | I n,d | and e ι p isbijective. Lemma 3.4. For all p ∈ Z and k ∈ Z we have ι p ( i ( k ) ) ∼ ( k − p ) ̟ n .Proof. We may and will suppose that 0 ≤ k < d since λ ∼ λ + d̟ n for any λ ∈ C n,d .We check that ι p ( i ( k ) ) = ( ( k − p ) ̟ n if 0 ≤ k ≤ d − n, − p̟ n + ( d − n ) ̟ d − k otherwise . If d − n < k < d , we see that ι p ( i ( k ) ) = sh d − k (( k − d − p ) ̟ n ) + ( d − n ) ̟ d − k + ( d − k ) ̟ n ∼ ( k − d − p ) ̟ n ∼ ( k − p ) ̟ n , which ends the proof. (cid:3) Therefore e ι p ( i ( p ) ) = 0 and X ( e ι p ( i ( − p +1 − n ) )) ≃ ¯ . Theorem 3.5. Let p ∈ Z . We equip the category e D ξ with the pivotal structure a p + d, − . The(super)fusion category e D ξ is a categorification of the modular datum defined by V n S and V n T : there exist a fourth root of unity ω and signs ( σ a ) ∈ {± } I n,d with σ i ( p ) = 1 such that S X ( e ι p ( a )) ,X ( e ι p ( b )) = ωσ a σ b ( ^ n S ) a,b and θ X ( e ι p ( a )) = ζ ∗ ( ^ n T ) a , where ζ ∗ = ζ n (1 − d )24 ξ −h ρ,ρ i− pn ( p + d ) − (2 p + d ) ( n ) .The (super)Grothendieck ring of e D ξ is then isomorphic to the ring defined by V n S withunit parameterized by i ( p ) .Proof. We use the formula of the renormalized S -matrix of e D ξ given in Theorem 2.12. Sincefor all a ∈ C n,d we have e ι p ( a ) ∼ ι p ( a ), there exist a sign η a ∈ {± } such that for all a, b ∈ C n,d , S X ( e ι p ( a )) ,X ( e ι p ( b )) = ωη a η b ξ h ι p ( a )+ ι p ( b ) , (2 p + d ) ̟ n i +2 pn ( p + n − P w ∈ W ( − l ( w ) ξ h w ( ι p ( a )+ ρ ) ,ι p ( b )+ ρ i √ d n . OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 21 Now, for any w ∈ W , we have h w ( ι p ( a ) + ρ ) , ι p ( b ) + ρ i = h ww ( n X i =1 a i ε i ) − p̟ n , w ( n X i =1 b i ε i ) − p̟ n i = n X i =1 a w − w − w ( i ) b i − p n X i =1 ( a i + b i ) + p n, so that S X ( e ι p ( a )) ,X ( e ι p ( b )) = ωη a η b ( − P ni =1 ( a i + b i ) P w ∈ W ( − l ( w ) Q ni =1 ξ a w ( i ) b i √ d n . We now set σ a = η a ( − P nj =1 a j . We hence find that S X ( e ι p ( a )) ,X ( e ι p ( b )) = ωσ a σ b ( V n S ) a,b . Bychanging the sign of every σ a if necessary, we have σ i ( p ) = 1.For the value of the twist, since θ p + d,X ( e ι p ( a )) = θ p + d,X ( ι p ( a )) , we find that θ p + d,X ( e ι p ( a )) = ξ h ι p ( a ) ,ι p ( a )+2 ρ +(2 p + d ) ̟ n i . But h ι p ( a ) , ι p ( a ) + 2 ρ + (2 p + d ) ̟ n i = h w ( n X i =1 a i ε i ) − ρ − p̟ n , w ( n X i =1 a i ε i ) + ρ + p̟ n + d̟ n i , and therefore θ p + d,X ( e ι p ( a )) = ξ −h ρ,ρ i− pn ( p + d ) − (2 p + d ) ( n ) ξ P ni =1 ( a i + da i ) , which leads to the desiredformula. (cid:3) As a corollary, we obtain a new proof of Proposition 3.1 since the signs σ a do not changethe integrality of the structure constants p N ca,b . Corollary 3.6. For any p and r , the fusion algebras A n,p and A n,r are isomorphic: thestructure of the fusion ring defined bu V n S does not depend on the choice of the specialelement of the form i ( p ) . Proof of Cuntz’ conjectures. Categorification has turned to be a powerful tool toprove positivity conjectures, and the categorical interpretation of the matrix V n S in termsof the category e D ξ will be crucial for proving the conjectures. Theorem 3.7. Conjecture 3.2 is true.Proof. We start first with the case n d [2]. The category e D ξ equipped with the pivotalstructure a p + d, − is a non-degenerate fusion category with simple objects X ( λ ) for λ ∈ e C n,d .The Verlinde formula asserts that the multiplicity of X ( γ ) in X ( α ) ⊗ X ( β ) is given by X κ ∈ e C n,d S α,κ S β,κ ¯ S γ,κ S ,κ = p N ca,b σ a σ b σ c , where e ι p ( a ) = α, e ι p ( b ) = β , and e ι p ( c ) = γ , the equality following from Theorem 3.5. Since amultiplicity in a fusion category is non-negative, we deduce that p N ca,b σ a σ b σ c is non-negativefor all a, b, c ∈ I n,d .We now turn to the case of n and d odd. Since the category e D ξ is a non-degeneratesuperfusion category, the same argument only prove that the structure constants are integers.But we have seen in Proposition 2.13 that in this case, the category D ξ , whose e D ξ is asupermodularization, is equivalent to D ξ, ⊠ sVect with D ξ, a non-degenerate braided fusion category. The modular invariants of D ξ, and e D ξ coincide up to signs, and the end of theproof is similar to the case n d [2].Finally, if both n and d are even, the category D ξ is slightly degenerate and since n > d ,Proposition 2.13 asserts that the category D ξ is not of the form D ξ, ⊠ sVect for D ξ, non-degenerate. Hence we cannot find a change of signs ( σ a ) a ∈ I n,d ∈ {± } I n,d such that for any a, b, c ∈ I n,d , the integer p N ca,b σ a σ b σ c is non-negative. (cid:3) Theorem 3.8. Conjecture 3.3 is true.Proof. Only the case of n and d even needs a comment. Since the fusion ring defined by V n S is isomorphic to the quotient of the Grothendieck ring Gr( D ξ ) by the ideal generated by[ ε ] + [ ], we easily check that the ring Gr( D ξ ) is isomorphic to the ring A ′ of Conjecture 3.3.Note that we crucially need that N ZX,Y N Z ⊗ εX,Y = 0 in Gr( D ξ ), which is true since ε sits in nontrivial degree for the Z / d Z -grading. (cid:3) Fourier matrices for G ( d, , n )In [9], Cuntz noticed that the Fourier matrices defined by Malle [18] can be expressedusing tensor products of exterior powers V n S . Since we constructed a categorification ofthese exterior powers using representations of the quantum enveloping algebra of gl n at aneven root of unity, we now explore the categorification of the Fourier matrices for G ( d, , n ).We fix d ≥ n , we will add a subscript n to thevarious objects considered in the previous sections. For example, we will denote the gl n -weight ρ by ρ n , the Weyl group W of gl n by W n and so on.4.1. Fourier matrices and exterior powers. We follow the presentation of [9, Section3]. Let m ∈ N , Y be a totally ordered set with md + 1 elements and π : Y → N be a map.Let w < . . . < w r be such that π ( Y ) = { w , . . . , w r } and n i = | π − ( w i ) | . Then we have P ri =1 n i = md + 1.We consider the set Ψ of maps f : Y → { , . . . , d − } such that f is strictly increasing on π − ( i ) for each i ∈ N . If f ∈ Ψ, we define a sign ε ( f ) ∈ {± } by ε ( f ) = ( − |{ ( y,y ′ ) ∈ Y × Y | y Recall the that the category D n,ξ − of Section 2.6 is obtained from C n,ξ − by adding isomorphisms and that the symmetriccenter of D n,ξ − is tensor generated by ε n = X (( d − n ) ω + ̟ n ) which satisfies ε ⊗ ε ≃ in D n,ξ − . We consider a Deligne tensor product of the categories D n,ξ − associated with therepresentations the quantum enveloping algebra of gl n at a root of unity.Let D n,ξ − = D n ,ξ − ⊠ . . . ⊠ D n r ,ξ − for n = ( n , . . . , n r ) such that 1 ≤ n i ≤ d and P ni =1 n i ≡ d ]. This category is a braided fusion category admitting many pivotal structures.Its simple objects are of the form X ( λ ) = X ( λ ) ⊠ · · · ⊠ X ( λ r ) with λ i ∈ C n i ,d . Its symmetriccenter has 2 r simple objects given by ε δ = ε ⊗ δ n ⊠ · · · ⊠ ε ⊗ δ r n r , where δ = ( δ , . . . , δ r ) ∈ { , } r and ε n i is the unique simple transparent object of D n i ,ξ − non-isomorphic to the unit object.The category D n,ξ − has a natural ( Z / d Z ) r -grading: a simple object X ( λ ) sits in degree( h λ i , ̟ n i i ) ≤ i ≤ r .Consider now E n,ξ − the full subcategory with simple objects X ( λ ) satisfying P ri =1 h λ i , ̟ n i i ≡ d ]. Thanks to the grading, it is easily seen that the category E n,ξ − is stable under the tensorproduct and is thus a braided fusion category.From the construction of the fusion datum of Section 4.1 and the results of Section 3, weexpect that the category E n,ξ − gives a categorification of the fusion datum of Section 4.1.We first determine the symmetric center of E n,ξ − in order to ensure that its S -matrix hasrank d Q ri =1 (cid:0) dn i (cid:1) .Since ε i is of degree d , the objects ε δ for δ ∈ { , } d are in the symmetric center of E n,ξ − ,and there are no other simple transparent objects: Proposition 4.2. The symmetric center of E n,ξ − is the full subcategory with simple objects ε δ for δ ∈ { , } d .Proof. We use the same strategy as in the proof of Proposition 2.7. The same proof showsthat if X is a transparent simple object, then for any simple object Y the morphism θ X ⊗ Y is a scalar multiple of the identity, where θ is the pivotal structure obtained from the tensorproduct of the pivotal structures θ , − . We now fix such a transparent simple object X ( λ ) and consider for 1 ≤ i ≤ r the simple object Y i = X ( µ ( i ) ) where µ ( i ) j = ( − ̟ n j if j = i,̟ − ̟ n j if j = i. Since P rj =1 n j ≡ d ], the object Y i is in E n,ξ − . As in the proof of Proposition 2.7, consideringdecomposition of the tensor product X ( λ ) ⊗ Y i shows that X ( λ i ) is invertible and therefore X ( λ ) is also invertible.We hence may and will suppose that for every 1 ≤ i ≤ r , there exists 1 ≤ l i ≤ n i and s i ∈ Z such that λ i = ( d − n i ) ̟ l i + s i ̟ n i . We aim to show that l i ≡ s i [ d ] for every 1 ≤ i ≤ r . Then X ( λ ) is transparent if and only if θ X ( λ ) ⊗ Y ( µ ) = θ X ( λ ) θ Y ( µ ) for any µ which is equivalent to r X j =1 h sh l j ( µ j ) + ( d − n j ) ̟ l j + s j ̟ n j , sh l j ( µ j ) + ( d − n j ) ̟ l j + s j ̟ n j + 2 ρ i ≡ r X j =1 h µ j , µ j + 2 ρ i + h ( d − n j ) ̟ l j + s j ̟ n j , ( d − n j ) ̟ l j + s j ̟ n j + 2 ρ i [2 d ] . As in the proof of Proposition 2.7, one may show that this is the equivalent to r X j =1 ( s j − l j ) h µ j , ̟ n j i ≡ d ] . We once again choose µ = µ ( i ) so that h µ ( i ) j , ̟ n j i = δ i,j − n j . Hence if X ( λ ) is transparent, weobtain that P rj =1 n j ( l j − s j ) + ( s i − l i ) ≡ d ]. But as X ( λ ) is in E n,ξ − , we have P rj =1 n j ( s j − l j ) ≡ d ] and hence ( s i − l i ) ≡ d ] for every 1 ≤ i ≤ r . This shows that X ( λ ) is isomorphicto an object of the form ε δ . (cid:3) It is readily seen that ε δ is in non trivial degree if there exist 1 ≤ i ≤ r such that δ i = 1.Therefore tensoring by ε δ has no fixed points on the set of simple objects. As inSection 2.6, we want to (super)modularize the category E n,ξ − and therefore we need that ε δ has a twist equal to 1 for the chosen pivotal structure and every δ ∈ { , } r . We thus equip D n i ,ξ − with a pivotal structure of the form a p i + d, − for some p i ∈ Z . The category E n,ξ − is then equipped with a pivotal structure that we will denote by a p + d, − ; the correspondingtwist will be denoted by θ p + d, − . Corollary 4.3. The symmetric center of E n,ξ − equipped with the pivotal structure a p + d, − is tensor generated by the objects ε δ for δ ∈ { , } r . Moreover, ε δ is of quantum dimension Q ≤ i ≤ rn i ≡ d [2] ( − δ i and of twist .The (super)modularization e E n,ξ − of E n,ξ − has its objects parameterized by the set e E n,d = ( λ ∈ e C n ,d × · · · × e C n r ,d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r X i =1 h λ i , ̟ n i i ≡ d ] ) . Note that the resulting category e E n,ξ − is a superfusion category as soon there exists i suchthat n i and d have the same parity. We summarize the results on the category e E n,ξ − and itsmodular invariants. OURIER MATRICES FOR G ( d, , n ) FROM QUANTUM GENERAL LINEAR GROUPS 25 Proposition 4.4. Recall that we endow the category the category e E n,ξ − with the pivotalstructure a p + d, − . Then there exists a fourth root of unity ω such that the renormalized S -matrix of e E n,ξ − is given by S λ,µ = ω √ d r Y i =1 ξ −h λ i + µ i , (2 p i + d ) ̟ ni i− p i n i ( p i + n i − P w ∈ W ni ( − l ( w ) ξ − h w ( λ i + ρ ni ) ,µ i + ρ ni i √ d n i for any λ, µ ∈ e E n,d . The twist on the simple object X ( λ ) is given by multiplication by ξ − P ri =1 h λ i ,λ i +2 ρ +(2 p i + d ) ̟ ni i .Proof. This follows immediately from Theorem 2.12. The extra √ d at the denominator comesfrom the fact we work with the modularization of the subcategory E n,ξ − and not with themodularization of the whole category D n,ξ − . Indeed, dim( E n,ξ − ) = dim( D n,ξ − ) /d thanksto the grading. (cid:3) Fourier matrix and eigenvalues of the Frobenius as modular invariants. Fi-nally, as expected, we recover the Fourier matrix S and the eigenvalues of the Frobenius T ofSection 4.1 from the category e E n,ξ − . We choose the integers p , . . . , p r as in Proposition 4.1,that is such that f defined by ( f ) i = i ( p i ) ∈ I n i ,d is in Ξ. This condition amounts to r X i =1 (cid:18) p i n i + (cid:18) n i (cid:19)(cid:19) ≡ m (cid:18) d (cid:19) [ d ] . Using the various maps e ι p from Section 3.3, we define a map e ι n,p : Ξ → e E n,d by e ι n,p ( f ) i = e ι n i ,p i ( f i ) , where f i ∈ I n i ,d is as in Section 4.1. Note that e ι n,p ( f ) is indeed in e E n,d since r X i =1 h e ι n i ,p i ( f i ) , ̟ n i = X y ∈ Y f ( y ) − r X i =1 (cid:18) p i n i + (cid:18) n i (cid:19)(cid:19) ≡ d ] , the last equality following from the fact that f ∈ Ξ and that the integers p , . . . , p r are chosensuch that f ∈ Ξ. The map e ι n,p is bijective since for every p the map e ι p is bijective and that | Ξ | = d Q ri =1 (cid:0) dn i (cid:1) = | e E n,d | . Theorem 4.5. We keep the above notations. The (super)category of e E n,ξ − is a categorifica-tion of the modular datum defined by S and T : there exist a fourth root of unity ω and signs ( σ f ) ∈ {± } Ξ with σ f = 1 such that S X ( e ι n,p ( f )) ,X ( e ι n,p ( g )) = ωσ f σ g S f,g and θ d +2 p,X ( e ι p ( a )) = Fr( f ) − T f . The (super)Grothendieck ring of e E n,ξ − is then isomorphic to the ring defined by S withunit parameterized by f .Proof. The proof is similar to the one of Theorem 3.5 using the modular invariants of e E n,ξ − given in Proposition 4.4, and is then omitted. (cid:3) As a corollary, we obtain an independent proof of the integrality of the structure constantsdefined by the matrix S , and moreover that the absolute value of these structure constantsalso define an associative ring.Finally, note that if we choose for f the special symbol as in [18, Bemerkung 2.25], thenit moreover satisfies Fr( f ) = 1 and the eigenvalues of the Frobenius coincide with the twistin e E n,ξ − .4.4. Ennola d -ality. Given an element f ∈ Ξ, Malle has defined a polynomial γ f ( q ) whichbehavior is similar to the degrees of the unipotent characters of a finite group of Lie type. Inparticular, a property similar to the Ennola duality exists, but ii is rather a d -ality. Thereexists a bijection E : Ξ → Ξ such that, up to a sign, the polynomials γ f ( ζq ) and γ E ( f ) ( q )coincide up to a sign. Therefore E d is the identity. This bijection is defined explicitly by Malle[18, Folgerung 3.11] in terms of d -symbols, and we give the translation in terms of functionsin Ξ. Let f ∈ Ξ. Its Ennola transform is the unique function E ( f ) ∈ Ξ such that E ( f ) i isgiven by reducing modulo 0 and sorting increasingly the set f ( π − ( w i )) + w i − P rk =1 w k n k .It is easily checked that E ( f ) indeed belongs to Ξ. Proposition 4.6. Let η be the invertible object X ( η ) with η i = ( w i − P rk =1 w k n k ) ̟ n i . Thenthe objects X ( e ι n,p ( f )) ⊗ η and X ( e ι n,p ( E ( f ))) are isomorphic in E n,ξ − : the Ennola d -ality isgiven by tensoring by an invertible object of trivial d -th tensor power.Proof. It is clear that η is indeed an object in E n,ξ − , that is that P ri =1 h η i , ̟ n i i ≡ d ].As X ( η i ) is a tensor power of det n i ,ξ − , it suffices to show that for every 1 ≤ i ≤ r we have ι n,p ( f ) + η i ∼ ι n,p ( E ( f )), which is immediate by definition of η . (cid:3) References [1] H. H. Andersen and C. Stroppel. Fusion rings for quantum groups. Algebr. Represent. Theory , 17(6):1869–1888, 2014.[2] B. Bakalov and A. Kirillov, Jr. Lectures on tensor categories and modular functors , volume 21 of UniversityLecture Series . American Mathematical Society, Providence, RI, 2001.[3] C. Bonnaf´e and R. Rouquier. An asymptotic cell category for cyclic groups. J. Algebra , 558:102–128, 2020.[4] M. Brou´e, G. Malle, and J. Michel. Towards spetses. I. Transform. Groups , 4(2-3):157–218, 1999. Dedicatedto the memory of Claude Chevalley.[5] M. Brou´e, G. Malle, and J. Michel. Split spetses for primitive reflection groups. Ast´erisque , (359):vi+146,2014.[6] A. Brugui`eres. 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