On a subfactor generalization of Wall's conjecture
aa r X i v : . [ m a t h . OA ] J un On a subfactor generalization of Wall’sconjecture
Robert Guralnick ∗ Department of MathematicsUniversity of Southern CaliforniaLos Angeles, California 90089-2532E-mail: [email protected]
Feng Xu † Department of MathematicsUniversity of California at RiversideRiverside, CA 92521E-mail: [email protected]
Abstract
In this paper we discuss a conjecture on intermediate subfactors which is ageneralization of Wall’s conjecture from the theory of finite groups. We explorespecial cases of this conjecture and present supporting evidence. In particularwe prove special cases of this conjecture related to some finite dimensionalKac Algebras of Izumi-Kosaki type which include relative version of Wall’sconjecture for solvable groups. ∗ Supported in part by NSF grants DMS- 0653873 and 1001962. † Supported in part by NSF grant DMS-0800521 and an academic senate grant from UCR. Introduction
Let M be a factor represented on a Hilbert space and N a subfactor of M which isirreducible, i.e., N ′ ∩ M = C . Let K be an intermediate von Neumann subalgebra forthe inclusion N ⊂ M. Note that K ′ ∩ K ⊂ N ′ ∩ M = C , K is automatically a factor.Hence the set of all intermediate subfactors for N ⊂ M forms a lattice under twonatural operations ∧ and ∨ defined by: K ∧ K = K ∩ K , K ∨ K = ( K ∪ K ) ′′ . The commutant map K → K ′ maps an intermediate subfactor N ⊂ K ⊂ M to M ′ ⊂ K ′ ⊂ N ′ . This map exchanges the two natural operations defined above.Let M ⊂ M be the Jones basic construction of N ⊂ M. Then M ⊂ M iscanonically isomorphic to M ′ ⊂ N ′ , and the lattice of intermediate subfactors for N ⊂ M is related to the lattice of intermediate subfactors for M ⊂ M by thecommutant map defined as above.Let G be a group and G be a subgroup of G . An interval sublattice [ G /G ] isthe lattice formed by all intermediate subgroups K, G ⊆ K ⊆ G . By cross product construction and Galois correspondence, every interval sublatticeof finite groups can be realized as intermediate subfactor lattice of finite index. Hencethe study of intermediate subfactor lattice of finite index is a natural generalization ofthe study of interval sublattice of finite groups. The study of intermediate subfactorshas been very active in recent years(cf. [4],[10], [18],[17], [16], [20], [28] and [26]for only a partial list). By a result of S. Popa (cf. [25]), if a subfactor N ⊂ M is irreducible and has finite index, then the set of intermediate subfactors between N and M is finite. This result was also independently proved by Y. Watatani (cf.[28]). In [28], Y. Watatani investigated the question of which finite lattices can berealized as intermediate subfactor lattices. Related questions were further studied byP. Grossman and V. F. R. Jones in [10] under certain conditions. As emphasizedin [10], even for a lattice consisting of six elements with shape a hexagon, it is notclear if it can be realized as intermediate subfactor lattice with finite index. Thisquestion has been solved recently by M. Aschbacher in [1] among other things. In[1], M. Aschbacher constructed a finite group G with a subgroup G such that theinterval sublattice [ G /G ] is a hexagon. The lattices that appear in [10, 28, 1] canall be realized as interval sublattice of finite groups. There are a number of oldproblems about interval sublattice of finite groups. It is therefore a natural programmeto investigate if these old problems have any generalizations to subfactor setting.The hope is that maybe subfactor theory can provide new perspective on these oldproblems.In [30] we consider the problem whether the very simple lattice M n consisting of alargest, a smallest and n pairwise incomparable elements can be realized as subfactorlattice. We showed in [30] all M n are realized as the lattice of intermediate subfactorsof a pair of hyperfinite type III factors with finite depth. Since it is conjectured thatinfinitely many M n can not be realized as interval sublattices of finite groups (cf. [3]2nd [24]), our result shows that if one is looking for obstructions for realizing finitelattice as lattice of intermediate subfactors with finite index, then the obstruction isvery different from what one may find in finite group theory.In 1961 G. E. Wall conjectured that the number of maximal subgroups of a finitegroup G is less than | G | , the order of G (cf. [27]). In the same paper he proved hisconjecture when G is solvable. See [19] for more recent result on Wall’s conjecture.Wall’s conjecture can be naturally generalized to a conjecture about maximalelements in the lattice of intermediate subfactors. What we mean by maximal elementsare those subfactors K = M, N with the property that if K is an intermediatesubfactor and K ⊂ K , then K = M or K. Minimal elements are defined similarlywhere N is not considered as an minimal element. When M is the cross product of N by a finite group G , the maximal elements correspond to maximal subgroups of G, and the order of G is the dimension of second higher relative commutant. Hence anatural generalization of Wall’s conjecture as proposed in [29] is the following: Conjecture 1.1.
Let N ⊂ M be an irreducible subfactor with finite index. Then thenumber of maximal intermediate subfactors is less than dimension of N ′ ∩ M (thedimension of second higher relative commutant of N ⊂ M ). We note that since maximal intermediate subfactors in N ⊂ M correspond to min-imal intermediate subfactors in M ⊂ M , and the dimension of second higher relativecommutant remains the same, the conjecture is equivalent to a similar conjecture asabove with maximal replaced by minimal.In [29], Conjecture 1.1 is verified for subfactors coming from certain conformalfield theories. These are subfactors not related to groups in general. In this paper weconsider those subfactors which are more closely related to groups and more generallyHopf algebras.If we take N and M to be cross products of a factor P by H and G with H asubgroup of G , then the minimal version of conjecture 1.1 in this case states thatthe number of minimal subgroups of G which strictly contain H is less than thenumber of double cosets of H in G . This follows from simple counting argument. Thenontrivial case is the maximal version of the above conjecture. In this case it gives ageneralization of Wall’s conjecture which we call relative version of Wall’s conjecture.The relative version of Wall’s conjecture states that the number of maximal subgroupsof G strictly containing a subgroup H is less than the number of double cosets of H in G. As a simple example when this can be proved, consider G = H × H, and D ∼ = H is a diagonal subgroup of G. Then the set of maximal subgroups of G containing D are in one to one correspondence with the set of maximal normal subgroups of H, and it is easy to check that the set of maximal normal subgroups has cardinality lessthan the number of irreducible representations of H. On the other hand the numberof double cosets of H in G is the same as the number of conjugacy classes of H, andthis is the same as the number of irreducible representations of H. So we have provedthe relative version of Wall’s conjecture in this case.In § G solvable. Wewill present two proofs. The first proof is motivated by an idea of V. F. R. Jones3hich is to seek linear independent vectors associated with minimal subfactors in thespace of second higher relative commutant. This proof is indirect but we hope thatthe idea will prove to be useful for more general case. We formulate a conjecture forgeneral subfactors (cf. Conjecture 2.1) which is stronger than Conjecture 1.1, and forsolvable groups this conjecture is proved in [29]. Here we modify the proof in [29] toprove a linear independence result (cf. Th. 2.7), and this result implies the relativeWall conjecture for solvable groups. The second proof is a more direct proof usingproperties of maximal subgroups of solvable groups.The cross product by finite group subfactor is a special case of depth 2 subfactor.If we take N ⊂ M to be depth 2, by [5],[22] such a subfactor comes from crossproduct by a finite dimensional *-Hopf algebra or Kac algebra A . By [16] or [22]the intermediate subfactors are in one to one correspondence to the set of left (orright) coideals of A . Then Conjecture 1.1 states that the number of maximal (resp.minimal) right coideals of A is less than the dimension of A . In § A of Izumi-Kosaki type with solvable groups as consideredin [14]. We also prove Conjecture 1.1 for the intermediate subfactors of Izumi-Kosakitype with solvable groups as considered in [14] which are not necessarily of depth 2(cf. Th. 3.13). Th. 3.13 generalizes Th. 2.9. It is interesting to note that the sametype of first cohomology problem encountered in Remark 2.8 also appears here but ina different way and solvability is once again used to ensure that the first cohomologygroup is trivial (cf. Lemma 3.6 and Lemma 3.7).We note that recently lattices of intermediate for other types of Kac algebrashave been obtained in [6]. Our conjecture can be verified in the examples of [6]where complete lattice of intermediate subfactors are determined. The maximal (orminimal) coideals are very few compared with the dimension of the Kac algebra inthese examples of [6].In § X × Y which does not contain either X nor Y. This lemma gives a proofof Wall’s conjecture for X × Y assuming that Wall’s conjecture is true for X and Y. We then propose a natural conjecture about tensor products of subfactors.At the end of this introduction let us consider a fusion algebra version of Conjecture1.1. Let ρ i ∈ End( M ) , i = 1 , ..., n be a finite system of irreducible sectors of a properlyinfinite factor M which is closed under fusion. Consider the Longo-Rehren subfactorassociated with such a system (cf. [21]). By [13], the intermediate subfactors arein one to one correspondence with the fusion subalgebras which are generated bya subset of simple objects ρ i , and Conjecture 1.1 states that the number of suchmaximal fusion subalgebras is bounded by n which is the number of simple objects.This motivates us to make the following conjecture: Conjecture 1.2.
Let F be a finite dimensional semisimple fusion algebra with n simple objects. Then the number of maximal fusion subalgebras which are generatedby a subset of the simple objects of F is less than n . If we take F to be the group algebra of G, then Conjecture 1.2 is equivalent toWall’s conjecture. 4f we take F to be the fusion algebra of representations of a finite group G , thenthe maximal fusion subalgebras are in one to one correspondence to minimal normalsubgroups of G, and the number of such subgroups are less than the number ofconjugacy classes of G , which is the same as the number of simple objects of F . Thisis a special case of a more general result of D. Nikshych and V. Ostrik, who provethat Conjecture 1.2 is true for commutative F [23].The second named author (F.X.) would like to thank Prof. V. F. R. Jones forhis encouragement and useful comments on conjecture 1.1 which inspired our firstproof presented in §
2, and for communications on numerical evidence supporting therelative version of Wall’s conjecture. F.X. would also like to thank Prof. D. Bischfor invitation to a conference on subfactors and fusion categories in Nashville wheresome of the results of this paper were discussed, and Professors Marie-Claude David,D. Nikshych and V. Ostrik for useful communications.
In this section we will prove Theorem 2.9, which confirms the relative version of Wall’sconjecture for solvable groups. We will give two proofs of this result. The first proofis motivated by the following conjecture, formulated as Conjecture A.1 in [29], whichcan be stated for general subfactors:
Conjecture 2.1.
Let N ⊂ M be an irreducible subfactor with finite Jones index,and let P i , ≤ i ≤ n be the set of minimal intermediate subfactors. Denote by e i ∈ N ′ ∩ M , ≤ i ≤ n the Jones projections e i from M onto P i and e N the Jonesprojections e N from M onto N. Then there are vectors ξ i , ξ ∈ N ′ ∩ M such that e i ξ i = ξ i , ≤ i ≤ n, e N ξ = ξ, and ξ i , ≤ i ≤ n, ξ are linearly independent. Remark 2.2.
We note that unlike conjecture 1.1, the conjecture above makes use ofthe algebra structure of N ′ ∩ M and therefore does not immediately imply the dualversion or if one replaces minimal by maximal. By definition conjecture 2.1 implies conjecture 1.1. In the case of subfactors fromgroups, it is easy to check that conjecture 2.1 is equivalent to:
Conjecture 2.3.
Let K i , ≤ i ≤ n be a set of maximal subgroups of G. Then thereare vectors ξ i ∈ l ( G ) , ≤ i ≤ n such that e G ξ i = 0 , ξ i are K i invariant and linearlyindependent. This conjecture is proved in [29] when G is solvable. It turns out a modificationof the proof presented in [29] gives a proof of a stronger statement. Let us makethe following stronger conjecture. First we need to introduce some notation. If H is a subgroup of G , let ℓ ( H ) = ℓ ( G, H ) be the permutation module C GH . Let ℓ ( H )denote the hyperplane of weight zero vectors in ℓ ( H ) (i.e. the complement to the1-dimensional G -fixed space on ℓ ( H )). 5 onjecture 2.4. Let K i , ≤ i ≤ n be a set of maximal subgroups of G. Set H = ∩ K i .Then there are vectors ξ i ∈ ℓ ( H ) , ≤ i ≤ n that are K i -invariant and linearlyindependent. In particular, this implies that n ≤ dim ℓ ( H ) H < | H/G \ H | . We will prove Conjecture 2.4 for solvable groups by modifying the arguments of[29]. We begin with some preparations that hold for all finite groups.
Lemma 2.5.
Suppose that K , . . . , K n are conjugate maximal subgroups of the finitegroup G . Then Conjecture 2.4 holds for { K , . . . , K n } . Proof
Set K = K . If n = 1, the result is obvious (in particular, if K is normal in G ). So assume that n >
1. Let H = ∩ K i . Of course, ℓ ( K ) is a submodule of ℓ ( H ).Let K , . . . , K m , m ≥ n be the set of all conjugates of K . Since K is not normal in G , K is self normalizing whence if we choose a permutation basis { v i | ≤ i ≤ m } for ℓ ( K ), then the stabilizers of the v i are precisely the K i . If m > n , then the vectors v i − v ∈ ℓ ( K ) , ≤ i ≤ n are clearly linearly independent (here v = P v i is fixedby G ). So it suffices to assume that m = n and so in particular, H is normal in G . So we may assume that H = 1. Let V be a nontrivial irreducible submoduleof ℓ ( K ). Then K does not act trivially on V . Note that, by Frobenius reciprocity,the multiplicity of V in ℓ ( K ) is precisely dim V K < dim V . Of course dim V is themultiplicity of V in ℓ ( H ). Thus, ℓ ( K ) ⊕ V is a submodule of ℓ ( H ). Now choosevectors v i − v , ≤ i < m as above and w m any fixed vector of K m in V . These areobviously linearly independent. (cid:4) Next we prove a reduction theorem for Conjecture 2.4. Note that the reductiondepends on the existence of the vectors and not just on cardinality.
Lemma 2.6.
Let S be a family of finite simple groups. Let F ( S ) denote the familyof all finite groups with all composition factors in S . Let K , . . . , K n be maximalsubgroups of the finite group G in F ( S ) and assume that Conjecture 2.4 fails with n | G | minimal. Then each K i has trivial core in G . In particular, G is a primitivepermutation group. Proof
Suppose that N is a nontrivial normal subgroup of G contained in K . Set H = ∩ K i . If each K i contains N , then ℓ ( H ) is a G/N -module and so
G/N, K /N, . . . , K n /N would give a counterexample to the conjecture.Reorder the K i so that N ≤ K i if and only if i ≤ s < n . Note that N K j = G for j > s . By the minimality of | G | n , we can choose v , . . . , v n ∈ ℓ ( H ) with K j v j = v j forall j such that { v , . . . , v s } and { v s +1 , . . . , v n } are linearly independent. It thus sufficesto show that spans of v , . . . , v s and v s +1 , . . . , v n have trivial intersection. Suppose that u is in this intersection.Since e N e K j = e G for j > s (since G = N K j ), it follows that 0 = e G v j = e N e K j v j = e N v j for j > s . Thus, e N u = 0. Since N fixes v i , i ≤ s , it follows that e N u = u . Thus, u = 0 and the result follows. (cid:4) Theorem 2.7.
Conjecture 2.4 is true for G solvable. roof Consider a counterexample with | G | n minimal. By Lemma 2.6, none of the K i contain a normal subgroup. It follows that G is a solvable primitive permutationgroup, whence G = AK where A is elementary abelian and K acts irreducibly on A . In particular, any maximal subgroup of G either contains A or is a complementto A . Since the core of each K i is trivial, G = AK i for each i . Since G is solvable, H ( K, A ) = 0, whence all of the K i are conjugate. Now apply Lemma 2.5 to completethe proof. (cid:4) Remark 2.8.
As we have seen, a minimal counterexample to Conjecture 2.4 wouldbe a primitive permutation groups, and the set of maximal subgroups must all havetrivial core. Such groups are classified by Aschbacher-O’Nan-Scott theorem (cf. § G is the semidirect product of an elementary abelian group V by K , and the action of K on V is irreducible. When G is not solvable, maximalsubgroups K of G with trivial core are not conjugates of K , and our proof as abovedoes not work. Such maximal subgroups are related to the first cohomology of K withcoefficients in V, and conjecture 2.4 implies that the order of this cohomology is lessthan | K | (cf. Question 12.2 of [12]). Unfortunately even though it is believed thatthe order of this cohomology is small (cf. [11]), the bound | K | has not been achievedyet. We give a second proof of Conjecture 2.4 for solvable groups which is not inductive.Let G be a solvable group. Let H ≤ G and let K , . . . , K r denote a maximalcollection of maximal subgroups of G containing H which are not conjugate. Let K ij , ≤ i ≤ r, ≤ j ≤ n i denote the set of all maximal subgroups of G containing H where K ij is conjugate to K i .It is easy to see that G = K i K j for i = j (cf [2]). Thus, Hom( ℓ ( K i ) , ℓ ( K j ) = 0if i = j . If K i is normal in G , set V i = 0. If not, let V i be a nontrivial irreduciblesubmodule of ℓ ( K i ) such that ℓ ( K i ) ⊕ V i embeds in ℓ ( H ) (as in the proof of Lemma2.5). Thus X := ⊕ i ( W i ⊕ V i ) embeds in ℓ ( H ) and as above, we can choose v ij in W i ⊕ V i linearly independent with K ij the stabilizer of v ij . (cid:4) Of course, this gives:
Theorem 2.9.
Let G be a finite solvable group. Let H be a subgroup of G . Then thenumber of maximal subgroups of G which contain H is less than | H/G \ H | . In this section we will prove Conjecture 1.1 for Kac algebras of Izumi-Kosaki typefor solvable groups. These Kac algebras are introduced in [14] and in more detailsin [15] by considering compositions of group type subfactors. Let us first recall somedefinitions from [14] to set up our notations. The reader is refereed to [15] for moredetails. Let G = N ⋊ H be semidirect product of two finite groups N, H.
For n ∈ , h ∈ H, we define n h := h − nh. Denote by L ( N ) the set of complex valued functionson N. For f ∈ L ( N ) , f h ( n ) := f ( h − nh ) , h ∈ H. Definition 3.1.
Denote by η h ( n , n ) , ξ n ( h , h ) U (1) valued cocycles as defined in § η h ( n , n ) η h ( n n , n ) = η h ( n , n n ) η h ( n , n ) , ξ n ( h h , h ) ξ n ( h , h ) = ξ n ( h , h h ) ξ n ( h , h ); Moreover, these cocycles verify the following Pentagon equation: η h ( n , n ) η h ( n h , n h ) η h ( n , n ) = ξ n n ( h , h ) ξ n ( h , h ) ξ n ( h , h ) and normalizations: η h ( e, n ) = η h ( n , e ) = ξ n ( e, h ) = ξ n ( h , e ) = η e ( n , n ) = 1 . For subfactor motivations for introducing these cocycles, we refer the reader to § Definition 3.2.
Kac algebras of Izumi-Kosaki type are defined as Hopf algebras A = L ( N ) ⋊ ξ H whose Hopf algebra structures are given in [14] as follows:(1) Algebra products: ( f ( n ) , h )( f ( n ) , h ) = ( f ( n ) f h ( n ) ξ n ( h , h ) , h h ) where f h ( n ) := f ( h − nh ); (2) Coproducts: ∆( n, h ) = X n η h ( nn − , n )( nn − , h ) ⊗ ( n , h ) (3) ∗ structure: ( f, h ) ∗ = ( f ξ ( h, h − ) h − , h − )The following two operators on L ( N ) will play an important role: Definition 3.3. ( L n,η h f )( m ) := f ( nm ) η h ( n, m ) , ( R n,η h f )( m ) := f ( mn ) η h ( n, m ) , ∀ n, m ∈ N, h ∈ H. The following lemma summarize the properties of these operators which followfrom definitions:
Lemma 3.4. L n ,η h L n ,η h = L n n ,η h η h ( n , n ) , R n ,η h R n ,η h = R n n ,η h η h ( n , n ) , L n ,η h R n ,η h = R n ,η h L n ,η h . A is of the form L A ⊂ L where L A is the fixed pointsubfactor of a factor L under the action of A as defined in § L A ⊂ L is of the form L B ⊂ L, where B is a right coidealof A , i.e., an ∗ subalgebra of A which is verifies that ∆( B ) ⊂ B ⊗ A . The following theorem gives a characterization of coideals of A : Theorem 3.5.
Let B be a right (resp. left) coideal of A . Then there are subgroups H ≤ H, N ≤ N and U (1) valued function λ : N × H → U (1) such that:(1) ∀ h ∈ H , hN h − = N ; λ ( n , h ) λ ( n , h ) = λ ( n n , h ) η h ( n , n ) , λ ( n, h ) λ ( h − nh , h ) ξ n ( h , h ) = λ ( n, h h )(1) and B = ⊕ h ∈ H ( C ( h ) , h ) where each C ( h ) ⊂ L ( N ) consists of functions f ∈ L ( N ) such that L n,η h f = λ ( n, h ) f (resp. R n,η h f = λ ( n, h ) f ) ∀ n ∈ N , h ∈ H .Conversely, any triple ( N , H , λ ) which verify the above conditions uniquely de-termine a coideal of A . Proof
We will prove the theorem for the case when B is a right coideal of A . Theremaining case is similar. We write elements of B as P h ( f h , h ) where f h ∈ L ( N ) . Wehave ∆ X h ( f h , h ) = X n ,h ( R n ,η h f h , h ) ⊗ ( n , h )Since B is a right coideal, it follows that for each fixed ( n , h ), ( R n ,η h f h , h ) ∈ B . Sowe have B = ⊕ h ( C ( h ) , h ) with C ( h ) a subspace of L ( N ) which is mapped by R n,h toitself. Since B is also an algebra, we have C ( h ) C ( h ) h ξ ( h , h ) ⊂ C ( h h ) (2)In particular C ( e ) is a subalgebra of L ( N ) which affords a right representation of N. It follows that there is a subgroup N ≤ N such that C ( e ) is the space of N -leftinvariant functions on N. Let N = S i N b i , ≤ i ≤ k with k = | N | / | N | be the leftcoset decompositions of N. Then δ Nb i is a basis of C ( e ) . Since B is an ∗ algebra, it follows that if ( f h , h ) ∈ B , then( f h , h ) ∗ ∈ ( C ( h − ) , h − ) , ( f h , h )( g h , h ) ∗ = ( f h ¯ g h , e ) ∈ ( C ( e ) , e ) . Let H := { h ∈ H | C ( h ) = 0 } . It follows easily from above that H is a subgroupof H. By equation (2) C ( e ) C ( h ) ⊂ C ( h ) . so it follows that C ( h ) = ⊕ i C ( h ) δ N b i . Assume that h ∈ H so C ( h ) = 0 . Since R b i ,η h maps C ( h ) to itself, we can assumethat C ( h ) δ N = 0 . Let f i = 0 ∈ C ( h ) δ N , i = 1 , f ¯ f ∈ C ( e ) δ N = C δ N . Weconclude that C ( h ) δ N is one dimensional, and by using operator R b − i ,η h , we concludethat C ( h ) δ N b i is one dimensional. Let f h = 0 ∈ C ( h ) δ N , then since R n ,η h maps C ( h ) δ N to itself, there is a function λ : N × H → U (1) such that R n ,η h f h = λ ( n , h ) f h .
9e can assume that f h ( e ) = 1 . Then we have( R n ,η h f h )( e ) = λ ( n , h ) = f h ( n ) . Equation (1) follows from Lemma 3.4 and equation (2). Let us show that C ( h ) , h ∈ H is the subspace of L ( N ) which verifies L n ,η h f = λ ( n , h ) f. We note that by definition 3.3 f h ( n ) = λ ( n , h ) verifies( L n ,η h f h )( m ) = λ ( n m, h ) η h ( n , m ) = λ ( n, h ) λ ( m, h ) = λ ( n, h ) f h ( m )where in the last equation we have used equation (1). Since C ( h ) is the linear span of R n,η h f h , by lemma 3.4 we have proved that for any f ∈ C ( h ) , L n ,η h f = λ ( n , h ) f. Onthe other hand by counting dimensions we conclude that C ( h ) , h ∈ H is the subspaceof L ( N ) which verifies L n ,η h f = λ ( n , h ) f. Let us show that ∀ h ∈ H , hN h − = N . By equation (2) we have f h ( n ) δ hN ⊂ C ( h ) δ N which is one dimensional, and it follows that δ hN = δ N for all h ∈ H , i.e., hN h − = N . Conversely for any triple ( N , H , λ ) which verify the conditions in theorem 3.5,we can simply define B := ⊕ h ∈ H ( C ( h ) , h ) where C ( h ) ⊂ L ( N ) consists of functions f ∈ L ( N ) such that L n,η h f = λ ( n, h ) f. We need to check that B is a right coideal.By inspection it is enough to check equation (2). By definition we need to check thatif f h i ∈ C ( h i ) , i = 1 , , then g ( n ) := f h ( n ) f h h ( n ) ξ n ( h , h ) ∈ C ( h h ) . So we need toshow that ( L n,η h h g )( m ) = λ ( n, h h ) g ( m ) , ∀ n ∈ N , m ∈ N. Since f h i ∈ C ( h i ) , i =1 , , we have f h i ( nm ) η h i ( n, m ) = λ ( n, h ) f h i ( m ) , i = 1 , . By using above equation and equation (1) it follows that ( L n,η h h g )( m ) = λ ( n, h h ) g ( m ) , ∀ n ∈ N , m ∈ N iff the following holds: η h ( n, m ) η h ( n h , m h ) η h h ( n, m ) = ξ nm ( h , h ) ξ n ( h , h ) ξ m ( h , h )which is the pentagon equation in definition 3.1. (cid:4) For a coideal B with ( N , H , λ ) as in Th. 3.5, we shall refer to ( N , H , λ ) as thetriple associated with B . We note that by Th. 3.5, such triple uniquely determine B . Moreover, suppose that the triples associated with B i are given by ( N i , H i , λ i ) , i = 1 , . Then B ⊂ B iff N ⊃ N , H ⊂ H and λ , λ agree on N × H . Lemma 3.6.
Let B be a right coideal of A as in Theorem 3.5 with triple ( N , H , λ ) . The the number of right coideals of A with the same ( N , H ) are given as follows:Let ˆ N be the set of homomorphisms from N to U (1) and form a group ˆ N ⋊ H . hen the right coideals of A with the same ( N , H ) are in one to one correspondencewith the set of cocycles from H to ˆ N , i.e., maps µ : H → ˆ N such that µ ( h ) µ ( h ) h = µ ( h h ) . Proof
Let B be a right coideal of A with triple ( N , H , λ ) as in Theorem 3.5. Let µ := λ /λ. By equation (1) we conclude that µ is a cocycle from H to ˆ N . Conversely,if µ is a cocycle from H to ˆ N , then B associated with the triple ( N , H , λµ ) is aright coideal of A by Theorem 3.5. (cid:4) Lemma 3.7.
Let H ⊂ H, N = { e } ⊂ N such that hN h − = N , ∀ h ∈ H . Let ˆ N be the set of homomorphisms from N to U (1) and form a group ˆ N ⋊ H . Assumethat H acts irreducibly on ˆ N and H is solvable. Then the number of cocycles from H to ˆ N , i.e., maps µ : H → ˆ N such that µ ( h ) µ ( h ) h = µ ( h h ) is less or equal to ( | N | − | H | . Proof If H acts trivially on ˆ N , since H acts irreducibly on ˆ N , it follows thatˆ N is an abelian group of prime order, and the number of cocycles from H to ˆ N isbounded by | H | . If H acts nontrivially on ˆ N , then H is a maximal subgroup ofˆ N ⋊ H with trivial core, and since H is solvable, all cocycles from H to ˆ N arecoboundaries by Th.16.1 of [8], and is bounded by | ˆ N | . Since | H | > (cid:4) Theorem 3.8.
Let A = L ( N ) ⋊ ξ H be Kac algebras of Izumi-Kosaki type as indefinition 3.2. Assume that N, H are solvable groups. Then the number of maximal(resp. minimal) right coideals is less than the dimension of A . Proof
Assume that B is a right coideal of A and let ( N , H , λ ) be the triple as inTh. 3.5. Let us first prove the minimal case. Since B ⊃ ( L ( N/N ) , e ) , if N = N, wemust have B = ( L ( N/N ) , e ) , and N must be maximal in N. The number of such N is less than | N | by Th. 2.9.If N = N, then H = e. Let Z p ≤ H be any minimal subgroups of H , then thetriple ( N, Z p , λ ) will give rise to a right coideal of A by Th. 3.5 which is contained in B . It follows that H = Z p . By Lemma 3.6, the number of such triple is bounded by | ˆ N | ≤ | N | . So minimal right coideal of A is bounded by the sum of number of maximalsubgroups of N and the product of the number of minimal subgroups of H by | N | , and it follows that the number of minimal right coideals is less than the dimension of A . Now assume that B is maximal. If N is trivial, then H is maximal in H, and byTh. 2.9 the number of maximal H is less than | H | . If N is nontrivial, then L h ∈ H ( L ( N ) , h ) ⊃ B , and it follows that H = H. Weclaim that N is generated by Ad H ( x ) for any nontrivial x ∈ N . In fact let N ′ ⊂ N
11e a subgroup generated by Ad H ( x ) for a nontrivial x ∈ N . Then the right coidealdetermined by the triple ( N ′ , H, λ ) contains B , and by maximality of B we have N ′ = N . It follows that H acts irreducibly on N / [ N , N ] , and therefore acts irreducibly onits dual ˆ N . By lemma 3.7 such B with fixed ( N , H ) is bounded by ( | N | − | H | . Notethat different N ’s intersect only at identity. It follows that the number of maximal B ’s is bounded by ( | H | −
1) + | H | ( | N | −
1) = | H || N | − . (cid:4) We consider the intermediate subfactors of L B ⊂ L corresponding to B as in Th.3.5. Lemma 3.9.
The dimension of second higher relative commutant associated with thesubfactor L B ⊂ L is given by X h ∈ H dim( C R ( h ) ∩ C L ( h )) where C R ( h ) ∩ C L ( h ) := { f ∈ L ( N ) , R n ,η h f = λ ( n , h ) f = L n ,η h f } . Proof
This follows from § (cid:4) Lemma 3.10.
Let B be as in Th. 3.5 with triple ( N , H , λ ) . Suppose that λ can beextended to N i ⊃ N such that the triple ( N i , H , λ ) , i = 1 , , ..., n gives a right coidealof A via Th. 3.5, and N i ∩ N j = N , ∀ i = j. Let k i be the number of double cosets of N in N i . Then dim( C R ( h ) ∩ C L ( h )) ≥ k −
1) + ... + ( k n − . Proof
On each double coset N bN of N i , we can define a function such that its valueon the double coset is simply the value of λ ( ., h ) and zero elsewhere. It is easy tocheck that these functions belong to C R ( h ) ∩ C L ( h ) , and they are linearly independentsince they have different support, and the lemma follows. (cid:4) The following two lemmas are straightforward consequences of definitions:
Lemma 3.11.
Let N ⊃ N . Then the number of homomorphisms from N to U (1) which takes value on N is bounded by the number of double cosets of N in N . Lemma 3.12.
Let N ⊂ N be a minimal extension of N which is Ad H invari-ant. Then the natural action of H on N /N [ N , N ] is irreducible. The dual of N /N [ N , N ] is abelian group of homomorphisms from N to U (1) which takes value on N . Theorem 3.13.
Let
B ⊂ A be a right coideal of A as in Th. 3.5. Then both minimalversion and maximal version of Conjecture 1.1 is true for subfactors L B ⊂ L when H, N are solvable. roof Let B i ⊂ B and ( N i , H i , λ i ) be the associated triple of B i as in Th. 3.5. Wehave N i ⊃ N , H i ⊂ H , and λ i agrees with λ on N × H i . Let us first prove the maximal case. Assume that B i ⊂ B , i = 1 , , , .., m is the listof maximal right coideals of A that is contained in B . Let ( N j , H j ) , j = 1 , , , ..., n be the list of different pairs of groups that are associated with B ′ i s as in Th. 3.5.If N j = N , then H j ⊂ H is maximal in H . The number of such maximal H j isless than | H | by Th. 2.9. If N j = N , since Ad H j ( N ) = N , it follows that H j = H , and N j is a minimal extension of N which is invariant under Ad H . Let k j be thenumber of double cosets of N in N j . Note that these N j ’s only intersect at N . ByLemma 3.12,Lemma 3.11 and Lemma 3.7 the number of such B i with fixed N j isbounded by ( k j − | H | . So m ≤ | H | − k −
1) + ... + ( k n − | H | and thetheorem follows from Lemma 3.9 and Lemma 3.10.Now assume that B i ⊂ B , i = 1 , , , .., m is the list of minimal right coideals of A that is contained in B . Let ( N j , H j ) , j = 1 , , , ..., p be the list of different pairsof groups that are associated with B ′ i s as in Th. 3.5. By considering ( L ( N/N j ) , e ) itfollows that if H j is trivial, then N j ⊂ N is a maximal subgroup, and the number ofsuch maximal subgroups is bounded by the double cosets of N in N by Th. 2.9. If H j is nontrivial, then N j = N, and it follows that H j has to be a minimal nontrivialsubgroup of H . The number of such subgroups of H is bounded by | H | − H isnot an abelian group of prime order, and 1 if H is an abelian group of prime order.For each fixed ( N, H j ) , the possible λ j ’s which agrees with λ on N × H j is clearlybounded by the number of homomorphisms from N to U (1) which vanishes on N ,and by Lemma 3.11 this number is bounded by the number of double cosets of N in N which is denoted by p . It follows that p ≤ p − | H | − p , and by Lemma3.9 we are done. (cid:4) Remark 3.14.
If we set H to be a trivial group in Th. 3.13, then we recover Th.2.9. Remark 3.15.
For subfactor L A ⊂ L B , we can map its intermediate subfactors tocertain right coideals of ˆ A which is the dual of A (cf. § L A ⊂ L B verifies themaximal and minimal version of Conjecture 1.1 when H, N is solvable.
Lemma 4.1.
Let G = X × Y be finite groups with | X | = x and | Y | = y . Thenthe number of maximal subgroups of G which contain neither X nor Y is at most ( x − y − (with equality if and only if X and Y are elementary abelian 2-groups). Proof
Let M be a maximal subgroup of G containing neither X nor Y . Let f : G → G/K be the natural homomorphism where K is the core of M in G . Then f ( X ) and f ( Y ) are normal nontrivial subgroups which commute in the primitive group G/K and moreover, they generate
G/K together. By the Aschbacher-O’Nan-Scott Theorem13although this can be proved easily in this case) this implies that either f ( X ) = f ( Y )has prime order p for some prime p or G/K = f ( X ) × f ( Y ) = S × S with S anonabelian simple group.Thus, passing to the quotient by the intersection of all the cores of such maximalsubgroups, we may assume that X and Y are direct products of simple groups. Write X = Q p X p × Q S X S where X p is the maximal elementary abelian p -quotient of X and X S is the maximal quotient of X that is a direct product of nonabelian simplegroups each isomorphic to S . Write Y in a similar manner. The previous paragraphshows that we may reduce to the case that either X and Y are each elementary abelian p -groups or are both direct products of a fixed nonabelian simple group S .In the first case, it is trivial to see that the total number of maximal subgroups is( xy − / ( p −
1) while the number of maximal subgroups containing either X or Y is ( x − / ( p −
1) + ( y − / ( p − X nor Y is ( x − y − / ( p − X = S a and Y = S b . If M is a maximal subgroup notcontaining X or Y , then M ∩ X is normal in X with X/ ( M ∩ X ) ∼ = S . Thus M is adirect factor of X isomorphic to S a − . There are a such factors. Thus, the number ofmaximal subgroups of X × Y not containing X or Y is abc where c is the number ofmaximal subgroups of S × S not containing either factor. This is precisely | Aut( S ) | (since any such maximal subgroup is { s, σ ( s ) | s ∈ S } where σ ∈ Aut( S )). Thus, inthis case the number of maximal subgroups not containing either factor is ab | Aut( S ) | .To complete the proof, we only need to know that | Aut( S ) | < ( | S | − .This is well known (and in fact much better inequalities can be shown). All suchexisting proofs depend upon the classification of finite simple groups. We note that theinequality we need follows from the fact that every finite nonabelian simple group canbe generated by two elements (note that if s ∈ S , then |{ σ ( s ) | σ ∈ Aut( S ) } < | S | − S – if s, t are generators, then any automorphism is determined by its imageson s, t whence the inequality). (cid:4) The following corollary follows immediately:
Corollary 4.2.
Let G = X × Y be finite groups such that both X and Y verify Wall’sconjecture, then G also verifies Wall’s conjecture. Based on Lemma 4.1, we propose the following tensor product conjecture:
Conjecture 4.3.
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