On a necessary condition for unitary categorification of fusion rings
aa r X i v : . [ m a t h . QA ] F e b ON A NECESSARY CONDITION FOR UNITARY CATEGORIFICATIONOF FUSION RINGS
PAVEL ETINGOF, DMITRI NIKSHYCH, AND VICTOR OSTRIK
Abstract.
In [LPW] Liu, Palcoux and Wu proved a remarkable necessary condition fora fusion ring to admit a unitary categorification, by constructing invariants of the fusionring that have to be positive if it is unitarily categorifiable. The main goal of this note isto provide a somewhat more direct proof of this result. In the last subsection we discussintegrality properties of the Liu-Palcoux-Wu invariants. Introduction
In [LPW] Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ringto be unitarily categorifiable, which can be used to disqualify many new fusion rings fromhaving such a categorification. They did so by constructing invariants of the fusion ring thathave to be positive if it is the Grothendieck ring of a unitary fusion category. The main goalof this note is to provide a somewhat more direct proof of this criterion, using the notion ofa formal codegree and the Drinfeld center of the underlying category. Thus this note doesnot contain any essentially new results, it is just our attempt to give an exposition of theresult of [LPW] in a somewhat more algebraic language. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariantsand their relation to the generalized Kaplansky 6th conjecture stating that in any sphericalfusion category, dimensions of objects divide the dimension of the category.
Acknowledgements.
We are very grateful to Sebastien Palcoux for useful discussions.P. E.’s work was partially supported by the NSF grant DMS - 1916120. The work of D. N.was partially supported by the NSF grant DMS-1800198. The work of V. O. was partiallysupported by the NSF grant DMS-1702251 and by the Russian Academic Excellence Project‘5-100’. 2.
Fusion rings
Let A be a fusion ring with a finite basis { b i } (where b = 1) and let τ : A → Z be thetrace function given by τ ( P i λ i b i ) := λ . There is an inner product ( a, b ) := τ ( ab ∗ ), for a, b ∈ A , with respect to which the basis b i is orthonormal. Let N ijm := ( b i b j , b m ) be thefusion coefficients and let d j be the Frobenius-Perron dimensions of the basis elements b j .Recall that N ijm = N jm ∗ i ∗ = N m ∗ ij ∗ = N j ∗ i ∗ m ∗ and d j = d j ∗ . For more details on fusion ringsand Frobenius-Perron dimensions we refer the reader to [EGNO, Chapter 3].We now give a somewhat shorter proof of [LPW, Proposition 8.1]. (This proposition willnot be used below.) Proposition 2.1. ( [LPW] , Proposition 8.1) We note that the paper [LPW] contains a nice diagrammatic proof of this result, see the proof ofProposition 7.7 there. i) P m N ijm ≤ min( d i , d j ) . (ii) N ijm ≤ d i d j /d m ; hence N ijm ≤ d m ( d i /d j ) − tt for any t ≥ . (iii) N ijm ≤ min( d i , d j , d m ) . (iv) P m N i i m N i i m ≤ d i p d i q for any distinct ≤ p, q ≤ .Proof. (i) P m N ijm = P m ( b i b j , b m ) = ( b i b j , b i b j ) = ( b i ∗ b i b j , b j ) ≤ FPdim( b i ∗ b i ) = d i . Simi-larly, P m N ijm ≤ d j .(ii) The first statement follows from the equality P m N ijm d m = d i d j . Using the symmetryproperties of N ijm , this implies that N ijm ≤ d m d i /d j and N ijm ≤ d m d j /d i . This implies thesecond statement, since for t ≥ − tt varies between 1 and − N ijm are matrix elements of the matrix of multiplication by b i in the orthonormal basis b j , so they do not exceed the norm d i of this matrix. The rest follows from the symmetryproperties of N ijm .(iv) P m N i i m N i i m = P m N i i m N i ∗ mi = ( b i ∗ b i b i , b i ) = ( b i b i b i ∗ , b i ). This is a matrixelement of each of the following operators on A : L ( b i ∗ ) L ( b i ); R ( b i ) R ( b i ); L ( b i ∗ ) R ( b i ); L ( b i ) L ( b i ); R ( b i ) R ( b i ∗ ); L ( b i ) R ( b i ∗ ), where L denotes left multiplication and R rightmultiplication. Thus it is at most the norms of these operators, as desired. (cid:3) Formal codegrees and the Drinfeld center
Recall that A C = A ⊗ Z C is a ∗ -algebra. Consider an irreducible representation( V, ρ : A C → End V ) of A C (which we will often shortly denote just by ρ ). The formalcodegree α ρ of ρ is the eigenvalue of the central element z ρ := P i Tr( ρ ( b i )) b ∗ i on ρ , analgebraic integer (the eigenvalues of this element on all the other irreducible representationsare 0), see e.g. [O1, Lemma 2.3]. Thus τ ( z ρ ) = dim ρ = α − ρ Tr ρ ( z ρ ). Hence τ ( a ) = X ρ α − ρ Tr ρ ( a ) , a ∈ A. In particular, α ρ > ρ , see also [O2, Remark 2.12].Let v , v ∈ V , f , f ∈ V ∗ . Lemma 3.1.
We have X i f ( ρ ( b i ) v ) f ( ρ ( b ∗ i ) v ) = α ρ f ( v ) f ( v ) . Proof.
It suffices to show that for any a , a ∈ A we have X i Tr ρ ( a b i )Tr ρ ( b ∗ i a ) = α ρ Tr ρ ( a a );then we can take a , a such that ρ ( a ) = v ⊗ f ∗ , ρ ( a ) = v ⊗ f ∗ in End V , which gives thedesired statement. But this is equivalent to the obvious relation X i Tr ρ ( a b i )( a , b i ) = Tr ρ ( a a ) , a , a ∈ A. (cid:3) Suppose now that C is a spherical fusion category that categorifies a fusion ring A . Recallthat irreducible (unitary) representations V of A C have the form V Z = Hom C ( , Z ) where Z is a simple object of the Drinfeld center Z ( C ) of C whose image in C (which we, abusing otation, will also denote by Z ) contains (see [O2, Theorem 2.13] and [S, Theorem 5.9]).Namely, the map ρ Z is constructed as follows. Note that V ∗ Z = Hom C ( Z, ) using thecomposition pairing. Given X ∈ C , v ∈ V Z , f ∈ V ∗ Z , we define ρ Z ( X ) by f ( ρ Z ( X ) v ) = Tr((1 X ⊗ f ) ◦ c X,Z ◦ ( v ⊗ X )) . where c X,Z : Z ⊗ X → X ⊗ Z is the central structure of Z . Using the identity c Z,X ⊗ Y = (1 X ⊗ c Z,Y ) ◦ ( c Z,X ⊗ Y ) , it is not hard to show that ρ Z is a representation. Note that ρ Z ( X ∗ ) = ρ Z ( X ) † (the adjointoperator) and ρ Z ∗ ∼ = ρ Z (the same action on the complex conjugate space). Theorem 3.2. ([O2, Theorem 2.13]) If ρ = ρ Z then one has α ρ = dim C dim Z . In particular, if [ Z : ] > then dim Z > . Corollary 3.3.
Under the assumptions of Lemma 3.1 X i f ( ρ ( b i ) v ) f ( ρ ( b ∗ i ) v ) = dim C dim Z f ( v ) f ( v ) . Positivity results
Recall that a unitary fusion category is a fusion category with a ∗ -structure ([T]; seealso [G], Subsection 2.1 for a full definition). A unitary categorification of a fusion ringis a realization of this ring as the Grothendieck ring of a unitary fusion category.Let ( V s , ρ s : A → End V s ) be a collection of irreducible (unitary) representations of A C and v s ∈ V s , s = 1 , ..., n . Theorem 4.1. ( [LPW] , Proposition 8.3) (i) If A admits a unitary categorification then wehave X i d n − i ( ρ ( b i ) v , v ) ... ( ρ n ( b i ) v n , v n ) ≥ . (ii) ( [LPW] , Corollary 8.5) If in (i) A is commutative then X i d n − i ρ ( b i ) ...ρ n ( b i ) ≥ . Example 4.2.
1. Let n = 1. Then Theorem 4.1 says that ( v, ρ ( P i d i b i ) v ) ≥
0. This holds(regardless of A being unitarily categorifiable) since R := P i d i b i is the regular element of A , hence Ra = FPdim( a ) R for any a ∈ A .2. Let n = 2. Then Theorem 4.1 says that P i ( ρ ( b i ) v , v )( ρ ( b i ) v , v ) ≥
0. Thisfollows (again regardless of A being unitarily categorifiable) from Lemma 3.1. Indeed, since ρ ( b ∗ i ) = ρ ( b i ) † , Lemma 3.1 implies that this sum is zero unless ρ ∼ = ρ , and X i ( ρ ( b i ) v , v )( ρ ( b i ) v , v ) = α ρ | ( v , v ) | . However, for n ≥
3, as shown in [LPW], unitary categorifiability of A is essential. Proof.
It suffices to prove (i) for n ≥
3. Let C be a unitary fusion category categorifying A . Recall ([EGNO], Section 9.5) that it has a canonical spherical structure in which thedimensions of simple objects X i are d i ; so let us endow C with this structure. Let Z , ..., Z n ∈Z ( C ), and consider the vector space Hom Z ( C ) ( , Z ⊗ ... ⊗ Z n ). Since C is unitary, this pace has a positive definite Hermitian inner product given by ( v, w ) = w † ◦ v . Now let Z , ..., Z n ∈ Z ( C ) be objects containing as objects of C . Let ( V i , ρ i ) be the correspondingrepresentations of A C , i = 1 , ..., n . We have a natural map φ : V ⊗ ... ⊗ V n → Hom Z ( C ) ( , Z ⊗ ... ⊗ Z n )given by the orthogonal projection of v ⊗ ... ⊗ v n ∈ Hom C ( , Z ⊗ ... ⊗ Z n ) to the spaceHom Z ( C ) ( , Z ⊗ ... ⊗ Z n ), where v i ∈ Hom( , Z i ). In other words, we may view v ⊗ ... ⊗ v n as an element of Hom Z ( C ) ( Z ⊗ ... ⊗ Z n , ) ∗ by taking composition, and φ ( v ⊗ ... ⊗ v n ) isthe corresponding element of Hom Z ( C ) ( , Z ⊗ ... ⊗ Z n ). This element may be viewed as a Z ( C )-morphism Z ∗ → Z ⊗ ... ⊗ Z n . Now, we haveHom C ( , Z ⊗ ... ⊗ Z n ) = ⊕ Z ∈ Irr Z ( C ) Hom C ( , Z ∗ ) ⊗ Hom Z ( C ) ( Z ∗ , Z ⊗ ... ⊗ Z n )= ⊕ Z ∈ Irr Z ( C ) Hom C ( Z, ) ⊗ Hom Z ( C ) ( Z ∗ , Z ⊗ ... ⊗ Z n ) . Let { e Z,j } is an orthonormal basis of V Z . Then we get(4.1) v ⊗ · · · ⊗ v n = X Z ∈ Irr Z ( C ) dim V Z X j =1 e † Z,j ⊗ φ ( e Z,j ⊗ v ⊗ ... ⊗ v n ) . Recall that c Z ⊗ Z ′ ,X = ( c Z,X ⊗ Z ′ ) ◦ (1 Z ⊗ c Z ′ ,X ) . Therefore, using (4.1), it is easy to see that1 d n − i ( ρ ( X i ) v , v ) ... ( ρ n ( X i ) v n , v n )= X Z ∈ Irr Z ( C ) dim Z dim V Z X j,k =1 ( ρ Z ( X i ) e Z,j , e
Z,k )( φ ( e Z,j ⊗ v ⊗ ... ⊗ v n ) , φ ( e Z,k ⊗ v ⊗ ... ⊗ v n ))= X Z ∈ Irr Z ( C ) dim Z dim V Z X j =1 ( φ ( e Z,j ⊗ v ⊗ ... ⊗ v n ) , φ ( ρ Z ∗ ( X i ) e Z,j ⊗ v ⊗ ... ⊗ v n )) . Let B Z : V Z → V Z be an operator whose matrix elements in the basis e Z,j are B Z,jk = ( φ ( e Z,j ⊗ v ⊗ ... ⊗ v n ) , φ ( e Z,k ⊗ v ⊗ ... ⊗ v n )) . Then 1 d n − i ( ρ ( X i ) v , v ) ... ( ρ n ( X i ) v n , v n ) = X Z ∈ Irr Z ( C ) dim Z dim V Z X j =1 ( B Z e Z,j , ρ Z ∗ ( X i ) e Z,j ) . Thus, X i d n − i ( ρ ( X i ) v , v )( ρ ( X i ) v , v ) ... ( ρ n ( X i ) v n , v n )= X i ( ρ ( X i ) v , v ) X Z ∈ Irr Z ( C ) dim Z dim V Z X j =1 ( B Z e Z,j , ρ Z ∗ ( X i ) e Z,j ) X i dim Z ( ρ Z ( X i ) v , v ) dim V Z X j =1 ( B Z e Z ,j , ρ Z ∗ ( X i ) e Z ,j )= dim V Z X j =1 dim Z ( B Z e Z ,j , v )( v , ρ Z ∗ ( X i ) e Z ,j )= ( B Z v , v ) = dim C · || φ ( v ⊗ ... ⊗ v n ) || , where for the last equality we used Lemma 3.1 and Theorem 3.2. Since C is unitary, so is itsDrinfeld center, so this squared norm is ≥
0, as desired. (cid:3)
The formula at the end of the proof of Theorem 4.1 can, in fact, be generalized to thesituation when the fusion category C is spherical but not assumed unitary or even Hermitian.Namely, let v i ∈ V i , f i ∈ V ∗ i , and let ψ : V ∗ ⊗ ... ⊗ V ∗ n → Hom( , Z ⊗ ... ⊗ Z n ) ∗ ∼ = Hom( Z ⊗ ... ⊗ Z n , )be the natural map. Proposition 4.3.
We have X i X i ) n − f ( ρ ( X i ) v ) ...f n ( ρ n ( X i ) v n ) = dim( C )( φ ( v ⊗ ... ⊗ v n ) , ψ ( f ⊗ ... ⊗ f n )) . where dim denotes the categorical dimensions. Consider the operator ψ ∗ φ ∈ End( V ⊗ ... ⊗ V n ). Proposition 4.3 immediately implies Corollary 4.4.
Let a , ..., a n ∈ A . Then X i X i ) n − Tr ρ ( a X i ) ... Tr ρ n ( a n X i ) = dim C ·
Tr ( ψ ∗ φ ◦ ( ρ ( a ) ⊗ ... ⊗ ρ n ( a n ))) . In particular, we have
Corollary 4.5. If V i are 1-dimensional then I n ( ρ , ..., ρ n ) := X i X i ) n − ρ ( X i ) ...ρ n ( X i ) = dim C · ( φ, ψ ) . Here we treat φ, ψ as vectors since the space V ⊗ ... ⊗ V n is 1-dimensional. Remark 4.6.
It is easy to see that the invariants I n satisfy the recursion I n ( ρ , ..., ρ n ) = X ρ α − ρ I n − ( ρ , ..., ρ n − , ρ ) I ( ρ, ρ n − , ρ n ) , n ≥ . (see [LPW]). This implies that the multiplication law on the complexified Grothendieckgroup of the category of A C -modules given by ρ ∗ ρ = X ρ α − ρ I ( ρ , ρ , ρ ) ρ is commutative and associative, giving this group the structure of a commutative Frobeniusalgebra. This is nothing but the dual fusion ring considered in [LPW]. . Integrality properties of spherical fusion categories
In this section we explore integrality properties for spherical fusion categories over C . Forsimplicity we will restrict ourselves to the case of commutative fusion rings.5.1. Isaacs criterion and Frobenius type.
Let C be a fusion category with fusion ring A . Let s ≥ Definition 5.1.
We say that C is s - Isaacs if for any character ρ : A → C and any simpleobject X ∈ C , the number λ s ( ρ, X ) := (dim C ) s (dim Z ρ ) − s ρ ( X )dim X is an algebraic integer.Since dim Z ρ divides dim C , if C is s -Isaacs then it is t -Isaacs for any t > s .The 0-Isaacs property will simply be called the Isaacs property . It was introduced in[LPR1, LPR2]. This definition was motivated by the following proposition.
Proposition 5.2.
Any ribbon fusion category is Isaacs.Proof.
In a ribbon category C we have ρ ( X ) = s ZρX dim Z ρ . where ( s ij ) is the S -matrix of theDrinfeld center Z ( C ). Thus λ ( ρ, X ) = s Z ρ X dim X , which is an algebraic integer (an eigenvalue of an integer matrix) by the Verlinde formula,see [EGNO], Corollary 8.14.4. (cid:3)
The following conjecture is a slight modification of Conjecture 2.5 of [LPR1] (and coincideswith it in the pseudounitary case and s = 0). Conjecture 5.3.
Any spherical fusion category is s -Isaacs for any s .Recall that C is said to be Frobenius type if dimensions of its simple objects divide dim C .The Kaplansky 6th conjecture for fusion categories states that any spherical fusion categoryis Frobenius type. Generalizing, let us say that a category C is s - Frobenius type for arational number s ≥ C ) s ; so 1-Frobeniustype is Frobenius type in the usual sense. Clearly, s -Frobenius type implies t -Frobenius typefor any t > s . Finally, it is clear that any Frobenius type category C is 1-Isaacs.The following proposition generalizes [LPR1], Proposition 2.6. Proposition 5.4. If C is s -Isaacs for s ≥ then it is s + -Frobenius type. In particular,if C is -Isaacs then it is Frobenius type.Proof. If C is s -Isaacs with s ≥ then the number X ρ λ s ( ρ, X ) λ s ( ρ, X )(dim Z ρ ) s − = X ρ (dim C ) s dim Z ρ ρ ( X ) ρ ( X )(dim X ) = (dim C ) s +1 (dim X ) is an algebraic integer, hence C is s + -Frobenius type, as claimed. (cid:3) Thus Conjecture 5.3 for s ≤ implies the Kaplansky 6th conjecture for spherical fusioncategories, while for larger s it implies weaker versions of this conjecture. .2. Integrality properties of I n . For any characters ρ i : A → C , i = 1 , ..., n , and arational number s ≥ J n,s ( ρ , ..., ρ n ) := (dim C ) ( n − s (dim Z ρ ... dim Z ρ n ) − s I n ( ρ , ..., ρ n ) . For example, J ,s ( ρ, η ) = dim C (dim Z ρ ) − s δ η,ρ . Theorem 5.5. C is s -Isaacs if and only if for any n ≥ and ρ i , i = 1 , ..., n , the number J n,s ( ρ ,...,ρ n )(dim Z ρ dim Z ρ ) − s is an algebraic integer.Proof. We have J n,s ( ρ , ..., ρ n )(dim Z ρ dim Z ρ ) − s = X X ρ ( X ) ρ ( X ) λ s ( ρ , X ) ...λ s ( ρ n , X ) , which proves the “only if” part. For the “if” part, consider the sum X η (dim Z η ) s J n,s ( ρ, ..., ρ, η )(dim Z ρ ) − s ) η ( Y ) == (dim C ) ( n − s +1 (dim Z ρ ) ( n − − s ) ρ ( Y ) n − dim( Y ) n − =dim C (dim Z ρ ) s − ρ ( Y ) λ s ( ρ, Y ) n − . This is an algebraic integer for all n , hence λ s ( ρ, Y ) is an algebraic integer, as claimed. (cid:3) Definition 5.6.
Let us say that C is strongly Isaacs if the number J n, ( ρ ,...,ρ n )dim C √ dim Z ρ dim Z ρ isan algebraic integer.It is easy to see that if C is strongly Isaacs then it is (0-)Isaacs. Theorem 5.7. (i) The category C = Rep( G ) for a finite group G is strongly Isaacs.(ii) Any modular category is strongly Isaacs.Proof. (i) The representations ρ i correspond to the conjugacy classes C i in G , and Z ρ i =Fun( C i ), so dim Z ρ i = | C i | . Also it is well known that in this case(5.1) J n, ( ρ , ..., ρ n )dim C = dim Z ρ ... dim Z ρ n dim C I n ( ρ , ..., ρ n ) = | S | , where S is the set of tuples ( g , ..., g n ) ∈ G n such that g ...g n = 1 and g i ∈ C i . The set S carries an action of G by conjugation, and the stabilizer of every element is contained inthe centralizer G i of the class C i for all i , so the size of each orbit is divisible by | C i | . Thisimplies that (5.1) is an integer and divisible by dim Z ρ i for all i , hence by p dim Z ρ dim Z ρ ,which yields the desired statement.(ii) Each ρ j corresponds to a simple object X j of C . A direct calculation using the Verlindeformula yields J n, ( ρ , ..., ρ n )dim C = dim X ... dim X n dim Hom( , X ⊗ ... ⊗ X n ) . Note also that Z ρ j = X j ⊠ X ∗ j , so dim Z ρ j = (dim X j ) . This implies the statement. (cid:3) uestion 5.8. Is any spherical fusion category strongly Isaacs? Is it true at least for ribboncategories? References [EGNO] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Vol. 205.
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
Email address : [email protected] Department of Mathematics and Statistics, University of New Hampshire, Durham, NH03824, USA
Email address : [email protected] Department of Mathematics, University of Oregon, Eugene, OR 97403, USALaboratory of Algebraic Geometry, National Research University Higher School ofEconomics, Moscow, Russia